Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Sharp maximal estimates for doubly oscillatory integrals

Author(s): Björn Gabriel Walther
Journal: Proc. Amer. Math. Soc. 130 (2002), 3641-3650.
MSC (1991): Primary 42B25, 42B99, 35L05, 35J10, 35Q40
Posted: May 1, 2002
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We study doubly oscillatory integrals

\begin{displaymath}\int_{\mathbf R^n} \, e^{i(\xi + y\vert\xi\vert + t\vert\xi\vert^ a)} \widehat f(\xi)\,d\xi \end{displaymath}

and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on $\mathbf{R}^n$.

References:

1.
L. Carleson, Some Analytic problems to related to Statistical Mechanics, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), 5-45, Lecture Notes in Math. 779, Springer, Berlin, 1980. MR 82j:82005

2.
H. P. Heinig; Sichun Wang, Maximal Function Estimates of Solutions to General Dispersive Partial Differential Equations, Trans. Amer. Math. Soc. 351 (1999), 79-108. MR 99c:35038

3.
L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, 1990. MR 91m:35001b; MR 85g:35002a

4.
E. Prestini, Radial functions and regularity of solutions to the Schrödinger equation, Monatsh. Math. 109 (1990), 135-143 MR 91j:35035

5.
J. L. Rubio de Francia, Maximal functions and Fourier transforms, Duke Math. J. 53 (1986), 395-404. MR 87j:42046

6.
P. Sjölin, Regularity of Solutions to the Schrödinger Equation, Duke Math. J. 55 (1987), 699-715. MR 88j:35026

7.
P. Sjölin, Radial functions and maximal estimates for solutions to the Schrödinger equation, J. Austral. Math. Soc. Ser. A 59 (1995), 134-142. MR 96d:42032

8.
P. Sjölin, $L ^ p$ maximal estimates for solutions to the Schrödinger equation, Math. Scand. 81 (1997), 35-68. MR 98j:35038

9.
C. Sogge; E. M. Stein, Averages of functions over hypersurfaces in $\mathbf{R}^ n$, Invent. Math. 82 (1985), 543-556. MR 87d:42030

10.
E. M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2174-2175. MR 54:8133a

11.
E. M. Stein, Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, No. 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993. MR 95c:42002

12.
E. M. Stein; G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971. MR 46:4102

13.
T. Tao; A. Vargas, A Bilinear Approach to Cone Multipliers II. Applications, Geom. Func. Anal. 10 (2000), 216-258. MR 2002e:42013

14.
L. Vega, Schrödinger Equations: Pointwise Convergence to the Initial Data, Proc. Amer. Math. Soc. 102 (1988), 874-878. MR 89d:35046

15.
B. G. Walther, Maximal Estimates for Oscillatory Integrals with Concave Phase, Contemp. Math. 189 (1995), 485-495. MR 96e:42024

16.
B. G. Walther, Norm Inequalities for Oscillatory Fourier Integrals, Doctoral thesis, TRITA-MAT-1998-MA-25, Royal Institute of Technology, Stockholm 1998.

17.
B. G. Walther, Some $L \sp p(L \sp \infty)$- and $L \sp 2(L \sp 2)$-estimates for oscillatory Fourier transforms, Analysis of Divergence (Orono, ME, 1997), 213-231. MR 2000j:00016; Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998. MR 2001e:42013

18.
B. G. Walther, A sharp weighted $L \sp 2$-estimate for the solution to the time-dependent Schrödinger equation, Ark. mat. 37 (1999), 381-393. MR 2000g:35029

19.
B. G. Walther, Higher Integrability for Maximal Oscillatory Fourier Integrals, Ann. Acad. Sci. Fenn. Ser. A I Math. 26 (2001), 189-204. MR 2002f:42021

20.
Sichun Wang, On the Maximal Operator associated with the Free Schrödinger Equation, Studia Math. 122 (1997), 167-182. MR 98f:42019

21.
Si Lei Wang, On the Weighted Estimate of the Solution associated with the Schrödinger equation, Proc. Amer. Math. Soc. 113 (1991), 87-92. MR 91k:35066

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42B25, 42B99, 35L05, 35J10, 35Q40

Retrieve articles in all Journals with MSC (1991): 42B25, 42B99, 35L05, 35J10, 35Q40

Additional Information:

Björn Gabriel Walther
Affiliation: Department of Mathematics, Royal Institute of Technology, SE -- 100 44 Stockholm, Sweden
Address at time of publication: Department of Mathematics, Brown University, Providence, Rhode Island 02912--1917
Email: WALTHER@Math.KTH.SE, WALTHER@Math.Brown.Edu

DOI: 10.1090/S0002-9939-02-06527-9
PII: S 0002-9939(02)06527-9
Keywords: Doubly oscillatory Fourier integrals, maximal estimates, wave equation, time-dependent Schr\"odinger equation
Received by editor(s): August 10, 2000
Received by editor(s) in revised form: July 19, 2001
Posted: May 1, 2002
Additional Notes: This paper is a revision of [16, Chapter 9]. The author would like to thank Professor {\it Per Sjölin,} Royal Institute of Technology, Stockholm, Sweden, for patience and support. The final draft was made during visits at Brown University, Providence, RI, USA, and Univerzita Komenského, Bratislava, Slovakia.
Communicated by: Christopher D. Sogge
Copyright of article: Copyright 2002, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google