# Sharp estimates for the gradient of the generalized Poisson integral for a half-space

## TL;DR

This paper derives explicit sharp estimates for the gradient of the generalized Poisson integral in a half-space, providing exact coefficients for specific cases and advancing understanding of boundary behavior in harmonic analysis.

## Contribution

It provides explicit formulas for the sharp coefficient in gradient estimates of the generalized Poisson integral, including special cases for p=1 and p=2.

## Key findings

- Explicit sharp coefficient formulas for the gradient estimate.
- Exact coefficient values for p=1 and p=2 cases.
- Enhanced understanding of boundary behavior in harmonic analysis.

## Abstract

A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function $f$ on ${\mathbb R}^n$ is obtained under the assumption that $f$ belongs to $L^p$. The explicit value of the coefficient is found for the cases $p=1$ and $p=2$.