Spectral multiplier theorems of H\"ormander type on Hardy and Lebesgue spaces

Let $X$ be a space of homogeneous type and let $L$ be an injective, non-negative, self-adjoint operator on $L^2(X)$ such that the semigroup generated by $-L$ fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator $F(L)$, initially defined on $H^1_L(X)\cap L^2(X)$, acts as a bounded linear operator on the Hardy space $H^1_L(X)$ associated with $L$ whenever $F$ is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish H\"ormander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators satisfying generalized Gaussian estimates in which the required differentiability order is relaxed compared to all known spectral multiplier results.