Spectral asymptotic formula for Bessel–Riesz transform commutators

Let Rλ ,j be the jth Bessel–Riesz transform, where n≥ 1, λ>0, and j=1,… ,n+1. In this article, we establish a Weyl-type asymptotic for [Rλ ,j,Mf], the commutator of Rλ ,j with the multiplication operator Mf, based on building a preliminary result that the endpoint weak Schatten norm of [Rλ ,j,Mf] can be characterized via homogeneous Sobolev norm Ẇ1,n+1(R₊ⁿ⁺¹) of the symbol f. Specifically, the asymptotic coefficient is equivalent to \|f\|_Ẇ1,n+1(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.