Endpoint estimates for commutators of singular integral operators
Let \(b\) be a locally integrable function on \(\mathbb{R}^n\) and let \(T\) be a Calderón-Zygmund singular integral operator. Consider the commutator operator \([b, T]\) defined for smooth functions \(f\) by \([b, T] f= bT(f)- T(bf)\). As is now well-known, \([b, T]\) is a bounded operator on \(L^p(\mathbb{R}^n)\), \(1< p< \infty\), when \(b\) is a BMO function. However, it is not of weak type \((1,1)\), as the author shows by an example. In place of weak \((1, 1)\) inequality, he shows the following: Let \(T\) and \(b\) be as above. Then, there exists a \(C> 0\) such that for each smooth function with compact support \[ |\{ y\in \mathbb{R}^n; |[b, T] f(y)|> \lambda\}|\leq C|b|_{\text{BMO}} \int {|f(y)|\over \lambda} \Biggl(1+ \log^+\Biggl({|f(y)|\over \lambda}\Biggr)\Biggr) dy,\qquad \lambda> 0. \] By imposing a condition on \(b\) stronger than BMO, he sharpens this estimate, replacing \(L\log L\) by \(L(\log L)^\varepsilon\), \(0< \varepsilon< 1\). He also show that \([b, T]\) is bounded from a modified Hardy space \(H^1_b\) to \(L^1\). Higher order commutators are also treated.
Calderón-Zygmund singular integral operator
scientific article; zbMATH DE number 764004