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arXiv · 2406.03297

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

Nick Lindemulder·Emiel Lorist·Floris Roodenburg·Mark Veraar
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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.

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Nick Lindemulder, Emiel Lorist, Floris Roodenburg, Mark Veraar. 2025-03-18. Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space. https://doi.org/10.1016/j.jfa.2025.110985

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