Nonlinear Maps between Besov and Sobolev spaces

Philip Brenner[1]; Peter Kumlin[2]

  • [1] IT-university of Göteborg, University of Gothenburg and Chalmers University of Technology SE-412 96 Göteborg Sweden
  • [2] Mathematical Sciences, University of Gothenburg and Chalmers University of Technology SE-412 96 Göteborg Sweden

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: 1, page 105-120
  • ISSN: 0240-2963

Abstract

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Our main result shows that for a large class of nonlinear local mappings between Besov and Sobolev space, interpolation is an exceptional low dimensional phenomenon. This extends previous results by Kumlin [13] from the case of analytic mappings to Lipschitz and Hölder continuous maps (Corollaries 1 and 2), and which go back to ideas of the late B.E.J. Dahlberg [8].

How to cite

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Brenner, Philip, and Kumlin, Peter. "Nonlinear Maps between Besov and Sobolev spaces." Annales de la faculté des sciences de Toulouse Mathématiques 19.1 (2010): 105-120. <http://eudml.org/doc/115841>.

@article{Brenner2010,
abstract = {Our main result shows that for a large class of nonlinear local mappings between Besov and Sobolev space, interpolation is an exceptional low dimensional phenomenon. This extends previous results by Kumlin [13] from the case of analytic mappings to Lipschitz and Hölder continuous maps (Corollaries 1 and 2), and which go back to ideas of the late B.E.J. Dahlberg [8].},
affiliation = {IT-university of Göteborg, University of Gothenburg and Chalmers University of Technology SE-412 96 Göteborg Sweden; Mathematical Sciences, University of Gothenburg and Chalmers University of Technology SE-412 96 Göteborg Sweden},
author = {Brenner, Philip, Kumlin, Peter},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {1},
number = {1},
pages = {105-120},
publisher = {Université Paul Sabatier, Toulouse},
title = {Nonlinear Maps between Besov and Sobolev spaces},
url = {http://eudml.org/doc/115841},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Brenner, Philip
AU - Kumlin, Peter
TI - Nonlinear Maps between Besov and Sobolev spaces
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/1//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 1
SP - 105
EP - 120
AB - Our main result shows that for a large class of nonlinear local mappings between Besov and Sobolev space, interpolation is an exceptional low dimensional phenomenon. This extends previous results by Kumlin [13] from the case of analytic mappings to Lipschitz and Hölder continuous maps (Corollaries 1 and 2), and which go back to ideas of the late B.E.J. Dahlberg [8].
LA - eng
UR - http://eudml.org/doc/115841
ER -

References

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  1. Bahouri (H.), Gerard (P.).— High frequency approximation of solutions to critical nonlinear equations, Amer. J. Math.121, p. 131-175 (1999). Zbl0919.35089MR1705001
  2. Bennet (C.), Sharpley (R.).— Interpolation of operators, Academic Press 1988. Zbl0647.46057MR928802
  3. Bergh (J.), Löfström (J.).— Interpolation spaces, Springer Verlag (1976). Zbl0344.46071
  4. Bourdaud (G.), Moussai (M.), Sickel (W.).— Towards sharp superposition theorems in Besov and Lizorkin-Triebel spaces, Nonlinear Analysis, (2007), doi: 10.1016/j.na.2007.02.035 Zbl1149.46026MR2404807
  5. Brenner (P.).— Space-time means and nonlinear Klein-Gordon Equations, Report, Department of Mathematics, Chalmers University of Technology, p. 1985-19. 
  6. Brenner (P.), Kumlin (P.).— On wave equations with supercritical nonlinearities, Arch. Math.74, p. 139-147 (2000). Zbl0971.35051MR1735230
  7. Brenner (P.), Thomée (V.), Wahlbin (L.B.).— Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Math. 434, Springer Verlag (1975). Zbl0294.35002MR461121
  8. Dahlberg (B.E.J.).— A note on Sobolev spaces, Proc. Symposia in Pure Math. XXV, p. 183-185, AMS (1979). Zbl0421.46027MR545257
  9. Gagliardo (E.).— Ulteriori Properta di alcune classi di funzioni in plui variabili, Ric. Mat., 8, 24-51 (1959). Zbl0199.44701MR109295
  10. Ginibre (J.), Velo (G.).— The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189, p. 487-505 (1985). Zbl0549.35108MR786279
  11. Heinz (E.), von Wahl (W.).— Zu einem Satz von F E Browder über nichtlinearen Wellengleichungen, Math. Z., 141, p. 33-45 (1975). Zbl0282.35068MR365257
  12. Hörmander (L.).— Lectures on nonlinear differential equations, Springer Verlag 1997. Zbl0881.35001MR1466700
  13. Kumlin (P.).— On mapping properties for some nonlinear operators related to hyperbolic problems, Göteborg 1985 (thesis) 
  14. Lebeau (G.).— Perte de régularité pour les équations des ondes sur-critiques, Bull. Soc. Math. France, 133, p. 145-157 , (2005). Zbl1071.35020MR2145023
  15. Lebeau (G.).— Nonlinear optics and supercritical wave equation, Bull. Soc. Math. Liege, 70, p. 267-306 (2001). Zbl1034.35137MR1813167
  16. Maligranda (L.).— Interpolation of locally Hölder operators, Studia Math., 78, p. 289-296 (1984). Zbl0566.46039MR782666
  17. Nirenberg (L.).— On elliptic partial differential equations, Ann. Scu. Norm. Sup. Pisa, 13:3, p. 115-162 (1959). Zbl0088.07601MR109940
  18. Peetre (J.).— Interpolation of Lipschitz operators and metric spaces, Mathematica (Cluj), 12:35, p. 325-334 (1970). Zbl0217.44504MR482280

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