Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Riesz transform, function spaces and their applications on infinite dimensional compact groups

Alexander Bendikov, Li Chen, Laurent Saloff-Coste

Abstract

On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(μ_t)_{t>0}$. We establish several regularity results of the solution to the Poisson equation $LU=F$, both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for $1\le p\le \infty$: $Λ_θ^p$, defined via the associated Markov semigroup, and $\mathrm L_θ^p$, defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of $Λ_θ^p$ space. In the distributional sense, we further show local regularity in the class of $\mathrm L_θ^{\infty}$ space. These results require some strong assumptions on $-L$. Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free $L^p$ ($1<p<\infty$) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.

Riesz transform, function spaces and their applications on infinite dimensional compact groups

Abstract

On a compact connected group , consider the infinitesimal generator of a central symmetric Gaussian convolution semigroup . We establish several regularity results of the solution to the Poisson equation , both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for : , defined via the associated Markov semigroup, and , defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of space. In the distributional sense, we further show local regularity in the class of space. These results require some strong assumptions on . Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free () boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.

Paper Structure

This paper contains 21 sections, 37 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. Infinite-dimensional torus.
  4. L1 differentiability of the heat semigroup
  5. Heat kernel estimates
  6. L1 differentiability of the heat semigroup
  7. Dimension-free Lp bounds of Riesz transforms
  8. Lp bounds of first and second order Riesz transforms
  9. Application on gradient bounds of the heat semigroup
  10. Lipschitz spaces
  11. Lipschitz spaces via the heat semigroup
  12. Riesz transforms on Lipschitz spaces
  13. Lipschitz spaces via distance
  14. Comparison of Lipschitz seminorms: the case 1p-infty
  15. Comparison of Lipschitz seminorms: the case 1infty
  16. ...and 6 more sections

Key Result

Theorem 2.4

Fix a projective basis $(Y_i)_{i\in I}$. Let $L=-\sum_{i,j\in I} a_{i,j} Y_iY_j$ with $A$ symmetric and positive-definite. Then the family $(X_i)_{i\in I}$ given by is a projective basis of $\mathfrak G$ and yields a square field decomposition $-L=\sum_{i\in I}X_i^2$.

Theorems & Definitions (86)

  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: BSC2002
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Definition 3.4
  • Theorem 3.5: BSC2000
  • Corollary 3.6: BSC2002
  • Remark 3.7
  • ...and 76 more