Abstract
In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of the \(A_2\) conjecture solved by Hytönen. Advances have greatly improved conceptual understanding of classical objects such as Calderón–Zygmund operators. However, plenty of operators do not fit into the class of Calderón–Zygmund operators and fail to be bounded on all \(L^p(w)\) spaces for \(p \in (1, \infty )\) and \(w \in A_p\). In this paper we develop Rubio de Francia extrapolation with quantitative bounds to investigate quantitative weighted inequalities for operators beyond the (multilinear) Calderón–Zygmund theory. We mainly establish a quantitative multilinear limited range extrapolation in terms of exponents \(p_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\) and weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i=1, \ldots , m\), which refines a result of Cruz-Uribe and Martell. We also present an extrapolation from multilinear operators to the corresponding commutators. Additionally, our result is quantitative and allows us to extend special quantitative estimates in the Banach space setting to the quasi-Banach space setting. Our proof is based on an off-diagonal extrapolation result with quantitative bounds. Finally, we present various applications to illustrate the utility of extrapolation by concentrating on quantitative weighted estimates for some typical multilinear operators such as bilinear Bochner–Riesz means, bilinear rough singular integrals, and multilinear Fourier multipliers. In the linear case, based on the Littlewood–Paley theory, we include weighted jump and variational inequalities for rough singular integrals.
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1 Introduction
1.1 Motivation and Main Results
The main goal of this paper is to establish multivariable Rubio de Francia extrapolation with quantitative bounds in order to investigate quantitative weighted inequalities for multilinear operators beyond the multilinear Calderón–Zygmund theory. We focus on the limited range extrapolation with exponents \(p_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\) and weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i=1, \ldots , m\), which is quite different from [76] for \(\textbf{w}=(w_1, \ldots , w_m) \in A_{\textbf{p}}\) (or general weights \(A_{\textbf{p}, \textbf{r}}\)). The main reason why we study it is that plenty of operators are beyond the Calderón–Zygmund theory so that they may not be bounded on all \(L^p(w)\) spaces for \(p \in (1, \infty )\) and \(w \in A_p\). This is the case for operators with the strong singularity, such as Bochner–Riesz means [6], rough singular integrals [89], Riesz transforms and square functions associated with second-order elliptic operators [3], operators associated with the Kato conjecture [4], and singular “non-integral” operators [9]. As well as the classes \(A_p\) are natural for the Calderón–Zygmund operators and characterize the weighted boundedness of Hardy–Littlewood maximal operators, the classes \(A_{\textbf{p}}\) are also the natural ones for multilinear Calderón–Zygmund operators and the multilinear Hardy–Littlewood maximal operators (cf. Theorem 2.10). In the multilinear setting, there are also many operators so that weighted inequalities holds for limited ranges. For multilinear Fourier multipliers, it is interesting that different forms of Sobolev regularity appear to determine whether product of scalar weights or multiple weights \(A_{\textbf{p}}\) could be used. Fujita and Tomita [43, 44] proved that whenever the symbol satisfies a product type Sobolev regularity, the weighted boundedness of multilinear Fourier multipliers holds for \(\textbf{w} \in A_{p_1/r_1} \times \cdots \times A_{p_m/r_m}\) but does not hold for \(\textbf{w} \in A_{(p_1/r_1, \ldots , p_m/r_m)}\), while the latter is valid under the classical Sobolev regularity. Other examples include strongly singular bilinear Calderón–Zygmund operators [7, Corollary 3.2], bilinear differential operators associated with fractional Leibniz rules [34, Theorem 1.1], bilinear pseudo-differential operators with symbols in the Hörmander classes [75, Remark 3.4], and so on.
The main contributions of this article are the following.
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Our first main result, Theorem 1.1, improves [30, Theorem 1.3] to an extrapolation with the quantitative weighted bounds, which in turn covers the multivariable extrapolation in [40, Theorem 6.1] and [47, Theorem 1.1] by taking \(\mathfrak {p}_i^-=1\) and \(\mathfrak {p}_i^+=\infty \), \(i=1, \ldots , m\).
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Our second main result, Theorem 1.2, establishes an extrapolation for commutators, which extends [8, Theorem 4.3] from the Banach range to the quasi-Banach range.
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We prove a limited range, off-diagonal extrapolation theorem with sharp weighted bounds (cf. Theorem 4.8), whose proof is distinct from and much simpler than that in [30, Theorem 1.8] because it only needs to define a Rubio de Francia iteration algorithm each time we consider the case \(q<q_0\) or \(q>q_0\). Thus, we not only refine [30, Theorem 1.8] to Theorem 4.8 with sharp bounds, but also remove the restriction \(\frac{1}{q_0} - \frac{1}{p_0} + \frac{1}{\mathfrak {p}_+} \ge 0\).
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Although our weights class is a special case of the class \(A_{\textbf{p}, \textbf{r}}\), Theorem 1.1 is independent of [76, Theorem 2.2], that is, one does not imply another one.
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When the exponents are greater than one, we can obtain quantitative \(A_p\) and off-diagonal extrapolation (cf. Theorems 4.1 and 4.5) by showing a “product-type embedding” theorem (cf. Theorems 4.2 and 4.6), respectively, which is quite different from the embedding technique used in [18, Proposition 3.18] to get extrapolation on general weighted Banach function spaces.
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Based on \(A_p\) extrapolation and interpolation, we present an extrapolation from weak type inequalities to strong type estimates (cf. Theorem 4.4). This allows us to obtain quantitative weighted strong estimates from weak (1, 1) type.
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This is the first time to use extrapolation to establish quantitative weighted norm inequalities for plenty of operators beyond the Calderón–Zygmund theory (cf. Sect. 5). The strong singularity of those operators leads the weights class to be \(A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i=1, \ldots , m\), instead of the more general class \(A_{\textbf{p}, \textbf{r}}\). It is totally novel to obtain quantitative estimates for those operators, although we do not show the sharpness, which goes beyond the scope of this article and will be our further topic.
In order to state our main results we need some notation. More definitions and notation are given in Sect. 2. Given \(1 \le \mathfrak {p}_- < \mathfrak {p}_+ \le \infty \) and \(p \in [\mathfrak {p}_-, \mathfrak {p}_+]\) with \(p \ne \infty \), considering Lemma 2.6, for any \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\), we define
where \(\tau _p:= \big (\frac{\mathfrak {p}_+}{p}\big )' \big (\frac{p}{\mathfrak {p}_-} -1\big ) +1\). Throughout this paper, given \(p_i, q_i \in [\mathfrak {p}_i^-, \mathfrak {p}_i^+]\), we always denote
Let \(\mathcal {F}\) denote a family of \((m+1)\)-tuples \((f, f_1, \ldots , f_m)\) of non-negative measurable functions. We would like to present an abstract methodology for extrapolation. We will see that extrapolation enables us to obtain vector-valued inequalities and weak-type estimates from extrapolation results immediately. In the current paper, we mainly apply this methodology to obtain quantitative weighted norm inequalities for plenty of operators.
Our first main result is formulated as follows.
Theorem 1.1
Given \(m \ge 1\), let \(\mathcal {F}\) be a family of extrapolation (m+1)-tuples. Let \(1 \le \mathfrak {p}_i^- < \mathfrak {p}_i^+ \le \infty \) for each \(i=1, \ldots , m\). Assume that for each \(i=1, \ldots , m\), there exists an exponent \(q_i \in (0, \infty )\) with \(q_i \in [\mathfrak {p}_i^-, \mathfrak {p}_i^+]\) such that for all weights \(v_i^{q_i} \in A_{q_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/q_i)'}\), \(i=1, \ldots , m\),
where \(\frac{1}{q} = \sum _{i=1}^m \frac{1}{q_i}\), \(v= \prod _{i=1}^m v_i\), and \(\Phi _i: [1, \infty ) \rightarrow [1, \infty )\) is an increasing function. Then for all exponents \(p_i \in (\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+})\) and all weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/p_i)'}\), \(i=1, \ldots , m\),
where \(\frac{1}{p} = \sum _{i=1}^m \frac{1}{p_i}\), \(w=\prod _{i=1}^m w_i\), \(\mathfrak {C}_i:= 2^{\max \{\frac{\tau _{p_i}}{p_i}, \frac{\tau '_{p_i}}{q_i}\}}\), and \(C_i\) depends only on n, \(p_i\), \(q_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\).
Moreover, for the same family of exponents and weights, and for all exponents \(r_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\),
for all \(\{(f^k, f^k_1, \cdots , f^k_m)\}_k \subset \mathcal {F}\), where \(\frac{1}{r}{=}\sum _{i{=}1}^m \frac{1}{r_i}\), \(\mathfrak {C}'_i{:=} 2^{\max \{\frac{\tau _{p_i}}{p_i}, \frac{\tau '_{p_i}}{r_i}\} + \max \{\frac{\tau _{r_i}}{r_i}, \frac{\tau '_{r_i}}{q_i}\}}\), and the constant \(C'_i\) depends only on n, \(p_i\), \(q_i\), \(r_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\).
As a result of Theorem 1.1 we can extend weighted estimates only valid in the Banach range to the quasi-Banach range. For example, weighted norm inequalities for the commutators of multilinear operators T with \({\text {BMO}}\) functions, more singular than operators T, were just proved in the case \(p \ge 1\) [8] since one used the trick of so-called Cauchy integral and Minkowski’s inequality. We will use Theorem 1.1 to deal with this problem and obtain a quantitative extrapolation from operators to the corresponding commutators with full ranges (cf. Theorem 1.2). Concerning the proof of Theorem 1.1, we borrow the ideas from [30, 40], which essentially reduce the multilinear problem to a linear extrapolation (cf. Theorem 4.8) by acting on one function at a time. In the linear case, the core of the proof is to obtain the quantitative bounds, which is due to the sharp weighted estimate (1.9) and sharp reverse Hölder’s inequality in Lemma 2.3.
In order to present an extrapolation theorem for commutators, let us introduce relevant notation and some definitions. Given a function \(b \in L^1_{{\text {loc}}}(\mathbb {R}^n)\), we say that \(b \in {\text {BMO}}\) if
where the supremum is taken over the collection of all cubes \(Q \subset \mathbb {R}^n\). Here and elsewhere, we write .
Let T be an operator from \(X_1 \times \cdots \times X_m\) into Y, where \(X_1, \ldots , X_m\) are some normed spaces and and Y is a quasi-normed space. Given \(\textbf{f}:= (f_1, \ldots ,f_m) \in X_1 \times \cdots \times X_m\), \({\textbf {b}}=(b_1, \ldots , b_m)\) of measurable functions, and \(k \in \mathbb {N}\), we define, whenever it makes sense, the k-th order commutator of T in the i-th entry of T as
where \(e_i\) is the basis of \(\mathbb {R}^n\) with the i-th component being 1 and other components being 0. Then, for a multi-index \(\alpha = (\alpha _1, \ldots , \alpha _m) \in \mathbb {N}^m\), we define
In particular, if T is an m-linear operator with a kernel representation of the form
then one can write \([T, {\textbf {b}}]_{\alpha }\) as
Our second main result is the following.
Theorem 1.2
Let T be an m-linear operator and let \(1 \le \mathfrak {p}_i^{-}<\mathfrak {p}_i^{+} \le \infty \), \(i=1, \ldots , m\), be such that \(\frac{1}{\mathfrak {p}_+}:= \sum _{i=1}^m \frac{1}{\mathfrak {p}_i^+}<1\). Assume that for each \(i=1, \ldots , m\), there exists an exponent \(q_i \in (0, \infty )\) with \(q_i \in {[}\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+}]\) such that for all weights \(v_i^{q_i} \in A_{q_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/q_i)'}\), \(i=1, \ldots , m\), we have
where \(\textbf{f}=(f_1, \ldots , f_m)\), \(\frac{1}{q} = \sum _{i=1}^m \frac{1}{q_i}\), \(v= \prod _{i=1}^m v_i\), and \(\Phi _i: [1, \infty ) \rightarrow [1, \infty )\) is an increasing function. Then for all exponents \(p_i \in (\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+})\), all weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/p_i)'}\), for all functions \({\textbf {b}}=(b_1,\ldots , b_m) \in {\text {BMO}}^m\), and for each multi-index \(\alpha \in \mathbb {N}^m\),
whenever \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), \(i=1, \ldots , m\), satisfy \(\frac{1}{s}:= \sum _{i=1}^m \frac{1}{s_i} \le 1\), where \(\frac{1}{p} = \sum _{i=1}^m \frac{1}{p_i}\), \(w=\prod _{i=1}^m w_i\), \({\widetilde{\Phi }}_i(t):= t^{\alpha _i \max \{1, \frac{1}{\tau _{s_i}-1}\}} \Phi _i(C_i \, t^{\gamma _i(s_i, q_i)})\), \(C_i\) depends only on n, \(s_i\), \(q_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), \(C'_i\) depends only on n, \(p_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), and \(C_0\) depends only on \(\alpha \), n, \(p_i\), \(q_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\).
Moreover, for the same family of exponents \(\textbf{p}\), weights \(\textbf{w}\), functions \({\textbf {b}}\), multi-index \(\alpha \), and for all exponents \(r_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\),
where \(\textbf{f}^k=(f_1^k, \ldots , f_m^k)\), \(\frac{1}{r} = \sum _{i=1}^m \frac{1}{r_i}\), C depends only on \(\alpha \), n, \(p_i\), \(q_i\), \(r_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), and \(C''_i\) depends only on n, \(p_i\), \(r_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\).
Remark 1.3
Let us see the existence of \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), \(i=1, \ldots , m\), satisfying \(\frac{1}{s}:= \sum _{i=1}^m \frac{1}{s_i} \le 1\). Indeed, by means of Theorem 1.1, the estimate (1.5) can be improved to all exponents \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), \(i=1, \ldots , m\). Given \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), \(i=1, \ldots , m\), there holds
This means that whenever \(\mathfrak {p}_+>1\), one can always choose \(s_i\) (for example, sufficiently close to \(\mathfrak {p}_i^+\)) such that \(\frac{1}{s} \le 1\).
To illustrate the existence, we present a special case:
where \(\frac{1}{\mathfrak {p}_{\pm }}:= \sum _{i=1}^m \frac{1}{\mathfrak {p}_i^{\pm }}\). In this scenario, picking
we easily verify that \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\) and
provided \(\mathfrak {p}_+>1\).
Remark 1.4
Let T be an m-linear operator. If the hypotheses (1.2) and (1.5) are assumed for T and all exponents \(q_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), then we will get better estimates. This means the following extrapolation: Assume that for all exponents \(p_i \in (\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+})\) and all weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/p_i)'}\), \(i=1, \ldots , m\),
where \(\frac{1}{p} = \sum _{i=1}^m \frac{1}{p_i}\) and \(w=\prod _{i=1}^m w_i\). Then for all exponents \(p_i, r_i \in (\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+})\) and all weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/p_i)'}\), \(i=1, \ldots , m\), we have
where \(\textbf{f}^k=(f_1^k, \ldots , f_m^k)\), \(\frac{1}{r} = \sum _{i=1}^m \frac{1}{r_i}\), \(C_0\) and \(C_i\) depend only on n, \(p_i\), \(r_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\).
Moreover, for the same family of exponents \(\textbf{p}\) and weights \(\textbf{w}\), for all functions \({\textbf {b}}=(b_1,\ldots , b_m) \in {\text {BMO}}^m\), and for each multi-index \(\alpha \in \mathbb {N}^m\), we have
and
whenever \(\frac{1}{s}:= \sum _{i=1}^m \frac{1}{s_i} \le 1\) with \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), where \({\widetilde{\Phi }}_i(t):= t^{\alpha _i \max \{1, \frac{1}{\tau _{s_i}-1}\}} \Phi _i(C_i \, t)\), \(C'_0\) depends only on \(\alpha \), n, \(p_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), \(C'_i\) depends only on n, \(p_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), and \(C''_0\) depends only on \(\alpha \), n, \(p_i\), \(r_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), and \(C''_i\) depends only on n, \(p_i\), \(r_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\). The proof is the same as that of Theorems 1.1 and 1.2. Details are left to the reader.
1.2 Historical Background
In the last two decades, it has been of great interest to obtain sharp weighted norm inequalities for operators T, which concerns estimates of the form
where the positive constant \(C_{n, p, T}\) depends only on n, p, and T, and the exponent \(\alpha _p(T)\) is optimal such that (1.8) holds. This kind of estimates gives the exact rate of growth of the weights norm. The first result was given by Buckley [10] for the Hardy–Littlewood maximal operator M that
and the exponent \(\frac{1}{p-1}\) is the best possible. The problem (1.8) for singular integrals gained new momentum from certain important applications to PDE. In the borderline case, a long-standing regularity problem for the solution of Beltrami equation on the plane was conjectured by Astala, Iwaniec, and Saksman [1], and first settled by Petermichl and Volberg [83] based on the sharp weighted estimate for the Ahlfors-Beurling operator B with \(\alpha _2(B)=1\). Then a question arose whether (1.8) with \(\alpha _2(T)=1\) holds for the general Calderón–Zygmund operators T, which is known as the \(A_2\) conjecture. Focusing on the critical case \(p=2\) results from a quantitative version of Rubio de Francia extrapolation due to Dragičević et al. [38].
Since then, many remarkable publications came to enrich the literature in this area. Petermichl [80] applied the method of Bellman function to obtain (1.8) for Hilbert transform H by showing \(\alpha _2(H)=1\). The same estimate holds for Riesz transforms \(R_j\) on \(\mathbb {R}^n\), see [81]. Later on, Lacey, Petermichl, and Reguera [62] investigated Haar shift operators \(S_{\tau }\) with parameter \(\tau \) in order to present a unified approach to obtain the sharp weighted estimates for B, H, and \(R_j\), by proving \(\alpha _2(S_{\tau })=1\) and noting that such three kinds of operators can be obtained by appropriate averaging of Haar shifts, see [39, 79, 82]. By means of local mean oscillation and extrapolation with sharp constants [38], Lerner [64] established the sharp estimates (1.8) for Littlewood–Paley operators S with \(\alpha _p(S)=\max \{\frac{1}{2}, \frac{1}{p-1}\}\), and Cruz-Uribe et al. [33] gave an alternative and simpler proof of (1.8) for B, H, and \(R_j\). In 2012, Hytönen [51] fully solved the \(A_2\) conjecture by showing a resulting representation of an arbitrary Calderón–Zygmund operator as an average of dyadic shifts over random dyadic systems. Significantly, it opened the study of dyadic analysis in the fields including the multilinear theory, the multiparameter theory, and the non-homogeneous theory. In particular, in terms of sharp weighted estimates, it promoted the development of sparse domination for varieties of operators. To sum up, there are three kinds of sparse domination: identities with suitable averaging, pointwise dominations, and bilinear forms. The specific type depends on the singularity of operators. For example, the Calderón–Zygmund operator [51] and Riesz potential [17] can be recovered from dyadic operators by averaging over dyadic grids. The pointwise sparse dominations hold for the Calderón–Zygmund operators [65] and the corresponding commutators [69], the multilinear Calderón–Zygmund operators [37], the multilinear pseudo-differential operators [20], and the multilinear Littlewood–Paley operators with minimal regularity [21]. Additionally, the sparse domination with a bilinear form goes to singular non-integral operators [9], Bochner–Riesz multipliers [6, 60], rough operators [27], and oscillatory integrals [63].
As aforementioned, one of the most useful and powerful tools in the weighted theory is the celebrated Rubio de Francia extrapolation theorem [84], which states that if a given operator T is bounded on \(L^{p_0}(w_0)\) for some \(p_0 \in [1, \infty )\) and for all \(w_0 \in A_{p_0}\), then T is bounded on \(L^p(w)\) for all \(p \in (1, \infty )\) and for all \(w \in A_p\). Indeed, extrapolation theorems allow us to reduce the general weighted \(L^p\) estimates for certain operators to a suitable case \(p=p_0\), for example, see [20] for the Coifman-Fefferman’s inequality for \(p_0 = 1\), [33, 51] for the Calderón–Zygmund operators for \(p_0 = 2\), [33, 64] for square functions for \(p_0=3\), and [61] for fractional integral operators for \(p_0 \in (1, n/\alpha )\) with \(0<\alpha <n\). Even more, the technique of extrapolation can refine some weighted estimates, see [31] for the Sawyer conjecture, [66, 67] for the weak Muckenhoupt–Wheeden conjecture, and [20, 77] for the local exponential decay estimates. Another interesting point is that by means of extrapolation, the vector-valued inequalities immediately follows from the corresponding scalar-valued estimates.
Over the years, Rubio de Francia’s result has been extended and complemented in different manners, see [32] and the references therein. Using the boundedness of the Hardy–Littlewood maximal operator instead of the Muckenhoupt weights, Cruz-Uribe and Wang [35] presented extrapolation in variable Lebesgue spaces, which was improved to generalized Orlicz spaces [29] and general Banach function spaces [18]. It is worth mentioning that the latter was stated in measure spaces and for general Muckenhoupt bases. This leads lots of applications, such as the well-posedness of the Dirichlet problem in the upper half-space whenever the boundary data belongs to different function spaces, the weighted boundedness of layer potential operators on domains, and the local Tb theorem for square functions in non-homogeneous spaces. Recently, a longstanding problem about extrapolation for multilinear Muckenhoupt classes of weights was solved by Li, Martell, and Ombrosi [71] by introducing some new multilinear Muckenhoupt classes \(A_{\textbf{p}, \textbf{r}}\) (cf. Definition 2.7), which contains the multivariable nature and is a generalization of the classes \(A_{\textbf{p}}\) introduced in [68] (cf. (2.31) below). Shortly afterwards, it was improved to the case with infinite exponents in [72] and with a quantitative bound in [76]. On the other hand, Hytönen and Lappas [53, 54] established a “compact version” of Rubio de Francia’s extrapolation theorem, which allows one to extrapolate the compactness of an operator from just one space to the full range of weighted spaces, provided that the operator is bounded. This result has been extended to the multilinear setting [19] by means of weighted interpolation for multilinear compact operators and weighted Fréchet–Kolmogorov characterization of compactness in the non-Banach case.
1.3 Structure of the Paper
In Sect. 2, we present some preliminaries and auxiliary results including the embedding and factorization of Muckenhoupt weights. Section 3 includes quantitative weighted estimates for various operators. Section 4 is devoted to showing Theorems 1.1 and 1.2 by means of a limited range off-diagonal extrapolation and extrapolation for commutators with Banach ranges. We also establish “product-type embedding” theorems to deduce quantitative \(A_p\) and off-diagonal extrapolation. In Sect. 5, we include many applications of Theorems 1.1 and 1.2. First, we give quantitative weighted norm inequalities for the bilinear Bochner–Riesz means of order \(\delta \) and commutators, where we utilize the \(A_{p_1} \times A_{p_2}\) weights when \(\delta \ge n-1/2\), and the \(A_{p_1/\mathfrak {p}_1^-} \cap RH_{(\mathfrak {p}_1^+/p_1)'} \times A_{p_2/\mathfrak {p}_2^-} \cap RH_{(\mathfrak {p}_2^+/p_2)'}\) weights when \(0<\delta <n-1/2\). The same weights conditions are used for the bilinear rough singular integrals for \(\Omega \in L^{\infty }(\mathbb {S}^{n-1})\) and \(L^q(\mathbb {S}^{n-1})\) with \(q \in (1, \infty )\), respectively. Additionally, under the minimal Sobolev regularity, we obtain the quantitative weighted bounds for the m-linear Fourier multipliers, the corresponding higher order commutators, and vector-valued inequalities, which only hold for product of scalar weights as mentioned before. Beyond that, after presenting quantitative weighted Littlewood–Paley theory, we establish weighted jump and variational inequalities for rough operators with \(\Omega \in L^q(\mathbb {S}^{n-1})\) with \(q \in (1, \infty )\). The proof also needs quantitative weighted estimates for rough singular integrals \(T_{\Omega }\) and rough maximal operators \(M_{\Omega }\), see Sect. 3. They contain many applications to Harmonic Analysis since variation inequalities not only immediately yield the pointwise convergence of the family of operators without using the Banach principle, but also can be used to measure the speed of convergence. Finally, we end up Sect. 5 with Riesz transforms associated to Schrödinger operators.
2 Preliminaries and Auxiliary Results
A measurable function w on \(\mathbb {R}^n\) is called a weight if \(0<w(x)<\infty \) for a.e. \(x \in \mathbb {R}^n\). For \(p \in (1, \infty )\), we define the Muckenhoupt class \(A_p\) as the collection of all weights w on \(\mathbb {R}^n\) satisfying
where the supremum is taken over all cubes \(Q \subset \mathbb {R}^n\). As for the case \(p=1\), we say that \(w\in A_1\) if
Then, we define \(A_{\infty }:=\bigcup _{p\ge 1}A_p\) and \([w]_{A_{\infty }}=\inf _{p>1} [w]_{A_p}\).
Given \(1 \le p \le \infty \) and \(0<q \le \infty \), we say that \(w \in A_{p,q}\) if it satisfies
where one has to replace the first term by \(\mathop {\mathrm {ess\,sup}}\limits _Q w\) when \(q=\infty \) and the second term by \(\mathop {\mathrm {ess\,sup}}\limits _Q w^{-1}\) when \(p=1\). One can easily check that \(w \in A_{p, q}\) if and only if \(w^q \in A_{1+q/p'}\) if and only if \(w^{-p'} \in A_{1+p'/q}\) with
If \(p=1\) and \(0<q<\infty \), then \(w \in A_{p, q}\) if and only if \(w^q \in A_1\) with \([w]_{A_{p, q}} = [w^q]_{A_1}^{\frac{1}{q}}\). If \(1<p \le \infty \) and \(q=\infty \), \(w \in A_{p, q}\) if and only if \(w^{-p'} \in A_1\) with \([w]_{A_{p, q}}=[w^{-p'}]_{A_1}^{\frac{1}{p'}}\).
For \(s\in (1,\infty ]\), the reverse Hölder class \(RH_s\) is the collection of all weights w such that
When \(s=\infty \), is understood as \((\mathop {\mathrm {ess\,sup}}\limits _{Q}w)\). Define \(RH_1:= \bigcup \limits _{1<s \le \infty } RH_s\). Then we see that \(RH_1=A_{\infty }\) (cf. [45, Theorem 7.3.3]).
2.1 Muchenhoupt Weights
The Hardy–Littlewood maximal operator M is defined by
where the supremum is taken over all cubes \(Q \subset \mathbb {R}^n\) containing x. We begin with the following estimate concerning the growth of \(C_{n, p}\) in (1.9) with respect to n and p.
Lemma 2.1
For any \(p \in (1, \infty )\) and \(w \in A_p\),
Proof
We follow the proof of [45, Theorem 7.1.9] to track the precise constants. Given a weight w, the centered weighted Hardy-Littlewood maximal operator \(M_w^c\) is defined by
where the supremum is taken over all cubes \(Q \subset \mathbb {R}^n\) centered at x. Let \(M^c\) denote \(M_w^c\) when \(w \equiv 1\). It was proved in [45, p.509] that
which together with interpolation theorem gives that for any weight w,
To proceed, we fix \(w \in A_p\) with \(p \in (1, \infty )\), and set \(\sigma :=w^{-\frac{1}{p-1}}\). As shown in [45, p. 508] that
which along with (2.3) in turn implies
The proof is complete. \(\square \)
Based on the weighted boundedness of Hardy–Littlewood maximal operator above, one can establish Rubio de Francia extrapolation theorem below, whose proof was contained in [32].
Theorem 2.2
For any \(p \in (1, \infty )\) and \(w \in A_p\), there exists an operator \(\mathcal {R}: L^p(w) \rightarrow L^p(w)\) such that for every non-negative function \(h \in L^p(w)\),
-
(a)
\(h \le \mathcal {R} h\);
-
(b)
\(\Vert \mathcal {R} h\Vert _{L^p(w)} \le 2 \Vert h\Vert _{L^p(w)}\);
-
(c)
\(\mathcal {R}h \in A_1\) with \([\mathcal {R} h]_{A_1} \le 2 \Vert M\Vert _{L^p(w) \rightarrow L^p(w)}\).
Let us recall the sharp reverse Hölder’s inequality.
Lemma 2.3
Let \(p \in (1, \infty )\) and \(w \in A_p\). Then there holds
for every cube Q, where
In particular, for any measurable subset \(E \subset Q\),
Proof
The estimate (2.4) was proved in [26, 55, 66]. Let us prove (2.6). If we set \(r:=1+\gamma _w\), then (2.4) implies that for any measurable subset \(E \subset Q\),
This shows (2.6). \(\square \)
Lemma 2.4
For any \(q \in (1, \infty )\) and \(v \in A_q\), there exist \(\gamma \in (0, 2^{-n-3})\) and \(q_0 \in (1, q)\) such that
Proof
Let \(q \in (1, \infty )\) and \(v \in A_q\). Then, \(v^{1-q'} \in A_{q'}\), and by Lemma 2.3,
and
for any cube \(Q \subset \mathbb {R}^n\), where
Setting
we see that
and use Jensen’s inequality and (2.8)–(2.10) to obtain
and
which immediately implies (2.7). \(\square \)
Lemma 2.5
The following properties hold:
-
(a)
Let \(1 \le p \le p_0 < \infty \). Then for any \(u \in A_p\) and \(v \in A_1\),
$$\begin{aligned} uv^{p-p_0} \in A_{p_0} \quad \text {with}\quad [u v^{p-p_0}]_{A_{p_0}} \le [u]_{A_p} [v]_{A_1}^{p_0-p}. \end{aligned}$$ -
(b)
Let \(1 \le q_0, q_1<\infty \). Then for any \(w_0 \in A_{q_0}\), \(w_1 \in A_{q_1}\), and \(\theta \in [0, 1]\),
$$\begin{aligned} {[}w]_{A_q} \le [w_0]_{A_{q_0}}^{(1-\theta ) \frac{q}{q_0}} {[}w_1]_{A_{q_1}}^{\theta \frac{q}{q_1}}, \end{aligned}$$where \(\frac{1}{q} = \frac{1-\theta }{q_0} + \frac{\theta }{q_1}\) and \(w^{\frac{1}{q}} = w_0^{\frac{1-\theta }{q_0}} w_1^{\frac{\theta }{q_1}}\). In particular, for any \(1 \le p_0< p<\infty \), \(u \in A_p\), and \(v \in A_1\),
$$\begin{aligned} u^{\frac{p_0-1}{p-1}} v^{\frac{p-p_0}{p-1}} \in A_{p_0} \quad \text { with }\quad \big [u^{\frac{p_0-1}{p-1}} v^{\frac{p-p_0}{p-1}} \big ]_{A_{p_0}} \le [u]_{A_p}^{\frac{p_0-1}{p-1}} {[}v]_{A_1}^{\frac{p-p_0}{p-1}}. \end{aligned}$$
Proof
We begin with showing part (a). Let \(u \in A_p\) and \(v \in A_1\). For each cube Q,
Set \(r=\frac{p'-1}{p'_0-1} = \frac{p_0-1}{p-1} \ge 1\). Then \(r'=\frac{p_0-1}{p_0-p}\), and by Hölder’s inequality,
Then it follows from (2.12) and (2.13) that \([u v^{p-p_0}]_{A_{p_0}} \le [u]_{A_p} [v]_{A_1}^{p_0-p}\).
Next, let us prove part (b). Note that \(\frac{1}{q} = \frac{1-\theta }{q_0} + \frac{\theta }{q_1}\), and then
Thus, Hölder’s inequality gives
and
By definition, (2.14), and (2.15), we immediately obtain \([w]_{A_q} \le [w_0]_{A_{q_0}}^{(1-\theta ) \frac{q}{q_0}} [w_1]_{A_{q_1}}^{\theta \frac{q}{q_1}}\). To conclude the proof, it suffices to pick
and note that \(w_0^{\frac{1-\theta }{q_0}} w_1^{\frac{\theta }{q_1}} =u^{\frac{p_0-1}{p_0(p-1)}} v^{\frac{p-p_0}{p_0(p-1)}} =w^{\frac{1}{p_0}} =w^{\frac{1}{q}}\), and
The proof is complete. \(\square \)
We sum up some of the properties of these classes in the following result.
Lemma 2.6
The following statements hold:
-
(a)
For any \(w_1, w_2 \in A_1\), \(w:=w_1^{1/s} w_{2}^{1-p} \in A_p \cap RH_s\) for all \(1 \le p < \infty \) and \(1<s \le \infty \). Moreover,
$$\begin{aligned} \max \{[w]_{A_p}, [w]_{RH_s} \} \le [w_1]_{A_1}^{\frac{1}{s}} {[}w_2]_{A_1}^{p-1}. \end{aligned}$$(2.16) -
(b)
Given \(1 \le p < \infty \) and \(1 \le s < \infty \), \(w \in A_p \cap RH_s\) if and only if \(w^s \in A_{\tau }\). Moreover,
$$\begin{aligned} {[}w^s]_{A_{\tau }} \le [w]_{A_p}^s [w]_{RH_s}^s \quad \text {and}\quad \max \big \{[w]_{A_p}^s, [w]_{RH_s}^s \big \} \le [w^s]_{A_{\tau }}, \end{aligned}$$(2.17)where \(\tau =s(p-1)+1\).
-
(c)
Let \(1 \le \mathfrak {p}_- < \mathfrak {p}_+ \le \infty \) and \(p \in (\mathfrak {p}_-, \mathfrak {p}_+)\). Then \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\) if and only if \(w^{-p'} \in A_{p'/\mathfrak {p}'_+} \cap RH_{(\mathfrak {p}'_-/p')'}\) with
$$\begin{aligned} {[}w^{p(\mathfrak {p}_+/p)'}]_{A_{\tau _p}} = {[}w^{-p'(\mathfrak {p}'_-/p')'}]_{A_{\tau '_p}}^{\tau _p -1}, \end{aligned}$$(2.18)where \(\tau _p= \big (\frac{\mathfrak {p}_+}{p}\big )' \big (\frac{p}{\mathfrak {p}_-} -1\big ) +1\).
-
(d)
Given \(1 \le \mathfrak {p}_- < \mathfrak {p}_+ \le \infty \), \(p \in (\mathfrak {p}_-, \mathfrak {p}_+)\), and \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\), there exists \({\widetilde{\mathfrak {p}}}_- \in (\mathfrak {p}_-, p)\) such that \(w^p \in A_{p/{\widetilde{\mathfrak {p}}}_-} \cap RH_{(\mathfrak {p}_+/p)'}\) with
$$\begin{aligned} {[}w^{p(\mathfrak {p}_+/p)'}]_{A_{{\widetilde{\tau }}_p}} \le 2^{\tau _p} {[}w^{p(\mathfrak {p}_+/p)'}]_{A_{\tau _p}} \quad \text { and }\quad \frac{\frac{1}{{\widetilde{\mathfrak {p}}}_-}}{\frac{1}{{\widetilde{\mathfrak {p}}}_-} - \frac{1}{p}} < (1+2^{-n-3}) \frac{\frac{1}{\mathfrak {p}_-}}{\frac{1}{\mathfrak {p}_-} - \frac{1}{p}}, \end{aligned}$$(2.19)where \(\tau _p= \big (\frac{\mathfrak {p}_+}{p}\big )' \big (\frac{p}{\mathfrak {p}_-} -1\big ) +1\) and \({\widetilde{\tau }}_p = \big (\frac{\mathfrak {p}_+}{p}\big )' \big (\frac{p}{{\widetilde{\mathfrak {p}}}_-} -1\big ) +1\).
Proof
Parts (a)–(c) are essentially contained in [3, 56]. We present a detailed proof to track the weight norms. To show (a), we fix \(1 \le p < \infty \), \(1<s \le \infty \), and let \(w_1, w_2 \in A_1\). By Jensen’s inequality,
and
when \(p=1\), the inequality (2.21) is replaced by
Then it follows from (2.20)–(2.22) that
Moreover, by definition and Jensen’s inequality, we have
when \(s=\infty \), the above still holds since is replaced by \(\mathop {\mathrm {ess\,sup}}\limits _Q w\). This means
Let us next show (b). Assume first that \(w \in A_p \cap RH_s\). Note that for any cube Q,
This implies
and hence,
On the other hand, assuming \(w^s \in A_{\tau }\), we deduce by Jensen’s inequality and (2.23),
and
which follows from
Hence, (b) follows from (2.24) and (2.27).
We turn to the proof of (c). One can check that
Then it follows that
and for any cube Q,
which implies
Finally, let us demonstrate (d). By part (b), there holds \(v:= w^{p(\mathfrak {p}_+/p)'} \in A_{\tau _p}\), which along with (2.7) and (2.11) applied to exponents \(q=\tau _p\) and \(q_0={\widetilde{\tau }}_p\), to arrive at the first estimate in (2.19) and
Moreover,
This proves the second estimate in (2.19) and completes the proof. \(\square \)
2.2 Multilinear Muckenhoupt Weights
The multilinear maximal operator is defined by
where the supremum is taken over all cubes Q containing x.
We are going to present the definition of the multilinear Muckenhoupt classes \(A_{\textbf{p}, \textbf{r}}\) introduced in [71, 72]. Given \(\textbf{p}=(p_1, \ldots , p_m)\) with \(1 \le p_1, \ldots , p_m \le \infty \) and \(\textbf{r}=(r_1, \ldots , r_{m+1})\) with \(1 \le r_1, \ldots , r_{m+1} < \infty \), we say that \(\textbf{r} \preceq \textbf{p}\) whenever
Analogously, we say that \(\textbf{r} \prec \textbf{p}\) if \(r_i<p_i\) for each \(i=1, \ldots , m\), and \(r'_{m+1}>p\).
Definition 2.7
Let \(\textbf{p}=(p_1,\ldots ,p_m)\) with \(1\le p_1, \ldots , p_m \le \infty \) and let \(\textbf{r}=(r_1, \ldots , r_{m+1})\) with \(1 \le r_1, \ldots , r_{m+1} < \infty \) such that \(\textbf{r} \preceq \textbf{p}\). Suppose that \(\textbf{w}=(w_1,\ldots ,w_m)\) and each \(w_i\) is a weight on \(\mathbb {R}^n\). We say that \(\textbf{w} \in A_{\textbf{p}, \textbf{r}}\) if
where \(\frac{1}{p} = \sum _{i=1}^m \frac{1}{p_i}\), \(w=\prod _{i=1}^m w_i\), and the supremum is taken over all cubes \(Q \subset \mathbb {R}^n\). When \(p=r'_{m+1}\), the term corresponding to w needs to be replaced by \(\mathop {\mathrm {ess\,sup}}\limits _Q w\) and, analogously, when \(p_i=r_i\), the term corresponding to \(w_i\) should be \(\mathop {\mathrm {ess\,sup}}\limits _Q w_i^{-1}\). Also, if \(p_i = \infty \), the term corresponding to \(w_i\) becomes . If \(p=\infty \), one will necessarily have \(r_{m+1} = 1\) and \(p_1= \cdots = p_m=\infty \), hence the term corresponding to w must be \(\mathop {\mathrm {ess\,sup}}\limits _Q w\) while the terms corresponding to \(w_i\) become . When \(r_{m+1} = 1\) and \(p<\infty \) the term corresponding to w needs to be replaced by .
Denote \(A_{\textbf{p}}:= A_{\textbf{p}, (1, \ldots , 1)}\) in Definition 2.7, that is,
where \(\frac{1}{p} = \sum _{i=1}^m \frac{1}{p_i}\) and \(w=\prod _{i=1}^m w_i\). We would like to observe that our definition of the classes \(A_{\textbf{p}}\) and \(A_{\textbf{p}, \textbf{r}}\) is slightly different to that in [68] and [71]. Essentially, they are the same since picking \(w_i=v_i^{p_i}\) for every \(i=1, \ldots , m\) in (2.30) and (2.31), we see that \(\textbf{v}=(v_1, \ldots , v_m)\) belongs to \(A_{\textbf{p}, \textbf{r}}\) in [71] and to \(A_{\textbf{p}}\) in [68], respectively.
Lemma 2.8
Let \(1 \le \mathfrak {p}_i^{-}<\mathfrak {p}_i^{+} \le \infty \), \(i=1, \ldots , m\). Assume that \(p_i \in [\mathfrak {p}_i^-, \mathfrak {p}_i^+]\) and \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i=1, \ldots , m\). Then \(\textbf{w}=(w_1, \ldots , w_m) \in A_{\textbf{q}, \textbf{r}}\) with
for any \(\textbf{q}=(q_1, \ldots , q_m)\) with \(1 \le q_1, \ldots , q_m < \infty \) and \(\textbf{r} = (r_1, \ldots , r_{m+1})\) with \(1 \le r_1, \ldots , r_{m+1} < \infty \) such that \(\textbf{r} \preceq \textbf{q}\), and
Proof
By Lemma 2.6 part (b), one has
where \(\tau _{p_i}:= \big (\frac{\mathfrak {p}_i^+}{p_i} \big )' \big (\frac{p_i}{\mathfrak {p}_i^-} -1 \big ) +1\), \(i=1, \ldots , m\). Set
Then it is easy to check that
which gives
On the other hand, let \(\textbf{q}=(q_1, \ldots , q_m)\) with \(1 \le q_1, \ldots , q_m < \infty \) and \(\textbf{r}=(r_1, \ldots , r_{m+1})\) with \(1\le r_1, \ldots , r_{m+1}<\infty \) such that \(\textbf{r} \preceq \textbf{q}\) and (2.32) holds. It follows from (2.32) and (2.33) that
Thus, writing \(w=\prod _{i=1}^m w_i\), we use (2.36) and Hölder’s inequality to obtain
As a consequence, collecting (2.33), (2.34), (2.35), and (2.37), we conclude that
This completes the proof. \(\square \)
Lemma 2.9
Let \(1 \le \mathfrak {p}_i^{-}<\mathfrak {p}_i^{+} \le \infty \), \(i=1, \ldots , m\). Assume that \(p_i \in [\mathfrak {p}_i^-, \mathfrak {p}_i^+]\) and \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i=1, \ldots , m\). Write \(w=\prod _{i=1}^m w_i\). Then \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\) with
where \(\frac{1}{p}= \frac{1}{p_1} + \cdots + \frac{1}{p_m}\) and \(\frac{1}{\mathfrak {p}_{\pm }} = \frac{1}{\mathfrak {p}_1^{\pm }} + \cdots + \frac{1}{\mathfrak {p}_m^{\pm }}\). In particular, if we take
then it follows
Proof
Set
Observe that
With (2.41)–(2.43) in hand, we use Hölder’s inequality to obtain
Analogously, we have
Then gathering (2.44) and (2.45), we arrive at
which immediately gives (2.38).
To proceed, we note that by (2.28) and (2.39),
which along with Lemma 2.6 part (c) and (2.38) yields
This shows (2.40). \(\square \)
2.3 Multilinear Calderón–Zygmund Operators
Given \(\delta >0\), we say that a function \(K: \mathbb {R}^{n(m+1)} {\setminus } \{x=y_1=\cdots =y_m\} \rightarrow \mathbb {C}\) is a \(\delta \)-Calderón–Zygmund kernel, if there exists a constant \(A>0\) such that
whenever \(|x-x'| \le \frac{1}{2} \max \limits _{1\le j \le m}|x-y_j|\), and for each \(i=1,\ldots ,m\),
whenever \(|y_i-y_i'| \le \frac{1}{2} \max \limits _{1\le j \le m}|x-y_j|\).
An m-linear operator \(T: \mathcal {S}(\mathbb {R}^n) \times \cdots \times \mathcal {S}(\mathbb {R}^n) \rightarrow \mathcal {S}'(\mathbb {R}^n)\) is called a \(\delta \)-Calderón–Zygmund operator if there exists a \(\delta \)-Calderón–Zygmund kernel K such that
whenever \(x \not \in \bigcap _{i=1}^m {\text {supp}}(f_i)\) and \(\textbf{f}=(f_1,\ldots ,f_m) \in \mathscr {C}_c^{\infty }(\mathbb {R}^n) \times \cdots \times \mathscr {C}_c^{\infty }(\mathbb {R}^n)\), and T can be boundedly extended from \(L^{q_1}(\mathbb {R}^n) \times \cdots \times L^{q_m}(\mathbb {R}^n)\) to \(L^q(\mathbb {R}^n)\) for some \(\frac{1}{q}=\frac{1}{q_1}+\cdots +\frac{1}{q_m}\) with \(1<q_1,\ldots ,q_m<\infty \).
Given a symbol \(\sigma \), the m-linear Fourier multiplier \(T_{\sigma }\) is defined by
for all \(f_i \in \mathcal {S}(\mathbb {R}^n)\), \(i=1,\ldots ,m\). The operator \(T_{\sigma }\) is called an m-linear Coifman–Meyer multiplier, if the symbol \(\sigma \in \mathscr {C}^s(\mathbb {R}^{nm} \setminus \{0\})\) satisfies
for each multi-indix \(\alpha =(\alpha _1,\ldots ,\alpha _m)\) with \(|\alpha |=\sum _{i=1}^m|\alpha _i| \le mn+1\).
It was shown in [48, Proposition 6] that Coifman–Meyer multipliers are examples of multilinear Calderón–Zygmund operators.
Below, the sharp weighted inequality for multilinear Calderón–Zygmund operators was given in [73, Theorem 1.4] with \(p \ge 1\) and extended to the case \(p<1\) in [76, Corollary 4.4].
Theorem 2.10
Let T be an m-linear Calderón–Zygmund operator. Then for all \(1< p_1, \ldots p_m < \infty \) and \(\textbf{w} \in A_{\textbf{p}}\),
where \(w=\prod _{i=1}^m w_i\) and \(\frac{1}{p}=\sum _{i=1}^m \frac{1}{p_i}\).
Theorem 2.11
[8, Theorem 4.3] Let T be an m-linear operator. Fix \(\theta _i >0\) and \(r_i \in (1, \infty )\), \(i=1, \ldots , m\). Let \(\frac{1}{p}=\sum _{i=1}^m \frac{1}{p_i} \le 1\) with \(1<p_1, \ldots , p_m<\infty \). Assume that there exist increasing functions \(\Psi _i: [1, \infty ) \rightarrow [0, \infty )\) such that for all \(v_i^{\theta _i} \in A_{r_i}\), \(i=1, \ldots , m\),
where \(v=\prod _{i=1}^m v_i\). Then, for all weights \(w_i^{\eta _i \theta _i} \in A_{s_i}\) with some \(\eta _i \in (1, \infty )\), for all \({\textbf {b}}= (b_1, \ldots , b_m) \in {\text {BMO}}^m\), and for each multi-index \(\alpha \in \mathbb {N}^m\),
where \(w=\prod _{i=1}^m w_i\), and \(\delta _i = \min \{1, r_i-1\}/(\eta '_i \theta _i)\), \(i=1, \ldots , m\).
Let us record Marcinkiewicz–Zygmund inequalities contained in [22, Proposition 5.3].
Lemma 2.12
Let \(0< p, q_1, \ldots , q_m< r < 2\) or \(r=2\) and \(0<p, q_1, \ldots , q_m <\infty \). Let \(\mu _1, \ldots , \mu _m\) and \(\nu \) be arbitrary \(\sigma \)-finite measures on \(\mathbb {R}^n\). Let T be an m-linear operator. Then, there exists a constant \(C>0\) such that the following estimates hold:
-
(i)
If T is bounded from \(L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m)\) to \(L^p(\nu )\), then
$$\begin{aligned} \bigg \Vert \bigg (\sum _{k_1, \ldots , k_m} |T(f^1_{k_1}, \ldots , f^m_{k_m})|^r \bigg )^{\frac{1}{r}}\bigg \Vert _{L^p(\nu )} \le C \Vert T\Vert \prod _{i=1}^m \bigg \Vert \bigg (\sum _{k_i} |f^i_{k_i}|^r \bigg )^{\frac{1}{r}}\bigg \Vert _{L^{q_i}(\mu _i)}, \end{aligned}$$where \(\Vert T\Vert := \Vert T\Vert _{L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m) \rightarrow L^p(\nu )}\).
-
(ii)
If T is bounded from \(L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m)\) to \(L^{p, \infty }(\nu )\), then
$$\begin{aligned} \bigg \Vert \bigg (\sum _{k_1, \ldots , k_m} |T(f^{1}_{k_1}, \ldots , f^m_{k_m})|^r \bigg )^{\frac{1}{r}}\bigg \Vert _{L^{p, \infty }(\nu )} \le C \Vert T\Vert _{\textrm{weak}} \prod _{i=1}^m \bigg \Vert \bigg (\sum _{k_i} |f^i_{k_i}|^r \bigg )^{\frac{1}{r}}\bigg \Vert _{L^{q_i}(\mu _i)}, \end{aligned}$$where \(\Vert T\Vert _\textrm{weak}:= \Vert T\Vert _{L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m) \rightarrow L^{p, \infty }(\nu )}\).
3 Quantitative Weighted Estimates
The goal of this section is to establish quantitative weighted estimates for (rough) maximal operators and singular integrals. We begin with the following interpolation result with change of measures due to Stein and Weiss [87], which plays an important role in dealing with weighted estimates.
Theorem 3.1
[87] Let \(p_0, p_1 \in [1, \infty ]\), and let \(w_0\) and \(w_1\) be weights. If the sublinear operator T satisfies
then for any \(\theta \in (0, 1)\),
where \(\frac{1}{p}=\frac{1-\theta }{p_0} + \frac{\theta }{p_1}\) and \(w^{\frac{1}{p}}=w_0^{\frac{1-\theta }{p_0}} w_1^{\frac{\theta }{p_1}}\).
The sharp maximal function\(M^{\#}\) is defined by
The following Fefferman-Stein inequality was shown in [16, Remark 1.9].
Lemma 3.2
For every \(p \in (0, \infty )\) and \(w \in A_{\infty }\),
whenever \(Mf \in L^p(w)\) or \(f \in L^{\infty }_c(\mathbb {R}^n)\).
We present a sharp weighted vector-valued Fefferman-Stein inequality.
Lemma 3.3
For any \(1< p, r < \infty \) and \(w \in A_p\),
Moreover, the exponent \(\max \{\frac{1}{r}, \frac{1}{p-1}\}\) is the best possible.
Proof
This inequality was given in [33, Theorem 1.12]. We here present a different proof. Let \(r \in (1, \infty )\). It was proved in [23, Theorem 1.11] that there exist \(3^n\) dyadic lattices \(\mathcal {D}_j\) and sparse families \(\mathcal {S}_j \subset \mathcal {D}_j\) such that
where
It follows from [13, Theorem2.1] that for all \(p \in (1, \infty )\) and \(w \in A_p\),
Thus, (3.1) and (3.3) imply the desired estimate. \(\square \)
Lemma 3.4
Let \(\mathscr {B}_1\) and \(\mathscr {B}_2\) be Banach spaces, and let \(\mathscr {L}(\mathscr {B}_1, \mathscr {B}_2)\) be the Banach space defined by all bounded linear operators from \(\mathscr {B}_1\) to \(\mathscr {B}_2\) with the operator norm \(\Vert \cdot \Vert _{\mathscr {L}(\mathscr {B}_1, \mathscr {B}_2)}\). Let T be a linear operator mapping \(\mathscr {B}_1\)-valued functions into \(\mathscr {B}_2\)-valued functions satisfying
-
(i)
T is bounded from \(L^2(\mathbb {R}^n, \mathscr {B}_1)\) into \(L^2(\mathbb {R}^n, \mathscr {B}_2)\).
-
(ii)
There exists a kernel function \(K(x) \in \mathscr {L}(\mathscr {B}_1, \mathscr {B}_2)\) such that
$$\begin{aligned} \Vert K(x-y) - K(x)\Vert _{\mathscr {L}(\mathscr {B}_1, \mathscr {B}_2)} \le C_K |y| |x|^{-n-1}, \quad 2|y| < |x|, \end{aligned}$$and for every \(f \in L^2(\mathbb {R}^n, \mathscr {B}_1)\) with compact support,
$$\begin{aligned} Tf(x) = \int _{\mathbb {R}^n} K(x-y) f(y) \, dy, \qquad \text {a.e. } x \not \in {\text {supp}}(f), \end{aligned}$$
Then for every \(p \in (1, \infty )\) and \(w \in A_p\),
Proof
It was shown in [85, p. 41–42] that
where \(C_T:= \Vert T\Vert _{L^2(\mathbb {R}^n, \mathscr {B}_1) \rightarrow L^2(\mathbb {R}^n, \mathscr {B}_2)} + C_K\). Then interpolating between the assumption (i) and (3.5) yields that for any \(1<r_0<r<2\),
where the implicit constant is independent of \(r_0\). As argued in the proof of (3.5), the inequality (3.6) implies for any \(1<r_0<r<2\),
Now let \(p \in (1, \infty )\) and \(w \in A_p\). Then, by Lemma 2.4, there exists \(\gamma \in (0, 2^{-n-3})\) and \( q_0 \in (1, p)\) such that \(q_0 = \frac{p}{1+\varepsilon }\), \(\frac{p-1}{p(1+\gamma )'}<\varepsilon <\frac{p-1}{(1+\gamma )'}\), \((1+\gamma )' \simeq [w]_{A_p}^{\max \{1, \frac{1}{p-1}\}}\), and \([w]_{A_{q_0}} \le 2^p [w]_{A_p}\). Set \(r:=p/q_0=1+\varepsilon \). If \(r \ge 2\), it follows from Lemma 3.2 and (3.5) that
since
If \(1<r<2\), we choose \(r_0=1+\frac{p-1}{p(1+\gamma )'}\) and invoke Lemma 3.2 and (3.7) to obtain that
This completes the proof. \(\square \)
Lemma 3.5
Let \(\varphi \in \mathcal {S}(\mathbb {R}^n)\) be such that \(\int _{\mathbb {R}^n} \varphi \, dx=0\) and \({\text {supp}}({\widehat{\varphi }}) \subset \{\xi \in \mathbb {R}^n: c_1 \le |\xi | \le c_2\}\) for some \(0<c_1<c_2 < \infty \). Set \(\varphi _k(x):= 2^{kn} \varphi (2^k x)\) for any \(k \in \mathbb {Z}\). Then for every \(p \in (1, \infty )\) and \(w \in A_p\),
If we assume in addition that \(\sum _{k \in \mathbb {Z}} |{\widehat{\varphi }}(2^{-k} \xi )|^2 = C_{\varphi }>0\) for all \(\xi \ne 0\), then
Proof
Since \(\varphi \in \mathcal {S}(\mathbb {R}^n)\), one can check that there exists \(C'_{\varphi }>0\) such that for any \(\beta \in (0, 1]\) and any \(y \in \mathbb {R}^n\), \(|\varphi (x)| \le C'_{\varphi } (1+|x|)^{-n-\beta }\) and
Recalling that \(\int _{\mathbb {R}^n} \varphi \, dx=0\), we see that \(\varphi /C'_{\varphi } \in \mathcal {C}_{\beta , 1}\), which is defined in [90, Definition 6.2]. Then by [90, Theorem 6.3],
where the implicit constant is independent of f and x. Thus, (3.8) follows from (3.12) and the sharp weighted estimate for \(G_{\beta }\) in [64, Theorem 1.1].
To show (3.9), we will use vector-valued singular integrals. By the support of \({\widehat{\varphi }}\), there exist \(j_0, j_1 \in \mathbb {N}\) such that \({\text {supp}}({\widehat{\varphi }}_{j+k}) \cap {\text {supp}}({\widehat{\varphi }}_j) = \emptyset \) whenever \(k \le -j_0-1\) or \(k \ge j_1+1\). This and Plancherel’s identity give
This means that the operator T defined by \(T(\{f_k\}_{k \in \mathbb {Z}}):= \sum _{k \in \mathbb {Z}} \varphi _k*f_k\), is a bounded linear operator from \(L^2(\mathbb {R}^n, \ell ^2)\) to \(L^2(\mathbb {R}^n)\), with the kernel \(K(x)=\{\varphi _k(x)\}_{k \in \mathbb {Z}}\) satisfying \(\Vert \nabla K(x)\Vert _{\mathscr {L}(\ell ^2, \mathbb {C})} \lesssim |x|^{-n-1}\) for all \(x \ne 0\). Hence, Lemma 3.4 implies (3.9).
Note that the inequality (3.10) is a consequence of Lemma 3.3 and that \(|\varphi _k * f_k| \lesssim Mf_k\) uniformly in \(k \in \mathbb {Z}\).
Finally, to get (3.11), we use Parseval’s identity and \(\sum _{k \in \mathbb {Z}} |{\widehat{\varphi }}_k(\xi )|^2=\sum _{k \in \mathbb {Z}} |{\widehat{\varphi }}(2^{-k} \xi )|^2 = C_{\varphi }\) to get that for any \(f, g \in L^2(\mathbb {R}^n)\),
Then it follows that for \(g \in \mathcal {S}(\mathbb {R}^n)\) with \(\Vert g\Vert _{L^{p'}(w^{1-p'})}=1\),
where we have used (3.8) in the last inequality. This gives at once (3.11). \(\square \)
Lemma 3.6
Given \(\varepsilon >0\) and a pairwise disjoint family of cubes \(\{Q_j\}\), we set
Then \(\Vert \mathfrak {M}_{\varepsilon }\Vert _{L^2(w)} \lesssim [w]_{A_2} w(\Omega )^{\frac{1}{2}}\) for any \(w \in A_2\).
Proof
Note that
which together with Lemma 3.3 gives that for any \(w \in A_2\),
where we have use the disjointness of \(\{Q_j\}\). This implies the desired estimate. \(\square \)
Given \(\Omega \in L^1(\mathbb {S}^{n-1})\), the rough maximal operator \(M_{\Omega }\) and singular integral \(T_{\Omega }\) are defined by
and
Theorem 3.7
Let \(q \in (1, \infty )\) and \(\Omega \in L^q(\mathbb {S}^{n-1})\) be such that \(\int _{\mathbb {S}^{n-1}} \Omega \, d\sigma =0\). Then for all \(p \in (q', \infty )\) and for all \(w \in A_{p/q'}\),
Moreover, the vector-valued inequality holds for \(q>2\):
Proof
By definition and Hölder’s inequality, one has
which together with (2.1) immediately gives (3.16). Then (3.18) is a consequence of (3.16), Theorem 1.1, and Remark 1.4.
To treat (3.17), we choose a radial nonnegative function \(\varphi \in \mathscr {C}_c^{\infty }(\mathbb {R}^n)\) such that \({\text {supp}}\varphi \subset \{|x|< 1/4\}\) and \(\int _{\mathbb {R}^n} \varphi \, dx =1\). Set \(\varphi _j(x):= 2^{-nj} \varphi (2^{-j} x)\) and \(\nu _j(x):= \frac{\Omega (x')}{|x|^n} {\textbf {1}}_{\{2^j \le |x|<2^{j+1}\}}(x)\) for each \(j \in \mathbb {Z}\). Define
Then,
It was proved in [89, p. 396] that for some \(\delta _0>0\),
where the implicit constants are independent of j.
On the other hand, it follows from [89, Lemma 2] that
which together with [89, Theorem 2] gives
In particular, (3.24) implies
and the interpolation theorem, (3.21), and (3.24) yield that for some \(\delta >0\),
Hence, by (3.22), (3.25), and [70, Theorem 1.2], we obtain that for any \(f \in L^{\infty }_c(\mathbb {R}^n)\) there exists a sparse family \(\mathcal {S}_j\) such that
where the dyadic operator \(\mathcal {A}_{\mathcal {S}}^{\gamma }\) is defined in (3.2) and the implicit constant is independent of j. Accordingly, we use (3.3), (3.27), and a density argument to arrive at
Now fix \(p \in (q', \infty )\) and \(w \in A_{p/q'}\). By Lemma 2.4, there exists \(\gamma \in (0, 1)\) such that
which along with (3.28) implies
In light of Theorem 3.1 with \(w_0 \equiv 1\), \(w_1=w^{1+\gamma }\), and \(\theta =\frac{1}{1+\gamma }\), interpolating between (3.26) and (3.29) gives
Note that \(1-\theta = \frac{1}{(1+\gamma )'}\) and \(e^{-t} < 2t^{-2}\) for any \(t >0\). As a consequence, (3.19), (3.28), and (3.30) imply
This completes the proof. \(\square \)
Theorem 3.8
Let \(q \in (1, \infty )\) and \(\Omega \in L^q(\mathbb {S}^{n-1})\). Then for all \(p \in (1, q)\) and for all \(w^{1-p'} \in A_{p'/q'}\),
Proof
Fix \(p \in (1, q)\) and \(w^{1-p'} \in A_{p'/q'}\). For \(j \in \mathbb {Z}\), set \(\nu _{\Omega , j}(x):= \frac{\Omega (x')}{|x|^n} {\textbf {1}}_{\{2^j \le |x|<2^{j+1}\}}(x)\). Define
If we set for any \(x' \in \mathbb {S}^{n-1}\), there there holds
Since \(w^{1-p'} \in A_{p'/q'} \subset A_{p'}\), we see that \(w \in A_p\) and by (2.1),
In order to estimate \({\textbf {S}}_{\Omega _0}\), we define a linear operator
Writing
one can verify that [89, Lemmas 1 and 2] hold for k and \(k^{(m)}\), with bounds independent of \(\varepsilon \). This means that \({\textbf {T}}_{\Omega _0}^{\varepsilon }\) behaves as \(T_{\Omega }\) in Theorem 3.7. Then by (3.17),
for any \(s \in (q', \infty )\) and \(v \in A_{s/q'}\). By duality, (3.34) implies
We would like to use (3.35) to bound \({\textbf {S}}_{\Omega _0}\). Let \(\{r_m(\cdot )\}_{m \in \mathbb {N}}\) be the system of Rademacher functions in [0, 1). By Khintchine’s inequality (cf. [45, p. 586]) and (3.35) applied to \(\varepsilon (t):= \{r_m(t)\}_{m \in \mathbb {Z}}\), we have
Therefore, (3.31) follows from (3.32), (3.33), and (3.36). \(\square \)
Lemma 3.9
Let \(\psi \in \mathscr {C}_c^{\infty }(\mathbb {R}^n)\) be a radial function such that \(0 \le \psi \le 1\), \({\text {supp}}\psi \subset \{1/2 \le |\xi | \le 2\}\) and \(\sum _{l \in \mathbb {Z}} \psi (2^{-l}\xi )^2 =1\) for \(|\xi | \ne 0\). Define the multiplier \(\Delta _l\) by \(\widehat{\Delta _l f}(\xi ) = \psi (2^{-l}\xi ) {\widehat{f}}(\xi )\). For \(j \in \mathbb {Z}\), set \(\nu _j(x):= \frac{\Omega (x')}{|x|^n} {\textbf {1}}_{\{2^j\le |x|<2^{j+1}\}}(x)\), where \(\Omega \) is the same as in Theorem 3.7. Then for all \(p \in (q', \infty )\) and \(w \in A_{p/q'}\),
Proof
Let \(p \in (q', \infty )\) and \(w \in A_{p/q'}\). Observe that
This and (3.16) yield
In light of Theorem 3.8, (3.38) implies that for any \(r \in (1, q)\) and \(v^{1-r'} \in A_{r'/q'}\),
which together with duality gives
Then, interpolating between (3.39) and (3.40), we obtain
Combining (3.10) with (3.8) and that \([w]_{A_p} \le [w]_{A_{p/q'}}\), this immediately implies (3.37). \(\square \)
4 Proof of Main Theorems
In this section, we will prove Theorems 1.1 and 1.2. The first step is to show Theorem 1.1, which will follow from Theorem 4.8, a limited rang, off-diagonal extrapolation with quantitative weights norms. Before proving the latter, we present some other quantitative extrapolation.
4.1 \(A_p\) Extrapolation
We begin with the \(A_p\) extrapolation with quantitative bounds.
Theorem 4.1
Let \(\mathcal {F}\) be a family of extrapolation pairs. Assume that there exist exponents \(p_0 \in [1, \infty ]\) such that for all weights \(v^{p_0} \in A_{p_0}\),
where \(\Phi : [1, \infty ) \rightarrow [1, \infty )\) is an increasing function. Then for all exponents \(p \in (1, \infty )\) and all weights \(w^p \in A_p\),
where \(C_p=3^{n(p'+8)(p_0-p)}\) if \(p<p_0\), and \(C_p=3^{n(p+8)}\) if \(p>p_0\).
Theorem 4.1 was shown in [38, 40] without the explicit constant \(C_p\). We restudy it by presenting a stronger result as follows.
Theorem 4.2
Let \(q \in [1, \infty ]\) and \(p \in (1, \infty )\). Then for any \(w^p \in A_p\), \(f \in L^p(w^p)\), and \(g \in L^{p'}(w^{-p'})\), there exists \(v^q \in A_q\) with \([v^q]_{A_q} \le C_p \, [w^p]_{A_p}^{\max \{1, \frac{q-1}{p-1}\}}\) such that
where \(C_p=3^{n(p'+8)(q-p)}\) if \(p<q\), and \(C_p=3^{n(p+8)}\) if \(p>q\).
Proof
Let \(p \in (1, \infty )\), \(w^p \in A_p\), \(f \in L^p(w^p)\), and \(g \in L^{p'}(w^{-p'})\). We may assume that f and g are nonnegative and non-trivial. Let us first consider the case \(p<q\). By \(w^p \in A_p\) and Theorem 2.2, there exists an operator \(\mathcal {R}: L^p(w^p) \rightarrow L^p(w^p)\) such that
Define
Then by Lemma 2.5, the last estimate in (4.4), and (2.1),
On the other hand, it follows from the first two estimate in (4.4) that
In view of \(p<q\), we set \(\frac{1}{r}:= \frac{1}{p} - \frac{1}{q}\). Then, \(\frac{1}{q'}=\frac{1}{p'} + \frac{1}{r}\), and by Hölder’s inequality,
Observe that
where the second estimate estimate in (4.4) was used in the last step. Now collecting (4.7)–(4.9), we obtain
This and (4.6) show (4.3) in the case \(p<q\).
Let us deal with the case \(q<p\), which is equivalent to \(p'<q'\). Also, \(w^p \in A_p\) is equivalent to \(w^{-p'} \in A_{p'}\). Note that \(g \in L^{p'}(w^{-p'})\) and \(f \in L^p(w^p)\). Invoking (4.3) for \(p'\), \(q'\), g, f, \(w^{-1}\) in place of p, q, f, g, and w, respectively, one can find a weight \(u^{q'} \in A_{q'}\) with
such that
Picking \(v=u^{-1}\) and using (4.10), we see that
and (4.11) can be rewritten as
This shows (4.3) in the case \(q<p\). \(\square \)
Proof of Theorem 4.1
Let \(p \in (1, \infty )\) and \(w^p \in A_p\). By duality,
Fix a nonnegative function \(h \in L^{p'}(w^{-p'})\) with \(\Vert hw^{-1}\Vert _{L^{p'}} = 1\). In view of Theorem 4.2, one can find a weight \(v^{p_0} \in A_{p_0}\) such that
where \(C_p=3^{n(p'+8)(p_0-p)}\) if \(p<p_0\), and \(C_p=3^{n(p+8)}\) if \(p>p_0\). Hence, by (4.1), (4.13), and (4.14),
which along with (4.12) yields at once (4.2) as desired. \(\square \)
Next, we would like to use Theorem 4.1 to get additional results.
Theorem 4.3
Let \(\mathcal {F}\) be a family of extrapolation pairs. Assume that there exist exponents \(p_0 \in (0, \infty )\) and \(q_0 \in [1, \infty )\) such that for all weights \(v \in A_{q_0}\),
where \(\Phi : [1, \infty ) \rightarrow [1, \infty )\) is an increasing function. Then for all exponents \(p \in (1, \infty )\) and all weights \(w \in A_p\),
where \(C_p=3^{n(p'+8)(q_0-p)}\) if \(p<q_0\), and \(C_p=3^{n(p+8)}\) if \(p>q_0\).
Proof
Set
Note that (4.15) implies that for all weights \(v \in A_{q_0}\),
for all \((F, G) \in {\widetilde{\mathcal {F}}}\). Then it follows from (4.17) and Theorem 4.1 with \(p_0\) replaced by \(q_0\) that for all exponent \(p \in (1, \infty )\) and for all weights \(w \in A_p\),
which can be rewritten as
where \(C_p=3^{n(p'+8)(q_0-p)}\) if \(p<q_0\), and \(C_p=3^{n(p+8)}\) if \(p>q_0\). This shows (4.16). \(\square \)
Theorem 4.4
Let T be a sublinear operator. Assume that there exists \(p_0 \in [1, \infty )\) such that for all \(v \in A_{p_0}\),
where \(\Phi : [1, \infty ) \rightarrow [1, \infty )\) is an increasing function. Then for all \(p \in (1, \infty )\) and for all \(w \in A_p\),
Proof
Given an arbitrary number \(\lambda >0\), we denote
The hypothesis (4.18) implies that for all weights \(v \in A_{p_0}\),
for all \((F_{\lambda }, G) \in \mathcal {F}_{\lambda }\). Thus, (4.21) means that (4.1) is satisfies for the family \(\mathcal {F}_{\lambda }\). Then Theorem 4.1 yields that for all exponents \(p \in (1, \infty )\) and all weights \(w \in A_p\),
where \(C_p=3^{n(p'+8)(p_0-p)}\) if \(p<p_0\), and \(C_p=3^{n(p+8)}\) if \(p>p_0\), which along with the arbitrariness of \(\lambda \) implies (4.19).
To prove (4.20), we fix \(q \in (1, \infty )\) and \(w \in A_q\). By Lemma 2.4, there exist \(\gamma \in (0, 1)\) and \(q_0 \in (1, q)\) so that
We may assume that \(\varepsilon <\frac{1}{2}\) since in this case (4.22) still holds. Choose \(q_1:= \frac{q}{1-\varepsilon } \in (q, 2q)\) such that \(\frac{1}{q}=\frac{1-\theta }{q_0} + \frac{\theta }{q_1}\) with \(\theta =\frac{1}{2}\). Then,
Then it follows from (4.22), (4.23), and (4.19) (with the exact constant \(C_p\), see the proof above) that
where
Additionally, by the choice of \(q_0\) and \(q_1\), and that \(\varepsilon <(q-1)/2\), we have
where \(C'_q\) depends only on n, \(p_0\), and q. Thus, invoking (4.27), we interpolate between (4.24) and (4.25) to conclude
This completes the proof. \(\square \)
4.2 Off-Diagonal Extrapolation
We next present a quantitative off-diagonal extrapolation below, which improves Theorem 4.1 to the limited range case.
Theorem 4.5
Let \(\mathcal {F}\) be a family of extrapolation pairs and \(1 \le \mathfrak {p}_- < \mathfrak {p}_+ \le \infty \). Assume that there exist exponents \(p_0 \in [\mathfrak {p}_-, \mathfrak {p}_+]\) and \(q_0 \in (1, \infty )\) such that for all weights \(v^{p_0} \in A_{p_0/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p_0)'}\),
where \(\Phi : [1, \infty ) \rightarrow [1, \infty )\) is an increasing function. Then for all exponents \(p \in (\mathfrak {p}_-, \mathfrak {p}_+)\) and \(q \in (1, \infty )\) satisfying \(\frac{1}{p} - \frac{1}{q} = \frac{1}{p_0} - \frac{1}{q_0}\), and all weights \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\),
To show Theorem 4.5, we present a more general result below.
Theorem 4.6
Let \(\beta \in (0, \infty )\), \(p_0, q_0 \in [1, \infty )\), \(p, q \in (1, \infty )\), and let \(r_0, r \in (\frac{1}{\beta }, \infty )\) be such that \(\frac{1}{q} - \frac{1}{q_0} = \frac{1}{r} - \frac{1}{r_0} = \frac{1}{p} - \frac{1}{p_0}\). Then for all weights \(w^r \in A_{r \beta }\) and for all functions \(f \in L^p(w^p)\) and \(g \in L^{q'}(w^{-q'})\), there exists a weight \(v^{r_0} \in A_{r_0 \beta }\) such that
Proof
Fix \(w^r \in A_{r \beta }\), \(f \in L^p(w^p)\), and \(g \in L^{q'}(w^{-q'})\). We first consider the case \(q<q_0\) (equivalently, \(p<p_0\) and \(r<r_0\)). Pick
By \(w^r \in A_{r \beta }\) and Theorem 2.2, there exists an operator \(\mathcal {R}: L^{r \beta }(w^r) \rightarrow L^{r \beta }(w^r)\) such that
Define
Then by Lemma 2.5 part (a), the last inequality in (4.33), and (2.1),
It follows from (4.32), (4.33), and (4.34) that
To proceed, we set \(\frac{1}{t}:= \frac{1}{q} - \frac{1}{q_0}\), equivalently \(\frac{1}{q'_0} = \frac{1}{q'} + \frac{1}{t}\). By Hölder’s inequality,
Now collecting (4.36), (4.37), and (4.38), we deduce that
provided \(\frac{1}{q} - \frac{1}{q_0} = \frac{1}{r} - \frac{1}{r_0} = \frac{1}{p} - \frac{1}{p_0}\). This shows the case \(q<q_0\).
Next let us deal with the case \(q>q_0\) (equivalently, \(p>p_0\) and \(r>r_0\)). Set
Recall that \(w^r \in A_{r \beta }\). Then we see that
Hence, the conclusion in the preceding case applied to the tuple \((q', p', s, q'_0, p'_0, s_0, g, f, w^{-1})\) in place of \((p, q, r, p_0, q_0, r_0, f, g, w)\) gives that there exists a weight \(u^{s_0} \in A_{s_0 \beta }\) so that
Note that by (4.40),
Pick \(v:= u^{-1}\). Then by (4.41), (4.42), and (4.44),
and (4.43) can be rewritten as
In the case \(q=q_0\), taking \(v:=w\), the conclusion is trivial. This completes the proof. \(\square \)
The following conclusion is a particular case of Theorem 4.6.
Theorem 4.7
Let \(1 \le \mathfrak {p}_- < \mathfrak {p}_+ \le \infty \), \(p_0 \in [\mathfrak {p}_-, \mathfrak {p}_+]\), \(p \in (\mathfrak {p}_-, \mathfrak {p}_+)\), and let \(q_0, q \in (1, \infty )\) be such that \(\frac{1}{q} - \frac{1}{q_0} = \frac{1}{p} - \frac{1}{p_0}\). Then for all weights \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\) and for all functions \(f \in L^p(w^p)\) and \(g \in L^{q'}(w^{-q'})\), there exists a weight \(v^{p_0} \in A_{p_0/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p_0)'}\) such that
Proof
Denote
Then one can check that
Let \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\), \(f \in L^p(w^p)\), and \(g \in L^{q'}(w^{-q'})\). Then it follows from Lemma 2.6 part (b) and (4.48) that \(w^r \in A_{r \beta }\), which together with Theorem 4.6 implies that there exists a weight \(v^{r_0} \in A_{r_0 \beta }\) such that
In view of (4.47), (4.48), and Lemma 2.6 part (b), we conclude from (4.49) and (4.50) that \(v^{p_0} \in A_{p_0/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p_0)'}\) so that (4.45) and (4.46) hold. \(\square \)
Let us see how we deduce Theorem 4.5 from Theorem 4.7.
Proof of Theorem 4.5
By duality,
Fix a nonnegative function \(h \in L^{q'}(w^{-q'})\) with \(\Vert hw^{-1}\Vert _{L^{q'}} = 1\). By Theorem 4.7, there exists a weight \(v^{p_0} \in A_{p_0/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p_0)'}\) such that
Then, in view of (4.52), we use (4.28) and (4.53) to obtain
This along with (4.51) gives at once (4.29) as desired. \(\square \)
4.3 Multilinear Extrapolation
If we use Theorem 4.5 to show Theorem 1.1, it requires all the exponents are Banach. Thus, we have to improve Theorem 4.5 to the non-Banach ranges as follows. But in this case, we cannot establish a “product-type embedding” as Theorem 4.7.
Theorem 4.8
Let \(\mathcal {F}\) be a family of extrapolation pairs and \(1 \le \mathfrak {p}_- < \mathfrak {p}_+ \le \infty \). Assume that there exist exponents \(p_0, q_0 \in (0, \infty )\) such that \(p_0 \in [\mathfrak {p}_-, \mathfrak {p}_+]\) and for all weights \(v^{p_0} \in A_{p_0/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p_0)'}\),
where \(\Phi : [1, \infty ) \rightarrow [1, \infty )\) is an increasing function. Then for all exponents \(p \in (\mathfrak {p}_-, \mathfrak {p}_+)\) and \(q \in (0, \infty )\) satisfying \(\frac{1}{p} - \frac{1}{q} = \frac{1}{p_0} - \frac{1}{q_0}\), and all weights \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\),
where the constant \(C_0\) depends only on n, p, \(p_0\), \(\mathfrak {p}_-\), and \(\mathfrak {p}_+\), and
Proof
Fix \(p \in (\mathfrak {p}_-, \mathfrak {p}_+)\) and \(q \in (0, \infty )\) satisfying \(\frac{1}{p} - \frac{1}{q} = \frac{1}{p_0} - \frac{1}{q_0}\), and let \(w^p \in A_{p/\mathfrak {p}_-} \cap RH_{(\mathfrak {p}_+/p)'}\). Fix \((f, g) \in \mathcal {F}\). Without loss of generality we may assume that \(0< \Vert gw\Vert _{L^p} < \infty \). Indeed, if \(\Vert gw\Vert _{L^p} = \infty \) there is nothing to prove, and if \(\Vert gw\Vert _{L^p} = 0\), then \(g=0\) a.e. and by (4.54) we see that \(f=0\) a.e., which trivially implies (4.55). We split the proof into two cases.
Case I: \(q<q_0\). Recall that \(\tau _t = \big (\frac{\mathfrak {p}_{+}}{t} \big )' \big (\frac{t}{\mathfrak {p}_-}-1)+1\) for any \(t \in [\mathfrak {p}_{-}, \mathfrak {p}_+]\). Obviously, \(\tau _t\) is an increasing function in t. Lemma 2.6 part (b) gives
Set
which along with (4.56) and Theorem 2.2 implies that there exists an operator \(\mathcal {R}: L^{\tau _p}(w^{p(\mathfrak {p}_+/p)'}) \rightarrow L^{\tau _p}(w^{p(\mathfrak {p}_+/p)'})\) such that
Then (4.57) and the second estimate in (4.58) yield
Assume first that \(p_0<\mathfrak {p}_+\). Pick
Considering \(p<p_0\), (4.56), and the last estimate in (4.58), we use Lemma 2.5 and (2.1) to get \(v^{p_0(\mathfrak {p}_+/p_0)'} \in A _{\tau _{p_0}}\) with
where the constant \(C_1\) depends only on n, p, \(p_0\), \(\mathfrak {p}_-\), and \(\mathfrak {p}_+\), which together with Lemma 2.6 part (b) implies
On the other hand, note that
provided
which also implies
By (4.57), the first estimate in (4.58), (4.63), and (4.66),
To proceed, we denote \(\frac{1}{r}:= \frac{1}{q} - \frac{1}{q_0}>0\). Then in light of (4.60), (4.63), and (4.64), it follows from Hölder’s inequality that
Furthermore, invoking (4.61), (4.62), and (4.54), we arrive at
where we have used (4.67), (4.59), and that \(\frac{1}{p} - \frac{1}{q} = \frac{1}{p_0} - \frac{1}{q_0}\).
Let us next treat the case \(p_0=\mathfrak {p}_+\). Choose \(v:= (\mathcal {R}h)^{\frac{1}{\mathfrak {p}_+} - \frac{1}{\mathfrak {p}_-}}\). Then it follows from Lemma 2.6 part (a) that
with
where we have used the last estimate in (4.58) and (2.1). In the current scenario,
In view of (4.57), (4.71), and (4.72), there holds
Hence, invoking (4.69)–(4.75), Hölder’s inequality, and (4.54), we deduce
where (4.59) was used in the last step.
Case II: \(q_0<q\). By Lemma 2.6 parts (b) and (c),
where \(s=p'(\mathfrak {p}'_-/p')'=\frac{1}{\frac{1}{\mathfrak {p}_-} - \frac{1}{p}}\). This and Theorem 2.2 yield that there exists an operator \(\mathcal {R}: L^{\tau '_p}(w^{-s}) \rightarrow L^{\tau '_p}(w^{-s})\) such that for any nonnegative function \({\widetilde{h}} \in L^{\tau '_p}(w^{-s})\),
Write \(\frac{1}{r}:= \frac{1}{q_0} - \frac{1}{q} = \frac{1}{p_0} - \frac{1}{p}>0\), equivalently, \(\frac{q}{q-q_0}=\frac{r}{q_0}\). By duality there exists a nonnegative function \(h \in L^{\frac{q}{q-q_0}}(w^q)\) with \(\Vert h\Vert _{L^{\frac{q}{q-q_0}}(w^q)} \le 1\) such that
Setting \(H:= \mathcal {R} \Big (h^{\frac{r}{\tau '_p q_0}} w^{\frac{s+q}{\tau '_p}} \Big )^{\frac{\tau '_p q_0}{r}} w^{-\frac{(s+q)q_0}{r}}\), we utilize (4.77) to obtain that \(h \le H\), and by (2.1) and (4.76),
provided that
which also gives
Now picking \(v:= w^{\frac{q}{q_0}} \, H^{\frac{1}{q_0}}\), we see that by (4.80)
To proceed, we observe that \(p_0< p <\mathfrak {p}_+\) and use (4.65) to deduce that
which in turn implies
Hence, it follows from (4.82) and (4.84) that
which along with (4.56), (4.79), and Lemma 2.5 part (b), yields
where the constant \(C_2\) depends only on n, p, \(p_0\), \(\mathfrak {p}_-\), and \(\mathfrak {p}_+\). By Lemma 2.6 part (b), this means that
With (4.78) and (4.86) in hand, the hypothesis (4.54) implies
where (4.81) and (4.85) were used in the last inequality. As a consequence, (4.55) follows at once from (4.68) and (4.87). \(\square \)
Proof of Theorem 1.1
Fix \(v_i^{q_i} \in A_{q_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/q_i)'}\), \(i=2, \ldots , m\). Set
By hypothesis (1.2), we see that for every \(v_1^{q_1} \in A_{q_1/\mathfrak {p}_1^-} \cap RH_{(\mathfrak {p}_1^+/q_1)'}\)
where \(\frac{1}{q} = \sum _{i=1}^m \frac{1}{q_i}\) and \(v=\prod _{i=1}^m v_i\). This verifies the hypothesis (4.54) for the family \(\mathcal {F}_1\). Then Theorem 4.8 implies that for every \(p_1 \in (\mathfrak {p}_1^-, \mathfrak {p}_1^+)\) and every \(w_1^{p_1} \in A_{p_1/\mathfrak {p}_1^-} \cap RH_{(\mathfrak {p}_1^+/p_1)'}\),
where \(\frac{1}{s_1} - \frac{1}{p_1} = \frac{1}{q} - \frac{1}{q_1}\),
Considering (4.88), we have
for all \((f, f_1, \ldots , f_m) \in \mathcal {F}\), for all \(p_1 \in (\mathfrak {p}_1^-, \mathfrak {p}_1^+)\), for all \(w_1^{p_1} \in A_{p_1/\mathfrak {p}_1^-} \cap RH_{(\mathfrak {p}_1^+/p_1)'}\), and for all \(v_i^{q_i} \in A_{q_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/q_i)'}\), \(i=2, \ldots , m\).
Now fix \(p_1 \in (\mathfrak {p}_1^-, \mathfrak {p}_1^+)\), \(w_1^{p_1} \in A_{p_1/\mathfrak {p}_1^-} \cap RH_{(\mathfrak {p}_1^+/p_1)'}\), and \(v_i^{q_i} \in A_{q_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/q_i)'}\), \(i=3, \ldots , m\). Set
It follows from (4.91) that for every \(v_2^{q_2} \in A_{q_2/\mathfrak {p}_2^-} \cap RH_{(\mathfrak {p}_2^+/q_2)'}\),
Invoking Theorem 4.8 applied to \(\mathcal {F}_2\), we have that for every \(p_2 \in (\mathfrak {p}_2^-, \mathfrak {p}_2^+)\) and every \(w_2^{p_2} \in A_{p_2/\mathfrak {p}_2^-} \cap RH_{(\mathfrak {p}_2^+/p_2)'}\),
where \(\frac{1}{s_2} - \frac{1}{p_2} = \frac{1}{s_1} - \frac{1}{q_2}\),
It follows from (4.92) that for every \(p_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), for every \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i=1, 2\), and for every \(v_j^{q_j} \in A_{q_j/\mathfrak {p}_j^-} \cap RH_{(\mathfrak {p}_j^+/q_j)'}\), \(j=3, \ldots , m\),
for all \((f, f_1, \ldots , f_m) \in \mathcal {F}\).
Inductively, one can show that for each \(k \in \{1, \ldots , m\}\), for every \(p_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), for every \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i \in \{1, \ldots , k\}\), and for every \(v_j^{q_j} \in A_{q_j/\mathfrak {p}_j^-} \cap RH_{(\mathfrak {p}_j^+/q_j)'}\), \(j=\{k+1, \ldots , m\}\),
for all \((f, f_1, \ldots , f_m) \in \mathcal {F}\), where \(s_0:= q\),
To conclude the proof, we take \(\frac{1}{s_m} = \frac{1}{p}:= \sum _{i=1}^m \frac{1}{p_i}\), and then (4.96) is satisfied. The inequality (4.95) immediately gives (1.3) as desired.
It remains to show the vector-valued inequality (1.4). Fix \(r_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), \(i=1, \ldots , m\), and set \(\frac{1}{r} = \sum _{i=1}^m \frac{1}{r_i}\). Given \(N \in \mathbb {N}\), we define
By (1.3), for all \((F, F_1, \ldots , F_m) \in \mathcal {F}_{\textbf{r}}^N\), and for all weights \(v_i^{r_i} \in A_{r_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/r_i)'}\), \(i=1, \ldots , m\),
where \( \mathfrak {C}_{i, 1}:= 2^{\max \{\frac{\tau _{r_i}}{r_i}, \frac{\tau '_{r_i}}{q_i}\}}\). This corresponds to (1.2) for the family \(\mathcal {F}_{\textbf{r}}^N\) and the exponent \(\textbf{r}=(r_1, \ldots , r_m)\). Then the estimate (1.3) applied to \(\mathcal {F}_{\textbf{r}}^N\) gives that for all exponents \(p_i \in (\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+})\) and all weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/p_i)'}\), \(i=1, \ldots , m\),
for all \((F, F_1, \ldots , F_m) \in \mathcal {F}_{\textbf{r}}^N\), where \(\mathfrak {C}_{i, 2}:= 2^{\max \{\frac{\tau _{p_i}}{p_i}, \frac{\tau '_{p_i}}{r_i}\}}\). The estimate (4.100) in turn implies
for all \(\{(f^k, f^k_1, \cdots , f^k_m)\}_k \subset \mathcal {F}\), where \(\mathfrak {C}'_i:= 2^{\max \{\frac{\tau _{p_i}}{p_i}, \frac{\tau '_{p_i}}{r_i}\} + \max \{\frac{\tau _{r_i}}{r_i}, \frac{\tau '_{r_i}}{q_i}\}}\), and the constant \(C'_i\) depends only on n, \(p_i\), \(q_i\), \(r_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\). Letting \(N \rightarrow \infty \), we conclude (1.4) as desired. \(\square \)
Proof of Theorem 1.2
Let \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), \(i=1, \ldots , m\), be such that \(\frac{1}{s}:= \sum _{i=1}^m \frac{1}{s_i} \le 1\). It follows from (1.5) and Theorem 1.1 that for all \(v_i^{s_i} \in A_{s_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/s_i)'}\), \(i=1, \ldots , m\),
where both \(C_0\) and \(C_i\) depend only on n, \(s_i\), \(q_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\).
Fix \({\textbf {b}}= (b_1, \ldots , b_m) \in {\text {BMO}}^m\) and multi-index \(\alpha \in \mathbb {N}^m\). Given \(v_i^{s_i} \in A_{s_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/s_i)'}\), \(i=1, \ldots , m\), in light of Lemma 2.6 part (b), we see that
which together with Lemma 2.4 yields that there exists \(\eta _i \in (1, 2)\) such that
Then in view of (4.102)–(4.104), Theorem 2.11 applied to \(p:=s \ge 1\), \(p_i:=s_i\), \(r_i:= \tau _{s_i}\), and \(\theta _i:=s_i(\mathfrak {p}_i^+/s_i)\), gives that for all \(v_i^{s_i} \in A_{s_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/s_i)'}\), \(i=1, \ldots , m\),
where \(C_i\) depends only on n, \(s_i\), \(q_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), and \(C_0\) depends only on the same parameters and additionally on \(\alpha \).
Observe that for each \(i=1, \ldots , m\),
Now with (4.105) and (4.106) in hand, we use Theorem 1.1 applied to \(s_i\) and \(C_0^{\frac{1}{m}} {\widetilde{\Phi }}_i\) in place of \(q_i\) and \(\Phi _i\) to deduce that for all exponents \(p_i, r_i \in (\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+})\) and for all weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/p_i)'}\), \(i=1, \ldots , m\),
where \(C_0\) depends only on \(\alpha \), n, \(p_i\), \(q_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), \(C'_i\) depends only on n, \(p_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), and
where \(\frac{1}{r} = \sum _{i=1}^m \frac{1}{r_i}\), C depends only on \(\alpha \), n, \(p_i\), \(q_i\), \(r_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\), and \(C''_i\) depends only on n, \(p_i\), \(r_i\), \(s_i\), \(\mathfrak {p}_i^-\), and \(\mathfrak {p}_i^+\). This completes the proof of Theorem 1.2. \(\square \)
5 Applications
This section is dedicated to using extrapolation to prove quantitative weighted inequalities for a variety of operators. This also shows that extrapolation theorems are useful and powerful.
5.1 Bilinear Bochner–Riesz Means
Given \(\delta \in \mathbb {R}\), the bilinear Bochner–Riesz means of order \(\delta \) is defined by
Theorem 5.1
Let \(n \ge 2\) and \(\delta \ge n-1/2\). Then for all \(p_i \in (1, \infty )\), for all \(w_i^{p_i} \in A_{p_i}\), for all \({\textbf {b}}=(b_1, b_2) \in {\text {BMO}}^2\), and for each multi-index \(\alpha \in \mathbb {N}^2\),
whenever \(\frac{1}{s}:= \frac{1}{s_1} + \frac{1}{s_2} \le 1\) with \(s_1, s_2 \in (1, \infty )\), where \(w=w_1 w_2\), \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}\),
and
Proof
Let us first consider the case \(\delta >n-1/2\). In this case, it was shown in [58, Lemma 3.1] that
where the implicit constant is independent of x, \(f_1\), and \(f_2\). Combining (5.3) with (2.1) and Hölder’s inequality, we obtain that for all \(p_i \in (1, \infty )\) and for all \(w_i^{p_i} \in A_{p_i}\),
where \(w=w_1 w_2\) and \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}\). Then it follows from (5.4), Theorem 1.2, and Remark 1.4 that for all \({\textbf {b}}=(b_1, b_2) \in {\text {BMO}}^2\) and for each multi-index \(\alpha \in \mathbb {N}^2\),
whenever \(s_1, s_2 \in (1, \infty )\) satisfy \(\frac{1}{s}:= \frac{1}{s_1} + \frac{1}{s_2} \le 1\).
Next, we turn to the case \(\delta =n-1/2\). Given \(\varepsilon _1 \in (0, \frac{1}{2})\), and \(\varepsilon _2>0\), we write
We first claim that for any \(u_1, u_2 \in A_2\),
where \(u=u_1 u_2\) and the constant \( \phi _1, \phi _2\) are non-negative function and \(\phi _2\) is increasing. Indeed, (5.6) can be obtained by following the proof of [58, Theorem 1.8]. We here only mention the difference:
provided the sharp estimate for the Hardy–Littlewood maximal operator in (2.1).
Now let \(v_1^2, v_2^2 \in A_2\), \(v:=v_1 v_2\), and by Lemma 2.4, there exists \(\gamma \in (0, 2^{-n-3})\) such that
Then, (5.6) applied to \(u_i=v_i^{2(1+\gamma )}\), \(i=1, 2\), gives the for any \(\theta \in (0, 1)\),
where (5.7) was used in the last step. Picking \(\theta =(1+\gamma )^{-1}\), \(\varepsilon _1=1/4\), and \(\varepsilon _2=(n-7/4) \gamma \), we utilize (5.5) and (5.8) to deduce that \(\delta (\theta )=n-1/2\) and
where we had used that \(\phi _1(\varepsilon _1)^{1-\theta } \phi _2(\varepsilon _2)^{\theta } \le \max \{1, \phi _1(1/4), \phi _2(n)\}\), and the implicit constant depends only on n.
Having proved (5.9) and invoking Theorems 1.1 and 1.2 applied to \(\mathfrak {p}_i^-=1\), \(\mathfrak {p}_i^+=\infty \), \(q_i=2\), and \(\Phi _i(t)=t\), we conclude (5.1) and (5.2). \(\square \)
The next result considers the case \(\delta <n-\frac{1}{2}\), which can be viewed as a complement of Theorem 5.1.
Theorem 5.2
Let \(n \ge 2\), \(0<\delta <n-\frac{1}{2}\), and \(0<\delta _1,\delta _2\le \frac{n}{2}\) be such that \(\delta _1+\delta _2<\delta \). Set \(\mathfrak {p}_1^-:= \frac{2n}{n+2\delta _1}\), \(\mathfrak {p}_2^-:= \frac{2n}{n+2\delta _2}\), and \(\mathfrak {p}_1^+ = \mathfrak {p}_2^+:=2\). Then for all \(w_i^2 \in A_{2/\mathfrak {p}_i^-}\cap RH_{(\mathfrak {p}_i^+/2)'}\), \(i=1, 2\),
Moreover, for all \(p_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\) and for all \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-}\cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i=1, 2\),
where \(w=w_1 w_2\) and \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}\).
Proof
We modify the proof of [74, Theorem 2] into the current setting. First, choose a nonnegative function \(\varphi \in \mathscr {C}_c^{\infty }(0,\,\infty )\) satisfying \({\text {supp}}\varphi \subset (\frac{1}{2},\,2)\) and \(\sum _{j \in \mathbb {Z}} \varphi (2^j s)=1\) for any \(s>0\). For each \(j \ge 0\), we define the bilinear operator
where
Here \(d\sigma \) is the surface measure on \(\mathbb {S}^{n-1}\). Then one has
Given \(j\ge 0\), let \(B_j=\{x\in \mathbb {R}^n: |x| < 2^{j(1+\gamma )}\}\) with \(\gamma >0\) chosen later, and split the kernel function \(K_j\) of \(T_j\) into four parts:
Letting \(T_j^\ell \) denote the bilinear operator with kernel \(K_j^\ell \), \(\ell =1,2,3,4\), we see that
Note that a straightforward calculation gives
and
In view of Theorem 1.1, it suffices to prove (5.10). Now let \(q_1=q_2=2\), \(v_1^2 \in A_{q_1/\mathfrak {p}_1^-} \cap RH_{(\mathfrak {p}_1^+/q_1)'} = A_{1+\frac{2\delta _1}{n}} \cap RH_{\infty }\), and \(v_2^2 \in A_{q_2/\mathfrak {p}_2^-} \cap RH_{(\mathfrak {p}_2^+/q_2)'} = A_{1+\frac{2\delta _2}{n}} \cap RH_{\infty }\). Considering (5.12)–(5.13), we are reduced to showing that there exists \(\varepsilon >0\) such that
To control \(T_j^4\), note that \(v_1^2 \in A_{1 + \frac{2\delta _1}{n}} \subset A_2\) and \(v_2^2 \in A_{1+\frac{2\delta _2}{n}} \subset A_2\) since \(\max \{n+2\delta _1, n+2\delta _2\} \le 2n\). Using (5.14), Cauchy–Schwarz inequality, (5.15), (2.1), and (1.1), we have
where in the second-to-last inequality we have picked \(N>0\) large enough so that \(N\gamma >(1+\gamma )n\), and then taken \(0<\varepsilon <N\gamma -(1+\gamma )n\). Similarly,
where we have chosen \(N>0\) sufficiently large so that \(N\gamma - 2(1+\gamma )n > \varepsilon \). Symmetrically to \(T_j^3(f_1, f_2)\), there holds
Finally, to prove (5.16) for \(T_j^1\), we proceed as follows. For fixed \(y\in \mathbb {R}^n\), set \(B_j(y, r)=\{x\in \mathbb {R}^n: |x-y|\le 2^{j(1+\gamma )} r\}\) with \(r>0\), and split \(f_1\) and \(f_2\) into three parts, respectively:
where
We should mention that each \(f_{1, i}\) and \(f_{2, i}\), \(i=1, 2, 3\), depend on the variable y. Let \(x\in B_j(y, \frac{1}{4})\). Since \(f_{1,3}\) is supported on \(\mathbb {R}^n\setminus B_j(y,\frac{5}{4})\), it follows from \(f_{1, 3}(x-y_1) \ne 0\) that \(|x-y_1-y| \ge \frac{5}{4} 2^{j(1+\gamma )}\), and so \(|y_1| \ge 2^{j(1+\gamma )}\). Noting that the kernel \(K_j^1\) is supported on \(B_j\times B_j\), we get \(T_j^1(f_{1, 3}, f_2)=0\). Similarly, \(T_j^1(f_1, f_{2, 3})=0\). Hence, for any \(x\in B_j(y, \frac{1}{4})\),
Since \(f_{1, 2}\) and \(f_{2, 2}\) are supported on \(B_j(y,\frac{5}{4})\setminus B_j(y,\frac{3}{4})\), it follows from \(f_{1, 2}(x-y_1) f_{2, 2}(x-y_2) \ne 0\) that \(|y_1| \ge 2^{j(1+\gamma )-1}\) and \(|y_2| \ge 2^{j(1+\gamma ) - 1}\). Then, repeating the proof of (5.17) yields
Since \(f_{1, 1}\) is supported on \(B_j(y,\frac{3}{4})\), it follows from \(f_{1, 1}(x-y_1) f_{2, 2}(x-y_2)\ne 0\) that \(|y_1| \le 2^{j(1+\gamma )}\) and \(|y_2| \ge 2^{j(1+\gamma )-1}\). Thus, we calculate much as in (5.18) to get
Symmetrically,
It remains to consider \(T_j^1(f_{1,1}, f_{2,1})\). Given \(\mathfrak {m}\in L^\infty (\mathbb {R})\), set \(T_{\mathfrak {m}} h:= \int _0^1 \mathfrak {m}(\lambda ) \, R_\lambda h \, \lambda ^{n-1} \, d\lambda \). Then [74, Lemma 3.1] states that
Let \(r=1+2\delta _2/n\). Then \(v_2^2 \in A_r \cap RH_{\infty }\). Using (5.23) and Hölder’s inequality, we have that for \(h \in L^2(v_2^2)\) with \({\text {supp}}h \subset B_j(y,\frac{3}{4})\),
where the definition (1.1) was used in the last inequality. Similarly,
Observe that
As argued in [74, (3.7)], we utilize (5.24)–(5.26) to get that for any fixed \(0<\kappa <\delta \),
provided choosing \(\kappa , \gamma , \varepsilon \) small enough so that \(\delta -\kappa -(1+\gamma )(\delta _1+\delta _2)>\varepsilon \). Summing (5.19)–(5.22) and (5.27) yields
Now, integrating the both sides of (5.28) with respect to y, using Cauchy–Schwarz inequality, and interchanging the order of integration, we conclude
This shows (5.16) for \(T_j^1\) and completes the whole proof. \(\square \)
5.2 Bilinear Rough Singular Integrals
Given \(\Omega \in L^q(\mathbb {S}^{2n-1})\) with \(1 \le q \le \infty \) and \(\int _{\mathbb {S}^{2n-1}} \Omega \, d\sigma =0\), we define the rough bilinear singular integral operator \(T_{\Omega }\) by
where the rough kernel \(K_{\Omega }\) is given by \(K_{\Omega }(y, z) = \frac{\Omega ((y, z)/|(y, z)|)}{|(y, z)|^{2n}}\).
A typical example of the rough bilinear operators is the Calderón commutator defined in [14] as
where a is the derivative of A. C. Calderón [15] established the boundedness of \(\mathcal {C}_a\) in the full range of exponents \(1<p_1, p_2< \infty \). It was shown in [14] that the Calderón commutator can be written as
where the kernel is given by
where \(e(t)=1\) if \(t>0\) and \(e(t)=0\) if \(t<0\). Observe that K(y, z) is odd and homogeneous of degree \(-2\) whose restriction on \(\mathbb {S}^1\) is \(\Omega (y,z)\). It is also easy to check that \(\Omega \) is odd and bounded, and hence Theorems 5.3–5.4 below can be applied to \(\mathcal {C}_a\).
Theorem 5.3
Let \(\Omega \in L^{\infty }(\mathbb {S}^{2n-1})\) and \(\int _{\mathbb {S}^{2n-1}} \Omega \, d\sigma =0\). Then for all \(p_i \in (1, \infty )\), for all \(w_i^{p_i} \in A_{p_i}\), for all \({\textbf {b}}=(b_1, b_2) \in {\text {BMO}}^2\), and for each multi-index \(\alpha \in \mathbb {N}^2\),
whenever \(\frac{1}{s}:= \frac{1}{s_1} + \frac{1}{s_2} \le 1\) with \(s_1, s_2 \in (1, \infty )\), where \(w=w_1 w_2\) and \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}\).
Proof
Picking \(r_1=r_2=r_3=1\) and \(p_1=p_2=q_1=q_2=2\), we see that (2.32) holds and \(p_i \in (1, \infty )\), \(i=1, 2\). Then Lemma 2.8 gives that
On the other hand, it was proved in [24] that for every \(\textbf{w}=(w_1, w_2) \in A_{(2, 2)}\),
where (5.31) was used in the last step. Thus, (5.29) and (5.30) follow at once from (5.32) and Theorems 1.1 and 1.2 applied to \(\mathfrak {p}_i^-=1\), \(\mathfrak {p}_i^+=\infty \), \(q_i=2\), \(\Phi _i(t)=t^{\frac{3}{2}}\). \(\square \)
Theorem 5.4
Let \(\Omega \in L^q(\mathbb {S}^{2n-1})\) with \(q>\frac{4}{3}\) and \(\int _{\mathbb {S}^{2n-1}} \Omega \, d\sigma =0\). Let \(\pi _q< \mathfrak {p}_i^- < \mathfrak {p}_i^+ \le \infty \), \(i=1, 2\), be such that \(\frac{1}{\pi '_q}< \frac{1}{\mathfrak {p}_+}:= \frac{1}{\mathfrak {p}_i^+} + \frac{1}{\mathfrak {p}_2^+} <1\), where \(\pi _q:= \max \big \{\frac{24n+3q-4}{8n+3q-4}, \frac{24n+q}{8n+q}\big \}\). Then for all \(p_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), for all \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), for all \({\textbf {b}}=(b_1, b_2) \in {\text {BMO}}^2\), and for each multi-index \(\alpha \in \mathbb {N}^2\),
whenever \(\frac{1}{s}:= \frac{1}{s_1} + \frac{1}{s_2} \le 1\) with \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), where \(w=w_1 w_2\), \(\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}\), \(\frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2}\),
Proof
By assumption, \(p_0:= \min \{\mathfrak {p}_1^-, \mathfrak {p}_2^-, \mathfrak {p}'_+\} > \pi _q\), which together with [49, Theorem 1.1] gives
for all \(f_1, f_2, f_3 \in \mathscr {C}_c^{\infty }(\mathbb {R}^n)\). This and Theorem 5.6 below imply (5.33) and (5.34) as desired. \(\square \)
Remark 5.5
In Theorem 5.3, the exponent \(\mathfrak {p}_i^- > \pi _q\) can be relaxed to \(\mathfrak {p}_i^- \ge \pi _q\), at the cost of a larger exponent appearing in (5.33) and (5.34). Indeed, to get the first inequality in (5.35), it requires that \(p_0\) is strictly greater than \(\pi _q\). When \(\mathfrak {p}_i^- = \pi _q\) and \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), Lemma 2.4 implies that there exists \(\mathfrak {p}_i^-< {\widetilde{\mathfrak {p}}}_i^- < p_i\) such that \(w_i^{p_i} \in A_{p_i/{\widetilde{\mathfrak {p}}}_i^-} \cap RH_{(\mathfrak {p}_i^+/p_i)'}\), \(i=1, 2\). Then \(p_0:= \min \{{\widetilde{\mathfrak {p}}}_1^-, {\widetilde{\mathfrak {p}}}_2^-, \mathfrak {p}'_+\} > \pi _q\). Combining this with Lemma 2.4 and the result in the case \(\mathfrak {p}_i^- > \pi _q\), we can formulate similar estimates as in Theorem 5.3. Details are left to the reader.
Recall that a family \(\mathcal {S}\) of cubes is called sparse if for every cube \(Q \in \mathcal {S}\), there exists \(E_Q \subset Q\) such that \(|E_Q | \ge \eta |Q|\) for some \(0<\eta <1\) and the collection \(\{E_Q\}_{Q \in \mathcal {S}}\) is pairwise disjoint.
Given a sparse family \(\mathcal {S}\) and \(\mathbf {\mathfrak {s}}=(\mathfrak {s}_1, \ldots , \mathfrak {s}_{m+1})\) with \(\mathfrak {s}_i \ge 1\), \(i=1, \dots , m+1\), we define the \((m+1)\)-sparse form
We are interested in those operators T that dominated by certain sparse form
for all \(f_1, \ldots , f_{m+1} \in \mathscr {C}_c^{\infty }(\mathbb {R}^n)\).
Theorem 5.6
Let \(1 \le \mathfrak {p}_i^- < \mathfrak {p}_i^+ \le \infty \), \(i=1, \dots , m\). Assume that the operator T satisfies (5.36) for the exponents \(\mathbf {\mathfrak {s}} = (\mathfrak {p}_1^-, \ldots , \mathfrak {p}_m^-, \mathfrak {p}'_+)\), where \(\frac{1}{\mathfrak {p}_+}:= \sum _{i=1}^m \frac{1}{\mathfrak {p}_i^+} < 1\). Then for all exponents \(p_i, r_i \in (\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+})\) and for all weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/p_i)'}\),
where
If in addition T is an m-linear linear operator, then for the same exponents and weights as above, for all \({\textbf {b}}=(b_1, \ldots , b_m) \in {\text {BMO}}^m\), and for each multi-index \(\alpha \), we have
and
whenever \(\frac{1}{s}:= \sum _{i=1}^m \frac{1}{s_i} \le 1\) with \(s_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\), where \(\Psi _i(t):= t^{\alpha _i \max \{1, \frac{1}{\tau _{s_i}-1}\} + \theta (\frac{1}{p_i} - \frac{1}{\mathfrak {p}_i^+})}\).
Proof
Let \(p_i \in (\mathfrak {p}_i^{-}, \mathfrak {p}_i^{+})\) and \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^{-}} \cap RH_{(\mathfrak {p}_i^{+}/p_i)'}\), \(i=1, \ldots , m\). By density, we may assume that \(f_1, \ldots , f_m \in \mathscr {C}_c^{\infty }(\mathbb {R}^n)\) in this sequel. By Lemma 2.8, one has \(\textbf{w} \in A_{\textbf{p}, \textbf{r}}\) with
Then it follows from (5.41) and [76, Corollary 4.2] that
Thus, (5.37) is a consequence of (5.41) and (5.42). With Theorem 1.1 and Remark 1.4 in hand, the estimate (5.37) in turn gives (5.38). Additionally, (5.39) and (5.40) follow from (5.37), Theorem 1.2, and Remark 1.4. The proof is complete. \(\square \)
We close the subsection with the following remark, which shows Theorems 1.1–1.2 and Theorem 5.6 contain a lot of applications. Details are left to the interested reader.
Remark 5.7
Now let us present some examples in terms of the hypothesis in Theorem 5.6.
-
In [9], Bernicot et al. established a bilinear sparse domination \(\Lambda _{\mathcal {S}, p_0, q'_0}\) for singular non-integral operators under certain assumptions. This verifies the hypothesis (5.36) for \(r_1=p_0\) and \(r_2=q'_0\). Note also that our extrapolation theorems above can be extended to spaces of homogeneous type since the corresponding sharp estimate for the Hardy–Littlewood operator (2.1) was established in [52, Proposition 7.13].
-
For Bochner–Riesz means \(\mathcal {B}^{\alpha }\) in \(\mathbb {R}^2\), the authors [6] proved a similar spare bilinear form to (5.36) with \(r_1=6/5\) and \(r_2=2\) whenever \(\alpha >1/6\). Much as before, one can not only recover [6, Theorem 1.2], but also obtain quantitative weighted estimates and vector-valued inequalities.
-
Bui et. al [12] studied the Schrödinger operator \(L=\Delta +V\) on \(\mathbb {R}^n\) with \(n \ge 3\), where \(V \in RH_q\) and \(q \in (n/2, n)\). Letting \(p_0=\big (\frac{1}{q}-\frac{1}{n}\big )^{-1}\) and K(x, y) be the kernel of the Riesz transform \(L^{-1/2}\nabla \), we see that K satisfies the Bui-Duong’s condition (cf. [12, Theorem 5.6]). The latter implies \(L^r\)-Hörmander condition (cf. [70, Proposition 3.2]). Then, combining the \(L^p\) bounds for \(\nabla L^{-1/2}\) with \(p \in (1, p_0]\) (cf. [86]) and the pointwise sparse domination in [70], we use a duality argument to conclude that there exists a sparse family \(\mathcal {S}\) such that \(|\langle \nabla L^{-1/2}f, g \rangle | \lesssim \Lambda _{\mathcal {S}, 1, p_0}(f, g)\). That is, the hypothesis (5.36) is satisfied for the Riesz transform \(\nabla L^{-1/2}\).
-
For the m-linear Calderón–Zygmund operators and the corresponding maximal truncation, pointwise sparse dominations were obtained in [28, 37], which immediately implies (5.36) with \(\textbf{r}=(1, \ldots , 1)\). Then one can improve Corollaries 8.2 and 8.3 in [47] to the quantitative weighted estimates.
-
Let \(1 \le r < \infty \) and \(\mathfrak {g}\) be the square function with the kernel \(K_t\) satisfies the m-linear \(L^r\)-Hörmander condition defined in [21]. Under the assumption that \(\mathfrak {g}\) is bounded from \(L^r(\mathbb {R}^n) \times \cdots \times L^r(\mathbb {R}^n)\) to \(L^{r/m, \infty }(\mathbb {R}^n)\), Cao and Yabuta [21] obtained a pointwise control of \(\mathfrak {g}\) by \(\Lambda _{\mathcal {S}, \textbf{r}}\), where \(\textbf{r}=(r, \ldots , r, 1)\). Then, the square function \(\mathfrak {g}\) verifies (5.36).
-
The operators satisfying (5.36) also include the discrete cubic Hilbert transform [36] and oscillatory integrals [63].
5.3 Multilinear Fourier Multipliers
Given \(s, m \in \mathbb {N}\), a function \(\sigma \in \mathscr {C}^s(\mathbb {R}^{nm} {\setminus } \{0\})\) is said to belong to \(\mathcal {M}^s(\mathbb {R}^{nm})\) if
for each multi-index \(\alpha =(\alpha _1, \ldots , \alpha _m)\) with \(\sum _{i=1}^m |\alpha _i| \le s\).
Given \(s \in \mathbb {R}\), the (usual) Sobolev space \(W^s(\mathbb {R}^{nm})\) is defined by the norm
where \({\widehat{f}}\) is the Fourier transform in all the variables. For \(\textbf{s}=(s_1,\ldots ,s_m) \in \mathbb {R}^m\), the Sobolev space of product type \(W^{\textbf{s}}(\mathbb {R}^{nm})\) consists of all \(f \in \mathcal {S}'(\mathbb {R}^{nm})\) such that
Given a function \(\sigma \) on \(\mathbb {R}^{nN}\), we set
where \(\Psi \in \mathcal {S}(\mathbb {R}^{nm})\) satisfy \({\text {supp}}\Psi \subset \{1/2 \le |\mathbf {\xi }| \le 2 \}\), and \(\sum _{k \in \mathbb {Z}} \Psi (2^{-k}\mathbf {\xi }) =1\) for all \(\mathbf {\xi } \in \mathbb {R}^{nm} {\setminus }\{0\}\). Denote
Then one has
For a bounded function \(\sigma \) on \(\mathbb {R}^{nm}\), the m-linear Fourier multiplier \(T_{\sigma }\) is defined by
for all \(f_1,\ldots f_m \in \mathcal {S}(\mathbb {R}^n)\).
Be means of extrapolation theorems, we improve Theorems 1.2 (i) and 6.2 in [43] to the weighted estimates with quantitative bounds. We can also establish the corresponding weighted estimates for the higher order commutators and vector-valued inequalities as follows.
Theorem 5.8
Let \(m \ge 2\), \(n/2 < s_i \le n\), \(i=1, \ldots , m\). Assume that \(\sigma \in \mathcal {W}^{\textbf{s}}(\mathbb {R}^{nm})\). Then for every \(p_i>n/s_i\), for every \(w_i^{p_i} \in A_{p_i s_i/n}\), \(i=1, \ldots , m\), for all \({\textbf {b}}=(b_1,\ldots , b_m) \in {\text {BMO}}^m\), and for each multi-index \(\alpha \in \mathbb {N}^m\),
where \(\frac{1}{p} = \sum _{i=1}^m \frac{1}{p_i}\) and \(w=\prod _{i=1}^m w_i\).
Moreover, for any \(r \in (n/s_i, 2]\), \(i=1, \ldots ,m\),
where
provided \(q_i \in (n/s_i, r)\), \(i=1, \ldots , m\).
Proof
We borrow some ideas from [43], but now we can give a proof without using the weighted Hardy space argument. Let \(\mathfrak {p}_i^-:= n/s_i\) and \(\mathfrak {p}_i^+:= \infty \) for each \(i=1, \ldots , m\). Let \(q=2\) and \(q_i=2m\) for \(1\le i \le m\). Then, \(q_i \in (\mathfrak {p}_i^-, \mathfrak {p}_i^+)\). Checking the proof of [43, Theorem 6.2], we can obtain that for any weight \(v_i^{q_i} \in A_{q_is_i/n}=A_{q_i/\mathfrak {p}_i^-} \cap RH_{(\mathfrak {p}_i^+/q_i)'}\), \(i=1, \ldots , m\),
Thus, (5.44) follows from (5.46) and Theorem 1.1 applied to \(\Phi _i(t) = t^{3/2}\).
Note that in the current scenario, \(\gamma _i(q_i, q_i)=1\), \(\tau _{q_i}=2ms_i/n\), and hence,
Then in view of (5.46), Theorem 1.2 applied to \(s_i=q_i=2m\) implies (5.45).
On the other hand, Lemma 2.12 and (5.44) give that for every \(q_i>n/s_i\), for every \(w_i^{q_i} \in A_{q_i s_i/n}\), \(i=1, \ldots , m\),
provided \(r=2\) or \(r \in (n/s_i, 2)\) and \(q_i \in (n/s_i, r)\), where \(\frac{1}{q}=\sum _{i=1}^m \frac{1}{q_i}\) and \(v=\prod _{i=1}^m v_i\). Therefore, the vector-valued inequalities above follow from Theorem 1.1 applied to \(\Phi _i(t) = t^{\frac{3}{2} \gamma _i(q_i, 2\,m)}\). \(\square \)
Theorem 5.9
Let \(1 \le r \le 2\) and \(s_1, s_2>1/r\). Let \(\sigma \) be a bounded function on \(\mathbb {R}^2\) satisfying
where \(\sigma _j\) is given in (5.43) with \(n=1\). Assume that \(1 \le \mathfrak {p}_1^{-}, \mathfrak {p}_2^{-}<\infty \) and \(\max \limits _{1 \le i \le 2}\frac{1}{s_i} < \min \limits _{1 \le i \le 2} \mathfrak {p}_i^{-}\). Then for all exponents \(p_i \in (\mathfrak {p}_i^{-}, \infty )\) and all weights \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-}\), \(i=1, 2\),
where \(\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2} \ge 1\) and \(w=w_1 w_2\).
Proof
We will use the same notation as [46]. By the same argument as [46, p. 970], we are deduced to showing the boundedness of \(T_{\sigma _1}\) and \(T_{\sigma _2}\), which satisfy
Here, \(\rho \in (1, 2)\) satisfies \(\max \limits _{i=1, 2}\frac{1}{s_i}< \rho < \min \{\mathfrak {p}_1^{-}, \mathfrak {p}_2^{-}, r\}\) if \(r>1\), and \(\rho =1\) if \(r=1\). The multiplier \(\Delta _j^{\theta }\) is defined by \(\widehat{\Delta _j^{\theta } f}={\widehat{\theta }}(2^{-j}\cdot ) {\widehat{f}}\), for each \(j \in \mathbb {Z}\), where \(\theta \in \mathcal {S}(\mathbb {R})\) satisfies \({\text {supp}}({\widehat{\theta }}) \subset \{ \xi \in \mathbb {R}: 1/c_0 \le |\xi | \le c_0 \}\), for some \(c_0>1\), and \(\sum _{j \in \mathbb {Z}} {\widehat{\theta }}(2^{-j}\xi )=C_{\theta }\) for all \(\xi \in \mathbb {R}\setminus \{0\}\). Considering the same property of \(\Delta _j^{\theta }\) and \(\Delta _j^{\zeta }\), we will suppress \(\theta \) and \(\zeta \) in this sequel.
Let \(w_i^{p_i} \in A_{p_i/\mathfrak {p}_i^-}\), \(i=1, 2\). By the choice of \(\rho \), we have
Let us control \(T_{\sigma _1}\) and \(T_{\sigma _2}\). Invoking (5.48)–(5.50), (2.1), and Lemma 3.3, we use Hölder’s inequality to conclude that
where the inequality (3.8) was used in the second-to-last step. To estimate \(T_{\sigma _1}\), we note that by Lemma 2.9,
since \(\frac{1}{\mathfrak {p}_-}:= \frac{1}{\mathfrak {p}_1^-} + \frac{1}{\mathfrak {p}_2^-} \le 2\) and \(p \le 1 \le 2\mathfrak {p}_-\). Therefore, in view of Lemma 5.10 applied to \(r=2\) and \(v=w^p\), (5.51), and (5.47), we proceed as above to obtain
This completes the proof. \(\square \)
In this subsection, we always choose \(\phi \in \mathcal {S}(\mathbb {R}^n)\) with \(\int _{\mathbb {R}^n} \phi \, dx=1\), and set \(\phi _t(x):= t^{-n} \phi (x/t)\) for any \(x \in \mathbb {R}^n\) and \(t>0\). And let \(\psi ,\,\Phi \in \mathcal {S}(\mathbb {R}^n)\) satisfy \(0 \le {\widehat{\psi }}(\xi )\le {\textbf {1}}_{\{1/2 \le |\xi | \le 2\}}\), \({\widehat{\psi }}(\xi ) \ge 0\) for \(1/2 \le |\xi | \le 2\), \(\sum _{j \in \mathbb {Z}} {\widehat{\psi }}(2^j\xi )=1\) for \(|\xi |\ne 0\), and \({\textbf {1}}_{\{1/2 \le |\xi | \le 2\}} \le {\widehat{\Phi }}(\xi ) \le {\textbf {1}}_{\{1/3 \le |\xi | \le 3\}}\). Denote \(\psi _j(x)=2^{-jn}\psi (x/2^j)\) and \(\Phi _j(x)=2^{-jn}\Phi (x/2^j)\) for each \(j \in \mathbb {Z}\).
Lemma 5.10
For all \(0<p \le 1 \le r \le 2\) and for all \(v \in A_r\),
Proof
It suffices to show the second inequality since \(|f(x)| \le \sup _{t>0} |\phi _t*f(x)|\) for all \(x \in \mathbb {R}^n\). By Lemma 5.11–5.12 below and estimates in [11, p. 588], we have
where we used Lemma 3.3 and that \(\lambda >\max \{\frac{nr}{p}, \frac{n}{2}\} = \frac{nr}{p}\), so \(s:= \frac{n}{\lambda }<\frac{p}{r}\) and \([v]_{A_{p/s}} \le [w]_r\). If we take \(\frac{n}{\lambda }=\frac{p(1-\varepsilon )}{r}\) for some \(\varepsilon \in (0, 1)\), then \(p-s=p-\frac{n}{\lambda } = p(1-\frac{1-\varepsilon }{r}) \ge p(1-(1 - \varepsilon ) ) = p \varepsilon \). This means \(\max \{\frac{1}{2}, \frac{1}{p-s}\} < \frac{1}{p\varepsilon }\). Consequently, taking \(\varepsilon =1/2\), we get the desired estimate. \(\square \)
We use the maximal operators \(N, N^+, N^*\) defined in [11]. Moreover, given a sequence \({\textbf {f}}=\{f_j\}\), a function u on \(\mathbb {R}^{n+1}_+\), and \(\alpha , \kappa >0\), we define
Lemma 5.11
For any \(p \in (0, \infty )\), \(r \in (1, \infty )\), and \(w \in A_r\),
Proof
The inequality (5.52) follows from the following
for all \(\eta >0\), where the first estimate is trivial and the second one is contained in [50]. The inequality (5.53) is a consequence of (5.52) and the pointwise estimate \(N^*{\textbf {f}} \lesssim {\widetilde{N}}^{**}_{\kappa } {\textbf {f}}\).
To show (5.54), we trace the proof of \(\Vert N {\textbf {f}}\Vert _{L^p(w)} \lesssim \Vert N^+ {\textbf {f}}\Vert _{L^p(w)}\) in [11]. Firstly, by (5.52) we have
Setting \({\widetilde{N}}_\mu u_i(x):= \sup _{t>0, |x-y|<\mu t} \big (\sum _{j\in \mathbb {Z}}|\phi _t^{(i)}*f_j(y)|^q \big )^{\frac{1}{q}}\), where \(\phi ^{(i)}=\frac{\partial \phi }{\partial x_i}\) and \(\mu >1\), we use (5.53) to get
Since \(r>1\) and \(w\in A_r\), Lemma 2.4 gives that \(r>\inf \{\rho >0: w\in A_\rho \}\). So, for \(s\in (0,1]\) with \(p/s = r> \inf \{\rho >0: w\in A_\rho \}\), and \(\delta >0\) satisfying \(\Gamma _\delta (y)\subset \Gamma _\mu (x)\) for all \((y,t) \in \Gamma _1(x)\), we get
Hence, taking \(L^{p/s}(w)\)-norm of both sides of the above, and using (5.55), we see that
Choosing \(\delta \) so small that \(C_2 \delta ^s<1/2\), we obtain
This completes the proof of (5.54). \(\square \)
Lemma 5.12
Then for any \(p \in (0, 1]\) and \(w \in A_2\),
Proof
Fix \(w \in A_2\) and \(\lambda >0\). It suffices to show
where the implicit constant is independent of \(\lambda \), and \(\Omega _{\lambda }:= \{N^* {\textbf {f}}>\lambda \}\) (cf. [85, p. 190]).
It follows from Whitney decomposition that one can find a pairwise disjoint family of cubes \(\{Q_j\}\) such that \(\Omega _{\lambda } = \bigcup _k Q_k\) and \({\text {dist}}(\mathbb {R}^n\backslash \Omega _{\lambda }, Q_k) \simeq \ell (Q_k)\). Then we choose a sequence of nonnegative functions \(\{\varphi _k\}_k\) such that \({\textbf {1}}_{\Omega _{\lambda }} = \sum _k \varphi _k\), with the following properties
Setting
we see that for all \(x \in \mathbb {R}^n\),
where \(x_j \in C_0 Q_j \cap (\mathbb {R}^n{\setminus } \Omega _{\lambda }) \ne \emptyset \) for all j and for some \(C_0>0\), which follows from the construction of Whitney decomposition of \(\Omega \).
Writing
and observing that
where \(\mathfrak {M}_1(x)\) is defined in (3.13), we invoke Lemma 3.6 to deduce
By Chebyshev’s inequality and (2.1),
To control the last term, we let T be the singular integral with \(\mathscr {L}(\ell ^2(\mathbb {Z}),{\mathbb {C}})\)-valued kernel \(\Phi =\{\Phi _j\}_{j \in \mathbb {Z}}\) defined by \(T({\textbf {g}}):= \sum _{j\in \mathbb {Z}} \Phi _j*g_j\) for good \(\ell ^2\)-valued functions \({\textbf {g}}=\{g_j\}_{j \in \mathbb {Z}}\). One can check that T is bounded from \(L^2(\mathbb {R}^n,\ell ^2)\) to \(L^2(\mathbb {R}^n, \ell ^2)\), \(\Vert \Phi \Vert _{\mathscr {L}(\ell ^2(\mathbb {Z}),{\mathbb {C}})}\lesssim |x|^{-n}\), and \(\Vert \nabla \Phi \Vert _{{\mathscr {L}}(\ell ^2(\mathbb {Z}),{\mathbb {C}})}\lesssim |x|^{-n-1}\) (cf. [88, p. 165]). Hence, this, Lemma 3.4, and (5.58) yield
As a consequence, (5.56) immediately follows from (5.57) and (5.59). \(\square \)
5.4 Weighted Jump Inequalities for Rough Operators
Let \(\mathcal {F}:= \{F_t(x)\}_{t>0}\) be a family of Lebesgue measurable functions defined on \(\mathbb {R}^n\). Given \(\lambda >0\), we introduce the \(\lambda \)-jump function \(N_\lambda ({\mathcal {F}})\) of \({\mathcal {F}}\), its value at x is the supremum over all N such that there exist \(s_1<t_1\le s_2<t_2\le \dotsc \le s_N<t_N\) with
Given \(\rho >0\), the value of the strong \(\rho \)-variation function \(\mathcal {V}_\rho ({\mathcal {F}})\) at x is defined by
where the supremum runs over all increasing sequences \(\{t_k\}_{k \ge 0}\).
Given \(\Omega \in L^1(\mathbb {S}^{n-1})\) and \(\varepsilon >0\), the truncated singular integral operator \(T_{\varepsilon }\) is defined by
The principal value singular integral and its maximal version are defined by
In this sequel, we write \(\mathcal {T}:= \{T_{\Omega , \varepsilon }\}_{\varepsilon >0}\).
Theorem 5.13
Let \(\rho >2\) and \(\Omega \in L^q(\mathbb {S}^{n-1})\) with \(q \in (1, \infty )\) be such that \(\int _{\mathbb {S}^{n-1}} \Omega \, d\sigma =0\). Then for all \(p \in (q', \infty )\) and for all \(w \in A_{p/q'}\),
where \(\mathbb {T} \in \big \{\sup \limits _{\lambda >0} \lambda \sqrt{N_{\lambda } \circ \mathcal {T}}, \mathcal {V}_{\rho } \circ \mathcal {T}, T_{\Omega , \#}\big \}\).
It suffices to show (5.60) for \(\mathbb {T}=\sup \limits _{\lambda >0} \lambda \sqrt{N_{\lambda } \circ \mathcal {T}}\), which immediately implies (5.60) for \(\mathbb {T} \in \{\mathcal {V}_{\rho } \circ \mathcal {T}, T_{\Omega , \#}\}\) since the following pointwise domination holds
provided that \(\ell ^{2,\infty }(\mathbb {N})\) embeds into \(\ell ^\rho (\mathbb {N})\) for all \(\rho >2\).
Let us turn to the proof of (5.60) for \(\mathbb {T}=\sup \limits _{\lambda >0} \lambda \sqrt{N_{\lambda } \circ \mathcal {T}}\). It was proved in [57, Lemma 1.3] that
where
Thus, we are reduced to proving
5.4.1 Dyadic Jump Estimates
We are going to show (5.61) in this subsection. Let \(\phi \in \mathcal {S}(\mathbb {R}^n)\) be a radial function such that \({\widehat{\phi }}(\xi )=1\) for \(|\xi | \le 2\) and \({\widehat{\phi }}(\xi )=0\) for \(|\xi |>4\). Define \({\widehat{\phi }}_k(\xi ) = {\widehat{\phi }}(2^k\xi )\) for each \(k \in \mathbb {Z}\). For each \(j \in \mathbb {Z}\), set \(\nu _j(x):= \frac{\Omega (x)}{|x|^n} {\textbf {1}}_{\{2^j\le |x|<2^{j+1}\}}(x)\). Then for any \(k \in \mathbb {Z}\),
where \(\delta _0\) is the Dirac measure at 0. Let \({\mathscr {T}}^if\) denote the family \(\{T^i_{k}f\}_{k \in \mathbb {Z}}\), \(i=1, 2, 3\). Hence, to get (5.61) it suffices to prove the following:
We begin with showing (5.63) for \(i=1\). Define
where \(\mathcal {D}_j\) is the family of dyadic cubes with sidelength \(2^j\).
Lemma 5.14
For any \(p \in (1, \infty )\) and \(w \in A_p\),
where \(\mathbb {T}f \in \big \{\big (\sum _{j \in \mathbb {Z}} |\mathbb {D}_j f|^2 \big )^{\frac{1}{2}},\, \sup _{\lambda >0} \lambda \sqrt{N_{\lambda }(\mathscr {E}f)}\big \}\).
Proof
For \(p=2\), the estimate (5.64) for dyadic operators is contained in [62], which established a sharp weighted inequality for the Haar shift operators. The general case is a consequence of the case \(p=2\) and Theorem 4.1. Then (5.64) for jump operators follows at once from (5.64) for \(\mathbb {T}f =\big (\sum _{j \in \mathbb {Z}} |\mathbb {D}_j f|^2 \big )^{\frac{1}{2}}\) and the proof of [59, Proposition 4.1]. \(\square \)
Define the square function as follows:
Lemma 5.15
For any \(w \in A_1\),
where the implicit constant is independent of \([w]_{A_1}\).
Proof
We claim that for all \(k, j \in \mathbb {Z}\),
for some \(\theta >0\), where the implicit constant and \(\theta \) are independent of k and j. To show (5.67), we first note that by [57, p. 6722], for any \(k \ge 0\),
which along with (2.1) gives
To control the case \(k<0\), we use the argument in [25, p. 2461–2463] and that
to see \(\mathcal {I}_{k, j}(x) = \sum _{d \ge 0} I_d(x)\), where for some \(\delta >0\),
Then summing these estimates up, we obtain (5.67) as desired.
Having shown (5.67), we use \(f(x) = \sum _{j \in \mathbb {Z}} \mathbb {D}_j f(x)\), a.e. \(x \in \mathbb {R}^n\), to deduce that
where we have used Minkowski’s inequality, (5.67), Cauchy–Schwarz inequality, and (5.64). This shows the first estimate in (5.66). Then, using the first inequality in (5.66) and Calderón–Zygmund decomposition as in [25, p. 2458–2460], we obtain the second estimate in (5.66). The proof is complete. \(\square \)
Lemma 5.16
Let \(\mathscr {U}\) be a family of operators given by \(\mathscr {U}f:=\{\phi _k * f\}_{k \in \mathbb {Z}}\). Then for all \(p \in (1, \infty )\) and for all \(w \in A_p\),
Proof
Since \(N_{\lambda }\) is subadditive,
where \(\mathscr {D}f:=\{\phi _k * f - \mathbb {E}_k f\}_{k \in \mathbb {Z}}\) and \(\mathscr {E}f:=\{\mathbb {E}_k f\}_{k \in \mathbb {Z}}\). Recall the square function in (5.65) and observe that \(\sup _{\lambda >0} \lambda \sqrt{N_{\lambda }(\mathscr {D}f)} \le \mathfrak {S} f\), which together with (5.66) and Theorem 4.4 applied to \(p_0=1\) implies
In view of (5.64) and (5.68), this gives at once the desired estimate. \(\square \)
Now using Lemma 5.16 and (3.17), we obtain
which shows (5.63) for \(i=1\).
For the term with \(\mathscr {T}^2\), it was shown in [25, p. 2453] that
where
with
It follows from Lemmas 3.5 and 3.9 that for any \(v \in A_{p/q'}\),
Then interpolating between (5.70) and (5.71) with \(v \equiv 1\) gives
On the other hand, for \(w \in A_{p/q'}\), by Lemma 2.4, there exists \(\gamma =\gamma _w \in (0, 1)\) such that
which along with (5.71) implies
Considering Theorem 3.1 with \(w_0 \equiv 1\), \(w_1=w^{1+\gamma }\), and \(\theta =\frac{1}{1+\gamma }\), we interpolate between (5.72) and (5.73) to arrive at
Note that \(e^{-t}<2t^{-2}\) for any \(t>0\), and
Similarly,
Hence, (5.69) and (5.74)–(5.76) imply
This shows (5.63) for \(i=2\).
To control the term with \(\mathscr {T}^3\), we note that by [25, p. 2456],
where
with
Analogously to (5.74), one has
and eventually,
This shows (5.63) for \(i=3\).
5.4.2 Short Variation Estimates
We will prove (5.62) in this subsection. As did in [25],
and for \(q<2\),
where
where \(\nu _{j, t}(x):= \frac{\Omega (x)}{|x|^n} {\textbf {1}}_{\{2^j t \le |x| \le 2^{j+1}\}}(x)\).
We claim that
Once (5.81) is obtained, we use (5.77), (5.78), and Stein-Weiss’s interpolation Theorem 3.1 as before to get
which shows (5.62) as desired.
It remains to demonstrate (5.81). If \(q>2\), we invoke (5.79), (3.18), (3.10), and (3.8) to deduce
To treat the case \(q<2\) (trivially, \(p>2\)), we observe that much as (3.37),
and
The latter, along with by Hölder’s inequality, Theorem 3.8 applied to \((p/2)'\) and \(\frac{q}{2-q}\) instead of p and q, (3.10), and (3.8), gives
Therefore, in the case \(q<2\), (5.81) follows from (5.80), (5.82), and (5.83). \(\square \)
5.5 Riesz Transforms Associated to Schrödinger Operators
Consider a real vector potential \(\textbf{a}=(a_1,\ldots , a_n)\) and an electric potential V. Assume that
Denote
We define the form Q by
with domain
Let us denote by A the self-adjoint operator associated with Q. Then A is given by the expression
and the domain of A is given by
Formally, we write
For convenience, denote
Duong et al. [41, 42] consecutively established the \(L^p\) boundedness of Riesz transform \(\mathcal {R}_k\) and its commutator \([\mathcal {R}_k, b]\), \(k=0, 1, \ldots , n\). More specifically, under the assumption (5.84), we have for any \(1<p<2\)
provided by \(b \in {\text {BMO}}\).
We would like to establish weighted version of (5.85) as follows.
Theorem 5.17
Assume that \(\textbf{a}\) and V satisfy (5.84). Let \(b \in {\text {BMO}}\). Then for every \(p \in (1, 2)\), for every weight \(w^p \in A_p \cap RH_{(2/p)'}\), and for every \(k=0, 1, \ldots , n\), both \(\mathcal {R}_k\) and \([\mathcal {R}_k, b]\) are bounded on \(L^p(w^p)\).
A particular case is the operator \(\mathscr {L}_V = -\Delta +V\), where \(V \in L^1_{{\text {loc}}}(\mathbb {R}^n)\) is a non-negative function. The \(L^2(\mathbb {R}^n)\) boundedness of \(\mathcal {R}_V:= \nabla \mathscr {L}_V^{-1/2}\) was given in [78, Theorem 8.1], while it was proved in [41] that \(\mathcal {R}_V\) is bounded from \(H_L^1(\mathbb {R}^n)\) to \(L^1(\mathbb {R}^n)\). Then the interpolation implies
However, (5.86) fails for general potentials \(V \in L^1_{{\text {loc}}}(\mathbb {R}^n)\) when \(p>2\), see [86]. Now Theorem 5.17 immediately implies the following weighted inequalities.
Theorem 5.18
Let \(\mathscr {L}_V=-\Delta +V\) with \(0 \le V \in L^1_{{\text {loc}}}(\mathbb {R}^n)\), and set \(\mathcal {R}_V:= \nabla \mathscr {L}_V^{-1/2}\). Then for any \(p \in (1, 2)\), for any \(w^p \in A_p \cap RH_{(2/p)'}\), and for any \(b \in {\text {BMO}}\), both \(\mathcal {R}_V\) and \([\mathcal {R}_V, b]\) are bounded on \(L^p(w^p)\).
The rest of this subsection is devoted to showing Theorem 5.17. For this purpose, we present two useful lemmas below.
Lemma 5.19
[3] Fix \(1<q \le \infty \), \(a \ge 1\) and \(w \in RH_{s'}\), \(1 \le s<\infty \). Assume that F, G, \(H_1\) and \(H_2\) are non-negative measurable functions on \(\mathbb {R}^n\) such that for each ball B there exist non-negative functions \(G_B\) and \(H_B\) with \(F(x) \le G_B(x) + H_B(x)\) for a.e. \(x \in B\) and for all \(x, {\bar{x}} \in B\),
Then for all \(p \in (0, q/s)\),
where the constant C depends only on n, a, p, q, and \([w]_{RH_{s'}}\).
To proceed, we introduce some notation. Given a ball B we set \(C_j(B):= 4B\) for \(j=1\) and \(C_j(B):= 2^{j+1}B {\setminus } 2^jB\) for \(j \ge 2\), and
Lemma 5.20
Let \(1\le q \le 2\) and B be a given ball and \(f \in L^q(\mathbb {R}^n)\) with \({\text {supp}}(f) \subseteq B\). Let \(\mathcal {A}_{r_B}=I-(I-e^{-r_B^2 A})^m\) with a given integer \(m \ge 1\). Then for all \(j \ge 1\) and \(k=0, 1, \ldots , n\),
and
where the implicit constants are independent of B, f, j and k.
Proof
We begin with showing (5.89). It follows from (3.1) and (3.2) in [42] that the kernel \(p_t(x,y)\) of \(e^{-tA}\) satisfies
Thus for all \(x \in C_j(B)\) and \(j \ge 2\), we have \(|x-y| \simeq 2^j r_B\) for any \(y \in B\) and
The above inequality also holds for \(j=1\). The desired estimate (5.89) immediately follows from (5.91) and the expansion
Now we turn to the proof of (5.90). Recalling that
one has
where \(g_{r}(t) = \sum _{\ell =0}^m (-1)^{\ell } C_m^{\ell } \frac{{\textbf {1}}_{\{t>\ell r^{2}\}}}{\sqrt{t-\ell r^{2}}}\). Now we claim that
Moreover, it was proved in [42, Proposition 3.1] that for any \(j \ge 2\), there exist positive constants \(c_1\) and \(c_2\) such that
which along with (5.92) gives
It remains to demonstrate (5.92). We will use the elementary estimates for \(g_r(t)\):
The first one is easy. The second one is an application of Taylor’s formula, see [2, Sec. 3]. Denote \(\alpha =4^j/c\). Then the inequality (5.94) gives that
Write \(\phi (s)=s^{-\frac{n}{2}} e^{-\frac{\alpha }{s}}\), \(s>0\). It is easy to see that \(\phi '(s)=s^{-\frac{n}{2}-2}e^{-\frac{\alpha }{s}}(\alpha -\frac{n}{2}s)\) and
Thus, by (5.93), changing variables and (5.96), we have for any \(0 \le \ell \le m\),
where the constant \(C_m\) depending only on m and n varies from line to line. Accordingly, the inequality (5.92) follows from (5.95) and (5.97). This completes the proof. \(\square \)
Proof of Theorem 5.17
Let \(p \in (1, 2)\) and \(w^p \in A_p \cap RH_{(2/p)'}\).We follow the ideas in [5]. Choose \(p_0\) and \(q_0\) such that \(1<p_0<p<q_0<2\) and \(w^p \in A_{p/p_0} \cap RH_{(q_0/p)'}\). This together with Lemma 2.6 part (c) gives that \(w^{-p'} \in A_r \cap RH_{s'}\), where \(r=p'/q'_0\), \(s=p'_0/p'\), and \(\tau _p=\big (\frac{q_0}{p}\big )' \big (\frac{p}{p_0}-1\big )+1\). Note that \(w^{-p'} \in \cap RH_{s'}\) implies \(w^{-p'} \in RH_{s'_0}\) for some \(s_0 \in (1, s)\).
Fix \(f \in L_c^{\infty }\) and a ball B with the radius \(r_B\). Write
where \(m \in \mathbb {N}\) is large enough. Observe that
We first control \(G_B\). By duality, there exists \(g \in L^{q_0}(B, dx/|B|)\) with norm 1 such that for all \(x \in B\),
where we have used (5.90). To estimate \(H_B\), we set \(q:=p'_0/q'_0\) and observe that by duality there exists \(h \in L^{p_0}(B, dx/|B|)\) with norm 1 such that for all \(x \in B\),
where (5.89) was used in the last step.
Consequently, (5.98) and (5.99) verify the hypotheses (5.87) with \(G(x) = M (|f|^{q'_0})(x)\) and \(H_1=H_2 \equiv 0\). Observe that \(r=p'/q'_0=q/s<q/s_0\). Then, invoking (5.88) applied to r, \(s_0\), and \(w^{-p'}\) in place of p, s, and w, respectively, we obtain
which together with duality yields the \(L^p(w^p)\)-boundedness of \(\mathcal {R}_k\). This along with Theorem 1.2 implies the \(L^p(w^p)\)-boundedness of \([\mathcal {R}_k, b]\). \(\square \)
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Acknowledgements
The first author acknowledges financial support from Spanish Ministry of Science and Innovation through the Ramón y Cajal 2021 (RYC2021-032600-I), through the “Severo Ochoa Programme for Centres of Excellence in R &D” (CEX2019-000904-S), and through PID2019-107914GB-I00, and from the Spanish National Research Council through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). The third author is supported by Natural Science Foundation of Henan (No. 232300421142). Part of this work was carried out while the first author was visiting the Hausdorff Institute for Mathematics, Bonn (Germany). The first author expresses his gratitude to this institution.
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Cao, M., Liu, H., Si, Z. et al. Limited Range Extrapolation with Quantitative Bounds and Applications. J Fourier Anal Appl 30, 7 (2024). https://doi.org/10.1007/s00041-023-10061-z
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DOI: https://doi.org/10.1007/s00041-023-10061-z
Keywords
- Rubio de Francia extrapolation
- Quantitative weighted estimates
- Bilinear Bochner–Riesz means
- Bilinear rough singular integrals
- Multilinear Fourier multipliers
- Weighted jump inequalities
- Littlewood–Paley theory