Fractional integrals on product manifolds
type: Item
instance of: scholarly article
title: Fractional integrals on product manifolds
zbMATH Open document ID: 1185.42009
author: Xinfeng Wu
author: Yong Ding
published in: Potential Analysis
publication date: 2009-05-04T00:00:00Z
review text: A mixed norm estimate is established for multi-parameter fractional integrals on product spaces. This result is then shown to imply boundedness of several particular kinds of fractional integrals, namely those of Nagel-Stein type on product manifolds, of Folland-Stein type on products of homogeneous groups and the discrete fractional integrals of Stein-Wainger type. Let \(\left( \mathcal{X}_{i},\mathcal{M}_{i},\mu _{i}\right) \), \( i=1,2\), be two measure spaces and define \(P=(p_{1},p_{2})\). Define the mixed norm space \[ \begin{multlined} L^{P}\left( \mathcal{X}_{1}\times \mathcal{X}_{2}\right) := \left\{f: \mathcal{X}_{1}\times \mathcal{X}_{2}\rightarrow \mathbb{R}^{1},f \text{ is } \mu _{1}\otimes \mu _{2}\text{-measurable},\right. \\ \left.\left\| f\right\| _{L^{P}\left( \mathcal{X}_{1}\times \mathcal{X}_{2}\right) }=\left\| \left\| f\right\| _{L^{p_{1}}\left( \mathcal{X}_{1},\mu _{1}\right) }\right\| _{L^{p_{2}}\left( \mathcal{X}_{2},\mu _{2}\right) }<\infty \right\},\end{multlined} \] For \(p_{2}=\infty \), \(1\leq r<\infty \), define the norm of \( L^{r,\infty }\left( \mathcal{X},\mu \right) \) to be \[ \left\| f\right\| _{L^{r,\infty }}:=\sup_{t>0}\left[ t^{r}\mu \left( \{x\in \mathcal{X}:\left| f(x)\right| >t\}\right) \right] ^{1/r}<\infty. \] Next define the two parameter integral operator \(T_{\alpha _{1},\alpha _{2}}\) by \[ T_{\alpha _{1},\alpha _{2}}f(x_{1},x_{2})=\int\nolimits_{\mathcal{X} _{2}}\int\nolimits_{\mathcal{X} _{1}}K(x_{1},y_{1};x_{2},y_{2})f(y_{1},y_{2})d\mu _{1}(y_{1})d\mu _{2}(y_{2}) , \] where \(K\) is a measurable function on \(\mathcal{X}_{1}\times \mathcal{X}_{2}\). (The indices \(\alpha _{i}\) refer to the conditions (a) - (d) given below). The authors' main result is the following theorem: Suppose \(0<\alpha _{1},\alpha _{2}<1\). If the kernel \(K\) satisfies the conditions: {\parindent=8mm \begin{itemize}\item[(a)]\(\sup_{y_{2}}\left\| \sup_{y_{1}}\left\| K\left( \cdot ,y_{1};\cdot ,y_{2}\right) \right\| _{L_{x_{1}}^{1/(1-\alph…
reviewed by: Caroline Sweezy
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 5550527
zbMATH Keywords: Multi-parameter fractional integral
zbMATH Keywords: mixed norm spaces
zbMATH Keywords: product spaces
MaRDI profile type: Publication
full work available at URL: s11118-009-9120-1
OpenAlex ID: W1976390679
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DOI: 10.1007/S11118-009-9120-1
description: scientific article; zbMATH DE number 5550527