Pointwise Multipliers on Spaces of Homogeneous Type in the Sense of Coifman and Weiss

Yanchang Han,Zongguang Liu,Fanghui Liao

doi:10.1155/2014/164624

Abstract

By applying the remarkable orthonormal basis constructed recently by Ausher and Hytönen on spaces of homogeneous type in the sense of Coifman and Weiss, pointwise multipliers of inhomogeneous Besov and Triebel-Lizorkin spaces are obtained. We make no additional assumptions on the quasi-metric or the doubling measure. Hence, the results of this paper extend earlier related results to a more general setting.

Highlights

The main purpose of this paper is to provide pointwise multipliers of inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss
By a pointwise multiplier from a function space A into another function space B, we meant that a function defines a bounded linear mapping from A into B by pointwise multiplication
Pointwise multipliers arise in many different areas of mathematical analysis and have many applications; for example, coefficients of differential operators and symbols of more general pseudodifferential operators may be considered as pointwise multipliers

Summary

Introduction

A natural question arises: whether pointwise multipliers still hold on spaces of homogeneous type in the sense of Coifman and Weiss with only the original quasi-metric and a doubling measure?. Auscher and Hytonen constructed an orthonormal basis with Holder regularity and exponential decay on spaces of homogeneous type [17] This result is remarkable since there are no additional assumptions other than those defining spaces of homogeneous type in the sense of Coifman and Weiss. In this paper, we will provide pointwise multipliers on spaces of homogeneous type in the sense of Coifman and Weiss with the original quasi-metric d and doubling measure μ. We briefly recall the orthonormal basis constructed in [17] and inhomogeneous Besov and Triebel-Lizorkin spaces obtained in [18] on spaces of homogeneous type in the sense of Coifman and Weiss.

Proof of Theorem 5

Findings

Proof of Theorem 6