# Gradient of solution of the Poisson equation in the unit ball and related operators

## TL;DR

This paper calculates the bounds of an integral operator related to the gradient of solutions to the Poisson equation in a unit ball, providing insights into its behavior in different function spaces.

## Contribution

It determines the exact $L^1 o L^1$ and $L^{f ext{infty}} o L^{ extbf ext{infty}}$ norms of the operator associated with the Poisson equation's gradient in the unit ball.

## Key findings

- Calculated $L^1 o L^1$ norm of the operator.
- Calculated $L^{ ext{infty}} o L^{ ext{infty}}$ norm of the operator.
- Provided explicit bounds for the operator in these norms.

## Abstract

In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.