Optimal semigroup estimates and functional calculus for the Laplacian on weighted Sobolev spaces

Floris Roodenburg

arXiv:2608.13314v1 Primary: math.AP Secondary: math.FA

Abstract

In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. These operators are studied on Sobolev spaces with power weights measuring the distance to the boundary. We prove optimal estimates for the resolvent operators and the corresponding heat semigroups. In addition, it is proved that the Dirichlet and Neumann Laplacians admit a bounded $H^\infty$-functional calculus on Sobolev spaces with certain compatibility conditions at the boundary. We show that these compatibility conditions cannot be omitted in general. The results in this paper are a direct extension of those obtained by Lindemulder, Lorist, the author, and Veraar in [J. Funct. Anal., 289(8):110985, 2025].

Disclosure

“nder the assumptions of Lemma 3.4. Thus the factor |λ|1−αbc,γ in Proposition 3.8 disap- pears. The rest of the arguments in Sections 4.1 and 4.2 can be used with minor modifica- tions to prove (4.7). AI disclosure statement. ChatGPT 5.6 by OpenAI was used to explore proof strategies for this paper. The paper was entirely written by the author, who takes full responsibility for the content. References [1] R. Denk and M. Dreher. Resolv”
PDF page 23

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T05:13:33.371848Z

Structural counts

Pages 23 pdf

Theorems 4 source

Lemmas 7 source

Propositions 7 source

Corollaries 0 source

Definitions 2 source

Displayed equations 143 source

Bibliography entries 13 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.