A Generalized Characterization of Commutators of Parabolic Singular Integrals

http://dx.doi.org/10.4153/CMB-1999-054-0
Canad. Math. Bull. 42(1999), 463-477

  Published:1999-12-01
 Printed: Dec 1999

Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1, \dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots \le\az_n$. Denote $|\az|=\az_1+\cdots+\az_n$. We characterize those functions $A(x)$ for which the parabolic Calder\'on commutator $$ T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n} K(x-y)[A(x)-A(y)]f(y)\,dy $$ is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{-|\az|-1}K(x)$, $K$ is smooth away from the origin and satisfies a certain cancellation property.