Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space

TL;DR

This paper establishes a bounded $H^ abla$-calculus for the Laplacian on weighted Sobolev spaces in the half-space, analyzing associated heat semigroups and maximal regularity, extending classical results to non-$A_p$ weights.

Contribution

It proves the bounded $H^ abla$-calculus for the Laplacian on weighted Sobolev spaces with non-$A_p$ weights and studies the properties of the heat semigroup and maximal regularity.

Findings

Bounded $H^ abla$-calculus on weighted Sobolev spaces.

Heat semigroup exhibits polynomial growth, unlike the $L^p$ case.

Maximal regularity results for the heat equation in weighted spaces.

Abstract

In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. We prove that this operator admits a bounded $H^\infty$-calculus on Sobolev spaces with power weights measuring the distance to the boundary. These weights do not necessarily belong to the class of Muckenhoupt $A_p$ weights. We additionally study the corresponding Dirichlet and Neumann heat semigroup. It is shown that these semigroups, in contrast to the $L^p$-case, have polynomial growth. Moreover, maximal regularity results for the heat equation are derived on inhomogeneous and homogeneous weighted Sobolev spaces.