$L^2$ continuity of the Calderón type commutator for the Littlewood-Paley operator with rough variable kernel

Abstract

For $b \in Lip(\mathbf{R}^{n})$, the Calderón type commutator for the Littlewood-Paley operator with variable kernel is defined by $$\mu_{\Omega,1;b}(f)(x) = \left( \int_{0}^{\infty}\left| \frac{1}{t^2}\int_{\vert x-y\vert \leq t}\frac{\Omega(x, x-y)}{\vert x - y \vert ^{n-1}} (b(x) - b(y))f(y) dy\right|^{2} \frac{dt}{t}\right)^{1/2}.$$ By giving a method based on Littlewood-Paley theory, Fourier transform and the spherical harmonic development, we prove the $L^{2}$ norm inequalities for the rough operators $\mu_{\Omega,1;b}$ with $\Omega(x,z^{\prime}) \in L^{\infty}(\mathbf{R}^{n} \times L^{q}(S^{n-1})\left(q \gt \frac{2(n-1)}{n} \right)$ satisfying certain cancellation conditions.