The Mixed Norm Spaces of Polyharmonic Functions

Abstract

Let $\Omega\subset\textbf R^n$ be a bounded domain with C 2 boundary. And let H k be the set of all polyharmonic functions f with order k on Ω. For 0<p, q≤∞ and ϕ a normal weight, the mixed-norm space $H_k^{p,q,\varphi } $ consists of all function f in H k for which the mixed-norm ||·|| p, q, ϕ <∞. The main result of the paper is the norm equivalence: $$ \|f\|_{p, q, \varphi} \simeq \sum_{j=0}^{m-1} |\nabla _j f (x_0)| + \|\nabla _m f \|_{p, q, r^{m}\varphi}, $$ where x 0 is a fixed point in Ω, m is a positive integer and $\nabla _j f$ is the jth gradient of f. A similar result for Bloch-type spaces is also obtained.