Sharp real-part theorems in the upper half-plane and similar estimates for harmonic functions

Explicit formulas for sharp coefficients in estimates of the modulus of an analytic function and its derivative in the upper half-plane are found. It is assumed that the boundary values of the real part of the function are in [L.sup.p]. As corollaries, sharp estimates for the modulus of the gradient...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2011-11, Vol.179 (1), p.144
Main Authors: Kresin, G, Maz'ya, V
Format: Article
Language:English
Online Access:Get full text
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Summary:Explicit formulas for sharp coefficients in estimates of the modulus of an analytic function and its derivative in the upper half-plane are found. It is assumed that the boundary values of the real part of the function are in [L.sup.p]. As corollaries, sharp estimates for the modulus of the gradient of a harmonic function in [R.sub.+.sup.2] are deduced. Besides, a representation for the best constant in the estimate of the modulus of the gradient of a harmonic function in [R.sub.+.sup.n] by the [L.sup.p]-norm of the boundary normal derivative is given, 1 [less than or equal to] p < [infinity]. This representation is formulated in terms of an optimization problem on the unit sphere which is solved for p [member of] [1, n]. Bibliography: 6 titles.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-011-0586-1