Convolution estimates related to space curves

Youngwoo Choi

doi:10.1186/1029-242x-2011-91

Abstract

Based on a uniform estimate of convolution operators with measures on a family of plane curves, we obtain optimal Lp -Lq boundedness of convolution operators with affine arclength measures supported on space curves satisfying a suitable condition. The result generalizes the previously known estimates.2000 Mathematics Subject Classifications: Primary 42B15; Secondary 42B20.

Highlights

Let I ⊂ R be an open interval and ψ : I ® R be a C3 function
The use of the affine arclength measure was suggested by Drury [2] to mitigate the effect of degeneracy and has been helpful to obtain uniform estimates
This article is organized as follows: a uniform estimate for convolution operators with measures supported on plane curves

Summary

Introduction

1 Introduction Let I ⊂ R be an open interval and ψ : I ® R be a C3 function. Let g : I ® R3 be the curve given by g(t) = (t, t2/2,ψ(t)), t Î I. The Lp - Lq mapping properties of the corresponding convolution operator Tσγ given by The use of the affine arclength measure was suggested by Drury [2] to mitigate the effect of degeneracy and has been helpful to obtain uniform estimates.

Results

Conclusion