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The first partial derivatives of generalized harmonic functions
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Paper
The first partial derivatives of generalized harmonic functions
2023
Overview
Suppose \\( ın R Z^-\\) such that \\(+>-1\\) and \\(1 p ınfty\\). Let \\(u=P_ [f]\\) be an \\(( )\\)-harmonic mapping on \\(D\\), the unit disc of \\(C\\), with the boundary \\(f\\) being absolutely continuous and \\(fın L^p(0,2)\\), where \\(f(e^i):=ddf(e^i)\\). In this paper, we investigate the membership of the partial derivatives \\(_z u\\) and \\(_zu\\) in the space \\(H_G^p(D)\\), the generalized Hardy space. We prove, if \\(+>0\\), then both \\(_z u\\) and \\(_zu\\) are in \\(H_G^p(D)\\). For \\(+<0\\), we show if \\(_z u\\) or \\(_zu ın H_G^1(D)\\) then \\(u=0\\) or \\(u\\) is a polyharmonic function.
Publisher
Cornell University Library, arXiv.org
Subject
