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

VÍCTOR ALMEIDA; ESTEFANÍA DALMASSO; LOURDES RODRÍGUEZ-MESA; JORGE J. BETANCOR

Congreso; Congreso Bienal de la Real Sociedad Matemática Española 2019; 2019

Real Sociedad Matemática Española (RSME)

In this paper we introduce variable exponent local Hardy spaces $h^{p(\cdot)}(\mathbb R^n)$ 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 $h^{p(\cdot)}(\mathbb R^n)$ by using an area square integral involving the semigroup $\{e^{-tL}\}_{t>0}$. A molecular characterization of $h^{p(\cdot)}(\mathbb R^n)$ is established. As an application of the molecular characterization we prove that $h^{p(\cdot)}(\mathbb R^n)$ coincides with the (global) Hardy space $H^{p(\cdot)}(\mathbb R^n)$ provided that $0$ does not belong to the spectrum of $L$. Also, we show that $h^{p(\cdot)}(\mathbb R^n)=H_{L+I}^{p(\cdot)}(\mathbb R^n)$.