H\"older classes via semigroups and Riesz transforms

TL;DR

This paper develops a semigroup-based framework for H"older classes, establishing boundedness of Riesz transforms under curvature conditions, and explores spectral multipliers and inequalities related to these classes.

Contribution

It introduces a new approach to H"older classes via semigroups, proves Riesz transform bounds under curvature conditions, and connects these to spectral multipliers and inequalities.

Findings

Riesz transforms are bounded between H"older classes on manifolds with nonnegative Ricci curvature.

A version of Morrey inequalities is equivalent to ultracontractivity, extending Sobolev inequalities.

Spectral multipliers include imaginary powers and smooth multipliers, with connections to Campanato's formula.

Abstract

We define H\"older classes $\Lambda_\alpha$ associated with a Markovian semigroup and prove that, when the semigroup satisfies the $\Gamma^2 \geq 0$ condition, the Riesz transforms are bounded between the H\"older classes. As a consequence, this bound holds in manifolds with nonnegative Ricci curvature. We also show, without the need for extra assumptions on the semigroup, that a version of the Morrey inequalities is equivalent to the ultracontractivity property. This result extends the semigroup approach to the Sobolev inequalities laid by Varopoulos. After that, we study certain families of operators between the homogeneous H\"older classes. One of these families is given by analytic spectral multipliers and includes the imaginary powers of the generator, the other, by smooth multipliers analogous to those in the Marcienkiewicz theorem. Lastly, we explore the connection between the H\"older norm and Campanato's formula for semigroups.