Isoperimetric type inequalities for harmonic functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:373

Article

Kalaj, David¹  Mestrovic, Romeo² 

[1] Univ Montenegro, Fac Nat Sci & Math, Podgorica 81000, Montenegro

[2] Univ Montenegro, Maritime Fac, Kotor 85330, Montenegro

关键词: Harmonic Bergman space;    Harmonic Hardy space;    Isoperimetric inequality;   

DOI  :  10.1016/j.jmaa.2010.08.009

来源: Elsevier

【 摘 要 】

For 0 < p < +infinity let h(p) be the harmonic Hardy space and let b(p) be the harmonic Bergman space of harmonic functions on the open unit disk U. Given 1 <= p < +infinity, denote by parallel to . parallel to(bP) and parallel to . parallel to(hp) the norms in the spaces b(p) and h(p), respectively. In this paper, we establish the harmonic h(p)-analogue of the known isoperimetric type inequality parallel to f parallel to(b2p) <= parallel to f parallel to(hp) where f is an arbitrary holomorphic function in the classical Hardy space H. We prove that for arbitrary p > 1, every function f is an element of h(p) satisfies the inequality parallel to f parallel to(b2p) <= a(p) parallel to f parallel to(hp), where a(p) > 1 is a suitable constant depending only on p. Furthermore, by using the Carleman inequality in the form parallel to f parallel to(b4) <= parallel to f parallel to(h2) with f is an element of H(2), we prove the following refinement of the above inequality for p = 2 parallel to f parallel to b(4) <= 4 root 1.5 + root 2 parallel to f parallel to(h2). (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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