Maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type

TL;DR

This paper develops new maximal function characterizations for local Hardy spaces on spaces of homogeneous type, extending classical results and introducing a novel Hardy space related to a critical function, with applications to Schrödinger operators.

Contribution

It introduces a new local Hardy space associated with a critical function and establishes maximal function characterizations for these spaces on spaces of homogeneous type.

Findings

Maximal function characterizations for local Hardy spaces are established.

The new Hardy space includes classical spaces as special cases.

Applications to Schrödinger operators on various geometric settings.

Abstract

Let $X$ be a space of homogeneous type and let $\mathfrak{L}$ be a nonnegative self-adjoint operator on $L^2(X)$ enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove the (local) nontangential and radial maximal function charaterization for the local Hardy spaces associated to $\mathfrak{L}$. This deduces the maximal function charaterization for local Hardy spaces in the sense of Coifman and Weiss provided that $\mathfrak{L}$ satisfies certain extra conditions. Secondly, we introduce the local Hardy space associated to the critical function $\rho$ which is motivated by the theory of Hardy spaces related to Schr\"odinger operators and includes the local Hardy spaces of Coifman and Weiss as a special case. Then we prove that these local Hardy spaces can be characterized by the the local nontangential and radial maximal function charaterization related to $\mathfrak{L}$ and $\rho$ and the global maximal function charaterizations associated to `perturbations' of $\mathfrak{L}$. As applications, we apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Shr\"odinger operators on manifolds to Shr\"odinger operators on connected and simply connected nilpotent Lie groups.