An elementary proof of sharp Sobolev embeddings

Malý, Jan; Pick, Luboš

doi:10.1090/S0002-9939-01-06060-9

An elementary proof of sharp Sobolev embeddings
HTML articles powered by AMS MathViewer

by Jan Malý and Luboš Pick PDF

Proc. Amer. Math. Soc. 130 (2002), 555-563 Request permission

Abstract:

We present an elementary unified and self-contained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Brézis and Wainger.

References

D. Bakry, T. Coulhon, M. Ledoux, and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), no. 4, 1033–1074. MR 1386760, DOI 10.1512/iumj.1995.44.2019
Colin Bennett, Ronald A. DeVore, and Robert Sharpley, Weak-$L^{\infty }$ and BMO, Ann. of Math. (2) 113 (1981), no. 3, 601–611. MR 621018, DOI 10.2307/2006999
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
Haïm Brézis and Stephen Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), no. 7, 773–789. MR 579997, DOI 10.1080/03605308008820154
Andrea Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J. 45 (1996), no. 1, 39–65. MR 1406683, DOI 10.1512/iumj.1996.45.1958
D. E. Edmunds, R. Kerman, and L. Pick, Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal. 170 (2000), no. 2, 307–355. MR 1740655, DOI 10.1006/jfan.1999.3508
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Kurt Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), no. 1, 77–102. MR 567435, DOI 10.7146/math.scand.a-11827
J. A. Hempel, G. R. Morris, and N. S. Trudinger, On the sharpness of a limiting case of the Sobolev imbedding theorem, Bull. Austral. Math. Soc. 3 (1970), 369–373. MR 280998, DOI 10.1017/S0004972700046074
Jan Malý and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI, 1997. MR 1461542, DOI 10.1090/surv/051
V. G. Maz’ya, A theorem on the multidimensional Schrödinger operator, (Russian), Izv. Akad. Nauk. 28 1964, 1145-1172.
V. G. Maz’ya, Sobolev Spaces, Springer, Berlin, 1975.
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
Jaak Peetre, Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 279–317 (French). MR 221282
S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR 165 (1965), 36–39 (Russian). MR 0192184
S. L. Sobolev, Applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by F. E. Browder. MR 0165337
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Luc Tartar, Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 3, 479–500 (English, with Italian summary). MR 1662313
Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. MR 0216286, DOI 10.1512/iumj.1968.17.17028
V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961), 805–808 (Russian). MR 0140822