EN
We show in two dimensions that if $Kf = ∫_{ℝ₊²} k(x,y)f(y)dy$, $k(x,y) = (e^{ix^{a}·y^{b}})/(|x-y|^{η})$, p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), $v_{p}(y) = y^{(p/p')(1̅-b/a)}$, then $||Kf||_{p} ≤ C||f||_{p,v_{p}}$ if η + α₁ + α₂ < 2, $α_{j} = 1 - b_{j}/a_{j}$, j = 1,2. Our methods apply in all dimensions and also for more general kernels.