Weighted weak $(1,1)$ and weighted $L\sp p$ estimates for oscillating kernels

Abstract

Weak type (1, 1) and strong type (p,p) inequalities are proved for operators defined by oscillating kernels. The techniques are sufficiently general to derive versions of these inequalities using weighted norms. 0. Introduction. Given a positive real number a > O, a 7& 1, define the oscillating kernel Ka by Ka(x) = (1 + |X|)-leilxla and consider the convolution operator Ka * f. In an earlier paper [2], we studied the boundedness properties of such operators on weighted LP spaces, 1 1, we define l/p f 1l P w = A E f(x)lpw(x) dx) R and say f E LP (R) if llfllP,w 1. The multiplier operator T, associated to Ka is defixled by (T:d) (() = 0(()1(l :/ e 1tl f((), where 0 is a C°° function defined by (i) 0(4) = (0 if :(: > 1/ when p < 0,