Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators

Let L be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder's continuity. Also assume that L has a bounded holomorphic functional calculus on L² (Rⁿ ). In this paper, we construct a frame decomposition for the functions belonging to the Hardy space H_L¹ (Rⁿ ) associated to L, and for functions in the Lebesgue spaces L^p , 1<p<∞. We then show that the corresponding H_L¹ (R ⁿ )-norm (resp. L^p (Rⁿ )-norm) of a function f in terms of the frame coefficients is equivalent to the H_L¹ (Rⁿ )-norm (resp. L^p (Rⁿ )-norm) of f. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space H_L¹ (Rⁿ ) under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of L.