Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Variational inequalities for the Ornstein--Uhlenbeck semigroup: the higher--dimensional case

Valentina Casarino, Paolo Ciatti, Peter Sjögren

TL;DR

The paper addresses the problem of establishing a weak type $(1,1)$ bound for the $\rho$-th order variation operator of the Ornstein–Uhlenbeck semigroup on $\mathbb{R}^n$ with invariant Gaussian measure $\gamma_\infty$, extending prior one-dimensional results to higher dimensions. The authors develop a robust approach combining precise Mehler kernel estimates, a local–global decomposition for small times, and vector-valued Calderón–Zygmund theory to prove the desired bound for $\varrho>2$, with an enhanced bound for large times. They also present a counterexample showing that the exponent $\varrho=2$ is sharp in the sense that the variation operator is not of strong nor weak type $(p,p)$ for any $p\in[1,\infty)$. The work advances the understanding of variational inequalities for diffusion semigroups in higher dimensions under nondoubling Gaussian-type settings, leveraging polar-coordinate techniques, ring decompositions, and kernel estimates through the Mehler kernel and CZ theory.

Abstract

We study the $\varrho$-th order variation seminorm of a general Ornstein--Uhlenbeck semigroup $\left(\mathcal H_t\right)_{t>0}$ in $\mathbb R^n$, taken with respect to $t$. We prove that this seminorm defines an operator of weak type $(1,1)$ with respect to the invariant measure when $\varrho> 2$. For large $t$, one has an enhanced version of the standard weak-type $(1,1)$ bound. For small $t$, the proof hinges on vector-valued Calderón--Zygmund techniques in the local region, and on the fact that the $t$ derivative of the integral kernel of $\mathcal H_t$ in the global region has a bounded number of zeros in $(0,1]$. A counterexample is given for $\varrho= 2$; in fact, we prove that the second order variation seminorm of $\left(\mathcal H_t\right)_{t>0}$, and therefore also the $\varrho$-th order variation seminorm for any $\varrho\in [1,2)$, is not of strong nor weak type $(p,p)$ for any $p \in [1,\infty)$ with respect to the invariant measure.

Variational inequalities for the Ornstein--Uhlenbeck semigroup: the higher--dimensional case

TL;DR

The paper addresses the problem of establishing a weak type bound for the -th order variation operator of the Ornstein–Uhlenbeck semigroup on with invariant Gaussian measure , extending prior one-dimensional results to higher dimensions. The authors develop a robust approach combining precise Mehler kernel estimates, a local–global decomposition for small times, and vector-valued Calderón–Zygmund theory to prove the desired bound for , with an enhanced bound for large times. They also present a counterexample showing that the exponent is sharp in the sense that the variation operator is not of strong nor weak type for any . The work advances the understanding of variational inequalities for diffusion semigroups in higher dimensions under nondoubling Gaussian-type settings, leveraging polar-coordinate techniques, ring decompositions, and kernel estimates through the Mehler kernel and CZ theory.

Abstract

We study the -th order variation seminorm of a general Ornstein--Uhlenbeck semigroup in , taken with respect to . We prove that this seminorm defines an operator of weak type with respect to the invariant measure when . For large , one has an enhanced version of the standard weak-type bound. For small , the proof hinges on vector-valued Calderón--Zygmund techniques in the local region, and on the fact that the derivative of the integral kernel of in the global region has a bounded number of zeros in . A counterexample is given for ; in fact, we prove that the second order variation seminorm of , and therefore also the -th order variation seminorm for any , is not of strong nor weak type for any with respect to the invariant measure.
Paper Structure (21 sections, 19 theorems, 163 equations)

This paper contains 21 sections, 19 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation
  4. The Mehler kernel
  5. Pointwise estimates for the Mehler kernel
  6. The variation seminorm
  7. Simplifications
  8. The variation of $\mathcal{H}_tf(x)$ in $[1,+\infty)$
  9. Local versus global region
  10. Splitting of $\mathbb{R}^n$ into rings
  11. The splitting of $\mathcal{H}_t$
  12. The variation of the global part
  13. The local part for small $t$
  14. Preparations
  15. Vector-valued Calderón--Zygmund operators
  16. ...and 6 more sections

Key Result

Theorem 1.1

For each $\varrho > 2$ the operator that maps $f \in L^1(\gamma_\infty)$ to the function where the $v(\varrho)$ seminorm is taken in the variable $t$, is of weak type $(1,1)$ with respect to the measure $\gamma_\infty$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • proof
  • Remark 4.4
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 23 more