A charaterization of commutators for parabolic singular integrals

In this paper, the authors give a characterization of the $L^p$-boundedness of the commutators for the parabolic singular integrals. More precisely, the authors prove that if $b\in \mathrm{BMO}_\varphi(\mathsf{R}^n,\rho)$, then the commutator $[b,T]$ is a bounded operator from $L^p(\mathsf{R}^n)$ to the Orlicz space $L_\psi(\mathsf{R}^n)$, where the kernel function $\Omega$ has no any smoothness on the unit sphere $S^{n-1}$. Conversely, if assuming on $\Omega$ a slight smoothness on $S^{n-1}$, then the boundedness of $[b,T]$ from $L^p(\mathsf{R}^n)$ to $L_\psi(\mathsf{R}^n)$ implies that $b\in \mathrm{BMO}_\varphi(\mathsf{R}^n,\rho)$. The results in this paper improve essentially and extend some known conclusions.

Weak-type (1, 1) estimates for parabolic singular integrals

We prove weak-type (1, 1) estimates for rough parabolic singular integrals on ℝ2 under the L log L condition on their kernels.

A Generalized Characterization of Commutators of Parabolic Singular Integrals

Let and , where λ > 0 and . Denote . We characterize those functions A(x) for which the parabolic Calderón commutatoris bounded on L2(ℝn), where , K is smooth away fromthe origin and satisfies a certain cancellation property.