Non-convolution Operators with Oscillating Kernels That Map

Here we consider the kernels Ω1(y, u) = K(y, u)ei|y−u|a for a > 1. We show that the operators Tƒ(y) = ∫ (Ω1(y + 12, u) − Ω1(y, u))ƒ(u) du map B(Rn) into itself. We also show that the operators ∫(Ω1(y, u)) ƒ(u) du map Lp into itself for p > 1.