ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 2, pp. 1–6.
Allerton Press, Inc., 2018.
Original Russian Text
R.A. Baladai, B.N. Khabibullin, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 2, pp. 3–9.
From Integral Estimates of Functions to Uniform Ones. II.
R. A. Baladai
and B. N. Khabibullin
Bashkir State University
ul. Z. Validi 32, Ufa, Bashkortostan, 450076 Russia
Received October 28, 2016
Abstract—In the theory of functions of complex variables, exact pointwise estimates of the func-
tions, obtained under certain integral constraints on their growth, are not common. As an example
of such estimates, we can mention the pointwise estimation of the module of a function from the
Fock space by its integral norm. Here we present a functional-analytic scheme for obtaining
such estimates and illustrate it on the examples of the classical Fock–Bargman and Bergman–
Djrbashian type spaces of holomorphic functions deﬁned on the n-dimensional complex space, balls,
Keywords: integral pre-norm, holomorphic function, automorphism, Fock space, Bergman
As usually, we denote by N, R,andC the set of all natural, real,andcomplex numbers;letλ
be the Lebesgue measure on C
, n ∈ N,withstandardeuclidean norm |z| :=
+ ···+ |z
) ∈ C
. For a nonempty (= ∅) connected open set, i.e., a domain, D ⊂ C
, we will
denote by Hol(D) the vector space (over C) of all holomorphic functions in D. For a pair of numbers
α ∈ (0, +∞) and p ∈ (0, +∞], ﬁniteness of the following pre-norms
for p =+∞, (1a)
for f ∈ Hol(C) deﬁnes the Fock–Bargman space F
Theorem F (, theorem 2.7, corollary 2.8). For every α ∈ (0, +∞), p ∈ (0, +∞], and a point z ∈ C
; moreover, the estimate f
is not improvable.
Theorem F can be easily generalized for the case C
, n>1; this is most likely done somewhere.
We only know some estimations without exact constants (, 2.1; ). Without claiming novelty,
in Example 1 below we give a sharp inequality for the Fock–Bargman space. To obtain such sharp
estimates, in Section 1 we suggest a simple functional-analytic construction (Theorem 1). It is
applicable not only to spaces of holomorphic functions, as it is done in Section 2 (Theorem 2). Here
we did not consider a possibility of application of Theorem 1 in “non-holomorphic” cases for explicitly
deﬁned spaces of functions. Examples 1–4 from Section 3 demonstrate applications of Theorem 2
to concrete spaces of Fock–Bargman and Bergman–Djrbashian type in C
, a ball, or a polydisk. In
particular, they essentially complement and develop the results from  (example 1).