Local Hardy spaces with variable exponents associated to non-negative self-adjoint operators satisfying Gaussian estimates

TL;DR

This paper introduces variable exponent local Hardy spaces linked to a non-negative self-adjoint operator with Gaussian estimates, providing molecular characterizations and demonstrating their equivalence to global variable Hardy spaces under certain spectral conditions.

Contribution

It develops a new framework for local Hardy spaces with variable exponents associated to operators, including molecular characterizations and equivalence results.

Findings

Established molecular characterization of the local Hardy spaces.

Proved the equivalence with global variable Hardy spaces when 0 is not in the spectrum.

Showed the local Hardy space coincides with the global space associated to L+I.

Abstract

In this paper we introduce variable exponent local Hardy spaces associated with a non-negative self-adjoint operator L. We define them by using an area square integral involving the heat semigroup associated to L. A molecular characterization is established and as an aplication of the molecular characterization we prove that our local Hardy space coincides with the (global) variable exponent Hardy space associated to L, provided that 0 does not belong to the spectrum of L. Also, we show that it coincides with the global variable exponent Hardy space associated to L+I.