Abstract
The main purpose of this paper is to establish the weighted \((L^{p},L^{q})\) inequalities of the oscillation and variation operators for the multilinear Calderón-Zygmund singular integral with a Lipschitz function.
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1 Introduction and results
Let K be a kernel on \(\mathbb{R}\times\mathbb{R}\setminus\{ (x,x): x\in\mathbb{R}\}\). Suppose that there exist two constants δ and C such that
We consider the family of operators \(T=\{T_{\epsilon}\}_{\epsilon>0}\) given by
A common method of measuring the speed of convergence of the family \(T_{\epsilon}\) is to consider the square functions
where \(\epsilon_{i}\) is a monotonically decreasing sequence which approaches 0. For convenience, other expressions have also been considered. Let \(\{t_{i}\}\) be a fixed sequence which decreases to zero. Following [1], the oscillation operator is defined as
and the ρ-variation operator is defined as
where the sup is taken over all sequences of real number \(\{ \epsilon_{i}\}\) decreasing to zero.
The oscillation and variation for some families of operators have been studied by many authors on probability, ergodic theory, and harmonic analysis; see [2–4]. Recently, some authors [5–8] researched the weighted estimates of the oscillation and variation operators for the commutators of singular integrals.
Let m be a positive integer, let b be a function on \(\mathbb{R}\), and 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 \(T_{\epsilon}\),
Note that when \(m=0\), \(T^{b}_{\epsilon}\) is just the commutator of \(T_{\epsilon}\) and b, which is denoted by \(T_{\epsilon,b}\), that is to say
However, when \(m>0\), \(T^{b}_{\epsilon}\) is a non-trivial generation of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [9–13]).
A locally integrable function b is said to be in Lipschitz space \(\mathrm{Lip}_{\beta}(\mathbb{R})\) if
where
In this paper, we will study the boundedness of oscillation and variation operators for the family of the multilinear singular integral related to a Lipschitz function defined by (1.5) in weighted Lebesgue space. Our main results are as follows.
Theorem 1.1
Suppose that \(K(x,y)\) satisfies (1.1)-(1.3), \(b^{(m)}\in\dot{\wedge}_{\beta}\), \(0<\beta\leq\delta<1\), where δ is the same as in (1.2). Let \(\rho>2\), \(T=\{T_{\epsilon}\} _{\epsilon>0}\) and \(T^{b}=\{T^{b}_{\epsilon}\}_{\epsilon>0}\) be given by (1.4) and (1.5), respectively. If \(\mathcal{O}(T)\) and \(\mathcal{V}_{\rho}(T)\) are bounded on \(L^{p_{0}}(\mathbb {R},dx)\) for some \(1< p_{0}<\infty\), then, for any \(1< p<1/\beta\) with \(1/q=1/p-\beta\), \(\omega\in A_{p,q}(\mathbb{R})\), \(\mathcal{O}(T^{b})\) and \(\mathcal{V}_{\rho}(T^{b})\) are bounded from \(L^{p}(\mathbb{R},\omega^{p} \,dx)\) into \(L^{q}(\mathbb{R},\omega^{q} \,dx)\).
Corollary 1.1
Suppose that \(K(x,y)\) satisfies (1.1)-(1.3), \(b\in\dot{\wedge}_{\beta}\), \(0<\beta\leq\delta<1\), where δ is the same as in (1.2). Let \(\rho>2\), \(T=\{T_{\epsilon}\} _{\epsilon>0}\) and \(T_{b}=\{T_{b,\epsilon}\}_{\epsilon>0}\) be given by (1.4) and (1.6), respectively. If \(\mathcal{O}(T)\) and \(\mathcal{V}_{\rho}(T)\) are bounded on \(L^{p_{0}}(\mathbb {R},dx)\) for some \(1< p_{0}<\infty\), then, for any \(1< p<1/\beta\) with \(1/q=1/p-\beta\), \(\omega\in A_{p,q}(\mathbb{R})\), \(\mathcal{O}(T_{b})\) and \(\mathcal{V}_{\rho}(T_{b})\) are bounded from \(L^{p}(\mathbb{R},\omega^{p} \,dx)\) into \(L^{q}(\mathbb{R},\omega^{q} \,dx)\).
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 Weight
A weight ω is a nonnegative, locally integrable function on \(\mathbb{R}\). The classical weight theories were introduced by Muckenhoupt and Wheeden in [14] and [15].
A weight ω is said to belong to the Muckenhoup class \(A_{p}(\mathbb{R})\) for \(1< p<\infty\), if there exists a constant C such that
for every interval I. The class \(A_{1}(\mathbb{R})\) is defined by replacing the above inequality with
When \(p=\infty\), we define \(A_{\infty}(\mathbb{R})=\bigcup_{1\leq p<\infty}A_{p}(\mathbb{R})\).
A weight \(\omega(x)\) is said to belong to the class \(A_{p,q}(\mathbb{R})\), \(1< p\leq q<\infty\), if
It is well known that if \(\omega\in A_{p.q}(\mathbb{R})\), then \(\omega^{q}\in A_{\infty}(\mathbb{R})\).
2.2 Function of \(\mathrm{Lip}_{\beta}(\mathbb{R})\)
The function of \(\mathrm{Lip}_{\beta}(\mathbb{R})\) has the following important properties.
Lemma 2.1
Let \(b\in \mathrm{Lip}_{\beta}(\mathbb{R})\). Then
-
(1)
\(1\leq p<\infty\)
$$\begin{aligned} \sup_{I}\frac{1}{ \vert I \vert ^{\beta}} \biggl(\frac{1}{ \vert I \vert } \int _{I} \bigl\vert b(x)-b_{I} \bigr\vert ^{p}\,dx \biggr)^{1/p}\leq C \Vert b \Vert _{\dot{\wedge}_{\beta}}; \end{aligned}$$ -
(2)
for any \(I_{1}\subset I_{2}\),
$$\begin{aligned} \frac{1}{ \vert I_{2} \vert } \int_{I_{2}} \bigl\vert b(y)-b_{I_{1}} \bigr\vert \,dy \lesssim\frac { \vert I_{2} \vert }{ \vert I_{1} \vert } \vert I_{2} \vert ^{\beta} \Vert b \Vert _{\dot{\wedge}_{\beta}}. \end{aligned}$$
2.3 Maximal function
We recall the definition of Hardy-Littlewood maximal operator and fractional maximal operator. The Hardy-Littlewood maximal operator is defined by
The fractional maximal function is defined as
for \(1\leq r<\infty\). In order to simplify the notation, we set \(M_{\beta}(f)(x)=M_{\beta,1}(f)(x)\).
Lemma 2.2
Let \(1< p<\infty\) and \(\omega\in A_{\infty}(\mathbb{R})\). Then
for all f such that the left hand side is finite.
Lemma 2.3
Suppose \(0<\beta<1\), \(1\leq r< p<1/\beta\), \(1/q=1/p-\beta\). If \(\omega\in A_{p,q}(\mathbb{R})\), then
2.4 Taylor series remainder
The following lemma gives an estimate on Taylor series remainder.
Lemma 2.4
[10] Let b be a function on \(\mathbb{R}\) and \(b^{(m)}\in L^{s}(\mathbb {R})\) for any \(s>1\). Then
where \(I_{x}^{y}\) is the interval \((x-5 \vert x-y \vert , x+5 \vert x-y \vert )\).
2.5 Oscillation and variation operators
We consider the operator
It is easy to check that
Following [4], we denote by E the mixed norm Banach space of two variable function h defined on \(\mathbb{R}\times\mathbb{N}\) such that
Given \(T=\{T_{\epsilon}\}_{\epsilon>0}\), where \(T_{\epsilon}\) defined as (1.4), 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 \(\mathcal{U}(T): f\rightarrow\mathcal{U}(T)f\) by
Then
On the other hand, 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 \(\mathcal {V}(T):f\rightarrow\mathcal{V}(T)f\) given by
Then
Next, 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 oscillation and variation operators.
Lemma 2.5
([5])
Suppose that \(K(x,y)\) satisfies (1.1)-(1.3), \(\rho >2\). Let \(T=\{T_{\epsilon}\}_{\epsilon>0}\) be given by (1.4). If \(O(T)\) and \(V_{\rho}(T)\) are bounded on \(L^{p_{0}}(R)\) for some \(1< p_{0}<\infty\), then, for any \(1< p<\infty\), \(\omega\in A_{p}(\mathbb{R})\),
and
3 The proof of main results
Note that if \(\omega\in A_{p,q}(\mathbb{R})\), then \(\omega^{q}\in A_{\infty}(\mathbb{R})\). By Lemma 2.2 and Lemma 2.3, we only need to prove
and
hold for any \(1< r<\infty\).
We will prove only inequality (3.1), since (3.2) can be obtained by a similar argument. Fix f and \(x_{0}\) with an interval \(I=(x_{0}-l,x_{0}+l)\). Write \(f=f_{1}+f_{2}=f\chi_{5I}+f\chi_{\mathbb{R}\setminus5I}\), and let
Then
Therefore
For \(x\in I\), \(k=0,-1,-2,\ldots\) , let \(E_{k}=\{y:2^{k-1}\cdot6l\leq \vert y-x \vert <2^{k}\cdot6l\}\), let \(I_{k}=\{y: \vert y-x \vert <2^{k}\cdot6l\}\), and let \({b}_{k}(z)=b(z)-\frac{1}{m!}(b^{(m)})_{I_{k}}z^{m}\). By [10] we have \(R_{m+1}({b};x,y)=R_{m+1}(b_{k};x,y)\) for any \(y\in E_{k}\).
By Lemma 2.5, we know \(\mathcal{O}'(T)\) is bounded on \(L^{u}(\mathbb{R})\) for \(u>1\). Then, using Hölder’s inequality, we deduce
Then
Since \({b}_{k}^{(m)}(y)=b^{(m)}(y)-(b^{(m)})_{I_{k}}\), then, applying Hölder’s inequality and Lemma 2.1, we get
We now estimate \(M_{2}\). For \(x\in I\), we have
For \(k=0,1,2,\ldots\) , let \(F_{k}=\{y:2^{k}\cdot4l\leq \vert y-x_{0} \vert <2^{k+1}\cdot4l\}\), let \(\widetilde{I}_{k}=\{y: \vert y-x_{0} \vert <2^{k}\cdot4l\}\), and let \(\widetilde{b}_{k}(z)=b(z)-\frac {1}{m!}(b^{(m)})_{\widetilde{I}_{k}}z^{m}\). Note that
By Minkowski’s inequalities and \(\Vert \{\chi_{\{ t_{i+1}< \vert x-y \vert <s\}}\}_{s\in J_{i}, i\in\mathbb{N}} \Vert _{E}\leq1\), we obtain
From the mean value theorem, there exists \(\eta\in I\) such that
For \(\eta, x\in I\), \(y\in F_{k}\), we have \(\vert y-x_{0} \vert \thickapprox \vert y-x \vert \thickapprox \vert y-\eta \vert \) and \(5 \vert y-\eta \vert \approx5 \vert y-x_{0} \vert \leq 2^{k+1}\cdot20l\). By Lemma 2.4 and Lemma 2.1 we get
Then
Since \(\vert K(x,y) \vert \leq C \vert x_{0}-y \vert ^{-1}\),
For \(N_{12}\), since \(x\in I\), \(y\in F_{k}\),
and
Thus
As for \(N_{13}\), due to
and noting \(\widetilde{b}_{k}^{(m)}(y)=b^{(m)}(y)-(b^{(m)})_{\widetilde {I}_{k}}\), we have
Notice
and by (1.2),
Similar to the estimates for \(N_{11}\), we have
Similar to the estimates for \(N_{13}\), we have
Then
Finally, let us estimate \(N_{2}\). Notice that the integral
will be non-zero in the following cases:
-
(i)
\(t_{i+1}< \vert x-y \vert <s\) and \(\vert x_{0}-y \vert \leq t_{i+1}\);
-
(ii)
\(t_{i+1}< \vert x-y \vert <s\) and \(\vert x_{0}-y \vert \geq s\);
-
(iii)
\(t_{i+1}< \vert x_{0}-y \vert <s\) and \(\vert x-y \vert \leq t_{i+1}\);
-
(iv)
\(t_{i+1}< \vert x_{0}-y \vert <s\) and \(\vert x-y \vert \geq s\).
In case (i) we have \(t_{i+1}< \vert x-y \vert \leq \vert x_{0}-x \vert + \vert x_{0}-y \vert <l+t_{i+1}\) as \(\vert x-x_{0} \vert < l\). Similarly, in case (iii) we have \(t_{i+1}< \vert x_{0}-y \vert <l+t_{i+1}\) as \(\vert x-x_{0} \vert < l\). In case (ii) we have \(s< \vert x_{0}-y \vert <l+s\) and in case (iv) we have \(s< \vert x-y \vert <l+s\). By (1.1) and taking \(1< t< r\), we have
Then
Notice
Choosing \(1< r< p\) with \(t=\sqrt{r}\), we have
But
and
Therefore
Similarly,
This completes the proof of (3.1). Hence, Theorem 1.1 is proved.
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Hu, Y., Wang, Y. Oscillation and variation inequalities for the multilinear singular integrals related to Lipschitz functions. J Inequal Appl 2017, 292 (2017). https://doi.org/10.1186/s13660-017-1568-8
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DOI: https://doi.org/10.1186/s13660-017-1568-8