## Generalized Poisson integral and sharp estimates for harmonic and biharmonic functions in the half-space. (arXiv:1709.03412v1 [math.AP])

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function $f$ on ${\mathbb R}^{n-1}$ is obtained under the assumption that $f$ belongs to $L^p$. It is assumed that the kernel of the integral depends on the parameters $\alpha$ and $\beta$. The explicit formulas for the sharp coefficients are found for the cases $p=1$, $p=2$ and for some values of $\alpha , \beta$ in the case $p=\infty$. 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.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function $f$ on ${\mathbb R}^{n-1}$ is obtained under the assumption that $f$ belongs to $L^p$. It is assumed that the kernel of the integral depends on the parameters $\alpha$ and $\beta$. The explicit formulas for the sharp coefficients are found for the cases $p=1$, $p=2$ and for some values of $\alpha , \beta$ in the case $p=\infty$. 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.