Abstract
Let \(S_{\alpha ,\psi }(f)\) be the square function defined by means of the cone in \({\mathbb R}^{n+1}_{+}\) of aperture \(\alpha \), and a standard kernel \(\psi \). Let \([w]_{A_p}\) denote the \(A_p\) characteristic of the weight \(w\). We show that for any \(1<p<\infty \) and \(\alpha \ge 1\),
For each fixed \(\alpha \) the dependence on \([w]_{A_p}\) is sharp. Also, on all class \(A_p\) the result is sharp in \(\alpha \). Previously this estimate was proved in the case \(\alpha =1\) using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on \(\alpha \). Hence we give a different proof suitable for all \(\alpha \ge 1\) and avoiding the notion of the intrinsic square function.
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1 Introduction
Let \(\psi \) be an integrable function, \(\int _{{\mathbb R}^n}\psi =0\), and, for some \(\varepsilon >0\),
Let \({\mathbb R}^{n+1}_+={\mathbb R}^{n}\times {\mathbb R}_{+}\) and \(\Gamma _{\alpha }(x)=\{(y,t)\in {\mathbb {R}}^{n+1}_+:|y-x|<\alpha t\}\). Set \(\psi _t(x)=t^{-n}\psi (x/t)\). Define the square function \(S_{\alpha ,\psi }(f)\) by
We drop the subscript \(\alpha \) if \(\alpha =1\).
Given a weight \(w\), define its \(A_p\) characteristic by
where the supremum is taken over all cubes \(Q\subset {\mathbb R}^n\).
It was proved in [13] that for any \(1<p<\infty \),
and this estimate is sharp in terms of \([w]_{A_p}\) (we also refer to [13] for a detailed history of closely related results).
Similarly one can show that
however, the sharp dependence on \(\alpha \) in this estimate cannot be determined by means of the approach from [13]. The aim of this paper is to find the sharp \(\gamma (\alpha )\) in (1.3).
Let us explain first why the method from [13] gives a rough estimate for \(\gamma (\alpha )\). The proof in [13] was based on the intrinsic square function \(G_{\alpha ,\beta }(f)\) by Wilson [19] defined as follows. For \(0<\beta \le 1\), let \({\mathcal C}_{\beta }\) be the family of functions supported in the unit ball with mean zero and such that for all \(x\) and \(x', |\varphi (x)-\varphi (x')|\le |x-x'|^{\beta }\). If \(f\in L^1_{\text {loc}}({\mathbb R}^n)\) and \((y,t)\in {\mathbb R}^{n+1}_+\), we define \( A_{\beta }(f)(y,t)=\sup _{\varphi \in {\mathcal C}_{\beta }}|f*\varphi _t(y)| \) and
Set \(G_{1,\beta }(f)=G_{\beta }(f)\).
The intrinsic square function has several interesting features (established in [19]). First, though \(G_{\beta }(f)\) is defined by means of kernels with uniform compact support, it pointwise dominates \(S_{\psi }(f)\). Also there is a pointwise relation between \(G_{\alpha ,\beta }(f)\) with different apertures:
Notice that for the usual square functions \(S_{\alpha ,\psi }(f)\) such a pointwise relation is not available.
In [13], (1.2) with \(G_{\beta }(f)\) instead of \(S_{\psi }(f)\) was obtained. Combining this with (1.4), we would obtain that one can take \(\gamma (\alpha )=\alpha ^{(3/2)n+\beta }\) in (1.3) assuming that \(\psi \in {\mathcal C}_{\beta }\). For non-compactly supported \(\psi \) some additional ideas from [19] can be used that lead to even worse estimate on \(\gamma (\alpha )\). Observe also that it is not clear to us whether (1.4) can be improved.
It is easy to see that the dependence \(\gamma (\alpha )=\alpha ^{(3/2)n+\beta }\) in (1.3) is far from the sharp one. For instance, it is obvious that the information on \(\beta \) should not appear in (1.3). All this indicates that the intrinsic square function approach is not suitable for our purposes in determining the sharp \(\gamma (\alpha )\).
Suppose we seek for \(\gamma (\alpha )\) in the form \(\gamma (\alpha )=\alpha ^{r}\). Then a simple observation shows that \(r\ge n\) for any \(1<p<\infty \). Indeed, consider the Littlewood–Paley function \(g_{\mu ,\psi }^*(f)\) defined by
Using the standard estimate
we obtain that (1.3) for some \(p=p_0\) and \(\gamma (\alpha )=\alpha ^{r_0}\) implies
This means that if \(\mu >2r_0/n\), then \(g_{\mu ,\psi }^*\) is bounded on \(L^{p_0}(w), w\in ~A_{p_0}\). From this, by the Rubio de Francia extrapolation theorem, \(g_{\mu ,\psi }^*\) is bounded on the unweighted \(L^p\) for any \(p>1\), whenever \(\mu >2r_0/n\). But it is well known [8] that \(g_{\mu ,\psi }^*\) is not bounded on \(L^p\) if \(1<\mu <2\) and \(1<p\le 2/\mu \). Hence, if \(r_0<n\), we would obtain a contradiction to the latter fact for \(p\) sufficiently close to \(1\).
Our main result shows that for any \(1<p<\infty \) one can take the optimal power growth \(\gamma (\alpha )=\alpha ^n\).
Theorem 1.1
For any \(1<p<\infty \) and for all \(1\le \alpha <\infty \),
By (1.5), we immediately obtain the following.
Corollary 1.2
Let \(\mu >2\). Then for any \(1<p<\infty \),
Observe that if \(\mu =2\), then \(g_{2,\psi }^*\) is also bounded on \(L^p(w)\) for \(w\in A_p\) (see [17]). However, the sharp dependence on \([w]_{A_p}\) in the corresponding \(L^p(w)\) inequality is unknown to us.
We emphasize that the growth \(\gamma (\alpha )=\alpha ^n\) is best possible in the weighted \(L^p(w)\) estimate for \(w\in A_p\). In the unweighted case a better dependence on \(\alpha \) is known, namely, \(\Vert S_{\alpha ,\psi }\Vert _{L^p}\le c_{p,n,\psi }\alpha ^{\frac{n}{\min (p,2)}},\) see [1, 18].
Some words about the proof of Theorem 1.1. As in [13], we use here the local mean oscillation decomposition. But in [13] we worked with the intrinsic square function, and due to the fact that this operator is defined by uniform compactly supported kernels, we arrived to the operator
where \(Q_j^k\) is a sparse family (see Sect. 2.2 for the definition of this notion) and \(\gamma >1\) (here we use the standard notations \(f_Q=\frac{1}{|Q|}\int _Qf\) and \(\gamma Q\) is the \(\gamma \)-fold concentric dilate of \(Q\)). This operator can be handled sufficiently easily.
Here we work with the square function \(S_{\alpha ,\psi }(f)\) directly, more precisely we consider its smooth variant \(\widetilde{S}_{\alpha ,\psi }(f)\). Applying the local mean oscillation decomposition to \(\widetilde{S}_{\alpha ,\psi }(f)\), we obtain that \(S_{\alpha ,\psi }(f)\) is essentially pointwise bounded by \(\alpha ^n{\mathcal B}(f)\), where
Observe that this pointwise aperture estimate is interesting in its own right. In order to handle \({\mathcal B}\), we use a mixture of ideas from recent papers on a simple proof of the \(A_2\) conjecture [14] and sharp weighted estimates for multilinear Calderón–Zygmund operators [5]. In particular, similarly to [14], we obtain the \(X^{(2)}\)-norm boundedness of \({\mathcal B}\) by \({\mathcal A}\) on an arbitrary Banach function space \(X\).
The paper is organized as follows. The next section contains some preliminary information. In Sect. 3, we obtain the main estimate, namely, the local mean oscillation estimate of \(\widetilde{S}_{\alpha ,\psi }(f)\). The proof of Theorem 1.1 is contained in Sect. 4. Section 5 contains some concluding remarks concerning the sharp aperture-weighted weak type estimates for \(S_{\alpha ,\psi }(f)\).
2 Preliminaries
2.1 A Weak Type \((1,1)\) Estimate for Square Functions
It is well known that the operator \(S_{\alpha ,\psi }\) is of weak type \((1,1)\). However, we could not find in the literature the sharp dependence on \(\alpha \) in the corresponding inequality. Hence we give below an argument based on general square functions.
For a measurable function \(F\) on \({\mathbb R}^{n+1}_+\) define
Lemma 2.1
For any \(\alpha \ge 1\),
Proof
We will use the following estimate, which can be found in [18, p. 315]: if \(\Omega \subset {\mathbb R}^n\) is an open set and
where \(M\) is the Hardy–Littlewood maximal operator, then
(observe that the definitions of \(S_{\alpha }(F)\) here and in [18] differ by the factor \(\alpha ^{n/2}\).)
Let \(\Omega _{\xi }=\{x:S_{1}(F)(x)>\xi \}\) and \(U_{\xi }=\{x:M\chi _{\Omega _{\xi }}(x)>1/2\alpha ^n\}\). Using the weak type \((1,1)\) estimate for \(M\), Chebyshev’s inequality, and the above estimate, we obtain
Further,
Combining this with the previous estimate gives
which proves (2.1). \(\square \)
Note that the sharp unweighted \(L^p\) estimates relating square functions of different apertures were obtained recently in [1].
By Lemma 2.1 and by the weak type \((1,1)\) estimate for \(S_{\psi }(f)\) [9],
2.2 Dyadic Grids and Sparse Families
Recall that the standard dyadic grid in \({\mathbb R}^n\) consists of the cubes
Denote the standard grid by \({\mathcal D}\).
By a general dyadic grid \({\fancyscript{D}}\) we mean a collection of cubes with the following properties: (i) for any \(Q\in {\fancyscript{D}}\) its sidelength \(\ell _Q\) is of the form \(2^k, k\in {\mathbb Z}\); (ii) \(Q\cap R\in \{Q,R,\emptyset \}\) for any \(Q,R\in {\fancyscript{D}}\); (iii) the cubes of a fixed sidelength \(2^k\) form a partition of \({\mathbb R}^n\).
Given a cube \(Q_0\), denote by \({\mathcal D}(Q_0)\) the set of all dyadic cubes with respect to \(Q_0\), that is, the cubes from \({\mathcal D}(Q_0)\) are formed by repeated subdivision of \(Q_0\) and each of its descendants into \(2^n\) congruent subcubes. Observe that if \(Q_0\in {\fancyscript{D}}\), then each cube from \({\mathcal D}(Q_0)\) will also belong to \({\fancyscript{D}}\).
We will use the following proposition from [10].
Proposition 2.2
There are \(2^n\) dyadic grids \({\fancyscript{D}}_{i}\) such that for any cube \(Q\subset {\mathbb R}^n\) there exists a cube \(Q_{i}\in {\fancyscript{D}}_{i}\) such that \(Q\subset Q_{i}\) and \(\ell _{Q_{i}}\le 6\ell _Q\).
We say that \(\{Q_j^k\}\) is a sparse family of cubes if: (i) the cubes \(Q_j^k\) are disjoint in \(j\), with \(k\) fixed; (ii) if \(\Omega _k=\cup _jQ_j^k\), then \(\Omega _{k+1}\subset ~\Omega _k\); (iii) \(|\Omega _{k+1}\cap Q_j^k|\le \frac{1}{2}|Q_j^k|\).
2.3 A “Local Mean Oscillation Decomposition”
The non-increasing rearrangement of a measurable function \(f\) on \({\mathbb R}^n\) is defined by
Given a measurable function \(f\) on \({\mathbb R}^n\) and a cube \(Q\), the local mean oscillation of \(f\) on \(Q\) is defined by
By a median value of \(f\) over \(Q\) we mean a possibly nonunique, real number \(m_f(Q)\) such that
It is easy to see that the set of all median values of \(f\) is either one point or a closed interval. In the latter case we will assume for the definiteness that \(m_f(Q)\) is the maximal median value. Observe that it follows from the definitions that
Given a cube \(Q_0\), the dyadic local sharp maximal function \(m^{\#,d}_{\lambda ;Q_0}f\) is defined by
The following theorem was proved in [15] (a very similar version can be found in [12]).
Theorem 2.3
Let \(f\) be a measurable function on \({\mathbb R}^n\) and let \(Q_0\) be a fixed cube. Then there exists a (possibly empty) sparse family of cubes \(Q_j^k\in {\mathcal D}(Q_0)\) such that for a.e. \(x\in Q_0\),
3 A Key Estimate
In this section we will obtain the main local mean oscillation estimate of \(S_{\alpha ,\psi }\). We consider a smooth version of \(S_{\alpha ,\psi }\) defined as follows. Let \(\Phi \) be a Schwartz function such that
Define
It is easy to see that
Hence, by (2.2),
Lemma 3.1
For any cube \(Q\subset {\mathbb R}^n\),
where \(\delta =\varepsilon \) from condition (1.1) if \(\varepsilon <1\), and \(\delta <1\) if \(\varepsilon =1\).
Proof
Given a cube \(Q\), let \(T(Q)=\{(y,t):y\in Q, 0<t<{\ell }_Q\},\) where \({\ell }_Q\) denotes the side length of \(Q\). For \(x\in Q\) we decompose \(\widetilde{S}_{\alpha ,\psi }(f)(x)^2\) into the sum of
and
Let us show first that
Using that \((a+b)^2\le 2(a^2+b^2)\), we get
Hence,
By (3.1),
Further, by (1.1), for \((y,t)\in T(2Q)\),
Hence, using Chebyshev’s inequality and that \(\int _{{\mathbb R}^n}\Phi \Big (\frac{x-y}{t\alpha }\Big )dx\le c_n(t\alpha )^n\), we have
By Hölder’s inequality,
Combining this with the previous estimate and with (3.5) and (3.4) proves (3.3).
Let \(x,x_0\in Q\), and let us estimate now \(|I_2(f)(x)-I_2(f)(x_0)|\). We have
Suppose \((y,t)\in T(2^{k+1}Q)\setminus T(2^kQ)\). If \(y\in 2^kQ\), then \(t\ge 2^k\ell _Q\). On the other hand, if \(y\in 2^{k+1}Q\setminus 2^kQ\), then for any \(x\in Q, |y-x|\ge \frac{2^k-1}{2}\ell _Q\). Hence, if \(t<\frac{2^k-1}{4\alpha }\ell _Q\), then \(|y-x|/\alpha t>2\) and \(|y-x_0|/\alpha t>2\), and therefore,
Assume that \(t\ge \frac{2^k-1}{4\alpha }\ell _Q\). This easily implies \(t\ge 2^{k-3}\ell _Q/\alpha \). Thus, using that
we get
Hence,
where
and
Let us first estimate \(J_1\). Using Minkowski’s integral inequality, we obtain
Since
we get
We turn to the estimate of \(J_2\). By (1.1), for \((y,t)\in T(2^{k+1}Q)\),
Therefore,
Combining the estimates for \(J_1\) and \(J_2\), we obtain
By Hölder’s inequality,
where
Therefore,
From this and from (3.3),
which completes the proof. \(\square \)
4 Proof of Theorem 1.1
4.1 Several Auxiliary Operators
Throughout this subsection we assume that \(f,g\ge 0\). Given a sparse family \({\mathcal S}=\{Q_j^k\}\subset {\fancyscript{D}}\), define
The following result was proved in [4].
Lemma 4.1
For any \(1<p<\infty \),
Given a sparse family \({\mathcal S}=\{Q_j^k\}\subset {\mathcal D}\), define
Applying Proposition 2.2, we decompose the family of cubes \(\{Q_j^k\}\) into \(2^n\) disjoint families \(F_{i}\) such that for any \(Q_j^k\in F_{i}\) there exists a cube \(P_{j,k}^{m,i}\in {\fancyscript{D}}_{i}\) such that \(2^mQ_j^k\subset P_{j,k}^{m,i}\) and \(\ell _{P_{j,k}^{m,i}}\le 6\ell _{2^mQ_j^k}\). Hence,
where
The following statement was obtained in [5].
Lemma 4.2
Suppose that the sum defining \({\fancyscript{M}}_{i,m}^{\mathcal S}(f,g)\) contains finitely many terms. Then there are at most \(2^n\) cubes \(Q_{\nu }\in {\fancyscript{D}}_{i}\) covering the support of \({\fancyscript{M}}_{i,m}^{\mathcal S}(f,g)\) so that for every \(Q_{\nu }\) there are two sparse families \({\mathcal S}_{i,1}\) and \({\mathcal S}_{i,2}\) from \({\fancyscript{D}}_{i}\) having the property that for a.e. \(x\in Q_{\nu }\),
Observe that the proof of Lemma 4.2 is based on Theorem 2.3 along with [14, Lemma 3.2]. Formally Lemma 4.2 follows from [5, Lemma 4.2] taking there \(m=2\) (which corresponds to a bilinear case) and \(l=m\), and from the subsequent argument in [5], Sect. 4.2].
Let \(X\) be a Banach function space, and let \(X'\) denote the associate space (see [2, Ch. 1]). Given a Banach function space \(X\), denote by \(X^{(2)}\) the space endowed with the norm
It is well known [16, Ch. 1] that \(X^{(2)}\) is also a Banach space.
Lemma 4.3
For any Banach function space \(X\),
Proof
By the standard argument, it suffices to prove the estimate for a finite partial sum \({\widetilde{\mathcal T}}^{\mathcal S}_{2,m}f\) from the series defining \({\mathcal T}^{\mathcal S}_{2,m}f\). Fix \({\mathcal S}\in {\mathcal D}\). By duality, there exists \(g\ge 0\) with \(\Vert g\Vert _{X'}=1\) such that
where the sum defining \({\fancyscript{M}}_{m}^{\mathcal S}(f,g)\) contains finitely many terms. By Lemma 4.2 and by Hölder’s inequality,
Summing up over \(Q_{\nu }\) and using (4.1), we obtain
This along with (4.2) completes the proof.
4.2 Proof of Theorem 1.1
Let \(Q\in {\mathcal D}\). By Lemma 3.1, for all \(x\in Q\),
Hence, applying Theorem 2.3 to \(\widetilde{S}_{\alpha ,\psi }(f)^2\), we get that there exists a sparse family \({\mathcal S}=\{Q_j^k\}\subset {\mathcal D}(Q)\) such that for a.e. \(x\in Q\),
Hence,
where
Assuming, for instance, that \(f\in L^1\), and using (2.3) and (3.1), we get
Therefore, letting \(Q\) tend to anyone of \(2^n\) quadrants and using Fatou’s lemma, by (4.3) we obtain
Combining Lemma 4.1 and Lemma 4.3 with \(X=L^{3/2}(w)\) yields
Hence, by the sharp version of the Rubio de Francia extrapolation theorem (see [6] or [7]),
Thus, applying this result along with Buckley’s estimate \(\Vert M\Vert _{L^p(w)}\le c_{n,p}[w]_{A_p}^{\frac{1}{p-1}}\) (see [3]) and (4.4), we get
and therefore, the proof is complete.
5 Concluding Remarks
In a recent work [11], the following weak type estimate was obtained for \(G_{\beta }(f)\) (and hence for \(S_{\psi }(f)\)): if \(1<p<3\), then
where \(\Phi _p(t)=1\) if \(1<p<2\) and \(\Phi _p(t)=1+\log t\) if \(p\ge 2\). The proof was based on the local mean oscillation decomposition technique along with the estimate
Since the space \(L^{p,\infty }(w)\) is normable if \(p>1\) (see, e.g., [2, p. 220]), combining Lemma 4.3 with \(X=L^{1+\varepsilon ,\infty }(w), \varepsilon >0,\) and (5.1) yields for \(2<p<3\) that
Hence, exactly as above, by (4.3) (and by the weak type estimate for \(M\) proved in [3]), we obtain
We emphasize that our approach does not allow to extend this estimate to \(1<p\le 2\). This is clearly related to the same problem with (5.2). The limitation \(2<p<3\) in (5.2) is due to Lemma 4.3 where the condition that \(X\) is a Banach function space was essential in the proof. This raises a natural question whether Lemma 4.3 holds under the condition that \(X\) is a quasi-Banach space. Observe that the same question can be asked regarding a recent estimate relating \(X\)-norms of Calderón–Zygmund and dyadic positive operators [15].
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I am very grateful to the referees for useful remarks and corrections.
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Communicated by Chris Heil.
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Lerner, A.K. On Sharp Aperture-Weighted Estimates for Square Functions. J Fourier Anal Appl 20, 784–800 (2014). https://doi.org/10.1007/s00041-014-9333-6
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DOI: https://doi.org/10.1007/s00041-014-9333-6