Geodesic
A second look on definition and equivalent norms of Sobolev spaces
Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 315-328

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MR Zbl
Sobolev's original definition of his spaces $L^{m,p}(\Omega)$ is revisited. It only assumed that $\Omega\subseteq\Bbb R^n$ is a domain. With elementary methods, essentially based on Poincare's inequality for balls (or cubes), the existence of intermediate derivates of functions $u\in L^{m,p}(\Omega)$ with respect to appropriate norms, and equivalence of these norms is proved.
Sobolev's original definition of his spaces $L^{m,p}(\Omega)$ is revisited. It only assumed that $\Omega\subseteq\Bbb R^n$ is a domain. With elementary methods, essentially based on Poincare's inequality for balls (or cubes), the existence of intermediate derivates of functions $u\in L^{m,p}(\Omega)$ with respect to appropriate norms, and equivalence of these norms is proved.
DOI : 10.21136/MB.1999.126243
Classification : 46E35
Keywords: Sobolev spaces; Poincaré’s inequality; existence of intermediate derivates 
Naumann, J.; Simader, C. G. A second look on definition and equivalent norms of Sobolev spaces. Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 315-328. doi: 10.21136/MB.1999.126243
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[1] Adams R. A.: Sobolev Spaces. Academic Press, Inc, Boston, 1978, | Zbl

[2] Besov O. V., Il'in V. P., Nikol'skij S. M.: Integral Representations of Functions and Imbedding Theorems. Engl. Transl: V.H. Winston & Sons, Washington; J. Wiley & Sons, New York, vol. I: 1978, vol. II: 1979, Izd. Nauka, Moskva, 1975. (In Russian.) | MR

[3] Burenkov V. I.: Sobolev Spaces on Domains. B. G. Teubner, Stuttgart, 1998. | MR | Zbl

[4] Gilbarg D., Trudinger N. S.: Elliptic Partial Differential Equations of Second Order. (2nd ed.), Springer-Verlag, Berlin, 1983. | MR | Zbl

[5] Kufner A., John O., Fučík S.: Function Spaces. Academia, Prague, 1977. | MR

[б] Maz'ja V. G.: Sobolev Spaces. Springer-Verlag, Berlin, 1985. | MR | Zbl

[7] Maz'ja V. G., Poborchij S. V.: Differentiable Functions on Bad Domains. World Scientific, Singapore, 1997. | MR

[8] Nečas J.: Les méthodes directes en théorie des équations elliptiques. Academia, Praha, 1967. | MR

[9] Nikoľskij S. M.: Approximation of Functions of Several Variables and Imbedding Theorems. Engl. transl: Springer-Verlag, Berlin 1975, Izd. Nauka, Moskva, 1969 (In Russian.). | MR

[10] Sobolev S. L.: On some estimates related to families of functions having derivatives which are square integrable. Dokl. Akad. Nauk 1 (1936), 267-270. (In Russian.)

[11] Sobolev S. L.: On a theorem in functional analysis. Mat. Sborn. 4 (1938), 471-497 (In Russian.); Engl. transl. Amer. Math. Soc. Transl. II Ser. 34 (1963), 39-68.

[12] Sobolev S. L.: Some Applications of Functional Analysis in Mathematical Physics. 1st ed.: LGU Leningrad, 1950; 2nd ed.: NGU Novosibirsk, 1962; Зrd ed.; Izd. Nauka, Moskva 1988. (In Russian.) Engl. transl.: Amer. Math. Soc. Providence R.I, 1963; German transl: Akademie-Verlag Berlin 1964. | MR

[13] Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. (2nd ed.), J. A.Barth Verlag, Beidelberg, 1995. | MR | Zbl

[14] Ziemer W. P.: Weakly Differentiable Functions. Springer-Verlag, New York, 1989. | MR | Zbl

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