For a(j), b(j) >= 1, j = 1,2,...,d, we prove that the operator K f(x) = f(R+)(d) k(x, y)f(y)dy maps L-p(R-+(d)) into itself for p = 1 + 1/r, where r = a(1)/b(1) = center dot center dot center dot = a(d)/b(d), and k(x, y) = phi(x, y)e(ig)(x,y), phi(x, y) satisfies (1.2) (e.g. phi(x, y) = vertical bar x - y vertical bar(i iota), iota real) and the phase g(x, y) = x(a) center dot y(b). We study operators with more general phases and for these operators we require that a(j), b(j) > 1, j = 1, 2,..., d, or a(l) = b(l) >= l for some l is an element of {1, 2,..., d}. (c) 2007 Elsevier Inc. All rights reserved.
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