# Dirichlet-to-Neumann and elliptic operators on C 1+$\kappa$ -domains: Poisson and Gaussian bounds

## TL;DR

This paper establishes Poisson upper bounds and gradient estimates for heat kernels and Green functions related to Dirichlet-to-Neumann and elliptic operators on C 1+$
abla$-domains, advancing understanding of boundary behavior and operator estimates.

## Contribution

It provides new Poisson bounds, gradient estimates, and L p-estimates for Dirichlet-to-Neumann and elliptic operators on domains with C 1+$
abla$ boundaries, extending previous results.

## Key findings

- Poisson upper bounds for heat kernels on C 1+$
abla$-domains
- Gradient estimates for Green functions up to the boundary
- L p-estimates for commutators of Dirichlet-to-Neumann operators

## Abstract

We prove Poisson upper bounds for the heat kernel of the Dirichlet-to-Neumann operator with variable H{\"o}lder coefficients when the underlying domain is bounded and has a C 1+$\kappa$-boundary for some $\kappa$ > 0. We also prove a number of other results such as gradient estimates for heat kernels and Green functions G of elliptic operators with possibly complex-valued coefficients. We establish H{\"o}lder continuity of $\nabla$ x $\nabla$ y G up to the boundary. These results are used to prove L p-estimates for commutators of Dirichlet-to-Neumann operators on the boundary of C 1+$\kappa$-domains. Such estimates are the keystone in our approach for the Poisson bounds.