Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Maximal function estimates of
solutions to general dispersive
partial differential equations


Authors: Hans P. Heinig and Sichun Wang
Journal: Trans. Amer. Math. Soc. 351 (1999), 79-108
MSC (1991): Primary 42B25; Secondary 42A45
DOI: https://doi.org/10.1090/S0002-9947-99-02116-9
MathSciNet review: 1458324
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $u(x,t)=(S_\Omega f)(x,t)$ be the solution of the general dispersive initial value problem:

\begin{displaymath}\partial _tu-i\Omega(D)u=0, \quad u(x,0)=f(x), \qquad (x,t)\in \mathbb{R}^n \times \mathbb{R}\end{displaymath}

and $S^{**}_\Omega f$ the global maximal operator of $S_\Omega f$. Sharp weighted $L^p$-esimates for $S^{**}_\Omega f$ with $f\in H_s(\mathbb{R}^n)$ are given for general phase functions $\Omega$.


References [Enhancements On Off] (What's this?)

  • 1. J. J. Benedetto, H. P. Heinig, and R. Johnson, Weighted Hardy spaces and the Laplace transform. II, Math. Nachr. 132 (1987), 29-55. MR 88m:44007
  • 2. J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer, New York, 1976. MR 58:2349
  • 3. J. Bourgain, A remark on Schrödinger operators, Israel J. Math. 77 (1992), 1-16. MR 93k:35071
  • 4. A. Carbery, Radial Fourier multipliers and associated maximal functions, North-Holland Math. Studies, vol. III, North-Holland, 1985, 49-55. MR 87i:42029
  • 5. L. Carleson, Some analytical problems related to statistical mechanics, Euclidean Harmonic Analysis, Lecture Notes in Math. 779 (1979), 5-45. MR 82j:82005
  • 6. P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1989), 413-446. MR 89d:35150
  • 7. M. Cowling, Pointwise behavior of solutions to Schrödinger equations, Harmonic Analysis, Lecture Notes in Math. 992 (1983), 83-90. MR 85c:34029
  • 8. B. E. J. Dahlberg and C. E. Kenig, A note on almost everywhere behavior of solutions to the Schrödinger equation, Harmonic Analysis, Lecture Notes in Math., 908 (1982), 205-209. MR 83f:35023
  • 9. H. P. Heinig, Weighted norm inequalities for classes of operators, Indiana Univ. Math. J. (4) 33 (1984), 573-582. MR 86c:42016
  • 10. L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, second ed., Springer, 1983. MR 85g:35002a
  • 11. C. E. Kenig and A. Ruiz, A strong type $(2,2)$ estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), 239-246. MR 85c:42010
  • 12. C. E. Kenig, D. Ponce, and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69. MR 92d:35081
  • 13. -, On the IVP for the non-linear Schrödinger equations, Contemp. Math., 189, Amer. Math. Soc., Providence, R.I., 1995. MR 96e:35071
  • 14. -, Small solutions to non-linear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 255-288. MR 94h:35238
  • 15. -, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contractions principle, Comm. Pure Appl. Math. 46 (1993), 527-620. MR 94h:35229
  • 16. E. Prestini, Radial functions and regularity of solutions to the Schrödinger equation, Monatsh. Math. 109 (1990), 135-143. MR 91j:35035
  • 17. W. Rudin, Principles of Mathematical Analysis, third ed., McGraw-Hill, 1976. MR 52:5893
  • 18. H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley, New York, 1985. MR 88k:42015b
  • 19. P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699-715. MR 88j:35026
  • 20. -, Radial functions and maximal estimates for solutions to the Schrödinger equation, J. Austral. Math. Soc. Ser. A. 59 (1995), 134-142. MR 96d:42032
  • 21. -, Global maximal estimates for solutions to the Schrödinger equation, Studia Math. 110 (1994), 105-114. MR 95e:35052
  • 22. -, $L^p$ maximal estimates for solutions to the Schrödinger equation, Private Communication, Aug. 1994.
  • 23. C. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, 105, Cambridge Univ. Press, 1993. MR 94c:35178
  • 24. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 44:7280
  • 25. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 46:4102
  • 26. -, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159-172. MR 19:1184d
  • 27. H. Triebel, Theory of Function Spaces, Monographs in Math., 78, Birkhäuser, 1983. MR 86j:46026
  • 28. L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878. MR 89d:35046
  • 29. B. Walther, Maximal estimates of oscillatory integrals with concave phase, Preprint, 25 (1994), Math. Dept., Uppsala Univ., Sweden.
  • 30. Sichun Wang, On the maximal operator associated with the Schrödinger equation, Studia Math. 122 (1997), 167-182. MR 98f:42019
  • 31. Silei Wang, On the weighted estimate of the solution associated with the Schrödinger equation, Proc. Amer. Math. Soc. 113 (1991), 87-92. MR 91k:35066
  • 32. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, 1922.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 42B25, 42A45

Retrieve articles in all journals with MSC (1991): 42B25, 42A45


Additional Information

Hans P. Heinig
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
Email: heinig@mcmail.cis.mcmaster.ca

Sichun Wang
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
Email: wangs@icarus.math.mcmaster.ca

DOI: https://doi.org/10.1090/S0002-9947-99-02116-9
Keywords: Dispersive PDE, free Schr\"odinger equation, phase functions, polynomials of principal type, regular zeroes, weighted $L^p$-spaces, Sobolev spaces
Received by editor(s): May 5, 1996
Received by editor(s) in revised form: July 1, 1996
Additional Notes: The research of the first author was supported in part by NSERC grant A-4837
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society