New weighted norm inequalities for certain classes of multilinear operators on Morrey-type spaces
type: Item
instance of: scholarly article
title: New weighted norm inequalities for certain classes of multilinear operators on Morrey-type spaces
zbMATH Open document ID: 1479.42057
author: Nan Zhao
author: Jiang Zhou
publication date: 2021-07-28T00:00:00Z
review text: In this paper, the authors investigate the boundedness of some multilinear operators on Morrey-type spaces. To be precise, let \(T\) be a multilinear operator defined on the \(m\)-fold product of Schwartz spaces and take values into the space of tempered distributions: \[T:\, \mathcal S(\mathbb R^n)\times\cdots\times \mathcal S(\mathbb R^n)\to \mathcal S^\prime(\mathbb R^n).\] Moreover, assume that \(T\) satisfies the following conditions. \par (i) There exists a function \(K\) defined off the diagonal \(x=y_1=\cdots=y_m\) in \((\mathbb R^n)^{m+1}\) such that for all \(x\notin \cap_{j=1}^m \mathrm{supp }f_j\), \[T(f_1, \ldots, f_m)(x)=\int_{(\mathbb R^n)^m} K(x, y_1,\dots, y_m)f_1(y_1)\cdots f_m(y_m)dy_1\dots dy_m.\] \par (ii) For any \(N\ge 0\), there exists a positive constant \(C\) such that \[|K(y_0, y_1, \dots, y_m)|\le \frac{C}{(\sum_{k,\,l=0}^m|y_k-y_l|)^{mn}(1+\sum_{k,\,l=0}^m|y_k-y_l|)^N}.\] \par (iii) For some \(\epsilon>0\) and any \(N\ge 0\), there exists a positive constant \(C\) such that \begin{align*} &|K(y_0, y_1, \dots,y_j,\dots, y_m)-K(y_0, y_1, \dots,y^\prime_j,\dots, y_m)|\\ &\quad\le \frac{C|y_j-y^\prime_j|^\epsilon}{(\sum_{k,\,l=0}^m|y_k-y_l|)^{mn+\epsilon}(1+\sum_{k,\,l=0}^m|y_k-y_l|)^N} \end{align*} provided that \(0\le j\le m\) and \(|y_j-y^\prime_j|\le \frac12\max_{0\le k\le m}|y_j-y_k|\). \par (iv) There exist \(1\le q_1,\ldots, q_m<\infty\) and \(\frac1q=\frac1{q_1}+\cdots+\frac1{q_m}\) such that \(T\) is bounded from \(L^{q_1}\times\cdots\times L^{q_m}\to L^q\). On the other hand, for any cube \(Q\) and \(\theta\ge0\), write \(\varphi(Q)=(1+r)\) and \(\varphi_\theta(Q)=(1+r)^\theta\), where \(r\) is the side-length of \(Q\). Let \(\vec p=(p_1,\ldots, p_m)\) such that \(1/p=1/{p_1}+\cdots+ 1/p_m\) with \(1\le p_1,\ldots, p_m<\infty\). Given \(\vec w=(w_1,\ldots, w_m)\) with each \(w_j\) being non-negative measurable function, let \[v_{\vec w}=\prod_{j=1}^mw_j^{p/p_j}.\] Write \(\vec w\in A^\theta_{\vec p}\), if \[\sup_Q\left(\frac1{\varp…
reviewed by: Dongyong Yang
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 7375763
zbMATH Keywords: $A^\theta_{\vec p}$ weights
zbMATH Keywords: multilinear operator
zbMATH Keywords: commutators
zbMATH Keywords: Morrey-type spaces
zbMATH Keywords: BMO$_\theta$ spaces
MaRDI profile type: Publication
full work available at URL: s10114-021-0251-7
OpenAlex ID: W3173814961
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DOI: 10.1007/S10114-021-0251-7
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 7375763