Spectral asymptotic formula of Bessel--Riesz commutator

Let $R_{\lambda ,j}$ be the $j$-th Bessel–Riesz transform, where $n\ge 1$, $\lambda >0$, and $j=1,\dots ,n+1$. 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 $\mathrm{\parallel }f{\mathrm{\parallel }}_{{\dot{W}}^{1,n+1}({\mathbb{R}}_{+}^{n+1})}\mathrm{.}$ 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.