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

Let L be a nonnegative self-adjoint operator on L²(X), 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(L) is bounded on HpL(X) for 0 < p ≤ 1, the Hardy space associated to operator L, when F is a suitable function.14 page(s