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The Lp-Lp mapping properties of convolution operators with the affine arclength measure on space curves

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

Youngwoo Choi
Youngwoo Choi
Affiliation:
Department of Mathematics Ajou UniversitySuwon 442-749Korea e-mail: [email protected]
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Abstract

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The Lp-improving properties of convolution operators with measures supported on space curves have been studied by various authors. If the underlying curve is non-degenerate, the convolution with the (Euclidean) arclength measure is a bounded operator from L3/2()3 into L2(3). Drury suggested that in case the underlying curve has degeneracies the appropriate measure to consider should be the affine arclength measure and the obtained a similar result for homogeneous curves t→(t, t2, tk), t >0 for k ≥ 4. This was further generalized by Pan to curves t → (t, tk, tt), t > 0 for l < k < l, k+l ≥ 5. In this article, we will extend Pan's result to (smooth) compact curves of finite type whose tangents never vanish. In addition, we give an example of a flat curve with the same mapping properties.

Keywords

convolutionaffine arclength measureflat curve

MSC classification

Secondary: 42B15: Multipliers 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 2 , October 2003 , pp. 247 - 262
Copyright
Copyright © Australian Mathematical Society 2003

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