Estimates of Oscillatory Integrals
by
Gary Sampson
Dept of math Auburn U., Auburn, Al 36849

Consider the operator (1) Kf(x)=∫_R² k(x, y)f(y)dy with (2) k(x, y)=f(x, y)e^{ig(x, y)}, g real-valued, with (3) g(x, y) = x^a·y^b +F_**(x^a, y^b) and x, y, a, b ∈ R² and a, b ≥ [`1]. Also we suppose for x, y ≥ [`1] that (4) |∂_x^a∂_y^bF_**(x, y)| ≤ C, for all a and b ∈ N². We prove that if a₁/b₁ = a₂/b₂, b₁ b₂ > 1 and a, b ≥ [`1], then ∥Kf∥_p ≤ C∥f∥_p for p=1+(b₁/a₁).

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Date received: December 16, 2007