We study the behavior of a multidimensional singular integral operator in function spaces defined by the conditions imposed on the generalized oscillation of a function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. R. Aliyeva, “Equivalent norms in spaces of mean oscillation,” Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 31, No. 4, 19–26 (2011).
N. K. Bari and S. B. Stechkin, “Best approximation and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obshch., 5, 483–552 (1956).
O. Blasco and M. A. Perez, “On functions of integrable mean oscillation,” Rev. Mat. Complut., 18, No. 2, 465–477 (2005).
S. Campanato, “Proprieta di hölderianita di alcune classi di funzioni,” Ann. Scuola Norm. Super. Pisa, 17, 175–188 (1963).
De R. Vore and R. Sharpley, “Maximal functions measuring smoothness,” Mem. Amer. Math. Soc., 47, No. 293, 1–115 (1984).
E. M. Dyn’kin and B. P. Osilenker, “Weighted estimates for singular integrals and their applications,” in: Itogi VINITI, Ser. Mat. Anal., 21 (1983), pp. 42–129.
Ch. Fefferman and E. M. Stein, “Hp spaces of several variables,” Acta Math., 129, No. 3-4, 137–193 (1972).
F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Comm. Pure Appl. Math., 14, 415–426 (1961).
G. N. Meyers, “Mean oscillation over cubes and Hölder continuity,” Proc. Amer. Math. Soc., 15, 717–721 (1964).
J. Peetre, “On the theory of Lp,λ spaces,” J. Funct. Anal., 4, 71–87 (1969).
R. M. Rzaev, “A multidimensional singular integral operator in spaces defined by conditions on the mean oscillation of functions,” Sov. Math. Dokl., 42, No. 2, 520–523 (1991).
R. M. Rzaev, “On boundedness of multidimensional singular integral operator in spaces \( {BMO}_{\varphi, \theta}^k \) and \( {H}_{\varphi, \theta}^k \),” Proc. Azerb. Math. Soc., 2, 164–175 (1996).
R. M. Rzaev, “A multidimensional singular integral operator in spaces defined by conditions on the kth order mean oscillation,” Dokl. Akad. Nauk, 356, No. 5, 602–604 (1997).
R. M. Rzaev, “On some maximal functions, measuring smoothness, and metric characteristics,” Trans. Acad. Sci. Azerb., 19, No. 5, 118–124 (1999).
R. M. Rzaev, “Some growth conditions for locally summable functions,” in: Abstr. Intern. Conf. Math. Mech., Baku, 147 (2006).
R. M. Rzaev and L. R. Aliyeva, “On local properties of functions and singular integrals in terms of the mean oscillation,” Centr. Europ. J. Math., 6, No. 4, 595–609 (2008).
R. M. Rzaev and L. R. Aliyeva, “Mean oscillation, Φ-oscillation, and harmonic oscillation,” Trans. Natl. Acad. Sci. Azerb., Ser. Phys.-Tech. Math. Sci., 30, No. 1, 167–176 (2010).
R. M. Rzaev, Z. Sh. Gakhramanova, and L. R. Alieva, “On generalized Besov and Campanato spaces,” Ukr. Mat. Zh., 69, No. 8, 1096–1106 (2017); English translation: Ukr. Math. J., 69, No. 8, 1275–1286 (2018).
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 (1970).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 9, pp. 1231–1244, September, 2021. Ukrainian DOI: 10.37863/umzh.v73i9.2278.
Rights and permissions
About this article
Cite this article
Rzaev, R.M., Aliyeva, L.R. & Huseinova, L.E. Singular Integral Operator in Spaces Defined by a Generalized Oscillation. Ukr Math J 73, 1428–1444 (2022). https://doi.org/10.1007/s11253-022-02003-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-022-02003-7