L²(${ℝ}^n$) boundedness for Calderón commutator with rough variable kernel

Abstract

For $b\in \mathrm{L}\mathrm{i}\mathrm{p}(R^n)$, the Calderón commutator with variable kernel is defined by

$$[b,{{\displaystyle T}}_{\mathrm{1}}]f(x)=p.v.{{\displaystyle \int }}_{R^n}{\displaystyle \frac{\mathrm{\Omega }(x,x-y)}{{\left|x-y\right|}^{n+\mathrm{1}}}}(b(x)-b(y))f(y)\mathrm{d}y.$$

In this paper, we establish the $L^{\mathrm{2}}(R^n)$ boundedness for $[b,{{\displaystyle T}}_{\mathrm{1}}]$ with $\Omega (x,z^{\text{'}})\in L^{\infty }(R^n)\times L^q(S^{n-\mathrm{1}})(q\rangle \mathrm{2}(n-\mathrm{1})/n)$ satisfying certain cancellation conditions. Moreover, the exponent $q\rangle \mathrm{2}(n-\mathrm{1})/n$ is optimal. Our main result improves a previous result of Calderón.

Keywords

Commutator / variable kernel / spherical harmonics function / Fourier transform

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