Geodesic
Hausdorff operators of special kind in $BMO$-type spaces and H\"older--Lipschitz spaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2020), pp. 8-21.

Voir la notice de l'article provenant de la source Math-Net.Ru

We show that some $BMO$-type spaces are invariant with respect to the Hausdorff operators of special kind (weighted Hardy-Cesaro operators). Also we obtain a sufficient and necessary conditions for such operators to be bounded in spaces of functions of generalized bounded variation. Finally, we study the invariance of Hölder–Lipschitz spaces with respect to these operators.
Keywords: Hausdorff operator, Hölder–Lipschitz spaces, $VMO(\mathbb R^n)$, $bmo(\mathbb R^n)$, functions of generalized bounded variation. 
@article{IVM_2020_12_a1,
     author = {S. S. Volosivets},
     title = {Hausdorff operators of special kind in $BMO$-type spaces and {H\"older--Lipschitz} spaces},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {8--21},
     publisher = {mathdoc},
     number = {12},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2020_12_a1/}
}
TY  - JOUR
AU  - S. S. Volosivets
TI  - Hausdorff operators of special kind in $BMO$-type spaces and H\"older--Lipschitz spaces
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2020
SP  - 8
EP  - 21
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2020_12_a1/
LA  - ru
ID  - IVM_2020_12_a1
ER  - 
%0 Journal Article
%A S. S. Volosivets
%T Hausdorff operators of special kind in $BMO$-type spaces and H\"older--Lipschitz spaces
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2020
%P 8-21
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2020_12_a1/
%G ru
%F IVM_2020_12_a1
S. S. Volosivets. Hausdorff operators of special kind in $BMO$-type spaces and H\"older--Lipschitz spaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2020), pp. 8-21. http://geodesic.mathdoc.fr/item/IVM_2020_12_a1/

[1] Liflyand E., Móricz F., “The Hausdorff operator is bounded on the real Hardy space $H^1(\mathbb R)$”, Proc. Amer. Math. Soc., 128:5 (2000), 1391–1396 | DOI | MR | Zbl

[2] Liflyand E., “Boundedness of multidimensional Hausdorff operators on $H^1(\mathbb R^n)$”, Acta Sci. Math. (Szeged), 74:3–4 (2008), 845–851 | MR | Zbl

[3] Andersen K. F., “Boundedness of Hausdorff operators on $L^p(\mathbb R^n)$, $H^1(\mathbb R^n)$, and $BMO(\mathbb R^n)$”, Acta Sci. Math. (Szeged), 69:1–2 (2003), 409–418 | MR | Zbl

[4] Giang D. V., Móricz F., “The Cesàro operator is bounded on the Hardy space”, Acta Sci. Math. (Szeged), 61:1–4 (1995), 535–544 | MR | Zbl

[5] Golubov B. I., “Ob ogranichennosti operatorov Khardi i Khardi-Littlvuda v prostranstvakh $Re H^1$ i $BMO$”, Matem. sb., 188:7 (1997), 93–106 | MR | Zbl

[6] Korenovskii A., Mean oscillations and equimeasurable rearrangements of functions, Springer, Berlin, 2007 | MR | Zbl

[7] Móricz F., “The harmonic Cesàro and Copson operators on the spaces $L^p$, $1\leq p\leq \infty$, $H^1$ and $BMO$”, Acta Sci Math (Szeged), 65:1–2 (1999), 293–310 | MR | Zbl

[8] Xiao J., “$L^p$ and $BMO$ bounds of weighted Hardy-Littlewood averages”, J. Math. Anal. Appl., 262 (2001), 660–666 | DOI | MR | Zbl

[9] Khardi G. G., Littlvud Dzh. E., Polia G., Neravenstva, 2-e izd., Inlit., M., 1948

[10] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[11] Fefferman C., Stein E. M., “$H^p$ spaces of several variables”, Acta Math., 129 (1972), 137–193 | DOI | MR | Zbl

[12] Goldberg D. A., “Local version of real Hardy spaces”, Duke Math. J., 46:1 (1979), 27–42 | DOI | MR | Zbl

[13] John F., Nirenberg L., “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426 | DOI | MR | Zbl

[14] Fan D., Lin X., “Hausdorff operator on real Hardy spaces”, Anal. (Berlin), 34:4 (2014), 319–337 | MR | Zbl

[15] Hung H. D., Ky L.D., Quang T. T., “Norm of the Hausdorff operator on the real Hardy space $H^1(\mathbb R)$”, Complex Anal. Oper. Theory, 12:2 (2018), 235–245 | DOI | MR | Zbl

[16] Sarason D., “Functions of vanishing mean oscillation”, Trans. Amer. Math. Soc., 207:2 (1975), 391–405 | DOI | MR | Zbl

[17] Stein E. M., Harmonic analysis: real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton, 1993 | MR | Zbl

[18] Dafni G., “Local $VMO$ and weak convergence on $h^1$”, Canad. Math. Bull., 45:1 (2002), 46–59 | DOI | MR | Zbl

[19] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[20] Volosivets S. S., “On Hardy and Hardy-Littlewood transforms in classes of functions with given majorant of modulus of continuity”, Acta Sci. Math. (Szeged), 75:1–2 (2009), 265–274 | MR | Zbl

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

[22] Waterman D., “On the summability of Fourier series of functions of $\Lambda$-bounded variation”, Stud. Math., 55:1 (1976), 87–95 | DOI | MR | Zbl

[23] Wiener N., “The quadratic variation of a function and its Fourier coefficients”, J. Math. and Phys., 3 (1924), 72–94 | DOI | Zbl

[24] Young L. C., “An inequality of Hölder type connected with Stieltjes integration”, Acta Math., 67 (1936), 251–282 | DOI | MR

[25] Terekhin A. P., “Priblizhenie funktsii ogranichennoi $p$-variatsii”, Izv. vuzov. Matem., 1965, no. 2, 171–187

[26] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR

[27] Krayukhin S. A., Volosivets S. S., “Functions of bounded $p$-variation and weighted integrability of Fourier transforms”, Acta Math. Hung., 159:2 (2019), 374–399 | DOI | MR | Zbl

[28] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984

[29] Ruan J., Fan D., “Hausdorff operators on the power weighted Hardy spaces”, J. Math. Anal. Appl., 433:1 (2016), 31–48 | DOI | MR | Zbl

[30] Volosivets S. S., “Convergence of series of Fourier coefficients of $p$-absolutely continuous functions”, Anal. Math., 26:1 (2000), 63–80 | DOI | MR | Zbl