Singular integrals with rough kernels along real-analytic submanifolds in \(\mathbb{R}^n\)
The authors continue their study of singular integral operators with rough kernels started in [\textit{D. Fan} and \textit{Y. Pan}, Am. J. Math. 119, No. 4, 799-839 (1997; Zbl 0899.42002)]. They consider the operator \(T_\Psi\) defined as \((T_\Psi f)(x)= \text{p.v. }\int_U f(x-\Psi(y))K(y)dy\), where \(\Psi\) is a mapping of \(U\subset R^d\) in \(R^n\) and \(K(y)=b(y){\Omega(y)\over|y|^n}\) (\(\int_{S^{n-1}}\Omega=0\)) is a Calderón-Zygmund kernel and obtain that for \(n\geq 2\) and bounded \(U\) for a real-analytic mapping \(\Psi\), if \(\Omega\in L^q\), for some \(q>1\) and \(\sup_R{1\over R}\int_0^R|b(t)|^\gamma dt<\infty\), then the operator \(T_\Psi\) is bounded on \(L^p\) for \(|{1\over p}-{1\over 2}|<\min\{{1\over 2},{1\over\gamma'}\}\).
scientific article; zbMATH DE number 1262347