Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

Abstract

Let $L$ be a nonnegative self-adjoint operator on $L^2\left(X\right)$, where $X$ is a space of homogeneous type. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier $F\left(L\right)$ is bounded on $H_L^p\left(X\right)$ for $0<p\le 1$, the Hardy space associated to operator $L$, when $F$ is a suitable function.