Remarks on spectral multiplier theorems on Hardy spaces associated with semigroups of operators

Let L be a non-negative, self-adjoint operator on L^2(\Omega), where (\Omega, d \mu) is a space of homogeneous type. Assume that the semigroup {T_t}_{t>0} generated by -L satisfies Gaussian bounds, or more generally Davies-Gaffney estimates. We say that f belongs to the Hardy space H^1_L if the square function S_h f(x)=(\iint_{\Gamma (x)} |t^2 L e^{-t^2 L} f(y)|^2 \frac{d\mu(y)}{\mu (B_d(x,t))} \frac{dt}{t})^{1/2} belongs to L^1(\Omega, d\mu), where \Gamma(x)={(y,t) \in \Omega \times (0,\infty): d(x,y)<t}. We prove spectral multiplier theorems for L on H^1_L.