  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 $L^p$, $1<p<\infty$. 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 $L$.
