Cauchy Problems for Parabolic Equations in Sobolev-Slobodeckii and H\"older Spaces on Uniformly Regular Riemannian Manifolds

TL;DR

This paper proves optimal regularity results for linear parabolic equations on Riemannian manifolds, extending classical flat-space results to curved spaces using advanced Fourier multiplier techniques.

Contribution

It introduces a unified approach for Sobolev-Slobodeckii and H"older spaces on manifolds, generalizing classical results to curved geometries with bounded geometry.

Findings

Established maximal regularity theorems for parabolic equations on manifolds.

Unified treatment of Sobolev-Slobodeckii and H"older spaces.

Recovered classical flat-space results as special cases.

Abstract

In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in earlier by the author. This allows for a unified treatment of Sobolev-Slobodeckii and little H\"older spaces. In the flat case we recover classical results for Petrowskii-parabolic Cauchy problems.