Mapping properties for oscillatory integrals in d-dimensions

For aj , bj 1, j = 1, 2, . . . , d, we prove that the operator Kf (x) = Rd + k(x, y)f (y)dy maps Lp(Rd +) into itself for p = 1 + 1 r , where r = a1 b1 = ··· = ad bd , and k(x, y) = ϕ(x, y)eig(x,y), ϕ(x, y) satisfies (1.2) (e.g. ϕ(x, y) = |x − y|iτ , τ real) and the phase g(x, y) = xa · yb. We study operators with more general phases and for these operators we require that aj , bj > 1, j = 1, 2, . . . , d, or al = bl 1 for some l ∈ {1, 2, . . . , d}. © 2007 Elsevier Inc. All rights reserved.