Frontiers of Mathematics in China >

2011 , Vol. 6 >Issue 1: 49 - 59

DOI: https://doi.org/10.1007/s11464-010-0088-3

RESEARCH ARTICLE

Oscillatory integrals on unit square along surfaces

  • Jiecheng CHEN 1,2 ,
  • Dashan FAN 3 ,
  • Huoxiong WU , 4 ,
  • Xiangrong ZHU 1,2
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  • 1. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
  • 2. Department of Mathematics, Zhejiang University, Hangzhou 310027, China
  • 3. Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
  • 4. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received date: 04 Jul 2010

Accepted date: 09 Oct 2010

Published date: 01 Feb 2011

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

Let Q2 = [0, 1]2 be the unit square in two-dimensional Euclidean space ℝ2. We study the Lp boundedness of the oscillatory integral operator Tα,β defined on the set (ℝ2+n) of Schwartz test functions by

Tα,βf(u,v,x)=∫Q2f(u-t,v-s,x-γ(t,s))t1+α1s1+α2eit-β1s-β2dtds,
where x∈ℝn, (u,v)∈ℝ2, (t,s,γ(t,s))=(t,s,tp1sq1,tp2sq2,⋯,tpnsqn) is a surface on ℝn+2, and β1>α1, β2>α2. Our results extend some known results on ℝ3.

Key words： Oscillatory integral; singular integral; unit square; surface; product space

Cite this article

Jiecheng CHEN , Dashan FAN , Huoxiong WU , Xiangrong ZHU . Oscillatory integrals on unit square along surfaces[J]. Frontiers of Mathematics in China, 2011 , 6(1) : 49 -59 . DOI: 10.1007/s11464-010-0088-3

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