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 $L^p$, $1<p<\infty$. 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 $L$.Comment: 37 pages, to appear in Journal of Approximation Theor

Equivalence of Littlewood-Paley square function and area function characterizations of weighted product Hardy spaces associated to operators

Let $L_{1}$ and $L_{2}$ be non-negative self-adjoint operators acting on $L^{2}(X_{1})$ and $L^{2}(X_{2})$, respectively, where $X_{1}$ and $X_{2}$ are spaces of homogeneous type. Assume that $L_{1}$ and $L_{2}$ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ associated to $L_{1}$ and $L_{2}$, for $p \in (0, \infty)$ and the weight $w$ belongs to the product Muckenhoupt class $A_{\infty}(X_{1} \times X_{2})$. Our main result is that the spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ introduced via area functions can be equivalently characterized by Littlewood-Paley $g$-functions, Littlewood-Paley $g^{\ast}_{\lambda_{1}, \lambda_{2}}$-functions, and Peetre type maximal functions, without any further assumptions beyond the Gaussian upper bounds on the heat kernels of $L_{1}$ and $L_{2}$. Our results are new even in the unweighted product setting.Comment: 18 page