Weak Type Bound for Oscillatory Singular Integrals

Abstract

Let $T_{P}f(x)=\int e^{iP(y)}K(y)f(x-y)dy$, where $K(y)$ is a smooth Calderón-Zygmund kernel on ${\mathbb{R}}^n$, and $P$ be a polynomial. The maximal truncations of $T_P$ satisfy the weak $L^1$ inequality, our proof simplifying and extending the argument of Chanillo and Christ for the weak type bound for $T_P$.

The Carleson measure argument is not sufficient to conclude the Theorem

References in corpus (1)

• On weighted norm inequalities for the Carleson and Walsh-Carleson operators