Real-Variable Theory of Local Variable Hardy Spaces

TL;DR

This paper develops a comprehensive real-variable framework for local variable Hardy spaces, including characterizations, atomic decompositions, dual space relations, and boundedness of key operators.

Contribution

It introduces new atomic and finite atomic decompositions for local variable Hardy spaces and explores their dual spaces and operator boundedness.

Findings

Established various real-variable characterizations of local variable Hardy spaces.

Proved atomic and finite atomic decompositions for these spaces.

Demonstrated boundedness of Calderón-Zygmund and fractional integrals on these spaces.

Abstract

In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterizations in terms of several local maximal functions. Next, the new atomic and the finite atomic decomposition for the local variable Hardy spaces are established. As an application, we also introduce the local variable Campanato space which is showed to be the dual space of the local variable Hardy spaces. Analogous to the homogeneous case, some equivalent definitions of the dual of local variable Hardy spaces are also considered. Finally, we show that the boundedness of inhomogeneous Calder\'on-Zygmund singular integrals and local fractional integrals on local variable Hardy spaces and their duals.