Spectral asymptotic formula of Bessel--Riesz commutator

TL;DR

This paper derives a Weyl-type asymptotic formula for the commutator of Bessel--Riesz transforms with multiplication operators, linking the asymptotic coefficient to a homogeneous Sobolev norm of the symbol.

Contribution

It introduces a novel asymptotic formula for Bessel--Riesz commutators, connecting their spectral behavior to Sobolev norms via Schur multiplier techniques.

Findings

Asymptotic coefficient equals Sobolev norm of the symbol

Characterization of endpoint weak Schatten norm through Sobolev space

Relation of Bessel--Riesz commutator to classical Riesz commutator

Abstract

Let $R_{\lambda,j}$ be the $j$-th Bessel--Riesz transform, where $n\geq 1$, $\lambda>0$, and $j=1,\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{\lambda,j}]$, the commutator of $R_{\lambda,j}$ with multiplication operator $M_f$, based on building a preliminary result that the endpoint weak Schatten norm of $[M_f,R_{\lambda,j}]$ can be characterised via homogeneous Sobolev norm $\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})$ of the symbol $f$. Specifically, the asymptotic coefficient is equivalent to $\|f\|_{\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})}.$ Our main strategy is to relate Bessel--Riesz commutator to classical Riesz commutator via Schur multipliers, and then to establish the boundedness of Schur multipliers.