RESEARCH ARTICLE

Oscillatory integrals on unit square along surfaces

  • Jiecheng CHEN 1,2 ,
  • Dashan FAN 3 ,
  • Huoxiong WU , 4 ,
  • Xiangrong ZHU 1,2
Expand
  • 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

Abstract

Let Q2 = [0, 1]2 be the unit square in two-dimensional Euclidean space 2. We study the Lp boundedness of the oscillatory integral operator Tα,β defined on the set (2+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 xn, (u,v)2, (t,s,γ(t,s))=(t,s,tp1sq1,tp2sq2,,tpnsqn) is a surface on n+2, and β1>α1, β2>α2. Our results extend some known results on 3.

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

1
Chen J, Fan D, Wang M, Zhu X. Lp bounds for oscillatory hyper Hilbert transforms along curves. Proc Amer Math Soc, 2008, 136: 3145-3153

DOI

2
Chen J, Fan D, Zhu X. Sharp L2 boundedness of the oscillatory hyper Hilbert transform along curves. Acta Math Sin (Engl Ser), 2010, 26(3): 653-658

DOI

3
Chandarana S. Lp bounds for hypersingular integral operators along curves. Pacific J Math, 1996, 175(2): 389-416

4
Fan D, Wu H. Certain oscillatory integrals on unit square and their applications. Sci in China, Ser A, 2008, 51(10): 1895-1903

DOI

5
Fefferman C, Stein E M. Hp spaces of several variables. Acta Math, 1972, 129: 137-193

DOI

6
Hirschman I I. Multiplier transforms, I. Duke Math J, 1956, 26: 222-242

7
Stein E M, Wainger S. Problems in harmonic analysis related to curvatures. Bull Amer Math Soc, 1978, 84: 1239-1295

DOI

8
Wainger S. Special Trigonometric Series in kDimension. Mem Amer Math Soc, No 59. Providence: AMS, 1965

9
Ye X. Boundedness of certain operators on function spaces.<DissertationTip/>, 2006

10
Zielinski M. Highly oscillatory singular integrals along curves.<DissertationTip/>, 1985

Options
Outlines

/