Math. Model. Nat. Phenom.
Volume 13, Number 4, 2018
|Number of page(s)||29|
|Published online||22 June 2018|
Generalized Poisson integral and sharp estimates for harmonic and biharmonic functions in the half-space
Department of Mathematics, Ariel University,
2 Department of Mathematical Sciences, University of Liverpool, M & O Building, Liverpool L69 3BX, UK
3 Department of Mathematics, Linköping University, 58183 Linköping, Sweden
4 RUDN University, 6 Miklukho-Maklay St., Moscow 117198, Russia
* Corresponding author: firstname.lastname@example.org
Accepted: 25 November 2017
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.
Mathematics Subject Classification: 31B10 / 31B05 / 31B30
Key words: Generalized Poisson integral / two-parametric kernel / sharp estimates / harmonic functions / biharmonic functions.
© EDP Sciences, 2018
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