Boundedness of Pseudo-Differential Operators on Weighted Hardy Spaces

Let \(d \in \{1,2,3, \ldots \}\) $d\in \{1,2,3,\dots \}$ , \(p \in (0,1]\) $p\in (0,1]$ and \(s \in [1,\infty )\) $s\in [1,\infty )$ . Let w be a Muckenhoupt weight of class \(A_s\) $A_s$ and \(H^p_w(\mathbb {R}^d)\) $H_w^p({\mathbb{R}}^d)$ the weighted Hardy space on \(\mathbb {R}^d\) ${\mathbb{R}}^d$ . Let \(m \in \mathbb {R}\) $m\in \mathbb{R}$ , \(\rho , \delta \in [0,1]\) $\rho ,\delta \in [0,1]$ and a belong to the Hörmander class \(S^m_{\rho ,\delta }\) $S_{\rho ,\delta }^m$ . Consider the pseudo-differential operator T associated with the symbol a. We prove the boundedness of T from \(H^p_w(\mathbb {R}^d)\) $H_w^p({\mathbb{R}}^d)$ to \(L^p_w(\mathbb {R}^d)\) $L_w^p({\mathbb{R}}^d)$ and on \(H^p_w(\mathbb {R}^d)\) $H_w^p({\mathbb{R}}^d)$ for certain values of the parameters p, s, m, \(\rho \) $\rho$ and \(\delta \) $\delta$ .