Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Abstract

Let $(X,d,\mu)$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that the semigroup $e^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimates. Let $H_L^p(X)$ be the Hardy space associated with $L$. We prove a Hörmander-type spectral multiplier theorem for $L$ on $H_L^p(X)$ for $0 < p <\infty:$ the operator $m(L)$ is bounded from $H_L^p(X)$ to $H_L^p(X)$ if the function $m$ possesses $s$ derivatives with suitable bounds and $s > n(1/p - 1/2)$ where $n$ is the "dimension" of $X$. By interpolation, $m(L)$ is bounded on $H_L^p(X)$ for all $0 < p < \infty$ if $m$ is infinitely differentiable with suitable bounds on its derivatives. We also obtain a spectral multiplier theorem on $L^p$ spaces with appropriate weights in the reverse Hölder class.