Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


On weighted $ L^2$ estimates for solutions of the wave equation

Authors: Youngwoo Koh and Ihyeok Seo
Journal: Proc. Amer. Math. Soc. 144 (2016), 3047-3061
MSC (2010): Primary 35B45; Secondary 35L05, 42B35
Published electronically: January 27, 2016
MathSciNet review: 3487235
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider weighted $ L^2$ integrability for solutions of the wave equation. For this, we obtain some weighed $ L^2$ estimates for the solutions with weights in Morrey-Campanato classes. Our method is based on a combination of bilinear interpolation and a localization argument which makes use of the Littlewood-Paley theorem and a property of Hardy-Littlewood maximal functions. We also apply the estimates to the problem of well-posedness for wave equations with potentials.

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

  • [1] J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz, and M. C. Vilela, Strichartz inequalities with weights in Morrey-Campanato classes, Collect. Math. 61 (2010), no. 1, 49-56. MR 2604858 (2011b:35191),
  • [2] J. A. Barceló, J. M. Bennett, A. Ruiz, and M. C. Vilela, Local smoothing for Kato potentials in three dimensions, Math. Nachr. 282 (2009), no. 10, 1391-1405. MR 2571701 (2010m:35078),
  • [3] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275 (58 #2349)
  • [4] Sagun Chanillo and Eric Sawyer, Unique continuation for $ \Delta +v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc. 318 (1990), no. 1, 275-300. MR 958886 (90f:35050),
  • [5] R. R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), no. 2, 249-254. MR 565349 (81b:42067),
  • [6] Daoyuan Fang and Chengbo Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math. 23 (2011), no. 1, 181-205. MR 2769870 (2012b:35208),
  • [7] Loukas Grafakos, Modern Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2014. MR 3243741
  • [8] Kunio Hidano, Jason Metcalfe, Hart F. Smith, Christopher D. Sogge, and Yi Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2789-2809. MR 2584618 (2012i:35243),
  • [9] Toshihiko Hoshiro, On weighted $ L^2$ estimates of solutions to wave equations, J. Anal. Math. 72 (1997), 127-140. MR 1482992 (99j:35120),
  • [10] Tosio Kato and Kenji Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), no. 4, 481-496. MR 1061120 (91i:47013),
  • [11] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955-980. MR 1646048 (2000d:35018)
  • [12] Douglas S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $ L^{p}$spaces, Trans. Amer. Math. Soc. 259 (1980), no. 1, 235-254. MR 561835 (80f:42013),
  • [13] Walter Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766-770. MR 0155146 (27 #5086)
  • [14] Cathleen S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A 306 (1968), 291-296. MR 0234136 (38 #2455)
  • [15] Alberto Ruiz and Luis Vega, Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J. 76 (1994), no. 3, 913-940. MR 1309336 (95m:35110),
  • [16] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813-874. MR 1175693 (94i:42024),
  • [17] I. Seo, From resolvent estimates to unique continuation for the Schrödinger equation, to appear in Trans. Amer. Math. Soc., arXiv:1401.0901.
  • [18] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43. With the assistance of Timothy S. Murphy. Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • [19] Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705-714. MR 0512086 (58 #23577)
  • [20] Mitsuru Sugimoto, Global smoothing properties of generalized Schrödinger equations, J. Anal. Math. 76 (1998), 191-204. MR 1676995 (2000a:35033),
  • [21] Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503903 (80i:46032b)
  • [22] M. C. Vilela, Regularity of solutions to the free Schrödinger equation with radial initial data, Illinois J. Math. 45 (2001), no. 2, 361-370. MR 1878609 (2002k:35061)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35B45, 35L05, 42B35

Retrieve articles in all journals with MSC (2010): 35B45, 35L05, 42B35

Additional Information

Youngwoo Koh
Affiliation: School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea

Ihyeok Seo
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

Keywords: Weighted estimates, wave equation, Morrey-Campanato class
Received by editor(s): January 9, 2015
Received by editor(s) in revised form: September 4, 2015
Published electronically: January 27, 2016
Additional Notes: The first author was supported by NRF grant 2012-008373.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society