Phillips-Type Theorems for Nikol'skii and Certain Other Function Spaces

S. S. Ajiev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 33-44.

Résumé

Analogues of the Phillips theorem are obtained for anisotropic Nikol'skii, Besov, and Lizorkin–Triebel spaces of functions defined on irregular domains (in particular, on open sets), as well as for the spaces, closely related to the above spaces, that are defined by local best approximations by polynomials in different metrics (including the spaces of DeVore and Sharpley and BMO).

Bibliographie

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

[2] Fichtenholtz G., Kantorovitch L., “Sur les opérations linéaires dans l'espace des fonctions bornées”, Stud. Math., 5 (1934), 69–98

[3] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 ; 2-е изд., 1996 | MR | Zbl

[4] Ilin V. P., “O nekotorykh svoistvakh funktsii iz prostranstva $W^l _{p,a,x}(G)$”, Zap. nauch. seminarov LOMI, 23, 1971, 33–40 | MR | Zbl

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

[8] DeVore R. A., Sharpley R. C., Maximal functions measuring smoothness, Mem. Amer. Math. Soc., 293, Amer. Math. Soc., Providence, RI, 1984 | MR

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

[10] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd. LGU, L., 1950; Наука, Новосибирск, 1962

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

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

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

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

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

[16] Phillips R. C., “On linear transformations”, Trans. Amer. Math. Soc., 48 (1940), 516–541 | DOI | MR | Zbl

[17] Rudin W., Functional analysis, 2nd ed., McGraw-Hill, N.Y., 1991 | MR | Zbl

[18] Adzhiev S. S., “Kharakterizatsii funktsionalnykh prostranstv $B_{p,q}^s(G)$, $L_{p,q}^s(G)$, $W_p^s(G)$ i nekotorykh drugikh. Prilozheniya”, Tr. MIAN, 227, 1999, 7–42 | MR | Zbl

[19] Adzhiev S. S., O nedopolnyaemykh podprostranstvakh i nekompaktnykh vlozheniyakh nekotorykh funktsionalnykh prostranstv, Dep. v VINITI. 02.06.00. No 1600-VOO, PME MFTI, M., 2000, 36 pp.