Kernel cancellation in a multiplication commutator

Statement

Suppose $T=cI+\operatorname{p.v.}K$ is a singular integral with $|K(x,y)|\le C|x-y|^{-n}$ , and let $a$ be bounded and Lipschitz. For compactly supported smooth $f$ , its commutator with multiplication is

[M_a,T]f(x)=a(x)Tf(x)-T(af)(x) =\int(a(x)-a(y))K(x,y)f(y)\,dy.

The integral is absolutely convergent, and the local term $cI$ cancels.

Gain at the diagonal

Lipschitz continuity gives $|a(x)-a(y)|\le C_a|x-y|$ , so the new kernel is bounded by $C C_a|x-y|^{1-n}$ near the diagonal, an integrable singularity. Boundedness of $a$ controls the remaining region. The identity follows first for truncated integrals and then by dominated convergence.