Oscillatory singular integral operators with variable Calderón-Zygmund kernels
The authors consider the variable oscillatory singular operator defined by \[ Kf(x)= \text{p.v. }\int e^{iP(x,y)}k(x,x-y)f(y) dy, \] where \(P(x,y)\) is a real polynomial and \(k(x,y)\) is Caldéron-Zygmund kernel in \(y\) which satisfies regularity conditions uniformly in \(x\) for a.e. \(x\). Then the operator and corresponding maximal operator are bounded on \(L^p\) for \(1<p<\infty\), and the norm depends only on the degree of the polynomial \(P(x,y)\), but not on the coefficients. The commutators in the Coifman-Rochberg-Weiss sense \[ [a,K]f(x)= \text{p.v. }\int e^{iP(x,y)}k(x,x-y)(a(x)-a(y))f(y)d y \] are also considered. For \(a\in \text{BMO}\) the commutator appears to be bounded on \(L^p\) for \(1<p<\infty\). This follows from a more general result about the variable oscillatory singular operators of order \(N\). The proofs use ideas of \textit{F. Ricci} and \textit{E. M. Stein} [J. Funct. Anal. 73, 179-194 (1987; Zbl 0622.42010)].
scientific article; zbMATH DE number 1406418