Superposition in homogeneous and vector valued Sobolev spaces
HTML articles powered by AMS MathViewer
- by Gérard Bourdaud PDF
- Trans. Amer. Math. Soc. 362 (2010), 6105-6130 Request permission
Abstract:
We give a sufficient condition on a function $f:\mathbb {R}^{k}\rightarrow \mathbb {R}$ so that it takes by superposition the homogeneous vector valued space $\dot {W}^{m}_{p}\cap \dot {W}^{1}_{mp}(\mathbb {R}^n, \mathbb {R}^k)$ into the corresponding real valued space, for integers $m,n,k$ such that $m\geq 2$, $k,n\geq 1$, and $p\in [1,+\infty [$. In case $k=1$, this condition also turns out to be necessary. For $k>1$, it is not proved to be necessary, but it is weaker than the conditions used till now, such as the continuity and boundedness of all derivatives up to order $m$.References
- David R. Adams, On the existence of capacitary strong type estimates in $R^{n}$, Ark. Mat. 14 (1976), no. 1, 125–140. MR 417774, DOI 10.1007/BF02385830
- David R. Adams and Michael Frazier, Composition operators on potential spaces, Proc. Amer. Math. Soc. 114 (1992), no. 1, 155–165. MR 1076570, DOI 10.1090/S0002-9939-1992-1076570-5
- Salah Eddine Allaoui, Remarques sur le calcul symbolique dans certains espaces de Besov à valeurs vectorielles, Ann. Math. Blaise Pascal 16 (2009), no. 2, 399–429 (French, with English and French summaries). MR 2568872
- A. Benedek and R. Panzone, The space $L^{p}$, with mixed norm, Duke Math. J. 28 (1961), 301–324. MR 126155
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- G. Bourdaud, Réalisations des espaces de Besov homogènes, Ark. Mat. 26 (1988), no. 1, 41–54 (French). MR 948279, DOI 10.1007/BF02386107
- Gérard Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. Math. 104 (1991), no. 2, 435–446 (French). MR 1098617, DOI 10.1007/BF01245083
- Gérard Bourdaud, Fonctions qui opèrent sur les espaces de Besov et de Triebel, Ann. Inst. H. Poincaré C Anal. Non Linéaire 10 (1993), no. 4, 413–422 (French, with English and French summaries). MR 1246460, DOI 10.1016/S0294-1449(16)30209-8
- Gérard Bourdaud, Superposition operators in Zygmund and BMO spaces, Function spaces, differential operators and nonlinear analysis (Teistungen, 2001) Birkhäuser, Basel, 2003, pp. 59–74. MR 1984163
- Gérard Bourdaud, Madani Moussai, and Winfried Sickel, An optimal symbolic calculus on Besov algebras, Ann. Inst. H. Poincaré C Anal. Non Linéaire 23 (2006), no. 6, 949–956 (English, with English and French summaries). MR 2271703, DOI 10.1016/j.anihpc.2006.06.001
- Gérard Bourdaud, Madani Moussai, and Winfried Sickel, Towards sharp superposition theorems in Besov and Lizorkin-Triebel spaces, Nonlinear Anal. 68 (2008), no. 10, 2889–2912. MR 2404807, DOI 10.1016/j.na.2007.02.035
- Haïm Brezis and Petru Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. 1 (2001), no. 4, 387–404. Dedicated to the memory of Tosio Kato. MR 1877265, DOI 10.1007/PL00001378
- Björn E. J. Dahlberg, A note on Sobolev spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 183–185. MR 545257
- Satoru Igari, Sur les fonctions qui opèrent sur l’espace $\hat A^{2}$, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 2, 525–536 (French). MR 188716
- Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition, Dover Publications, Inc., New York, 1976. MR 0422992
- Moshe Marcus and Victor J. Mizel, Complete characterization of functions which act, via superposition, on Sobolev spaces, Trans. Amer. Math. Soc. 251 (1979), 187–218. MR 531975, DOI 10.1090/S0002-9947-1979-0531975-1
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- V. Maz′ya and T. Shaposhnikova, An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces, J. Evol. Equ. 2 (2002), no. 1, 113–125. MR 1890884, DOI 10.1007/s00028-002-8082-1
- L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
- Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319, DOI 10.1515/9783110812411
Additional Information
- Gérard Bourdaud
- Affiliation: Institut de Mathématiques de Jussieu, Équipe d’Analyse Fonctionnelle, Université Paris Diderot, 175 rue du Chevaleret, 75013 Paris, France
- Email: bourdaud@math.jussieu.fr
- Received by editor(s): August 26, 2008
- Received by editor(s) in revised form: June 25, 2009
- Published electronically: June 16, 2010
- © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 6105-6130
- MSC (2000): Primary 46E35, 47H30
- DOI: https://doi.org/10.1090/S0002-9947-2010-05150-5
- MathSciNet review: 2661510