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 
 
 

 

Estimates for radial solutions to the wave equation


Authors: Javier Duoandikoetxea, Adela Moyua and Osane Oruetxebarria
Journal: Proc. Amer. Math. Soc. 144 (2016), 1543-1552
MSC (2010): Primary 35L05; Secondary 35B05
DOI: https://doi.org/10.1090/proc/12767
Published electronically: December 21, 2015
MathSciNet review: 3451231
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Cauchy problem for the wave equation with null initial position and radial initial velocity $ \psi $. With the solution $ u$ of this problem we define the operator $ S(\psi )=\sup _{t>0}t^{-1}\vert u(x,t)\vert$. Using simple one-dimensional operators to bound pointwise the operator $ S$ we obtain weighted $ L^p$ estimates with power weights. Even for the unweighted estimates the result for dimension higher than 3 differs from the one with general functions.


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

  • [1] Leonardo Colzani, Time decay and regularity of solutions to the wave equation and convergence of Fourier expansions, J. Fourier Anal. Appl. 9 (2003), no. 1, 49-66. MR 1953072 (2003k:35142), https://doi.org/10.1007/s00041-003-0002-4
  • [2] Leonardo Colzani, Alice Cominardi, and Krzysztof Stempak, Radial solutions to the wave equation, Ann. Mat. Pura Appl. (4) 181 (2002), no. 1, 25-54 (English, with English and Italian summaries). MR 1895024 (2003c:35101), https://doi.org/10.1007/s102310100025
  • [3] Javier Duoandikoetxea, Adela Moyua, and Osane Oruetxebarria, The spherical maximal operator on radial functions, J. Math. Anal. Appl. 387 (2012), no. 2, 655-666. MR 2853134 (2012m:42017), https://doi.org/10.1016/j.jmaa.2011.09.028
  • [4] Javier Duoandikoetxea, Francisco J. Martín-Reyes, and Sheldy Ombrosi, Calderón weights as Muckenhoupt weights, Indiana Univ. Math. J. 62 (2013), no. 3, 891-910. MR 3164849, https://doi.org/10.1512/iumj.2013.62.4971
  • [5] Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943 (2011c:35002)
  • [6] Steve Hofmann, Singular integrals with power weights, Proc. Amer. Math. Soc. 110 (1990), no. 2, 343-353. MR 1021899 (90m:42026), https://doi.org/10.2307/2048076
  • [7] Detlef Müller and Andreas Seeger, Inequalities for spherically symmetric solutions of the wave equation, Math. Z. 218 (1995), no. 3, 417-426. MR 1324536 (96e:35093), https://doi.org/10.1007/BF02571912
  • [8] Fernando Soria and Guido Weiss, A remark on singular integrals and power weights, Indiana Univ. Math. J. 43 (1994), no. 1, 187-204. MR 1275458 (95g:42028), https://doi.org/10.1512/iumj.1994.43.43009
  • [9] Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174-2175. MR 0420116 (54 #8133a)
  • [10] Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239-1295. MR 508453 (80k:42023), https://doi.org/10.1090/S0002-9904-1978-14554-6
  • [11] Alberto Torchinsky, The Fourier transform and the wave equation, Amer. Math. Monthly 118 (2011), no. 7, 599-609. MR 2826451 (2012h:42016), https://doi.org/10.4169/amer.math.monthly.118.07.599

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35L05, 35B05

Retrieve articles in all journals with MSC (2010): 35L05, 35B05


Additional Information

Javier Duoandikoetxea
Affiliation: Departamento de Matemáticas, Universidad del País Vasco/Euskal Herriko Unibertsitatea, Apartado 644, 48080 Bilbao, Spain
Email: javier.duoandikoetxea@ehu.eus

Adela Moyua
Affiliation: Departamento de Matemáticas, Universidad del País Vasco/Euskal Herriko Unibertsitatea, Apartado 644, 48080 Bilbao, Spain

Osane Oruetxebarria
Affiliation: Departamento de Matemáticas, Universidad del País Vasco/Euskal Herriko Unibertsitatea, Apartado 644, 48080 Bilbao, Spain
Email: osane.oruetxebarria@ehu.eus

DOI: https://doi.org/10.1090/proc/12767
Keywords: Wave equation, radial solutions, weighted inequalities
Received by editor(s): October 16, 2014
Received by editor(s) in revised form: February 25, 2015
Published electronically: December 21, 2015
Additional Notes: This work was supported by grant MTM2011-24054 of the Ministerio de Economía y Competitividad (Spain) and grant IT-641-13 of the Basque Gouvernment
Adela Moyua (1956-2013). The author passed away during the production of this work
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society