Abstract :
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.
