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The Mixed Norm Spaces of Polyharmonic Functions

Authors:

Zhangjian Hu Miroslav Pavlović Xuejun Zhang

Institution:

(1) Department of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang, 313000, People’s Republic of China;(2) Matematicki fakultet, Studentski trg 16, 11001 Belgrade, p.p. 550, Serbia;(3) College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410006, People’s Republic of China

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. This research is partially supported by the National Natural Science Foundation of China (10471039), the MNZŽS Grant No. 144010 (Serbia), and the Natural Science Foundation of Zhejiang Province (M103104).

Keywords:

Polyharmonic functions Mixed norm spaces Normal weights

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