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

G. Kresin, V. Maz'ya

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Abstract

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 Lp. As corollaries, sharp estimates for the modulus of the gradient of a harmonic function in ℝ2+ are deduced. Besides, a representation for the best constant in the estimate of the modulus of the gradient of a harmonic function in ℝn+ by the Lp-norm of the boundary normal derivative is given, 1 ≤ p < ∞. This representation is formulated in terms of an optimization problem on the unit sphere which is solved for p ∈ [1, n]. Bibliography: 6 titles.

Original languageEnglish
Pages (from-to)144-163
Number of pages20
JournalJournal of Mathematical Sciences
Volume179
Issue number1
DOIs
StatePublished - Nov 2011

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