Boundedness of Pseudo-Differential Operators on Weighted Hardy Spaces

Abstract

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