Riesz transforms and Sobolev spaces associated to the partial harmonic oscillator

TL;DR

This paper develops Sobolev spaces linked to the partial harmonic oscillator, defines fractional powers and Riesz transforms, and proves their boundedness, leading to various classical inequalities in these potential spaces.

Contribution

It introduces new Sobolev spaces associated with the partial harmonic oscillator and establishes the boundedness of Riesz transforms within this framework.

Findings

Riesz transforms are bounded on classical Sobolev spaces.

Defined Sobolev spaces via fractional powers of the operator.

Derived classical inequalities in the potential spaces associated to the operator.

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 $$\AH=-\partial_{\rho}^2-\Delta_x+|x|^2,$$ 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 $L_{\AH}^{\alpha, p}$.