Sparse bounds for oscillatory and random singular integrals

Michael Lacey, Scott Spencer
2017 New York Journal of Mathematics New York J. Math   unpublished
Let TP f (x) = e iP (y) K(y)f (x − y) dy, where K(y) is a smooth Calderón-Zygmund kernel on R n , and P be a polynomial. We show that there is a sparse bound for the bilinear form TP f, g. This in turn easily implies Ap 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.