Boundedness on Triebel–Lizorkin spaces for the Calderón commutator with rough kernel

Liang, Rong ; Tao, Xiangxing 

Canadian mathematical bulletin, Tome 68 (2025) no. 4, pp. 1069-1081

Résumé

In this article, the authors consider the boundedness on Triebel–Lizorkin spaces for the d-dimensional Calderón commutator defined by $$ \begin{align*}T_{\Omega,a}f(x)=\mathrm{p.\,v.}\int_{\mathbb{R}^d}\frac{\Omega(x-y)}{|x-y|^{d+1}}\big(a(x)-a(y)\big)f(y){d}y,\end{align*} $$where $\Omega $ is homogeneous of degree zero, integrable on $S^{d-1}$ and has a vanishing moment of order one, a is a Lipschitz function on $\mathbb {R}^d$. The authors proved that if $$ \begin{align*}\sup_{\zeta\in S^{d-1}}\int_{S^{d-1}}|\Omega(\theta)|\log ^{\beta} \big(\frac{1}{|\theta\cdot\zeta|}\big)d\theta<\infty\end{align*} $$with $\beta \in (1,\,\infty )$, then $T_{\Omega ,a}$ is bounded on Triebel–Lizorkin spaces $\dot {F}_{p}^{0,q}(\mathbb {R}^d)$ for $1+\frac {1}{2\beta -1}<p,q<2\beta $.

Détail

DOI : 10.4153/S0008439525000268

Mots-clés : Calderón commutator, Triebel–Lizorkin space, Littlewood–Paley theory, Calderón reproducing formula