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 H"older’s continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2({\mathbb{R}}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_L^1({\mathbb{R}}^n)$ associated to $L$, and for functions in the Lebesgue spaces $L^p$, $1<p<\mathrm{\infty }$. We then show that the corresponding $H_L^1({\mathbb{R}}^n)$-norm (resp. $L^p({\mathbb{R}}^n)$-norm) of a function $f$ in terms of the frame coefficients is equivalent to the $H_L^1({\mathbb{R}}^n)$-norm (resp. $L^p({\mathbb{R}}^n)$-norm) of $f$. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space $H_L^1({\mathbb{R}}^n)$ under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of $L$.