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Journal of the London Mathematical Society 2000 61(1):275-285; doi:10.1112/S002461079900825X
© 2000 by London Mathematical Society
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© The London Mathematical Society

Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in Rn

Youngwoo Choi

Department of Mathematics, Ajou University Suwon 442-749, Korea, youngwoo{at}madang.ajou.ac.kr

Received 4 December 1997. Revision received 9 November 1998.

McMichael proved that the convolution with the (euclidean) arclength measure supported on the curve t ↦ (t, t2, ..., tn), 0 < t < 1, maps Lp(Rn) boundedly into Lp'(Rn) if and only if 2n(n+1)/(n2+n+2) ≤ p ≤ 2. In proving this, a uniform estimate on damping oscillatory integrals with polynomial phase was crucial. In this paper, a remarkably simple proof of the same estimate on oscillatory integrals is presented. In addition, it is shown that the convolution operator with the affine arclength measure on any polynomial curve in Rn maps Lp(Rn) boundedly into Lp'(Rn) if p = 2n(n+1)/(n2+n+2).


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