Weighted \(L^ p\)-boundedness for higher order commutators of oscillatory singular integrals
The authors study the weighted \(L^p\)-boundedness for higher-order commutators of oscillatory singular integrals defined by \[ T^m_bf(x)=\text{p.v. }\int e^{iP(x,y)}{\Omega(x-y)\over|x-y|^n} h(|x-y|)[b(x)- b(y)]^mf(y)dy, \] where \(P(x,y)\) is a real polynomial on \(\mathbb{R}^n\times\mathbb{R}^n\), \(\Omega\in L\log^+L(S^{n-1})\), homogeneous of degree zero, \(b(r)\in\text{BMO}(\mathbb{R}_+)\), \(h(r)\in\text{BV}(\mathbb{R}_+)\), i.e., a bounded variation function on \(\mathbb{R}_+\), and \(m\in\mathbb{Z}_+\). The main result in this paper gives a necessary and sufficient condition so that \(T^m_b\) is bounded on \(L^p(\omega)\), \(1<p<\infty\), for any real nontrivial polynomial \(P(x,y)\), where \(L^p(\omega)\) denotes the weighted \(L^p\) space with certain weight \(\omega(x)\). In order to obtain the above result we also discuss the weighted \(L^p\)-boundedness of \(M^m_{\Omega,b}\), a rough maximal operator related to higher-order commutators, defined by \[ M^m_{\Omega,b}f(x)= \sup_{r>0} {1\over r^n} \int_{|x-y|<r}|\Omega(x- y)||b(x)- b(y)|^m|f(y)|dy. \]
scientific article; zbMATH DE number 1044122