OSCILLATORY INTEGRALS AND NEWTON POLYHEDRA
JOONIL KIM
~ be a vector polynomial of two variables. Given Ij = [0, 1] or [0, ∞), we
Abstract. Let P
talk about the largest number µ that the oscillatory integrals satisfy
Z
~
eihξ,P (x,y)i Ψ(x, y)dxdy = O(|ξ|−µ ) as |ξ| → ∞
I1 ×I2
for all Ψ in a certain class of C ∞ functions. This number µ is described in terms of some
~ . We next discuss about the cases
generalized notion of Newton polyhedra associated with P
of operators T λ where T λ f (x) is defined as an integral over the global domain (−∞, ∞):
Z ∞
T λ f (x) =
eiλP (x,y) f (y)dy where f is in Schwartz class.
−∞
Department of Mathematics, Yonsei University, Seoul 120-729, Korea
E-mail address: jikim7030@yonsei.ac.kr
2000 Mathematics Subject Classification. Primary 42B20, 42B25.
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