Generalized Poisson integral and sharp estimates for harmonic and biharmonic functions in the half-space

G. Kresin, V. Maz'ya

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Abstract

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on Rn-1 is obtained under the assumption that f belongs to Lp. It is assumed that the kernel of the integral depends on the parameters α and β. The explicit formulas for the sharp coefficients are found for the cases p = 1, p = 2 and for some values of α, β in the case p = ∞. Conditions ensuring the validity of some analogues of the Khavinson's conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space.

Original languageEnglish
Article number2018032
JournalMathematical Modelling of Natural Phenomena
Volume13
Issue number4
DOIs
StatePublished - 2018

Keywords

  • Biharmonic functions
  • Generalized Poisson integral
  • Harmonic functions
  • Sharp estimates
  • Two-parametric kernel

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