Characterizations of $B^s_{p,q}(G),L^s_{p,q}(G),W^s_p(G)$ and certain other function spaces. Applications

S. S. Ajiev

Informatics and Automation, Investigations in the theory of differentiable functions of many variables and its applications. Part 18, Tome 227 (1999), pp. 7-42.

Résumé

The anisotropic Sobolev, Nikol'skii, Besov, and Lizorkin–Triebel spaces are considered on irregular domains (in particular, on open sets) as well as closely related spaces defined by the best local polynomial approximation in different metrics. The questions of the equivalence of norms, dependence of the space structure on the defining parameters, boundedness of pointwise multipliers, description of maximal subalgebras, and also the questions concerning the reflexivity, translation continuity, and comleteness of these spaces are studied.

Bibliographie

[1] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[2] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, 2-e izd., Nauka. Fizmatlit, M., 1996, 480 pp. | MR

[3] Ilin V. P., “Funktsionalnye prostranstva $\mathcal L^{\lambda,a;b_s}_{r,p,\theta}(G)$”, Zap. nauch. seminarov LOMI, 30, 1972, 51–75 | MR | Zbl

[4] Besov O. V., “O prostranstvakh Soboleva–Liuvillya i Lizorkina–Tribelya na oblasti”, Tr. MIAN, 192, 1990, 20–34 | MR

[5] Triebel H., “Local approximation spaces”, Ztschr. Anal. und Anwend., 8:3 (1989), 261–288 | MR | Zbl

[6] Brudnyi Yu. A., “Prostranstva, opredelyaemye s pomoschyu lokalnykh priblizhenii”, Tr. Mosk. mat. o-va, 24, 1971, 70–132 | MR

[7] Jonsson A., Wallin H., Function spaces on subset of $\mathbb R^n$, Pt 1, Math. Repts, 2, Acad. publ., Harwood, 1984, 222 pp. | MR

[8] Calderón A. P., “Estimates for singular integral operators in terms of maximal functions”, Stud. math., 44 (1972), 563–582 | MR | Zbl

[9] De Vore R. A., Sharpley R. C., Maximal functions measuring smoothness, Mem. Amer. Math. Soc., 293, Amer. Math. Soc., Providence, R.I., 1984 | MR

[10] Seeger A., “A note on Triebel–Lizorkin spaces”, Approximation and function spaces (Warsaw, 1986), Banach Center Publ., 22, PWN, Warsaw, 1989, 391–400 | MR

[11] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd. LGU, L., 1950; Соболев С. Л., Некоторые применения функционального анализа в математической физике, Наука, Новосибирск, 1962 | MR

[12] Ilin V. P., “Integralnye predstavleniya differentsiruemykh funktsii i ikh primenenie k voprosam prodolzheniya funktsii klassov $W_p^l(G)$”, Sib. mat. zhurn., 8:3 (1967), 573–586

[13] Besov O. V., “Integralnye predstavleniya funktsii i teoremy vlozheniya dlya oblasti s usloviem gibkogo roga”, Tr. MIAN, 170, 1984, 12–30 | MR | Zbl

[14] Bojarski B., “Sharp maximal operator of fractional order and Sobolev imbedding inequalities”, Bull. Pol. Acad. Sci. Ser. Math., 33 (1985), 7–16 | MR | Zbl

[15] Kokilashvili V. M., “Maksimalnye funktsii v vesovykh prostranstvakh”, Tr. Tbil. mat. in-ta, 65, 1980, 110–121 | MR | Zbl

[16] Kalyabin G. A., “Teoremy o prodolzhenii, multiplikatorakh i diffeomorfizmakh dlya obobschennykh klassov Soboleva–Liuvillya v oblastyakh s lipshitsevoi granitsei”, Tr. MIAN, 172, 1985, 173–186 | MR | Zbl

[17] Triebel H., “Approximation number and entropy numbers of imbedding of fractional Besov–Sobolev spaces in Orlich spaces”, Proc. London Math. Soc., ser. 3, 66 (1993), 589–618 | DOI | MR | Zbl

[18] Sickel W., Triebel H., “Hölder inequalities and sharp imbeddings in function spaces of $B_{p,q}^s$ and $L_{p,q}^s$ type”, Ztschr. Anal. und Anwend., 14 (1995), 105–140 | MR | Zbl

[19] Sickel W., “On pointwise multipliers for $F_{p,q}^s(\mathbb R^n)$ in case $\sigma _{p,q}/p$”, Ann. Mat. Pura ed Appl., 176 (1999), 209–250 | DOI | MR | Zbl

[20] Franke J., “On the spaces $F_{p,q}^s$ of Triebel–Lizorkin type. Pointwise multipliers and spaces on domaines”, Math. Nachr., 125 (1986), 29–68 | MR | Zbl

[21] Spanne S., “Some function space, defined using the mean oscillation over cubes”, Ann. Scuola Norm. Super. Pisa., ser. 3, 19:4 (1965), 593–608 | MR | Zbl

[22] Calderón A. P., Scott R., “Sobolev type inequalities for $p>0$”, Stud. math., 62 (1978), 75–92 | MR | Zbl

[23] Campanato S., “Propriétá di inclusione per spazi di Morrey”, Ric. Mat., 12 (1963), 67–86 | MR | Zbl

[24] Campanato S., “Propriétá di una famiglia di spazi funzionali”, Ann. Scuola Norm. Super. Pisa, 18 (1964), 137–160 | MR | Zbl

[25] Peetre J., “On the theory of $L_{p,\lambda}$ spaces”, J. Funct. Anal., 4 (1969), 71–87 | DOI | MR | Zbl

[26] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR

[27] Mazya V. G., Shaposhnikova T. O., Multiplikatory v prostranstvakh differentsiruemykh funktsii, Izd-vo LGU, L., 1986, 404 s. | MR

[28] Triebel H., Theory of function spaces, Acad. Verlagsges., Geest Portig, Leipzig, 1983 ; Triebel H., Theory of function spaces, Birkhäuser, Boston, 1983 ; Tribel Kh., Teoriya funktsionalnykh prostranstv, Mir, M., 1986 | MR | Zbl | MR | Zbl

[29] Besov O. V., “Interpolyatsiya prostranstv differentsiruemykh funktsii na oblasti”, Tr. MIAN, 214, 1997, 59–82 | Zbl

[30] Dorronsoro J. R., “A characterization of potential spaces”, Proc. Amer. Math. Soc., 95 (1985), 21–31 | DOI | MR | Zbl

[31] Adzhiev S. S., “Kharakterizatsiya funktsionalnykh prostranstv $B_{p,q}^s(G)$, $L_{p,q}^s(G)$, $W_p^s(G)$ i vlozheniya v $BMO(G)$”, Tr. MIAN, 214, 1997, 7–24 | MR | Zbl