Geodesic
On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems
Funkcionalʹnyj analiz i ego priloženiâ, Tome 53 (2019) no. 2, pp. 42-58.

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

Let $M$ be a subharmonic function on a domain $D\subset \mathbb C^n$ with Riesz measure $\nu_M$, ${\mathsf Z} \subset D$. As was shown in the first of the preceding articles, if there exists a holomorphic function $ f\neq 0 $ on $D$, $f ({\mathsf Z}) = 0$, $|f|\leq \exp M$ on $D$, then there is some scale of integral uniform estimates from above of the distribution of the set $\mathsf Z$ in terms of $\nu_M$. In this article we show that for $n = 1$ this result is “almost invertible”. From such scale estimates of the distribution of points of the sequence ${\mathsf Z}:= \{{\mathsf z} _k \}_{k = 1,2, \dots} \subset D \subset \mathbb C$ by $\nu_M$ it follows that there exists a nonzero holomorphic function $f$ in $D$, $f (\mathsf Z) =0$, $|f| \leq \exp M^{\uparrow}$ on $D$, where the function $ M^{\uparrow} \geq M$ on $D$ is constructed by averaging of $M$ in rapidly convergent disks as we approach the boundary of the domain $D$ with some possible additive logarithmic component associated with the rate of narrowing of these disks.
Keywords: holomorphic function, sequence of zeros, subharmonic function, Jensen measure, test function
Mots-clés : balayage. 
@article{FAA_2019_53_2_a3,
     author = {B. N. Khabibullin and F. B. Khabibullin},
     title = {On the {Distribution} of {Zero} {Sets} of {Holomorphic} {Functions.} {III.} {Inversion} {Theorems}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {42--58},
     publisher = {mathdoc},
     volume = {53},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2019_53_2_a3/}
}
TY  - JOUR
AU  - B. N. Khabibullin
AU  - F. B. Khabibullin
TI  - On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2019
SP  - 42
EP  - 58
VL  - 53
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2019_53_2_a3/
LA  - ru
ID  - FAA_2019_53_2_a3
ER  - 
%0 Journal Article
%A B. N. Khabibullin
%A F. B. Khabibullin
%T On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2019
%P 42-58
%V 53
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2019_53_2_a3/
%G ru
%F FAA_2019_53_2_a3
B. N. Khabibullin; F. B. Khabibullin. On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems. Funkcionalʹnyj analiz i ego priloženiâ, Tome 53 (2019) no. 2, pp. 42-58. http://geodesic.mathdoc.fr/item/FAA_2019_53_2_a3/

[1] B. N. Khabibullin, A. P. Rozit, “K raspredeleniyu nulevykh mnozhestv golomorfnykh funktsii”, Funkts. analiz i ego pril., 52:1 (2018), 26–42 | DOI | MR | Zbl

[2] E. B. Menshikova, B. N. Khabibullin, “K raspredeleniyu nulevykh mnozhestv golomorfnykh funktsii. II”, Funkts. analiz i ego pril., 53:1 (2019), 84–87 | DOI | MR | Zbl

[3] Th. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[4] N. S. Landkof, Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR

[5] B. N. Khabibullin, “Posledovatelnosti nulei golomorfnykh funktsii, predstavlenie meromorfnykh funktsii i garmonicheskie minoranty”, Matem. sb., 198:2 (2007), 121–160 | DOI | Zbl

[6] B. N. Khabibullin, Polnota sistem eksponent i mnozhestva edinstvennosti, izd. 4-e dopolnennoe, RITs BashGU, Ufa, 2012

[7] J. Bliedtner, W. Hansen, Potential Theory. An Analytic and Probabilistic Approach to Balayage, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1986 | MR | Zbl

[8] N. Burbaki, Integrirovanie. Mery, integrirovanie mer, Nauka, M., 1967 | MR

[9] B. N. Khabibullin, N. R. Tamindarova, “Uniqueness theorems for subharmonic and holomorphic functions of several variables on a domain”, Azerb. J. Math., 7:1 (2017), 70–79 | MR

[10] B. N. Khabibullin, N. R Tamindarova, “Subharmonic test functions and the distribution of zero sets of holomorphic functions”, Lobachevskii J. Math., 38:1 (2017), 38–43 | DOI | MR | Zbl

[11] B. N. Khabibullin, Z. F. Abdullina, A. P. Rozit, “Teorema edinstvennosti i subgarmonicheskie testovye funktsii”, Algebra i analiz, 30:2 (2018), 318–334

[12] U. Kheiman, P. Kennedi, Subgarmonicheskie funktsii, Mir, M., 1980

[13] B. N. Khabibullin, “Posledovatelnosti needinstvennosti dlya vesovykh prostranstv golomorfnykh funktsii”, Izv. vuzov. Matem., 4 (2015), 75–84 | Zbl

[14] B. N. Khabibullin, F. B. Khabibullin, “O mnozhestvakh needinstvennosti dlya prostranstv golomorfnykh funktsii”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 4(35) (2016), 108–115

[15] B. N. Khabibullin, “Kriterii (sub-)garmonichnosti i prodolzhenie (sub-)garmonicheskikh funktsii”, Sib. matem. zhurn., 44:4 (2003), 905–925 | MR | Zbl

[16] B. N. Khabibullin, T. Yu. Baiguskarov, “Logarifm modulya golomorfnoi funktsii kak minoranta dlya subgarmonicheskoi funktsii”, Matem. zametki, 99:4 (2016), 588–602 | DOI | Zbl

[17] B. N. Khabibullin, “Nuli golomorfnykh funktsii s ogranicheniyami na rost v oblasti”, Issledovaniya po matematicheskomu analizu, Matematicheskii forum (Itogi nauki. Yug Rossii), 3, Vladikavkazskii nauchnyi tsentr RAN i RSO–A, Vladikavkaz, 2009, 282–291

[18] W. Hansen, I. Netuka, “Convexity properties of harmonic measures”, Adv. Math., 218:4 (2008), 1181–1223 | DOI | MR | Zbl

[19] S. G. Krantz, Function Theory of Several Complex Variables, 2nd ed., Amer. Math. Soc, Providence, RI, 2001 | MR | Zbl

[20] S. Bu, W. Schachermayer W., “Approximation of Jensen measures by image measures under holomorphic functions and applications”, Trans. Amer. Math. Soc., 331:2 (1992), 585–608 | DOI | MR | Zbl

[21] E. A. Poletsky, “Holomorphic currents”, Indiana Univ. Math. J., 42:1 (1993), 85–144 | DOI | MR | Zbl

[22] E. A. Poletsky, “Disk envelopes of functions II”, J. Funct. Anal., 163:1 (1999), 111–132 | DOI | MR | Zbl

[23] B. J. Cole, T. J. Ransford, “Jensen measures and harmonic measures”, J. Reine Angew. Math., 541 (2001), 29–53 | MR | Zbl