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Riesz transforms and Sobolev spaces associated to the partial harmonic oscillator

Xiaoyan Su, Ying Wang, Guixiang Xu

TL;DR

The paper develops a comprehensive framework for Sobolev-type spaces and harmonic analysis associated with the partial harmonic oscillator $H_{ extup{par}}=-\partial_\rho^2-\Delta_x+|x|^2$. It defines fractional powers via the heat kernel, establishes $L^p$-boundedness for negative powers, and constructs Riesz transforms with boundedness on classical Sobolev spaces using a new adapted symbol calculus. It introduces and proves equivalence of Sobolev spaces $W^{k,p}_{H_{ extup{par}}}$ with potential spaces $L^{k,p}_{H_{ extup{par}}}$, and derives a suite of integral inequalities (Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg–Sobolev, Hardy) in the $H_{ extup{par}}$-context. The results advance a microlocal, heat-kernel-driven analysis for partial oscillators and yield refined functional-analytic tools for PDEs involving $H_{ extup{par}}$, with potential extensions to more general partial harmonic operators.

Abstract

In this paper, our goal is to establish the Sobolev space associated to the partial harmonic oscillator. Based on its heat kernel estimate, we firstly give the definition of the fractional powers of the partial harmonic oscillator $$\AH=-\partial_ρ^2-Δ_x+|x|^2,$$ and show that its negative powers are well defined on $L^p(\mathbb R^{d+1})$ for $p\in [1,\infty]$. We then define associated Riesz transforms and show that they are bounded on classical Sobolev spaces by the calculus of symbols. Secondly, by a factorization of the operator $\AH$, we define two families of Sobolev spaces with positive integer indices, and show the equivalence between them by the boundedness of Riesz transforms. Moreover, the adapted symbolic calculus also implies the boundedness of Riesz type transforms on the Sobolev spaces associated to the partial harmonic oscillator $\AH$. Lastly, as applications of our results, we obtain the revised Hardy-Littlewood-Sobolev inequality, the Gagliardo-Nirenberg-Sobolev inequality, and Hardy's inequality in the potential space $L_{\AH}^{α, p}$.

Riesz transforms and Sobolev spaces associated to the partial harmonic oscillator

TL;DR

The paper develops a comprehensive framework for Sobolev-type spaces and harmonic analysis associated with the partial harmonic oscillator . It defines fractional powers via the heat kernel, establishes -boundedness for negative powers, and constructs Riesz transforms with boundedness on classical Sobolev spaces using a new adapted symbol calculus. It introduces and proves equivalence of Sobolev spaces with potential spaces , and derives a suite of integral inequalities (Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg–Sobolev, Hardy) in the -context. The results advance a microlocal, heat-kernel-driven analysis for partial oscillators and yield refined functional-analytic tools for PDEs involving , with potential extensions to more general partial harmonic operators.

Abstract

In this paper, our goal is to establish the Sobolev space associated to the partial harmonic oscillator. Based on its heat kernel estimate, we firstly give the definition of the fractional powers of the partial harmonic oscillator and show that its negative powers are well defined on for . We then define associated Riesz transforms and show that they are bounded on classical Sobolev spaces by the calculus of symbols. Secondly, by a factorization of the operator , we define two families of Sobolev spaces with positive integer indices, and show the equivalence between them by the boundedness of Riesz transforms. Moreover, the adapted symbolic calculus also implies the boundedness of Riesz type transforms on the Sobolev spaces associated to the partial harmonic oscillator . Lastly, as applications of our results, we obtain the revised Hardy-Littlewood-Sobolev inequality, the Gagliardo-Nirenberg-Sobolev inequality, and Hardy's inequality in the potential space .
Paper Structure (16 sections, 16 theorems, 141 equations)

This paper contains 16 sections, 16 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Hermite functions
  5. Symbol class
  6. Fractional powers of the operator $H_{\textup{par}}$
  7. Functional Calculus for the operator $H_{\textup{par}}$
  8. Fractional powers of the operator $H_{\textup{par}}$ and the heat semigroup
  9. Properties of negative fractional powers of $H_{\textup{par}}$
  10. Symbols for fractional powers $H_{\textup{par}}^{\alpha}$
  11. Riesz transforms and symbols
  12. Sobolev spaces associated to the partial harmonic oscillator
  13. Some Integral Inequalities adapted to the operator $H_{\textup{par}}$
  14. Hardy--Littlewood--Sobolev inequality.
  15. Gagliardo--Nirenberg--Sobolev inequality
  16. ...and 1 more sections

Key Result

Proposition 2.3

Let $\alpha<0$. The symbols of the negative fractional powers $H^{\alpha}$ of the Hermite operators belong to $\Gamma^{2\alpha}$ .

Theorems & Definitions (37)

  • Definition 2.1: TaylorTaylor91
  • Definition 2.2: NicolaRodino
  • Proposition 2.3: Thangavelu18
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 27 more