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Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in R n
Author(s) -
Choi Youngwoo
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079900825x
Subject(s) - measure (data warehouse) , polynomial , mathematics , convolution (computer science) , affine transformation , operator (biology) , mathematical analysis , euclidean geometry , pure mathematics , combinatorics , geometry , computer science , chemistry , biochemistry , repressor , database , machine learning , artificial neural network , transcription factor , gene
McMichael proved that the convolution with the (euclidean) arclength measure supported on the curve t ↦ ( t , t 2 , …, t n ), 0 < t < 1, maps L p (R n ) boundedly into L p ′ (R n ) if and only if 2 n ( n +1)/( n 2 + 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 R n maps L p (R n ) boundedly into L p ′ (R n ) if p = 2 n ( n +1)/( n 2 + n +2).