Operators of Almost Hadamard-Type and the Hardy--Littlewood Operator in the Space of Entire Functions of Several Complex Variables

O. A. Ivanova ; S. N. Melikhov

Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 52-64.

Résumé

We introduce the class of operators of almost Hadamard type, that is, linear continuous operators that act on a locally convex space containing all polynomials and have the property that the homogeneous polynomials of any given degree form an invariant subspace. The Hadamard type (diagonal) operators, for which each monomial is an eigenvector, are a special case of operators of almost Hadamard type. The operators of almost Hadamard type are studied in the space of all entire functions of several complex variables. The results are used to describe all linear operators continuous in this space and commuting in it with the multidimensional analog of the Hardy–Littlewood operator.

Détail

Keywords: operator of Hadamard type, Hardy–Littlewood operator, space of entire functions.

Bibliographie

[1] S. S. Linchuk, “Diagonalnye operatory v prostranstvakh analiticheskikh funktsii i ikh prilozheniya”, Aktualnye voprosy teorii funktsii, Izd-vo RGU, Rostov-na-Donu, 1987, 118–121

[2] A. V. Bratischev, “O lineinykh operatorakh, simvol kotorykh yavlyaetsya funktsiei proizvedeniya svoikh argumentov”, Dokl. AN, 365:1 (1999), 9–12

[3] A. V. Bratischev, “Ob operatorakh obobschennogo differentsirovaniya Gelfonda–Leonteva”, Kompleksnyi analiz, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 153, VINITI RAN, M., 2018, 29–54 | MR

[4] M. Trybula, “Hadamard multipliers on spaces of holomorphic functions”, Integral Equations Operator Theory, 88:2 (2015), 249–268 | DOI | MR

[5] P. Domański, M. Langenbruch, “Representation of multipliers on spaces of real analytic functions”, Analysis (Munich), 32:2 (2012), 137–162 | DOI | MR

[6] P. Domański, M. Langenbruch, “Algebra of multipliers on the space of real analytic functions of one variable”, Studia Math., 212 (2012), 155–171 | DOI | MR

[7] P. Domański, M. Langenbruch, “Hadamard multipliers on spaces of real analytic functions”, Adv. Math., 240 (2013), 575–612 | DOI | MR

[8] P. Domański, M. Langenbruch, “Hadamard type operators on spaces of real analytic functions in several variables”, J. Funct. Anal., 269 (2015), 3868–3913 | DOI | MR

[9] P. Domański, M. Langenbruch, D. Vogt, “Multiplier projections on spaces of real analytic functions in several variables”, Complex Var. Elliptic Equ., 62:2 (2017), 241–268 | MR

[10] D. Vogt, “Hadamard type operators on spaces of smooth functions”, Math. Nachr., 288 (2015), 353–361 | DOI | MR

[11] D. Vogt, “Hadamard operators on $\mathcal D'(\mathbb R^N)$”, Studia Math., 237 (2017), 137–152 | DOI | MR

[12] D. Vogt, “Hadamard operators on $\mathcal D'(\Omega)$”, Math. Nachr., 290 (2017), 1374–1380 | DOI | MR

[13] D. Vogt, “$\mathcal E'$ as an algebra by multiplicative convolution”, Funct. Approx. Comment. Math., 59:1 (2018), 117–128 | MR

[14] D. Vogt, Hadamard Type Operators on Temperate Distributions, 2018, arXiv: 1812.06299

[15] P. Domański, M. Langenbruch, “Surjectivity of Hadamard type operators on spaces of smooth functions”, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 113:2 (2019), 1625–1676 | DOI | MR

[16] B. I. Golubov, “Ob ogranichennosti operatorov Khardi i Khardi–Litlvuda v prostranstvakh $\operatorname {Re}H^1$ i $\mathrm {BMO}$”, Matem. sb., 188:7 (1997), 93–106 | DOI | MR | Zbl

[17] S. Stevich, “Kompaktnost operatora Khardi–Littlvuda na prostranstvakh garmonicheskikh funktsii”, Sib. matem. zhurn., 50:1 (2009), 205–221 | MR

[18] S. S. Volosivets, “Modifitsirovannye operatory Khardi i Khardi–Littlvuda i ikh povedenie v razlichnykh prostranstvakh”, Izv. RAN. Ser. matem., 75:1 (2011), 29–52 | DOI | MR | Zbl

[19] G. Gao, Y. Zhong, “Some estimates of Hardy operators and their commutators on Morrey-Herz spaces”, J. Math. Inequal., 11 (2017), 49–58 | DOI | MR

[20] S. S. Volosivets, B. I. Golubov, “Drobnye modifitsirovannye operatory Khardi i Khardi—Littlvuda v prostranstvakh Morri—Gertsa i ikh kommutatory v vesovykh prostranstvakh”, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 171, VINITI RAN, M., 2019, 70–77 | DOI

[21] S. S. Volosivets, B. I. Golubov, “Drobnye modifitsirovannye operatory Khardi i Khardi–Littlvuda i ikh kommutatory”, Izv. vuzov. Matem., 2019, no. 9, 16–26 | DOI

[22] Kh. Shefer, Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR | Zbl

[23] B. V. Shabat, Vvedenie v kompleksnyi analiz, Nauka, M., 1969 | MR

[24] V. Feller, Vvedenie v teoriyu veroyatnostei i ee prilozheniya, T. 1, Mir, M., 1984 | MR | Zbl

[25] R. Edvards, Funktsionalnyi analiz. Teoriya i prilozheniya, Mir, M., 1969 | Zbl

[26] V. V. Zharinov, “Kompaktnye semeistva LVP i prostranstva FS i DFS”, UMN, 34:4 (208) (1979), 97–131 | MR | Zbl

[27] L. Khermander, Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR | Zbl

[28] A. P. Robertson, V. Dzh. Robertson, Topologicheskie vektornye prostranstva, Mir, M., 1967 | MR

[29] A. A. Goldberg, “Elementarnye zamechaniya o formulakh dlya opredeleniya poryadka i tipa tselykh funktsii mnogikh peremennykh”, Dokl. AN Arm. SSR, 29 (1959), 145–151 | MR

[30] L. I. Ronkin, Vvedenie v teoriyu tselykh funktsii mnogikh peremennykh, Nauka, M., 1971 | MR

[31] A. V. Abanin, Yu. F. Korobeinik, “Diagonalnye operatory v prostranstvakh chislovykh semeistv. Prilozheniya k ryadam Dirikhle”, Aktualnye voprosy matematicheskogo analiza, Izd-vo RGU, Rostov-na-Donu, 1978, 8–17