Singular integral operator in spaces defined by a generalized oscillation

  • R. M. Rzaev Azerbaijan. holding ped. un-t, Baku
  • L. R. Aliyeva Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan, Baku
  • L. E. Huseinova Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan, Baku

Abstract

We study the behavior of a multidimensional singular integral operator in the function spaces de ned by the conditions imposed on generalized oscillation of a function.

Author Biography

R. M. Rzaev, Azerbaijan. holding ped. un-t, Baku



References

L. R. Aliyeva, Equivalent norms in spaces of mean oscillation, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. and Math. Sci., 31, № 4, 19 – 26 (2011).

N. K. Bari, S. B. Stechkin, Best approximation and differential properties of two conjugate functions (in Russian), Tr. Mosk. Mat. Obshch., 5, 483 – 552 (1956).

O. Blasco, M. A. Perez, On functions of integrable mean oscillation, Rev. Mat. Complut., 18, № 2, 465 – 477 (2005), https://doi.org/10.5209/rev_REMA.2005.v18.n2.16696 DOI: https://doi.org/10.5209/rev_REMA.2005.v18.n2.16696

S. Campanato, Proprieta di h¨olderianita di alcune classi di funzioni, Ann. Scuola Norm. Super. Pisa, 17, 175 – 188 (1963).

De R. Vore, R. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc., 47, № 293, 1 – 115 (1984), https://doi.org/10.1090/memo/0293 DOI: https://doi.org/10.1090/memo/0293

E. M. Dynʹkin, B. P. Osilenker, Weighted estimates for singular integrals and their applications (in Russian), Itogi Nauki i Tekh. Ser. Mat. Anal., 21, 42 – 129 (1983).

Ch. Fefferman, E. M. Stein, $Hsp{p}$ spaces of several variables, Acta Math., 129, № 3-4, 137 – 193 (1972), https://doi.org/10.1007/BF02392215 DOI: https://doi.org/10.1007/BF02392215

F. John, L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure and Appl. Math., 14, 415 – 426 (1961), https://doi.org/10.1002/cpa.3160140317 DOI: https://doi.org/10.1002/cpa.3160140317

G. N. Meyers, Mean oscillation over cubes and H¨older continuity, Proc. Amer. Math. Soc., 15, 717 – 721 (1964), https://doi.org/10.2307/2034586 DOI: https://doi.org/10.1090/S0002-9939-1964-0168712-3

J. Peetre, On the theory of ${cal L}sb{p},,sb{lambda }$ spaces, J. Funct. Anal., 4, 71 – 87 (1969), https://doi.org/10.1016/0022-1236(69)90022-6 DOI: https://doi.org/10.1016/0022-1236(69)90022-6

R. M. Rzaev, A multidimensional singular integral operator in spaces defined by conditions on the mean oscillation of functions, Sov. Math. Dokl., 42, № 2, 520 – 523 (1991).

R. M. Rzaev, On boundedness of multidimensional singular integral operator in spaces

${rm BMO}^k_{phi,theta}$ and $H^k_{phi,theta}$ (in Russian), Proc. Azerb. Math. Soc., 2, 164 – 175 (1996).

R. M. Rzaev, A multidimensional singular integral operator in spaces defined by conditions on the k-th order mean oscillation (in Russian), Dokl. Akad. Nauk, 356, № 5, 602 – 604 (1997).

R. M. Rzaev, On some maximal functions, measuring smoothness, and metric characteristics, Trans. Acad. Sci. Azerb., 19, № 5, 118 – 124 (1999).

R. M. Rzaev, Some growth conditions for locally summable functions, Abstracts Intern. Conf. Math. and Mech., Baku, 147 (2006).

R. M. Rzaev, L. R. Aliyeva, On local properties of functions and singular integrals in terms of the mean oscillation, Cent. Eur. J. Math., 6, № 4, 595 – 609 (2008), https://doi.org/10.2478/s11533-008-0046-4 DOI: https://doi.org/10.2478/s11533-008-0046-4

R. M. Rzaev, L. R. Aliyeva, Mean oscillation, $Phi$ -oscillation and harmonic oscillation, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. and Math. Sci., 30, № 1, 167 – 176 (2010).

R. M. Rzaev, Z. Sh. Gakhramanova, L. R. Alieva, On generalized Besov and Campanato spaces, Ukr. Math. J., 69, № 8, 1275 – 1286 (2018), https://doi.org/10.1007/s11253-017-1430-7 DOI: https://doi.org/10.1007/s11253-017-1430-7

S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Super. Pisa, 19, 593 – 608 (1965).

E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ (1970). DOI: https://doi.org/10.1515/9781400883882

Published
16.09.2021
How to Cite
Rzaev, R. M., L. R. Aliyeva, and L. E. Huseinova. “Singular Integral Operator in Spaces Defined by a Generalized Oscillation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 9, Sept. 2021, pp. 1231 -44, doi:10.37863/umzh.v73i9.2278.
Section
Research articles