Nonlinear Maps between Besov and Sobolev spaces
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 1, pp. 105-120.

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].

Notre résultat principal est que pour une grande famille d’applications non linéaires entre espaces de Besov et de Sobolev, l’interpolation est un phénomène propre aux petites dimensions. Ceci prolonge des résultats obtenus précédemment par Kumlin [13] pour des applications analytiques au cas d’applications Hölder continues ou encore Lipschitz (Corollaires 1 and 2), et qui remontent aux idées de B.E.J. Dahlberg [8].

DOI: 10.5802/afst.1238
Brenner, Philip 1; Kumlin, Peter 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
@article{AFST_2010_6_19_1_105_0,
     author = {Brenner, Philip and Kumlin, Peter},
     title = {Nonlinear {Maps} between {Besov} and {Sobolev} spaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {105--120},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
     number = {1},
     year = {2010},
     doi = {10.5802/afst.1238},
     mrnumber = {2597783},
     zbl = {1195.46018},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1238/}
}
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
PY  - 2010
DA  - 2010///
SP  - 105
EP  - 120
VL  - Ser. 6, 19
IS  - 1
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://archive.numdam.org/articles/10.5802/afst.1238/
UR  - https://www.ams.org/mathscinet-getitem?mr=2597783
UR  - https://zbmath.org/?q=an%3A1195.46018
UR  - https://doi.org/10.5802/afst.1238
DO  - 10.5802/afst.1238
LA  - en
ID  - AFST_2010_6_19_1_105_0
ER  - 
%0 Journal Article
%A Brenner, Philip
%A Kumlin, Peter
%T Nonlinear Maps between Besov and Sobolev spaces
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2010
%P 105-120
%V Ser. 6, 19
%N 1
%I Université Paul Sabatier, Institut de mathématiques
%C Toulouse
%U https://doi.org/10.5802/afst.1238
%R 10.5802/afst.1238
%G en
%F AFST_2010_6_19_1_105_0
Brenner, Philip; Kumlin, Peter. Nonlinear Maps between Besov and Sobolev spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 1, pp. 105-120. doi : 10.5802/afst.1238. http://archive.numdam.org/articles/10.5802/afst.1238/

[1] Bahouri (H.), Gerard (P.).— High frequency approximation of solutions to critical nonlinear equations, Amer. J. Math. 121, p. 131-175 (1999). | MR | Zbl

[2] Bennet (C.), Sharpley (R.).— Interpolation of operators, Academic Press 1988. | MR | Zbl

[3] Bergh (J.), Löfström (J.).— Interpolation spaces, Springer Verlag (1976). | Zbl

[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 | MR | Zbl

[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). | MR | Zbl

[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). | MR | Zbl

[8] Dahlberg (B.E.J.).— A note on Sobolev spaces, Proc. Symposia in Pure Math. XXV, p. 183-185, AMS (1979). | MR | Zbl

[9] Gagliardo (E.).— Ulteriori Properta di alcune classi di funzioni in plui variabili, Ric. Mat., 8, 24-51 (1959). | MR | Zbl

[10] Ginibre (J.), Velo (G.).— The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189, p. 487-505 (1985). | MR | Zbl

[11] Heinz (E.), von Wahl (W.).— Zu einem Satz von F E Browder über nichtlinearen Wellengleichungen, Math. Z., 141, p. 33-45 (1975). | MR | Zbl

[12] Hörmander (L.).— Lectures on nonlinear differential equations, Springer Verlag 1997. | MR

[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). | Numdam | MR | Zbl

[15] Lebeau (G.).— Nonlinear optics and supercritical wave equation, Bull. Soc. Math. Liege, 70, p. 267-306 (2001). | MR | Zbl

[16] Maligranda (L.).— Interpolation of locally Hölder operators, Studia Math., 78, p. 289-296 (1984). | MR | Zbl

[17] Nirenberg (L.).— On elliptic partial differential equations, Ann. Scu. Norm. Sup. Pisa, 13:3, p. 115-162 (1959). | Numdam | MR | Zbl

[18] Peetre (J.).— Interpolation of Lipschitz operators and metric spaces, Mathematica (Cluj), 12:35, p. 325-334 (1970). | MR | Zbl

Cited by Sources: