Notes on limits of Sobolev spaces and the continuity of interpolation scales
HTML articles powered by AMS MathViewer
- by Mario Milman PDF
- Trans. Amer. Math. Soc. 357 (2005), 3425-3442 Request permission
Abstract:
We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz’ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Jesús Bastero, Mario Milman, and Francisco J. Ruiz, On sharp reiteration theorems and weighted norm inequalities, Studia Math. 142 (2000), no. 1, 7–24. MR 1792286, DOI 10.4064/sm-142-1-7-24
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations (J. L. Menaldi, E. Rofman and A. Sulem, eds.) a volume in honour of A. Bensoussan’s 60$^{th}$ birthday, IOS Press, 2001, pp. 439-455.
- Kh. Brezis, How to recognize constant functions. A connection with Sobolev spaces, Uspekhi Mat. Nauk 57 (2002), no. 4(346), 59–74 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 4, 693–708. MR 1942116, DOI 10.1070/RM2002v057n04ABEH000533
- Yu. A. Brudnyĭ and N. Ya. Krugljak, Interpolation functors and interpolation spaces. Vol. I, North-Holland Mathematical Library, vol. 47, North-Holland Publishing Co., Amsterdam, 1991. Translated from the Russian by Natalie Wadhwa; With a preface by Jaak Peetre. MR 1107298
- A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
- Michael Cwikel, Monotonicity properties of interpolation spaces, Ark. Mat. 14 (1976), no. 2, 213–236. MR 442714, DOI 10.1007/BF02385836
- Michael Cwikel, Complex interpolation spaces, a discrete definition and reiteration, Indiana Univ. Math. J. 27 (1978), no. 6, 1005–1009. MR 511254, DOI 10.1512/iumj.1978.27.27068
- Michael Cwikel and Svante Janson, Real and complex interpolation methods for finite and infinite families of Banach spaces, Adv. in Math. 66 (1987), no. 3, 234–290. MR 915856, DOI 10.1016/0001-8708(87)90036-3
- Michael Cwikel, Nigel Kalton, Mario Milman, and Richard Rochberg, A unified theory of commutator estimates for a class of interpolation methods, Adv. Math. 169 (2002), no. 2, 241–312. MR 1926224, DOI 10.1006/aima.2001.2061
- M. Cwikel, M. Milman, and Y. Sagher, Complex interpolation of some quasi-Banach spaces, J. Funct. Anal. 65 (1986), no. 3, 339–347. MR 826431, DOI 10.1016/0022-1236(86)90023-6
- Michael Cwikel, Per G. Nilsson, and Gideon Schechtman, Interpolation of weighted Banach lattices. A characterization of relatively decomposable Banach lattices, Mem. Amer. Math. Soc. 165 (2003), no. 787, vi+127. MR 1996919, DOI 10.1090/memo/0787
- Michael Cwikel and Amal Sharif, Complex interpolation spaces generated by the Gagliardo completion of an arbitrary Banach couple, Interpolation spaces and related topics (Haifa, 1990) Israel Math. Conf. Proc., vol. 5, Bar-Ilan Univ., Ramat Gan, 1992, pp. 57–59. MR 1206491
- Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
- M. E. Gomez and M. Milman, Extrapolation spaces and almost-everywhere convergence of singular integrals, J. London Math. Soc. (2) 34 (1986), no. 2, 305–316. MR 856514, DOI 10.1112/jlms/s2-34.2.305
- Svante Janson, Minimal and maximal methods of interpolation, J. Functional Analysis 44 (1981), no. 1, 50–73. MR 638294, DOI 10.1016/0022-1236(81)90004-5
- Björn Jawerth and Mario Milman, Extrapolation theory with applications, Mem. Amer. Math. Soc. 89 (1991), no. 440, iv+82. MR 1046185, DOI 10.1090/memo/0440
- H. Johnen and K. Scherer, On the equivalence of the $K$-functional and moduli of continuity and some applications, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977, pp. 119–140. MR 0487423
- Nigel Kalton and Marius Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc. 350 (1998), no. 10, 3903–3922. MR 1443193, DOI 10.1090/S0002-9947-98-02008-X
- Nigel J. Kalton and Mikhail I. Ostrovskii, Distances between Banach spaces, Forum Math. 11 (1999), no. 1, 17–48. MR 1673915, DOI 10.1515/form.11.1.17
- G. Karadzhov and M. Milman, Extrapolation theory: new results and applications, J. Approx. Theory 133 (2005), 38-99.
- V. I. Kolyada and A. Lerner, On limiting embeddings of Besov spaces, preprint.
- S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
- Natan Krugljak and Mario Milman, A distance between orbits that controls commutator estimates and invertibility of operators, Adv. Math. 182 (2004), no. 1, 78–123. MR 2028497, DOI 10.1016/S0001-8708(03)00074-4
- V. Maz′ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002), no. 2, 230–238. MR 1940355, DOI 10.1006/jfan.2002.3955
- Jaak Peetre, Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 279–317 (French). MR 221282
- Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
- Jaak Peetre, A counterexample connected with Gagliardo’s trace theorem, Comment. Math. Special Issue 2 (1979), 277–282. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. MR 552011
- Richard Rochberg, Function theoretic results for complex interpolation families of Banach spaces, Trans. Amer. Math. Soc. 284 (1984), no. 2, 745–758. MR 743742, DOI 10.1090/S0002-9947-1984-0743742-6
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580
Additional Information
- Mario Milman
- Affiliation: Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431
- Email: extrapol@bellsouth.net
- Received by editor(s): July 8, 2003
- Published electronically: April 27, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 3425-3442
- MSC (2000): Primary 46E30, 46M35, 26D10
- DOI: https://doi.org/10.1090/S0002-9947-05-03937-1
- MathSciNet review: 2146631