DAMPING OSCILLATORY INTEGRALS WITH POLYNOMIAL PHASE AND CONVOLUTION OPERATORS WITH THE AFFINE ARCLENGTH MEASURE ON POLYNOMIAL CURVES IN ℝⁿ

Abstract

McMichael proved that the convolution with the (euclidean) arclength measure supported on the curve t [xrarr ] (t, t², …, tⁿ), 0 < t < 1, maps L^p(ℝⁿ) boundedly into L^p′(ℝⁿ) if and only if 2n(n+1)/(n²+n+2) [les ] p [les ] 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 L^p(ℝⁿ) boundedly into L^p′(ℝⁿ) if p = 2n(n+1)/(n²+n+2).