Zygmund-type estimates for fractional integration and differentiation operators of variable order

B. G. Vakulov ; E. S. Kochurov ; N. G. Samko

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2011), pp. 25-34.

Résumé

We consider non-standard generalized Hölder spaces of functions defined on a segment of the real axis, whose local continuity modulus has a majorant varying from point to point. We establish some properties of fractional integration operators of variable order acting from variable generalized Hölder spaces to those with a “better” majorant, as well as properties of fractional differentiation operators of variable order acting from the same spaces to those with a “worse” majorant.

Détail

Keywords: fractional integration operators, fractional differentiation operators, generalized continuity modulus, generalized Hölder spaces.

Bibliographie

[1] Guseinov A. I., Mukhtarov Kh. Sh., Vvedenie v teoriyu nelineinykh singulyarnykh uravnenii, Nauka, M., 1980 | MR

[2] Ginzburg A. I., Karapetyants N. K., “Drobnoe integrodifferentsirovanie v gëlderovskikh klassakh peremennogo poryadka”, Dokl. RAN, 339:4 (1994), 439–441 | Zbl

[3] Karapetyants N. K., Ginzburg A. I., “Fractional integrodifferentiation in Hölder classes of arbitrary order”, Georg. Math. J., 2:2 (1995), 141–150 | DOI | MR | Zbl

[4] Ross B., Samko S. G., “Fractional integration operator of variable order in the Hölder spaces $H^{\lambda(x)}$”, Intern. J. Math. Math. Sci., 18:4 (1995), 777–788 | DOI | MR | Zbl

[5] Samko S. G., “Differentiation and integration of variable order and the spaces $L^{p(x)}$”, Contemporary Math., 212 (1998), 203–219 | MR | Zbl

[6] Vakulov B. G., “Sfericheskie potentsaly v vesovykh prostranstvakh Gëldera peremennogo poryadka”, Dokl. RAN, 400:1 (2005), 7–10 | MR

[7] Vakulov B. G., “Sfericheskie operatory tipa potentsiala v vesovykh prostranstvakh Gëldera peremennogo poryadka”, Vladikavkazskii matem. zhurn., 7:2 (2005), 26–40 | MR

[8] Vakulov B. G., “Operatory sfericheskoi svertki so stepenno-logarifmicheskim yadrom v prostranstvakh obobschennoi peremennoi gëlderovosti”, Izv. vuzov. Severo-Kavkazskii region. Estestvennye nauki, 2006, no. 1, 7–10 | Zbl

[9] Vakulov B. G., “Operatory sfericheskoi svertki v prostranstvakh peremennoi gëlderovosti”, Matem. zametki, 80:5 (2006), 683–695 | MR | Zbl

[10] Vakulov B. G., “Spherical potentials of complex order in the variable Hölder spaces”, Integral Trans. Spec. Funct., 16:5–6 (2005), 489–497 | DOI | MR | Zbl

[11] Vakulov B. G., Samko N. G., Samko S. G., “Operatory tipa potentsiala i gipersingulyarnye integraly v prostranstvakh Gëldera peremennogo poryadka na odnorodnykh prostranstvakh”, Izv. vuzov. Severo-Kavkazskii region. Estestvennye nauki. Aktualnye probl. matem. gidrodinamiki, 2009, Spetsvypusk, 40–45

[12] Karapetyants N. K., Samko N. G., “Weighted theorems on fractional integrals in the generalized Hölder spaces $H_0^w(\varrho)$ via the indices $m_w$ and $M_w$”, Fract. Calc. Appl. Anal., 7:4 (2004), 437–458 | MR | Zbl

[13] Bari N. K., Stechkin S. B., “Nailuchshie priblizheniya i differentsialnye svoistva dvukh sopryazhennykh funktsii”, Tr. Mosk. matem. o-va, 5, 1956, 483–522 | MR | Zbl

[14] Samko S. G., Murdaev Kh. M., “Weighted Zygmund estimates for fractional differentiation and integration, and their applications”, Proc. of the Steklov Institute of Math., 3, 1989, 233–235

[15] Maligranda L., Orlicz spaces and interpolation, Departamento de Matemática. Universidade Estadual de Campinas, Campinas SP Brazil, 1989 | MR | Zbl

[16] Matuszewska W., Orlicz W., “On some classes of functions with regard to their orders of growth”, Studia Math., 26 (1965), 11–24 | MR | Zbl

[17] Samko N. G., “Singular integral operators in weighted spaces with generalized Hölder condition”, Proc. A. Razmadze Math. Inst. Tbilisi, 120, 1999, 107–134 | MR | Zbl

[18] Samko N. G., “Criterion of Fredholmness of singular operators with piece-wise continuous coefficients in the generalized Hölder spaces with weight”, Proc. of IWOTA' 2000 (Setembro 12–15, 2000, Faro, Portugal), Operator Theory: Advances and Applications, 142, Birkhauser, 2002, 363–376 | MR

[19] Samko N. G., “On compactness of integral operators with a generalized weak singularity in weighted spaces of continuous functions with a given continuity modulus”, Proc. A. Razmadze Math. Inst.. Tbilisi, 136, 2004, 91–113 | MR | Zbl

[20] Samko N. G., “On non-equilibrated almost monotonic functions of the Zygmund–Bari–Stechkin class”, Real Anal. Exch., 30:2 (2005), 727–745 | MR

[21] Samko N. G., “Singular integral operators in weighted spaces of continuous functions with an oscillating continuity modulus and oscillating weights”, Proc. of IWOTA (Newcastle, July 2004), Operator Theory: Advances and Applications, 171, Birkhauser, 2006, 323–347 | MR | Zbl

[22] Samko N. G., “Singular integral operators in weighted spaces of continuous functions with non-equilibrated continuity modulus”, Mathem. Nachrichten, 279:12 (2006), 1359–1375 | DOI | MR | Zbl

[23] Samko N. G., “Parameter depending Bari–Stechkin classes and local dimensions of measure metric spaces”, Proc. A. Razmadze Math. Inst. Tbilisi, 145, 2007, 122–129 | MR | Zbl

[24] Samko N. G., “Parameter depending almost monotonic functions and their applications to dimensions in metric measure spaces”, J. Function Spaces Appl., 7:1 (2009), 61–89 | MR | Zbl