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

Abstract

In this paper we introduce variable exponent local Hardy spaces $hLp$ associated with a non-negative self-adjoint operator $L$. We assume that, for every $t>0$, the operator $e^{-tL}$ has an integral representation whose kernel satisfies a Gaussian upper bound. We define $hLp$ by using an area square integral involving the semigroup ${e^{-tL}}_{t>0}$. A molecular characterization of $hLp$ is established. As an application of the molecular characterization we prove that $hLp$ coincides with the (global) Hardy space $HLp$ provided that $0$ does not belong to the spectrum of $L$. Also, we show that $hLp=H_{L+I}^{p(cdot)}(mathbb R^n)$.