Scaled cutoff commutator estimate

Statement

Let $n\ge2$ , let $T=cI+\operatorname{p.v.}K$ have kernel bounded by $C|x-y|^{-n}$ , and let $a_R(x)=a(x/R)$ with $a$ bounded and Lipschitz. For

1<q<\frac n{n-1},

the multiplication commutator extends from smooth compactly supported inputs to $L^1$ , and

\|[M_{a_R},T]f\|_q\le C_{a,n,q}R^{-n+n/q}\|f\|_1.

The constant also depends on the kernel bound and is independent of $R>0$ and $f$ .

Integrable majorant

The commutator kernel is dominated by

k_R(z)=C|z|^{-n}\min(|z|/R,1).

Its $L^q$ norm is finite at zero if $q<n/(n-1)$ , and at infinity if $q>1$ . Rescaling $z=Rz'$ gives $\|k_R\|_q=C R^{-n+n/q}$ . Young's convolution inequality proves the result and its unique extension. In $n=3$ , choosing $q=4/3$ gives $C R^{-3/4}\|f\|_1$ .

For an $L^1$ input the notation denotes this integral extension; individual terms $a_RT f$ and $T(a_Rf)$ need their own interpretation.