Derivative bounds and continuity of maximal commutators
Volume 266 / 2022
Studia Mathematica 266 (2022), 93-119
MSC: Primary 42B25; Secondary 46E35.
DOI: 10.4064/sm210920-25-10
Published online: 20 April 2022
Abstract
We study the regularity properties of maximal commutators acting on $W^{1,1}$-functions. More precisely, let $\mathfrak {M}_{b}$ be the one-dimensional maximal commutator with symbol $b$. Under the condition that $b\in W^{1,1}(\mathbb {R})$ and $\|b’\|_{L^\infty (\mathbb {R})} \lt \infty $, we prove that the map $f\mapsto (\mathfrak {M}_{b}f)’$ is bounded and continuous from $W^{1,1}(\mathbb {R})$ to $L^{q}(\mathbb {R})$ for any $q\in (1,\infty )$.
