A characterization of multipliers in the Hedberg--Netrusov spaces

A. I. Parfenov

Matematičeskie trudy, Tome 14 (2011) no. 1, pp. 158-194.

Résumé

We expose a criterion of the membership of a function in the space of pointwise multipliers between the Hedberg–Netrusov spaces. The criterion is stated in terms of a pair of discrete weighted inequalities. Connections of the criterion with the problem of rectifiability of Lipschitz surfaces are established.

Bibliographie

[1] Bak Dzh.-G., Shkalikov A. A., “Multiplikatory v dualnykh sobolevskikh prostranstvakh i operatory Shrëdingera s potentsialami-raspredeleniyami”, Matem. zametki, 71:5 (2002), 643–651 | MR | Zbl

[2] Berg I., Lëfstrëm I., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[3] Brudnyi Yu. A., “Prostranstva, opredelyaemye s pomoschyu lokalnykh priblizhenii”, Tr. Mosk. matem. obschestva, 24, 1971, 69–132 | MR | Zbl

[4] Brudnyi Yu. A., Ganzburg M. I., “Ob odnoi ekstremalnoi zadache dlya mnogochlenov $n$ peremennykh”, Izv. AN SSSR. Ser. matem., 37:2 (1973), 344–355 | MR | Zbl

[5] Golovkin K. K., Solonnikov V. A., “Ob otsenkakh operatorov svertki”, Zap. nauchn. sem. LOMI / Mat. in-t im. V. A. Steklova. Leningr. otd-nie, 7, Nauka, M.–L., 1968, 6–86 | MR | Zbl

[6] Gulisashvili A. B., “O multiplikatorakh v prostranstvakh Besova”, Zap. nauchn. sem.LOMI / Mat. in-t im. V. A. Steklova. Leningr. otd-nie, 135, Nauka, M.–L., 1984, 36–50 | MR | Zbl

[7] Kalyabin G. A., “Potochechnye multiplikatory v nekotorykh prostranstvakh Soboleva, soderzhaschikh neogranichennye funktsii”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam, Ch. 16, Tr. MIRAN, 204, Nauka, M., 1993, 160–165 | MR | Zbl

[8] Kokilashvili V. M., “Maksimalnye neravenstva i multiplikatory v vesovykh prostranstvakh Lizorkina–Tribelya”, Dokl. AN SSSR, 239:1 (1978), 42–45 | MR | Zbl

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

[10] Neiman-zade M. I., Shkalikov A. A., “Operatory Shrëdingera s singulyarnymi potentsialami iz prostranstv multiplikatorov”, Matem. zametki, 66:5 (1999), 723–733 | MR | Zbl

[11] Netrusov Yu. V., “Mnozhestva osobennostei funktsii iz prostranstv tipa Besova i Lizorkina–Tribelya”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam, Ch. 13, Tr. MIAN SSSR, 187, Nauka, M., 1989, 162–177 | MR | Zbl

[12] Netrusov Yu. V., “Teoremy o sledakh i multiplikatorakh dlya funktsii iz prostranstv Lizorkina–Tribelya”, Zap. nauchn. sem. POMI, 200, 1992, 132–138 | Zbl

[13] Parfenov A. I., “Kriterii raspryamlyaemosti lipshitsevoi poverkhnosti po Lizorkinu–Tribelyu. I”, Matem. tr., 12:1 (2009), 144–204 | MR

[14] Parfenov A. I., “Kriterii raspryamlyaemosti lipshitsevoi poverkhnosti po Lizorkinu–Tribelyu. II”, Matem. tr., 12:2 (2009), 139–159 | MR

[15] Parfenov A. I., “Kriterii raspryamlyaemosti lipshitsevoi poverkhnosti po Lizorkinu–Tribelyu. III”, Matem. tr., 13:2 (2010), 139–178 | MR

[16] Rychkov V. S., “Potochechnye multiplikatory v vesovykh prostranstvakh Soboleva na polupryamoi”, Matem. zametki, 56:1 (1994), 78–87 | MR | Zbl

[17] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR | Zbl

[18] Tribel Kh., Teoriya funktsionalnykh prostranstv, Mir, M., 1986 | MR | Zbl

[19] Andersen K. F., John R. T., “Weighted inequalities for vector-valued maximal functions and singular integrals”, Studia Math., 69:1 (1980), 19–31 | MR | Zbl

[20] Bownik M., “Duality and interpolation of anisotropic Triebel–Lizorkin spaces”, Math. Z., 259:1 (2008), 131–169 | DOI | MR | Zbl

[21] Brudnyi Y., “Sobolev spaces and their relatives: local polynomial approximation approach”, Sobolev Spaces in Mathematics, v. II, Int. Math. Ser. (N.Y.), 9, Springer, New York; Tamara Rozhkovskaya Publisher, Novosibirsk, 2009, 31–68 | MR | Zbl

[22] DeVore R. A., Popov V. A., “Interpolation of Besov spaces”, Trans. Amer. Math. Soc., 305:1 (1988), 397–414 | DOI | MR | Zbl

[23] Frazier M., Jawerth B., “A discrete transform and decompositions of distribution spaces”, J. Funct. Anal., 93:1 (1990), 34–170 | DOI | MR | Zbl

[24] Hedberg L. I., Netrusov Yu., An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation, Mem. Amer. Math. Soc., 188, no. 882, 2007 | MR

[25] Maz'ya V., Mitrea M., Shaposhnikova T., “The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients”, J. d'Analyse Math., 110:1 (2010), 167–239 | DOI | MR

[26] Maz'ya V. G., Shaposhnikova T. O., Theory of Sobolev Multipliers. With Applications to Differential and Integral Operators, Grundlehren der Mathematischen Wissenschaften, 337, Springer-Verlag, Berlin–Heidelberg, 2009 | MR

[27] Muckenhoupt B., “Weighted norm inequalities for the Hardy maximal function”, Trans. Amer. Math. Soc., 165 (1972), 207–226 | DOI | MR | Zbl

[28] Rychkov V. S., “Littlewood–Paley theory and function spaces with $A_p^\mathrm{loc}$ weights”, Math. Nachr., 224:1 (2001), 145–180 | 3.0.CO;2-2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[29] Triebel H., Theory of Function Spaces II, Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, etc., 1992 | MR | Zbl

[30] Triebel H., Theory of Function Spaces, Monographs in Mathematics III, 100, Birkhäuser Verlag, Basel, etc., 2006 | MR | Zbl

[31] Verbitsky I. E., “Imbedding and multiplier theorems for discrete Littlewood–Paley spaces”, Pacific J. Math., 176:2 (1996), 529–556 | MR | Zbl