Singular integrals along submanifolds of finite type
Let \(n\in \mathbb N\), \(n\geq 2\), and \(K(y)=\Omega(y)/|y|^n\) be a Calderón-Zygmund kernel on \(\mathbb R^n\). Let \(B(0, 1)\) be the unit ball centered at the origin in \(\mathbb R^n\). For \(d\in \mathbb N\) and a \(C^\infty\) mapping \(\Phi:B(0,1)\to\mathbb R^d\), define the singular integral operator \(T_\Phi\) on \(\mathbb R^d\) by \(T_\Phi(x)=\text{p.v.}\int_{B(0,1)}f(x-\Phi(y))K(y) dy\). The authors' main theorem is as follows: Suppose (i) \(\Phi\) is of finite type at \(0\), i.e., for each unit vector \(\eta\in \mathbb R^d\), there is a multi-index \(\alpha\) with \(|\alpha|\geq 1\) so that \(\partial_x^\alpha[\Phi(x)\cdot \eta]|_{x=0}\neq 0\), and (ii) \(\Omega\in L^q(S^{n-1})\) for some \(q>1\). Then \(T_\Phi\) is a bounded operator from \(L^p(\mathbb R^d)\) to itself for \(1<p<\infty\). They also give the \(L^p\) boundedness for the corresponding maximal truncated singular integrals. Their result is an improvement of E.~M.~Stein's one; Stein proved his result under the assumptions (i) and \(\Omega\in C^1(S^{n-1})\).
scientific article; zbMATH DE number 1548259