$L^{p}$ type mapping estimates for oscillatory integrals in higher dimensions
We show in two dimensions that if Kf = T R2 + k(x, y)f(y) dy, k(x, y) = e a ·yb/|x − y|, p = 4/(2 + η), a ≥ b ≥ 1̄ = (1, 1), vp(y) = y (p/p′)(1̄−b/a), then ‖Kf‖p ≤ C‖f‖p,vp if η +α1 +α2 < 2, αj = 1−bj/aj , j = 1, 2. Our methods apply in all dimensions and also for more general kernels. 0. Introduction. Our purpose is to study mapping properties of the operators given by (0.1) Kf = Ka,bf(x) = \ R d + k(x, y)f(y) dy with x ∈ R+, i.e. x = (x1, . . . , xd) with xj ≥ 0 for 1 ≤ j ≤ d. Here we are forced to consider mappings from Lv into L p w where w(x), v(y) are non-negative, measurable functions (generally referred to as weights) that are positive a.e., and for the most part the weights will be power weights. For the kernel, we take (0.2) k(x, y) = φ(x, y)eig(x,y) with g(x, y) = xa · yb = dj=1 x aj j y bj j , aj , bj ≥ 1, and φ(x, y) satisfies (0.4) below; we also set |x|a · |y|b = ∑d j=1 |xj|j |yj |bj . The model case occurs when k(x, y) = |x− y|−ηeixa·yb , 0 ≤ η < 2−α1 −α2, and αj = 1 − bj/aj , j = 1, 2. We also consider the case where k(x, y) = |x − y|−ηeixa·ybγ(x − y), 0 ≤ η < 2, where γ ∈ C∞(R2) and γ(x) = 0 if |x| ≤ 1, γ(x) = 1 if |x| ≥ 2, and 0 ≤ γ(x) ≤ 1. We mostly work in d = 2 dimensions, and since most of the arguments are iteration type these results can be extended to higher dimensions. We shall give a more detailed explanation of this at the end of Section 0. Our aim is thus to prove an (Lv, L p w) estimate for the operators defined in (0.1) and (0.2), where φ(x, y) satisfies (0.4), a1 ≥ b1 ≥ 1, a2 ≥ b2 ≥ 1, 2000 Mathematics Subject Classification: 42B20, 46B70, 47G10.