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.
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We consider operators of the form $(Ωf)(y) = ʃ_{-∞}^∞ Ω(y,u)f(u)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h ∈ L^∞$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space $Ḃ^{0,1}_1$ (= B) into itself. In particular, all operators with $h(y) = e^{i|y|^a}$, a > 0, a ≠ 1, map B into itself.