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Let $L$ be the generator of an analytic semigroup whose kernels satisfyGaussian upper bounds and H\"older's continuity. Also assume that $L$ has abounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper,we construct a frame decomposition for the functions belonging to the Hardyspace $H_{L}^{1}(\mathbb{R}^n)$ associated to $L$, and for functions in theLebesgue spaces $L^p$, $1<p<\infty$. 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 anapplication of the frame decomposition, we establish the radial maximalsemigroup characterization of the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ underthe extra condition of Gaussian upper bounds on the gradient of the heatkernels of $L$.

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