Sparse Bounds for Oscillatory and Random 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. We show that there is a sparse bound for the bilinear form $⟨T_{P}f,g⟩$. This in turn easily implies $A_p$ inequalities. The method of proof is applied in a random discrete setting, yielding the first weighted inequalities for operators defined on sparse sets of integers.

14 pages. To appear in NYJM