Weighted \(L^q\) estimates for derivatives of weighted \(H^p\) functions (Q1282048)

The authors characterize the nonnegative measures \(\mu\) on \({\mathbb{R}}^{d+1}_+\) and the weights \(\nu\) on \({\mathbb{R}}^d\) so that the following inequality holds for a given pair \(p,q\) of indices, \(0<p,q< \infty,\) and all harmonic functions \(u(x,y)\) on \({\mathbb{R}}^{d+1}_+\), \[ \left(\int_{{\mathbb{R}}^{d+1}_+}| D^\beta u(x,y)| ^q d\mu(x,y)\right)^{1/q} \leq C\| u\| _{H^p(\nu, {\mathbb{R}}^d)} \] with a constant \(C\) that is independent of \(u\), where \(\beta\) is a multi-index of order \(m\in {\mathbb{N}}\), \(D^\beta\) denotes a combination of ordinary partial derivatives in \(x\) or \(y\) and \(H^p(\nu, {\mathbb{R}}^d)\) is the weighted Hardy space. If \(0<p<q<\infty\) and \(\nu\) is a doubling weight on \({\mathbb{R}}^d\), the characterization is given by some inequality involving the volumes on \(\mu\) of the balls in \({\mathbb{R}}^{d+1}_+\) with respect to the hyperbolic metric and the volumes on \(\nu\) of the usual balls in \({\mathbb{R}}^d\); while if \(0<q\leq p<\infty\) and \(\nu\) is a Fefferman-Muckenhoupt \(A_\infty\) weight on \({\mathbb{R}}^d\), the characterization is given by means of the weighted versions of the tent spaces of Coifman, Meyer and Stein, and an auxiliary function involving the volumes on \(\mu\) of the balls in \({\mathbb{R}}^{d+1}_+\) with respect to the hyperbolic metric and the volumes on \(\nu\) of the usual balls in \({\mathbb{R}}^d.\)