Abstract
Suppose that the kernel K satisfies a certain Hörmander type condition. Let b be a function satisfying \(D^{\alpha}b\in BMO(\mathbb{R}^{n})\) for \(\vert \alpha \vert =m\), and let \(T^{b}=\{T^{b}_{\epsilon}\}_{\epsilon>0}\) be a family of multilinear singular integral operators, i.e.,
The main purpose of this paper is to establish the weighted \(L^{p}\)-boundedness of the variation operator and the oscillation operator for \(T^{b}\).
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1 Introduction and results
Let K be a singular kernel in \(\mathbb{R}^{n}\) and satisfy
where C is a fixed constant. Consider the family of operators \(T=\{T_{\epsilon}\}_{\epsilon>0}\), where
The ρ-variation operator is defined by
where the supremum is taken over all sequences of real numbers \(\{ \epsilon_{i}\}\) decreasing to zero. The oscillation operator is defined by
where \(\{t_{i}\}\) is a fixed sequence which decreases to zero.
Variation and oscillation can be used to measure the speed of convergence of certain convergent families of operators. The operators of variation and oscillation have attracted many researchers’ attention in probability, ergodic theory, and harmonic analysis. Bourgain [1] obtained variation inequality for the ergodic averages of a dynamic system, his work has launched a new research direction in harmonic analysis. Campbell, Jones, Reinhold, and Wierdl in [2] established the \(L^{p}\)-boundedness of variation operator and oscillation operator for the Hilbert transform. Recently, many other publications have enriched this research direction [3–7].
Let \(1\leq r\leq\infty, m\in\mathbb{N}\cup\{0\}\). We say that the kernel K satisfies the \(H_{r,m}\)-Hörmander condition if there exist the constants \(c\geq1\) and \(C_{r,m}>0\) such that, for all \(y\in\mathbb{R}^{n}\) and \(R>c \vert x \vert \),
if \(r<\infty\), and
in the case \(r=\infty\).
We notice that \(H_{r,0}\) is \(L^{r}\)-Hörmander condition, which was studied in depth in [8].
Let K satisfy (1) and \(H_{r,1}\)-Hörmander condition, and let \(T_{b}=\{T_{\epsilon,b}\}_{\epsilon>0}\), where \(T_{\epsilon ,b}\) is the commutator of \(T_{\epsilon}\) and b,
Suppose \(r>1,\rho>2\), and \(b\in BMO(\mathbb{R}^{n})\). Zhang and Wu proved in [9] that, if \(V_{\rho}(T)\) and \(\mathcal {O}(T)\) are bounded on \(L^{p_{0}}(\mathbb{R}^{n})\) for some \(p_{0}>1\), then \(V_{\rho}(T_{b})\) and \(\mathcal{O}(T_{b}) \) are bounded on \(L^{p}(\mathbb{R}^{n}, \omega)\) for any \(\max\{r',p_{0}\}< p<\infty, \omega \in A_{p/\max\{r',p_{0}\}}(\mathbb{R}^{n})\).
Given m is a positive integer, and b is a function on \(\mathbb{R}^{n}\). Let \(R_{m+1}(b;x,y)\) be the \(m+1\)th Taylor series remainder of b at x expander about y, i.e.,
We consider the family of operators \(T^{b}=\{T^{b}_{\epsilon}\}_{\epsilon >0}\), where \(T^{b}_{\epsilon}\) are the multilinear singular integral operators of \(T_{\epsilon}\) as follows:
Note that when \(m=0, T^{b}_{\epsilon}\) is just the commutator of \(T_{\epsilon}\) and b. However, when \(m>0, T^{b}_{\epsilon}\) is a non-trivial generation of the commutator. It is well known that multilinear operators have been widely studied by many authors (see [10–14]).
In [15], Hu and Wang established the weighted \((L^{p},L^{q})\) inequalities of the variation and oscillation operators for the multilinear Calderón–Zygmund singular integral with a Lipschitz function in \(\mathbb{R}\). In this paper, if K satisfies (1) and \(H_{r,1}\)-Hörmander condition, we will study the bounded behaviors of variation and oscillation operators for the family of the multilinear singular integrals defined by (4) in \(L^{p}(\mathbb{R}^{n},\omega(x)\,dx)\) when \(D^{\alpha}b\in BMO(\mathbb{R}^{n})\) for \(\vert \alpha \vert =m\). Our main results can be formulated as follows.
Theorem 1
Let K satisfy (1), \(\rho>2\), and let \(T=\{T_{\epsilon}\}_{\epsilon>0}\) and \(T^{b}=\{T^{b}_{\epsilon}\} _{\epsilon>0}\) be given by (2) and (4), respectively. Suppose that \(K\in H_{r,1}\),\(V_{\rho}(T)\), and \(\mathcal{O}(T) \) are bounded on \(L^{p_{0}}(\mathbb{R}^{n})\) for some \(p_{0}>1\), and \(D^{\alpha}b\in BMO(\mathbb{R}^{n})\) for \(\vert \alpha \vert =m\), then \(V_{\rho}(T^{b})\) and \(\mathcal{O}(T^{b}) \) are bounded on \(L^{p}(\mathbb {R}^{n},\omega)\) for any \(\max\{r',p_{0}\}< p<\infty, \omega\in A_{p/\max\{ r',p_{0}\}}(\mathbb{R}^{n})\).
Corollary 1
([9])
Let K satisfy (1), \(\rho>2\), and let \(T=\{T_{\epsilon}\} _{\epsilon>0}\) and \(T_{b}=\{T_{\epsilon,b}\}_{\epsilon>0}\) be given by (2) and (3), respectively. Suppose that \(K\in H_{r,1}\),\(V_{\rho}(T)\) and \(\mathcal{O}(T) \) are bounded on \(L^{p_{0}}(\mathbb{R}^{n})\) for some \(p_{0}>1\), and \(b\in BMO(\mathbb{R}^{n})\), then \(V_{\rho}(T_{b})\) and \(\mathcal{O}(T_{b}) \) are bounded on \(L^{p}(\mathbb{R}^{n},\omega)\) for any \(\max\{r',p_{0}\} < p<\infty, \omega\in A_{p/\max\{r',p_{0}\}}(\mathbb{R}^{n})\).
In this paper, we shall use the symbol \(A\lesssim B\) to indicate that there exists a universal positive constant C, independent of all important parameters, such that \(A\leq CB\). \(A\thickapprox B\) means that \(A\lesssim B\) and \(B\lesssim A\).
2 Some preliminaries
2.1 \(A_{p}(\mathbb{R}^{n})\) weight
A weight ω belongs to \(A_{p}(\mathbb{R}^{n})\) for \(1< p<\infty\) if there exists a constant C such that, for every ball \(B\subset\mathbb{R}^{n} \),
where \(p^{\prime}\) is the dual of p such that \(1/p+1/p^{\prime}=1\). A weight ω belongs to \(A_{1}(\mathbb{R}^{n})\) if
2.2 Function of \(BMO(\mathbb{R}^{n})\)
Following [16], a locally integrable function b is said to be in \(BMO(\mathbb{R}^{n})\) if
where
The function of \(BMO(\mathbb{R}^{n})\) has the following property.
Lemma 1
Suppose \(b\in BMO(\mathbb{R}^{n}),B_{k}=2^{k}B, k\in\mathbb{N}\cup\{0\}\). Then, for \(1\leq p<\infty\),
2.3 Maximal function
The Hardy–Littlewood maximal operator is defined by
We also define the maximal function
for \(1\leq r<\infty\). It is well known that \(M_{r}\) is bounded on \(L^{p}(\mathbb{R}^{n},\omega)\) for \(r< p<\infty\) and \(\omega\in A_{p}(\mathbb{R}^{n})\) (see [17]).
2.4 Taylor series remainder
By definition, it is obvious that
The following lemma gives an estimate on Taylor series remainder.
Lemma 2
([11])
Let b be a function on \(\mathbb{R}^{n}\) with mth order derivatives in \(L^{q}(\mathbb{R})\) for some \(q>n\). Then
where \(Q(x,y)\) is the cube centered at x and having diameter \(5\sqrt{n} \vert x-y \vert \).
2.5 Variation and oscillation operators
Following [2], let \(\Theta=\{\beta: \beta=\{\epsilon_{i}\},\epsilon _{i}\in\mathbb{R},\epsilon_{i}\searrow0\}\). We denote by \(F_{\rho}\) the mixed norm space of two variable functions \(g(i,\beta)\) such that
We also consider the \(F_{\rho}\)-valued operator \(\mathcal {V}(T):f\rightarrow\mathcal{V}(T)f\) by
This implies that
On the other hand, we consider the operator
It is easy to check that
We denote by E the mixed norm Banach space of two-variable function h defined on \(\mathbb{R}\times\mathbb{N}\) such that
For a fixed decreasing sequence \(\{t_{i}\}\) with \(t_{i}\searrow0\), let \(J_{i}=(t_{i+1},t_{i}]\) and define the E-valued operator \(\mathcal{U}(T): f\rightarrow\mathcal{U}(T)f\) given by
Then
Let B be a Banach space and φ be a B-valued function, we define the sharp maximal operator as follows:
Then
and
Finally, let us recall some results about the boundedness of \(V_{\rho}(T)\) and \(\mathcal{O}(T) \).
Lemma 3
([9])
Let K satisfy (1), \(\rho>2\), \(T=\{T_{\epsilon}\} _{\epsilon>0}\) be given by (2). Suppose that K satisfies (1) and \(K\in H_{r,0}\)-Hörmander condition, \(V_{\rho}(T)\) and \(\mathcal{O}(T) \) are bounded on \(L^{p_{0}}(\mathbb{R}^{n})\) for some \(1< p_{0}<\infty\). Then \(V_{\rho}(T)\) and \(\mathcal{O}(T) \) are bounded on \(L^{p}(\mathbb {R}^{n})\) for any \(\max\{r',p_{0}\}< p<\infty\).
3 Proof of Theorem 1
By the fact that \(M_{s}\) is bounded on \(L^{p}(\mathbb{R}^{n},\omega)\) for \(1\leq s< p<\infty\) and \(\omega\in A_{p}(\mathbb{R}^{n})\), we need to prove
and
for any \(s> \max\{r',p_{0}\}\).
We will only need to prove (5), since (6) can be obtained by a similar argument. To prove (5), it suffices to prove that for any fixed \(x_{0}\in\mathbb{R}^{n}\) and some \({F_{\rho}}\)-valued constant \(c_{1}\), such that for every ball \(B=B(x_{0},l)\) with radius l, centered at \(x_{0}\), the following inequality
holds.
We write \(f=f_{1}+f_{2}=f\chi_{10B}+f\chi_{\mathbb{R}^{n}\setminus10B}\). Then
Let \(x_{1}\) be a point at the boundary of 2B, and let
Then
For any \(x\in B, k\in\mathbb{Z}\), let \(E_{k}=\{y:2^{k}\cdot3l\leq \vert y-x \vert <2^{k+1}\cdot3l\}\), let \(F_{k}=\{y: \vert y-x \vert <2^{k+1}\cdot3l\}\), and let
By [11] we have \(R_{m+1}({b};x,y)=R_{m+1}(b_{k};x,y)\) for any \(y\in E_{k}\). From Lemma 3, we know \(V_{\rho}(T)\) is bounded on \(L^{u}(\mathbb{R}^{n})\) for \(\max\{r',p_{0}\}< u< s\). Then, using Hölder’s inequality, we deduce
For any \(y\in E_{k}\), by Lemma 2 and Lemma 1,
Then we have
By \(D^{\alpha}b_{k}(y)=D^{\alpha}b(y)-(D^{\alpha}b)_{F_{k}}\), applying Hölder’s inequality and Lemma 2, we get
We now estimate \(M_{2}\). We write
By Minkowski’s inequality and \(\Vert \{\epsilon _{i+1}< \vert x-y \vert <\epsilon_{i}\}_{\beta=\{\epsilon_{i}\}\in\Theta} \Vert _{F_{\rho}}\leq1\), we obtain
Applying the formula ([11] p. 448), we have
Then, by Lemmas 1 and 2, we have
and
So
By (1) we have
Then
For \(y\in E_{k}, x\in B\) and \(x_{1}\) being a point at the boundary of 2B, we get \(\vert x_{1}-y \vert \thickapprox \vert x-y \vert \). Thus
and
Then
As for \(N_{13}\), due to
for \(\vert \alpha \vert =m\), and \(D^{\alpha}{b}_{k}(y)=D^{\alpha}{b}(y)-(D^{\alpha}{b})_{F_{k}}\), we have
Let us estimate \(N_{14}\) now. For \(y\in(10B)^{c}\), we have \(\vert y-x_{1} \vert \geq \vert y-x_{0} \vert - \vert x_{0}-x_{1} \vert >8l\). For \(k=1,2,\ldots\) , let \(\widetilde{E}_{k}=\{y:2^{k}\cdot 3l\leq \vert y-x_{1} \vert <2^{k+1}\cdot3l\}\), let \(\widetilde{F}_{k}=\{y: \vert y-x_{1} \vert <2^{k+1}\cdot3l\}\), and let
Note that \(Q(x,y)\subset2\sqrt{n}Q(x_{1},y)\), then for \(x\in B, y\in \widetilde{E}_{k}\) we have
So
Then
Taking \(R=3l\), then \(\vert x-x_{1} \vert < R\). By Hölder’s inequality and \(K\in H_{r,1}\subset H_{r,0} \), we have
and
Finally, let us estimate \(N_{2}\). Note that the integral
will only be non-zero if either \(\chi_{\{\epsilon_{i+1}< \vert x-y \vert <\epsilon _{i}\}}(y)=1\) and \(\chi_{\{\epsilon_{i+1}< \vert x_{1}-y \vert <\epsilon_{i}\}}(y)=0\) or vice versa. That means the integral will only be non-zero in the following cases:
-
(i)
\(\epsilon_{i+1}< \vert x-y \vert <\epsilon_{i}\) and \(\vert x_{1}-y \vert \leq \epsilon_{i+1}\);
-
(ii)
\(\epsilon_{i+1}< \vert x-y \vert <\epsilon_{i}\) and \(\vert x_{1}-y \vert \geq \epsilon_{i}\);
-
(iii)
\(\epsilon_{i+1}< \vert x_{1}-y \vert <\epsilon_{i}\) and \(\vert x-y \vert \leq \epsilon_{i+1}\);
-
(iv)
\(\epsilon_{i+1}< \vert x_{1}-y \vert <\epsilon_{i}\) and \(\vert x-y \vert \geq \epsilon_{i}\).
In case (i) we observe that \(\epsilon_{i+1}< \vert x-y \vert \leq \vert x_{1}-x \vert + \vert x_{1}-y \vert <3l+\epsilon_{i+1}\) as \(\vert x-x_{0} \vert < l\). Similarly, in case (iii) we have \(\epsilon_{i+1}< \vert x_{1}-y \vert <3l+\epsilon _{i+1}\) as \(\vert x-x_{0} \vert < l\). In case (ii) we have \(\epsilon_{i}< \vert x_{1}-y \vert <3l+\epsilon_{i}\), and in case (iv) we have \(\epsilon_{i}< \vert x-y \vert <3l+\epsilon_{i}\). By (1) we have
It is easy to check that \(\vert x-y \vert \geq9l\), \(\vert x_{1}-y \vert \geq8l\), and \(\frac{8}{11} \vert x-y \vert \leq \vert x_{1}-y \vert \leq\frac{4}{3} \vert x-y \vert \) for \(x\in B,y\in (10B)^{c}\); moreover, if \(3l\geq\epsilon_{i+1},i\in\mathbb{N}\), we have
and
this means \(P_{1}=P_{3}=0\). Similarly, \(P_{2}=P_{4}=0\) for \(3l\geq\epsilon _{i}, i\in\mathbb{N}\). By Hölder’s inequality with t satisfying \(1< t<\sqrt{\min(r',\rho)}\), we get
and
Note that for \(3l<\epsilon_{i+1}\), we have \((3l+\epsilon _{i+1})^{n}-(\epsilon_{i+1})^{n}\lesssim(\epsilon_{i+1})^{n-1}l\). Then
Similarly, we have
and
Then
A straight computation deduces that
Similar to the estimates for \(N_{13}\), for \(x\in B, y\in\widetilde {E}_{k}\), we have
Then
However,
and
Then
Similarly,
This completes the proof of Theorem 1.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11701207).
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Xia, Y. Variation and oscillation for the multilinear singular integrals satisfying Hörmander type conditions. J Inequal Appl 2018, 93 (2018). https://doi.org/10.1186/s13660-018-1689-8
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DOI: https://doi.org/10.1186/s13660-018-1689-8