On Extremal Problems in Certain New Bergman Type Spaces in Some Bounded Domains in ℂ^𝑛

Abstract

Based on recent results on boundedness of Bergman projection with positive Bergman kernel in analytic spaces in various types of domains in ℂ^𝑛, we extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains, and minimal bounded homogeneous domains.

1. Introduction

The goal of this paper is to add several new results for distances in analytic Bergman type spaces of functions of several variables. It turns out that our distance theorem we proved before in case of unit disk, a sharp result under certain natural additional condition, is valid also in various domains and various Bergman type analytic spaces. Namely, we look at analytic Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains. These analytic spaces act as direct extensions of well-known analytic Bergman spaces in the unit disk. These analytic spaces are relatively new and we will include some basic facts on them in our paper. They will also be needed for proofs of our assertions partially. We will start this paper with two sharp results on distances in Bergman type spaces in two domains: Siegel domains of second type and in minimal homogeneous domains in ℂ^𝑛. Then one side estimates for distance function in Lie ball and bounded symmetric domains of tube type will be given based directly on recent advances related to boundedness of Bergman type projections in Bergman type spaces in these type domains.

Our intention in this paper is the same as in our previous papers on this topic. Namely, we collect some facts from earlier investigation concerning Bergman projection with positive Bergman kernel and Bergman kernel and use them for our purposes in estimates of dist_𝑌(𝑓,𝒳) function (distance function).

Following our previous papers [1, 2] we can easily obtain a sharp result for distance function. We need only several tools and the following scheme.

First we need an embedding of our quazinormed analytic space (in any domain) into another one (𝑋⊂𝑌); this immediately poses a problem of dist_𝑌(𝑓,𝑋)=inf_𝑔∈𝑋‖𝑓−𝑔‖_𝑌 for all 𝑓∈𝑌⧵𝑋. Then we need the Bergman reproducing formula for all 𝑓 function from 𝑌 space. Then, finally, we use the boundedness of Bergman type projections with |𝐾(𝑧,𝑤)| positive kernel acting from 𝑋 to 𝑋 together with Forelli-Rudin type sharp estimates of Bergman kernel. These three tools were used in general Siegel domain of second type, polydisk, and unit ball in [1–4] (see also various references there). We continue to use these tools providing new sharp (and not sharp) results in various spaces of analytic functions in this paper.

Note that our theorem on Siegel domains was formulated in [5] without proof. We provide the complete proof here. We also note that various problems, related to Bergman type projections, are applied in many problems in function theory (see, e.g., [6] and references there).

First we provide a known result in the unit disk with complete proof taken from our previous papers [1, 2]. In the unit disk case all arguments here are short and transparent and are based on several tools like Forelli-Rudin type estimate and estimates for Bergman type projections with positive Bergman kernel. Then we will see arguing similarly as in unit disk and we will easily complete the proof of more complicated cases. The complete formulations of our last theorems will be given, but sometimes sketches of proofs will be added and details of proofs of higher-dimensional cases will be left to readers.

Note that it is easy to see that our assertions may have various applications in approximation theory; for example see [6] and references there.

The base of our proofs is properties of Bergman projection in various domains given in [7–12]. The estimates of Bergman kernel from [7–12] are also playing an important role below in our proofs. Note that arguments we use below are very close to arguments which were used before in [1, 2, 13]. As a result we alert the reader that the exposition is sketchy sometimes.

Throughout the paper, we write 𝐶 (sometimes with indexes) to denote a positive constant which might be different at each occurrence (even in a chain of inequalities) but is independent of the functions or variables being discussed.

The notation 𝐴≍𝐵 means that there is a positive constant 𝐶, such that 𝐵/𝐶≤𝐴≤𝐶𝐵. We will write for two expressions 𝐴≲𝐵 if there is a positive constant 𝐶 such that 𝐴<𝐶𝐵.

This paper can be viewed as continuation of [13] where various other cases (domains) were also considered. In [13] the base of all our proofs in complex domains in higher dimension was the Bergman reproducing formula, while here all our assertions are based on some recent results on boundedness of Bergman projections with positive Bergman kernel in Bergman spaces in such type domains.

2. Notations, Definitions, and Preliminaries

We will need various definitions and assertions for formulations of main results. These are assertions on various types of domains we consider in this paper and analytic functions on them.

We denote by 𝐁_𝑛 the unit ball in ℂ^𝑛. As usual, we denote by 𝐻(𝐁_𝑛) the space of all holomorphic functions in 𝐁_𝑛. For 1≤𝑝<+∞ and 𝛼>−1, denote by 𝐻^𝑝_𝛼(𝐁_𝑛)  (or 𝐴^𝑝_𝛼(𝐁_𝑛)) the space of all functions 𝑓 holomorphic in 𝐁_𝑛 and satisfying the condition _𝐁𝑛𝑓(𝑤)^𝑝1−|𝑤|^2𝛼𝑑𝜈(𝑤)<+∞, (1) where 𝑑𝜈 is the Lebesgue measure in ℂ^𝑛.

Further, for a complex number 𝛽 with Re𝛽>−1, put 𝑐_𝑛𝛽=Γ𝑛+1+𝛽𝜋^𝑛.⋅Γ1+𝛽 (2)

Let 𝐴^∞_𝛼(𝐁_𝑛)={𝑓∈𝐻(𝐁_𝑛)∶sup|𝑓(𝑧)|(1−|𝑧|)^𝛼<∞}, 𝛼≥0.

The following theorem is well-known and it has many applications in various problems in function theory (see [14]).

Theorem A. Assume that 1≤𝑝<+∞, 𝛼>−1 and that the complex number 𝛽 satisfies the condition Re𝛽≥𝛼,𝑝=1Re𝛽>𝛼+1𝑝−1,1<𝑝<∞. (3) Then each function 𝑓∈𝐴^𝑝_𝛼(𝐁_𝑛) admits the following integral representations: 𝑓(𝑧)=𝑐_𝑛𝛽⋅_𝐁𝑛𝑓(𝑤)1−|𝑤|^2𝛽(1−⟨𝑧,𝑤⟩)^𝑛+1+𝛽𝑑𝜈(𝑤),𝑧∈𝐁_𝑛,𝑓(0)=𝑐_𝑛𝛽⋅_𝐁𝑛𝑓(𝑤)1−|𝑤|^2𝛽(1−⟨𝑧,𝑤⟩)^𝑛+1+𝛽𝑑𝜈(𝑤),𝑧∈𝐁_𝑛, (4)(5) where ⟨⋅,⋅⟩ is the Hermitian inner product in ℂ^𝑛.

For 𝑛>1, the theorem was proved in [15] (when 𝛼=0) and in [16] (when 𝛼>−1, 𝑛=1).

These integral representation theorems were the core of our approach for estimates of distance function in our recent papers (see [1, 2, 13, 17]) and we will see the same in this paper.

We will start this section with various known assertions on analytic function spaces on Siegel domains of second type. Next we will continue adding some lemmas on each Bergman type analytic space on each domain in higher dimension which we will consider in this paper. We alert the reader that some assertions below will serve as introductory material and will not be used during the proof to make the reading of paper more convenient.

Let 𝐷 be a usual homogeneous Siegel domain of second type. Let 𝑑𝜈 denote the Lebesgue measure on 𝐷 (for all other bounded domains in this paper we will also use the same notation below) and let as usual 𝐻(𝐷) be the space of holomorphic functions on 𝐷 endowed as usual with the topology of uniform convergence on compact subsets of 𝐷.

The Bergman projection 𝑃 of 𝐷 is as usual the orthogonal projection of 𝐿²(𝐷,𝑑𝜈) onto its subspace 𝐴²(𝐷) consisting of holomorphic functions. Moreover it is known that 𝑃 is the integral operator defined on 𝐿²(𝐷,𝑑𝜈) by the Bergman kernel 𝐵(𝑧,𝜁) which for 𝐷 was computed for example in [18, 19].

Let 𝑟 be a real number, for example. We fix it. Since 𝐷 is homogeneous, the function 𝜁→𝐵(𝜁,𝜁) does not vanish on 𝐷; we can set 𝐿^𝑝,𝑟(𝐷)=𝐿^𝑝𝐷,𝐵^−𝑟(𝜁,𝜁)𝑑𝜈(𝜁),0<𝑝<∞. (6) Let 𝑝 be an arbitrary positive number. The weighted Bergman space is defined as usual by 𝐴^𝑝,𝑟(𝐷)=𝐿^𝑝,𝑟(𝐷)∩𝐻(𝐷). We put 𝐴^𝑝,0=𝐴^𝑝(𝐷).

The so-called weighted Bergman projection 𝑃_𝜀 is the orthogonal projection of 𝐿^2,𝜀(𝐷) onto 𝐴^2,𝜀(𝐷). This fact can be found in [8, 10]. It is proved in [8, 10] that there exists a real number 𝜀_𝐷<0 such that 𝐴^2,𝜀(𝐷)={0} if 𝜀≤𝜀_𝐷 and that for 𝜀>𝜀_𝐷. 𝑃_𝜀 is the integral operator defined on 𝐿^2,𝜀(𝐷) by the weighted Bergman kernel 𝐶_𝜀𝐵^1+𝜀(𝜁,𝑧). In all our work we will assume that 𝜀>𝜀_𝐷.

The norm ‖⋅‖_𝑝,𝑟 of 𝐴^𝑝,𝑟(𝐷) with 𝑟>𝜀_𝐷 is defined by 𝑓_𝑝,𝑟=_𝐷𝑓(𝑧)^𝑝𝐵^−𝑟(𝑧,𝑧)𝑑𝜈(𝑧)^1/𝑝,𝑓∈𝐴^𝑝,𝑟(𝐷). (7) We need some assertions (see [8, 10]).

Note the exact expression of Bergman kernel for these domains can be seen in [8, 10, 18].

We denote by 𝑏((𝜏₁,𝜏₂),(𝜏₃,𝜏₄)) Bergman kernel for the Siegel domain of the second type, which differs from 𝐵(𝑧,𝜁) Bergman kernel by constant. We will use it in text also.

Lemma A. Let ℎ∈𝐿^∞(𝐷). Take 𝜌>𝜌₀ for large fixed 𝜌₀. Then the function 𝑧⟶𝐺(𝑧)=_𝐷𝐵^1+𝜌(𝑧,𝜁)ℎ(𝜁)𝑑𝜈(𝜁) (8) satisfies the estimate sup_𝑧∈𝐷|𝐺(𝑧)|𝐵^−𝜌(𝑧,𝑧)≤𝐶‖ℎ‖_∞ and 𝐺∈𝐻(𝐷).

Lemma B. For each 𝜌 sufficiently large and for each 𝐺∈𝐻(𝐷) such that sup_𝑧∈𝐷|𝐺(𝑧)||𝐵^−𝜌(𝑧,𝑧)|<∞ one has the reproducing formula 𝐺(𝜁)=𝐶_𝜌𝐷𝐵^1+𝜌(𝜁,𝑧)𝐺(𝑧)𝐵^−𝜌(𝑧,𝑧)𝑑𝜈(𝑧),𝑧∈𝐷. (9)

We will need for our theorems some basic facts for Siegel domains of second type. We denote by 𝑑_𝑖, 𝑞_𝑖, and 𝑛_𝑖 parameters of a Siegel domains of second type (see [4, 5, 8, 10]). We will use usual operations between two vectors for such parameters below in our text.

The following lemma is complete analogue of so-called Forelly-Rudin type estimates for our Siegel domains of second type (see [8, 10]).

Lemma C. Let 𝛼 and 𝜀 be in ℝ^𝑙, (𝜁,𝑣)∈𝐷. Then for 𝜀_𝑖>(𝑛_𝑖+2)/2(2𝑑−𝑞)_𝑖 and 𝛼_𝑖−𝜀_𝑖>𝑛_𝑖/(−2)(2𝑑−𝑞)_𝑖, 𝑖=1,…,𝑙_𝐷𝐵^1+𝛼((𝜁,𝑣),(𝑧,𝑢))𝐵^−𝜀((𝑧,𝑢),(𝑧,𝑢))𝑑𝑣(𝑧,𝑢)=𝐶_𝛼,𝜀𝐵^𝛼−𝜀((𝜁,𝑣),(𝜁,𝑣)). (10)

Lemma D. Let 𝑟 be a vector of ℝ^𝑙 such that 𝑟_𝑖>(𝑛_𝑖+2)/(−2)(2𝑑−𝑞)_𝑖 for all 𝑖=1,…,𝑙 and 𝑝 is a real number such that 1≤𝑝<min{(𝑛_𝑖−2(2𝑑−𝑞)_𝑖(1+𝑟_𝑖))/𝑛_𝑖}. Then for all 𝜀∈ℝ^𝑙 such that 𝜀_𝑖>(𝑛_𝑖+2)/(2(2𝑑−𝑞)_𝑖)((𝑝−1)/𝑝)+(𝑟_𝑖/𝑝), 𝑖=1,…,𝑙∶ 𝑃_𝜀𝑓=𝑓, 𝑓∈𝐴^𝑝,𝑞.

We list in Lemma E other properties of Bergman kernel. The last estimate in assertion below is an embedding theorem which connect so-called growth spaces with Bergman spaces. This allows to pose a distance problem (see also the complete analogue of this result in other simpler domains in [1, 17]).

Lemma E. Let 𝛼∈ℝ^𝑙, 𝛼_𝑗≤0, 𝑖=1,…,𝑙. Then |𝑏^𝛼((𝜁,𝑣),(𝑧,𝑢))|≤𝑐_𝛼𝑏^𝛼((𝜁,𝑣),(𝜁,𝑣)) and |𝑏^𝛼((𝜁,𝑣)+(𝜁^′,𝑣^′),(𝑧,𝑢)+(𝑧^′,𝑢^′))|≤𝑐_𝛼𝑏^𝛼((𝜁,𝑣),(𝜁,𝑣)) for all (𝜁,𝑣), (𝜁^′,𝑣^′), (𝑧,𝑢), (𝑧^′,𝑢^′) in 𝐷. For all 𝑓∈𝐴^𝑝,𝑟(𝐷), 𝑝>0𝑓(𝑧,𝑢)^𝑝≤𝐶𝑏^1+𝑟((𝑧,𝑢),(𝑧,𝑢))𝑓^𝑝_𝑝,𝑟. (11)

The following result concerns the boundedness of Bergman type projection with positive Bergman kernel in weighted Bergman spaces. Note that this fact is classical in simpler domains and it has also many applications in analytic function theory.

Proposition A. Let 𝜀 and 𝑟 be in ℝ^𝑙 such that 𝜀_𝑖>1/(2𝑑−𝑞)_𝑖 and 𝑟_𝑖>(𝑛_𝑖+2)/2(2𝑑−𝑞)_𝑖, 𝑖=1,…,𝑙. Then 𝑃_𝜀 is bounded from 𝐿^𝑝,𝑟(𝐷) into 𝐴^𝑙,𝑟(𝐷) if max_{𝑖=1,…,𝑙}1,2𝑛_𝑖+2−22𝑑−𝑞_𝑖𝑟_𝑖𝑛_𝑖+2−22𝑑−𝑞_𝑖𝜀_𝑖<𝑝<min_{𝑖=1,…,𝑙}2𝑛_𝑖+2−22𝑑−𝑞_𝑖𝑟_𝑖𝑛_𝑖. (12)

The following assertion provides integral representation for a certain so-called analytic “growth space” on Siegel domains of the second type.

Proposition B. Let 𝑟 and 𝜀 be two vectors of ℝ^𝑙 such that 𝜀_𝑖>𝑛_𝑖/(−2)(2𝑑−𝑞)_𝑖, 𝑟_𝑖>((𝑛_𝑖+2)/2(2𝑑−𝑞)_𝑖)+𝜀_𝑖, 𝑖=1,…,𝑙. Let 𝐺 be in 𝐻(𝐷) such that sup_𝑧∈𝐷|𝐺(𝑧)|𝐵^−𝜀(𝑧,𝑧)<∞; (13) then 𝑃𝑟𝐺=𝐺.

The following result explains the structure of functions from Bergman spaces on Siegel domains of second type. It is an extension of a classical theorem on atomic decomposition of Bergman spaces in the unit disk on a complex plane.

Proposition C. Let 𝐷⊂ℂ^𝑁 be a symmetric Siegel domain of second type, 𝑝∈(2𝑁/(2𝑁+1),1), 𝑧∈ℝ^𝑙, 𝑟_𝑗>(𝑛_𝑖+2)/2(2𝑑−𝑞)_𝑖. Then there are two constants 𝐶=𝐶(𝑝,𝑟) and 𝐶₁=𝐶₁(𝑝,𝑟) such that for every 𝑓∈𝐴^𝑝,𝑟(𝐷) there exists an 𝑙^𝑝 sequence {𝜆_𝑖} such that 𝑓(𝑧)=^∞_𝑖=0𝜆_𝑖𝑏^𝛼/𝑝𝑧,𝑧_𝑖𝑏^{(1+𝑟−𝛼)/𝑝}𝑧_𝑗,𝑧_𝑗, (14) where {𝑧_𝑖} is a lattice in 𝐷 and the following estimate holds: 𝐶𝑓^𝑝_𝑝,𝑟≤𝜀𝜆_𝑖^𝑝≤𝐶𝑓^𝑝_𝑝,𝑟. (15)

We add some basic facts on minimal bounded homogeneous domains and we will use them partially in our paper (see [11, 12]).

Let 𝐷 be a bounded domain in ℂ^𝑛. We say that 𝐷 is a minimal domain with a center 𝑡∈𝐷 if the following condition is satisfied: for every biholomorphism 𝜓∶𝐷→𝐷^′ with det𝐽(𝜓,𝑡)=1, (𝐽s is the complex Jacobian of the map 𝜓), we have Vol𝐷^′≥Vol(𝐷). (16)

Let now us denote by 𝐾_𝐷 the Bergman kernel of 𝐷, that is, the reproducing kernel of 𝐿²(𝐷). It is known that 𝐷 is a minimal domain with a center 𝑡 if and only if 𝐾_𝐷(𝑧,𝑡)=𝐾_𝐷(𝑡,𝑡) for any 𝑧∈𝐷, (see [20] and references there).

From [21], Proposition 3.6, we see that 𝐷 is a minimal domain with a center 𝑡 if and only if 𝐾_𝐷()=1𝑧,𝑡Vol(𝐷), (17) for any 𝑧∈𝐷.

Every bounded homogeneous domain is biholomorphic to a representative bounded homogeneous domain.

Therefore, every bounded homogeneous domain is biholomorphic to a minimal bounded homogeneous domain (see [11, 12]).

For any 𝑧∈𝒰 and 𝜌>0, let 𝐁𝑧,𝜌=𝑤∈𝒰∶𝛽(𝑧,𝑤)≤𝜌 (18) be the Bergman metric disk with center 𝑧 and radius 𝜌, where 𝛽 denotes the Bergman distance on 𝒰, (see [11, 12]).

We fix a minimal bounded homogeneous domain 𝒰 with a center 𝑡. For a bounded linear operator 𝑇 on 𝐴²_𝛼(𝒰) (this is standard analytic part of standard 𝐿² space in 𝒰) the Berezin symbol 𝑇 of 𝑇 is defined by 𝑇(𝑧)=𝑇𝑘_𝑧,𝑘_𝑧(𝑧∈𝒰), (19) where 𝑘_𝑧 is a normalized Bergman kernel in Bergman space 𝐴²_𝛼 in minimal bounded homogeneous domain 𝒰. For a Borel measure 𝜇 on 𝒰, we define a function 𝜇 on 𝒰 by 𝜇(𝑧)=_𝒰𝑘_𝑧(𝑤)²𝑑𝜇(𝑤), (20) which is called the Berezin symbol of the measure 𝜇. Since 𝐾_𝒰(𝑧,𝑤) is a bounded function on 𝐁(𝑡,𝜌)×𝒰, 𝜇 is a continuous function if 𝜇 is finite.

We will provide some basic facts for a minimal bounded homogeneous domains.

Lemma F (see [20]). There exists a constant 𝑀_𝜌 such that 𝑀_𝜌⁻¹≤𝑘_𝑎(𝑧)²𝑉𝑜𝑙𝐁𝑎,𝜌≤𝑀_𝜌, (21) for all 𝑎∈𝒰 and 𝑧∈𝐁(𝑎,𝜌).

Lemma G (see [20]). There exists a sequence {𝑤_𝑗}⊂𝒰 satisfying the following conditions. (1)𝒰=⋃^∞_𝑗=1𝐁(𝑤_𝑗,𝜌). (2)𝐁(𝑤_𝑖,𝜌/4)∩𝐁(𝑤_𝑗,𝜌/4)=0.(3)There exists a positive integer 𝑁 such that each point 𝑧∈𝒰 belongs to at most 𝑁 of the sets 𝐁(𝑤_𝑗,2𝜌).

Lemma H (see [20]). There exists a constant 𝐶 such that 𝑓(𝑎)^𝑝≤𝐶𝑉𝑜𝑙𝐁𝑎,𝜌_{𝐁(𝑎,𝜌)}𝑓(𝑧)^𝑝𝑑𝜈(𝑧) (22) for all 𝑓∈𝐻(𝒰), 𝑝≥1, and 𝑎∈𝒰, where 𝐻(𝒰) is a space of analytic functions in 𝒰.

Theorem B (see [22]). Take any 𝜌>0. Then, there exists 𝐶_𝜌>0 such that 𝐶_𝜌⁻¹≤𝐾_𝒰(𝑧,𝑎)𝐾_𝒰(𝑎,𝑎)≤𝐶_𝜌, (23) for all 𝑧,𝑎∈𝒰 with 𝛽_𝒰(𝑧,𝑎)≤𝜌, where 𝛽_𝒰 means the Bergman distance on 𝒰.

Authors in [22] introduce certain equivariant holomorphic maps 𝜃_𝑛𝑗∶𝒰→𝒰^′ for 𝑗=1,…,𝑟 (:=rank 𝒰) from 𝒰 into the Siegel disk 𝒰_𝑛 of rank 𝑛_𝑗. Authors in [22] obtain the following formula for the description of 𝐾_𝒰.

Theorem C (see [22]). There exist integers 𝑠₁,…,𝑠_𝑟 such that 𝐾_𝒰(𝑧,𝑤)=𝑉𝑜𝑙(𝒰)^𝑟−1_𝑗=1det𝐼_𝑛𝑗−𝜃_𝑛𝑗(𝑧)𝜃_𝑛𝑗(𝑤)^−𝑠𝑗 (24) for 𝑧,𝑤∈𝒰. Recall that the Bergman kernel 𝐾_𝒰𝑚 of the Siegel disk 𝒰_𝑚 is given by 𝐾_𝒰𝑚(𝑧,𝑤)=𝑉𝑜𝑙𝒰_𝑚⁻¹det𝐼_𝑚−𝑧𝑧^−(𝑚+1). (25)

We will denote the weighted reproducing Bergman kernel for weighted Bergman 𝐴²_𝛼 spaces in this type domains below simply as 𝐾_𝛼 omitting index 𝒰.

Note from lemmas above (see [11]) that we have sup𝑓(𝑧)𝐾(𝑧,𝑧)^𝑡<𝐶𝑓_𝐴2𝛽,where𝑡=−𝛽−12. (26)

This allows putting distance problems for these domains which we solve in Theorem 8.

We need now some preliminaries for Bergman spaces in Lie ball.

Let 𝐷 denote each of the following domains in ℂ^𝑛, 𝑛≥3:(1)the tube Ω=ℝ^𝑛+𝑖Γ over the spherical cone Γ=𝑦₁,…,𝑦_𝑛∈ℝ^𝑛∶𝑦₁>0,𝑦₁𝑦₂−𝑦²₃−⋯−𝑦²_𝑛>0, (27)(2)the Lie ball 𝜔=𝑧∈ℂ^𝑛∶^𝑛_𝑗=1𝑧²_𝑗<1,1−2|𝑧|²+^𝑛_𝑗=1𝑧²_𝑗²>0. (28)

Obviously, the first domain is unbounded while the second one is bounded. It is well- known that they are biholomorphically equivalent and, in Elie Cartans classification of bounded symmetric domains [23], they are representatives of class IV (according to Huas numbering [24]). We are interested in bounded Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains in this paper.

To make the exposition easier, we remind the readers about the basic definitions again.

Let 𝐻(𝐷) denote the space of holomorphic functions in 𝐷 domain and let, as above, 𝑑𝜈 be Lebesgue measure in ℂ^𝑛. For every 𝑝≥1, the Bergman space 𝐴^𝑝(𝐷) is defined by 𝐴^𝑝(𝐷)=𝐻(𝐷)∩𝐿^𝑝(𝐷,𝑑𝜈). For every 𝑓∈𝐴^𝑝(𝐷), we set ‖𝑓‖_{𝐴𝑝(𝐷)}=‖𝑓‖_{𝐿𝑝(𝐷,𝑑𝜈)} for 𝑝≥1; this is a norm under which 𝐴^𝑝(𝐷) is a Banach space. The Bergman projection 𝑃_𝐷 of 𝐷 is the orthogonal projection of the Hilbert space 𝐿²(𝐷,𝑑𝜈) onto its closed subspace 𝐴²(𝐷). Moreover, 𝑃_𝐷 is the integral operator associated with a kernel 𝐵_𝐷(⋅,⋅) called the Bergman kernel of 𝐷. Finally, let 𝑃^∗_𝐷 denote the integral operator associated with the positive kernel |𝐵_𝐷(⋅,⋅)|.

The following results were proved in [7].

Theorem D (see [7]). For every 𝑝∈(1,(3𝑛−2)/2𝑛]∪[(3𝑛−2)/(𝑛−2),∞), the Bergman projection 𝑃_𝐷 is unbounded on 𝐿^𝑝(𝐷,𝑑𝜈).

Theorem E (see [7]). Let 𝑝≥1. The operator 𝑃^∗_𝐷 is bounded on 𝐿^𝑝(𝐷,𝑑𝜈) if and only if 𝑝∈((2𝑛−2)/𝑛,(2𝑛−2)/(𝑛−2)). Furthermore, the Bergman projection 𝑃_𝐷 is bounded from 𝐿^𝑝(𝐷,𝑑𝜈) to 𝐴^𝑝(𝐷) when 𝑝∈((2𝑛−2)/𝑛,(2𝑛−2)/(𝑛−2)).

For the tube domain Ω some of these results were announced in [25]. The question whether 𝑃_𝐷 is bounded on 𝐿^𝑝(𝐷,𝑑𝜈) when 𝑝 belongs to ((3𝑛−2)/2𝑛,(2𝑛−2)/𝑛]∪[(2𝑛−2)/(𝑛−2),(3𝑛−2)/(𝑛−2)) remains open. The case of all homogeneous Siegel domains of second type has recently been considered by 𝐷. Bekolle and A. Temgoua Kagou. They proved that there is a range of 𝑝, around 2, where the Bergman projection is bounded in 𝐿^𝑝, while there is a range of 𝑝, around 1 and ∞, where it is unbounded (see [8]). In all cases the critical result is not known.

We add some basic facts on Bergman kernel on these domains.

Proposition D (see [7]). The Bergman kernel 𝐵_Ω(𝑠,𝑧) of Ω is given by 𝐵_Ω(𝜁,𝑧)=𝑐_𝑛𝜁₁−𝑧₁𝜁₂−𝑧₂−^𝑛_𝑗=3𝜁_𝑗−𝑧_𝑗^2−𝑛, (29) where 𝜁=(𝜁₁,…,𝜁_𝑛), 𝑧=(𝑧₁,…,𝑧_𝑛)∈Ω.

Definition 1 (see [7]). Let 𝑘(𝑡,𝑦) denote the positive kernel defined on the cone Γ 𝑘(𝑡,𝑦)=[(𝑡₁+𝑦₁)(𝑡₂+𝑦₂)−∑^𝑛_𝑗=3(𝑡_𝑗+𝑦_𝑗)²]^−𝑛/2, 𝑡=(𝑡₁,…,𝑡_𝑛), 𝑦=(𝑦₁,…,𝑦_𝑛), 𝑦∈Γ.

Proposition E (see [7]). For each 𝑝≥1, there exists a constant 𝐶_𝑝 such that, for all 𝑦∈Γ and 𝜉=𝑠+𝑖𝑡∈Ω, _ℝ𝑛𝐵_Ω𝜁,𝑥+𝑖𝑦^𝑝𝑑𝑥=𝐶_𝑝𝑘𝑡,𝑦^2𝑝−1. (30) Moreover, there exists a constant 𝑐_𝑝 such that, for each 𝑦∈Γ such that |𝑦|<1/100 and each 𝜁=𝑠+𝑖𝑡∈Ω such that |𝜁|<1/100, _{𝐼×⋯×𝐼}𝐵_Ω𝜁,𝑥+𝑖𝑦^𝑝𝑑𝑥≥𝐶_𝑝𝑘𝑡,𝑦^2𝑝−1, (31) where 𝐼 denotes the interval [−1,1].

Let 𝑧=Φ(𝑧^′) be the linear fractional mapping from 𝜔 onto Ω which is given in [23]. In particular, we assume that Φ(0)=𝑒, where 𝑒=(𝑖,𝑖,0,…,0) and Φ is holomorphic outside 𝑍={𝑧∈ℂ^𝑛∶𝑄(𝑧)=0}, where 𝑄 is a polynomial such that 𝑄(0)=1. In view of the change of variables formula, the Bergman kernel 𝐵_𝜔(𝜁^′,𝑧^′) of 𝜔 has the following expression in terms of that of Ω: 𝐵_𝜔𝜁^′,𝑧^′=𝐵_ΩΦ𝜁^′,Φ𝑧^′𝐽Φ𝜁^′𝐽Φ𝑧^′, (32) where 𝐽s is the complex Jacobian of the map Φ. On the other hand, since 𝜔 is a circular domain, for each real number 𝜃, 𝐵_𝜔𝑒^𝑖𝜃𝜁^′,𝑒^𝑖𝜃𝑧^′=𝐵_𝜔𝜁^′,𝑧^′ (33) and thus, there exists a constant 𝐶 such that 𝐵_𝜔(𝜁^′,0)=𝐶 for each 𝜁^′∈𝜔. Hence, from (32), we get 𝐽Φ𝜁^′=𝐶^′𝐵_𝜔(Φ𝜁^′,𝑒⁻¹. (34) The following property of Bergman kernel is vital.

Lemma I (see [7]). For all 𝑧 and 𝜁 in Ω, 𝐵_𝜔(𝜁,𝑧)≤𝐵_𝜔(𝑧,𝑧). (35)

We add some basic facts on bounded symmetric domains of tube type from [23].

Let 𝐷 be an irreducible bounded symmetric domain of tube type in ℂ^𝑛. That is, 𝐷 is conformally equivalent to a tube domain 𝑇_Ω=ℝ^𝑛+𝑖Ω over a symmetric cone Ω in ℝ^𝑛. Irreducible symmetric cones are completely classified (see [26]), being either light-cones Λ_𝑛=𝑦₁,𝑦^′∈ℝ^𝑛∶𝑦₁>𝑦^′,𝑛≥3, (36) or cones of positive-definite symmetric or hermitian matrices, namely, Sym₊(𝑟,ℝ),Her₊(𝑟,ℂ),Her₊(𝑟,ℍ),Her₊(3,𝕆). (37) We write 𝑟 for the rank of the cone (which in light-cones is 𝑟=2) and Δ for the associated determinant function (which in light-cones is the Lorentz form Δ(𝑦)=𝑦²₁−|𝑦^′|²).

An important open question in these domains, 𝐷 and 𝑇_Ω, concerns the 𝐿^𝑝 boundedness of the associated Bergman projections, that is, the orthogonal projection 𝑃 mapping 𝐿² into the subspace of holomorphic functions 𝐴². In contrast with Cauchy-Szego projections (which are not bounded in 𝐿^𝑝 for any 𝑝≠2, if 𝑛>1), the 𝐿^𝑝-boundedness of Bergman projections has been conjectured in a small interval around 𝑝=2, namely, 1+𝑛−𝑟2𝑛<𝑝<1+2𝑛.𝑛−𝑟 (38) At the moment, positive results are only known to hold in a proper subinterval 1+𝑛−𝑟2𝑛−𝑟<𝑝<1+2𝑛−𝑟,𝑛−𝑟 (39) with a small improvement over this range in the case of light-cones.

Also, we are interested in applying the transference principle to the family of weighted Bergman projections in 𝐷 and 𝑇_Ω. Using the notation in the text from [26], Chapter XIII, these operators are defined for 𝜈>(2𝑛/𝑟)−1 by 𝑃_𝜈𝑓(𝑧)=_𝐷𝐵^𝐷_𝜈(𝑧,𝑤)𝑓(𝑤)𝑑𝜇_𝜈(𝑤)𝒫,𝑧∈𝐷,_𝜈𝑔(𝜁)=_𝑇Ω𝐵^𝑇Ω_𝜈𝜁,𝜂𝑔𝜂𝑑𝜆_𝜈𝜂,𝜁∈𝑇_Ω, (40) where the Bergman kernels and their associated measures have the explicit expressions 𝐵^𝐷_𝜈(𝑧,𝑤)=𝑐_𝜈ℎ(𝑧,𝑤)^−𝜈,𝑑𝜇_𝜈(𝑤)=ℎ(𝑤)^{𝜈−(2𝑛/𝑟)}𝐵𝑑𝑤,_𝑇Ω𝜈𝜁,𝜂=𝑐^′_𝜈Δ𝜁−𝜂^−𝜈,𝑑𝜆_𝜈𝜂=Δℑ𝑚𝜂^{𝜈−(2𝑛/𝑟)}𝑑𝜂 (41) for certain constants 𝑐_𝜈, 𝑐^′_𝜈. Here ℎ(𝑧,𝑤) denotes the unique polynomial (holomorphic in 𝑧 and antiholomorphic in 𝑤) such that ℎ(𝑧)∶=ℎ(𝑧,𝑧) is 𝑈-invariant and ℎ(𝑥)=Δ(𝑒−𝑥²), 𝑥∈ℝ^𝑛.

The Bergman kernels in these two domains are related by the formula 𝐵^𝐷_𝜈(𝑧,𝑤)=𝑐_𝜈̃𝐵^𝑇Ω_𝜈(Φ(𝑧),Φ(𝑤))𝐽_Φ(𝑧)𝐽_Φ(𝑤)^{𝜈/(2𝑛/𝑟)}. (42)

Corollary 2 (see [9]). Let 𝜈>(2𝑛/𝑟)−1 and 𝑝,𝑞∈[1,∞]. Then the following are equivalent:(a)𝑃_𝜈 is bounded from 𝐿^𝑝(𝐷,𝑑𝜇_𝜈)→𝐿^𝑞(𝐷,𝑑𝜇_𝜈);(b)𝒫^𝐁_𝜈 is bounded from 𝐿^𝑝(𝑇_Ω,𝑑𝜆_𝜈)→𝐿^𝑞(𝑇_Ω,𝑑𝜆_𝜈).

Remark 3. The statements in the corollary can only hold if 𝑝≥𝑞.

The transference principle also applies to the positive operators 𝑃⁺_𝜈𝑓(𝑧)=_𝐷𝐵^𝐷_𝜈(𝑧,𝑤)𝑓(𝑤)𝑑𝜇_𝜈(𝑤),𝒫⁺_𝜈𝑔(𝜁)=_𝑇Ω𝐵^𝑇Ω_𝜈𝜁,𝜂𝑔𝜂𝑑𝜆_𝜈𝜂. (43) In this case we can even state a stronger result. We consider a new operator, acting on functions in Ω by 𝑓⟼𝒬𝑓𝑦=𝜒_𝐁𝑦_Ω∩𝔹𝑓(𝑢)Δ𝑦+𝑢^{𝜈−(𝑛/𝑟)}𝑑𝜆_𝜈(𝑢),𝑦∈Ω. (44) Here, with a slight abuse of notation, we still write 𝑑𝜆_𝜈 for the measure Δ^{𝜈−(2𝑛/𝑟)}(𝑢)𝑑𝑢 in Ω and 𝐁 for the closed unit ball in ℝ^𝑛. We write 𝐿_𝜈^𝑝,𝑝(𝑇_Ω)=𝐿^𝑝_𝑦(Ω,𝜆_𝜈;𝐿^𝑝_𝑥(ℝ^𝑛)), that is, the space with mixed norm given by ‖𝐹‖_𝐿𝜈^𝑝,𝑝=_Ωℝ𝑛𝐹𝑥+𝑖𝑦^𝑝𝑑𝑥^𝑝𝑝/𝑑𝜆_𝜈(𝑦)^1/𝑝, (45) where 𝑝,𝑝∈(0,∞].

Proposition F (see [9]). Let 𝜈>(2𝑛/𝑟)−1 and 1≤𝑞≤𝑝≤∞. The following are equivalent:(a)𝑃⁺_𝜈 is bounded from 𝐿^𝑝(𝐷,𝑑𝜇_𝜈)→𝐿^𝑞(𝐷,𝑑𝜇_𝜈);(b)𝒫_𝜈^+,𝔹 is bounded from 𝐿^𝑝(𝑇_Ω,𝑑𝜆_𝜈)→𝐿^𝑞(𝑇_Ω,𝑑𝜆_𝜈);(c)𝒫_𝜈^+,𝔹 is bounded from 𝐿_𝜈^𝑝,𝑝(𝑇_Ω)→𝐿_𝜈^𝑞,𝑞(𝑇_Ω) for all 1≤𝑞≤𝑝≤∞;(d)𝒬 is bounded from 𝐿^𝑝(Ω,𝑑𝜆_𝜈)→𝐿^𝑞(Ω,𝑑𝜆_𝜈).

The Bergman projection 𝑃₀ for a bounded domain in 𝐂^𝑛, along with the associated Bergman kernel function, has proved fundamental for the theory of boundary behavior of holomohic mappings (see [27, 28] and their references).

It is well-known that on smooth bounded strongly pseudoconvex domains the Bergman projection preserves each Sobolev space 𝑊^𝑟, 𝑟≥0. The same is true more generally for those smooth bounded weakly pseudoconvex domains which admit subelliptic estimates for the 𝜕-Neumann problem (see [27, 28]).

By the mean value property for 𝑛≥1 we have for any bounded 𝐷 domain ℂ^𝑛𝑓(𝑧)≤𝐶𝑓^𝑝_𝐴𝑝0⋅(dist(𝑧,𝜕𝐷))^−𝑛,𝐴^𝑝₀=𝑓∈𝐻(𝐷)∶_𝐷𝑓(𝑧)^𝑝𝑑𝛿(𝑧)<∞,0<𝑝<∞, (46) where 𝑧∈𝐷 and 𝛿(𝑧) is Lebesgues measure on 𝐷 and dist is Euclidean distance from 𝑧 to the boundary of 𝐷.

This immediately pose a dist problem in all such very general 𝐷 domains. We give one side estimates based on recent results on Bergman projection in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains.

We follow also our sharp recent result in unit disk. Note if bounded 𝐷 domain has smooth 𝐶² boundary, then 𝑓(𝑧)dist(𝑧,𝜕𝐷)^{(𝑛+1)/𝑝}≤𝐶𝑓_𝐴^𝑝₀. (47) A bounded domain in ℂ^𝑛, Ω⊂ℂ^𝑛 is called smooth if there is a 𝐶^∞ defining function 𝜌∶ℂ^𝑛→ℝ such that Ω={𝑧∶𝜌<0} the boundary 𝑏Ω={𝑧∶𝜌(𝑧)=0} and the gradient of 𝜌 does not vanish in 𝑏Ω (see [7, 12, 14, 23, 29]).

Inequality similar to (47) also is valid for weighted Bergman spaces (see [30]).

3. On Distance Function in Bergman Type Spaces in Certain Domains in Higher Dimension

In this section we provide main results of the paper. Based fully on preliminaries of previous section, the plan of this section is the following: first we formulate a result in the unit disk and then repeat arguments we provided in proof of that theorem in various situations in Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains. Since all proofs are short the repetitions of arguments which are needed in higher dimension will be omitted; sometimes sketches will be given.

Our results on Siegel domains and minimal homogeneous domains are sharp.

Let 𝐔 be, as usual, the unit disk on the complex plane, and let 𝑑𝑚₂(𝑧) be the normalized Lebesgue measure on 𝐔. Let 𝐻(𝐔) be the space of all analytic functions on the unit disk 𝐔. For 𝑓∈𝐻(𝐔) and 𝑓(𝑧)=∑_𝑘𝑎_𝑘𝑧^𝑘, define the fractional derivative of the function 𝑓 as usual in the following manner: 𝐷^𝛼𝑓(𝑧)=^∞_𝑘=0(𝑘+1)^𝛼𝑎_𝑘𝑧^𝑘,𝛼∈ℝ. (48) We will write 𝐷𝑓(𝑧) if 𝛼=1. Obviously, for all 𝛼∈ℝ, 𝐷^𝛼𝑓∈𝐻(𝐔) if 𝑓∈𝐻(𝐔).

For 𝑘>𝑠, 0<𝑝, 𝑞≤∞, the weighted analytic Besov space 𝐴_𝑠^𝑞,𝑝(𝐔) is the class of analytic functions satisfying 𝑓^𝑞_{𝐴𝑠𝑞,𝑝}=¹_0𝐓𝐷^𝑘𝑓(𝑟𝜉)^𝑝𝑑𝜉^𝑞/𝑝(1−𝑟)^{(𝑘−𝑠)𝑞−1}𝑑𝑟<∞, (49) where 𝐓={𝜉∶|𝜉|=1} is circle and 𝑑𝜉 is the Lebesgue measure on the circle 𝐓.

We denote by 𝐴^𝑞_𝑠(𝐔) the 𝐴_𝑠^𝑞,𝑞(𝐔) analytic Besov spaces in the unit disk for all real 𝑠 numbers. Note also that for 𝑠<0 we have that these spaces are 𝐴^𝑞_{−𝑠𝑞−1}(𝐔) Bergman spaces according to definition above for unit ball and we will use this notation below for all negative 𝑠 numbers in Besov spaces.

It is well-known that 𝐴^𝑞_{−𝑠𝑞−1}(𝐔)⊂𝐴^∞_−𝑡(𝐔), 𝑡=𝑠−(1/𝑞), 𝑡<0, 𝑠<0.

Let further Ω^𝑘_𝜀,−𝑡={𝑧∈𝐔∶|𝐷^𝑘𝑓(𝑧)|(1−|𝑧|²)^𝜀≥−𝑡}, 𝜀≥0, 𝑡<0, Ω⁰_𝜀,−𝑡=Ω_𝜀,−𝑡.

It is easy to note that, based on previous section results, the complete analogues of the embedding we just provided are valid also in Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains. We leave this easy task to readers. This allows posing a dist problem in each space we consider in this paper.

In the following theorem (see [1]) we calculate distances from a weighted Bloch class to Bergman spaces for 𝑞≤1. We will see that almost each argument below is also valid not only in unit disk but based on preliminaries in previous section in Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains. So this theorem is very typical for us though it is known.

Theorem 4. Let 0<𝑞≤1, 𝑠<0, 𝑡≤𝑠−(1/𝑞), 𝛽>((1−𝑠𝑞)/𝑞)−2, and 𝛽>−1−𝑡. Let 𝑓∈𝐴^∞_−𝑡. Then the following are equivalent: (a)𝑙₁=𝑑𝑖𝑠𝑡_𝐴^∞_−𝑡(𝑓,𝐴^𝑞_{−𝑠𝑞−1}); (b)𝑙₂=inf{𝜀>0∶∫_𝐔(∫_{Ω𝜀,−𝑡(𝑓)}((1−|𝑤|)^𝛽+𝑡/|1−𝑧𝑤|^2+𝛽)𝑑𝑚₂(𝑤))^𝑞(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧)<∞}.

Proof. First we show that 𝑙₁≤𝐶𝑙₂. For 𝛽>−1−𝑡, we have 𝑓(𝑧)=𝐶𝛽_{𝐔⧵Ω𝜀,−𝑡}𝑓(𝑤)(1−|𝑤|)^𝛽(1−𝑤𝑧)^𝛽+2𝑑𝑚₂(𝑤)+_{Ω𝜀,−𝑡}𝑓(𝑤)(1−|𝑤|)^𝛽(1−𝑤𝑧)^𝛽+2𝑑𝑚₂(𝑤)=𝑓₁(𝑧)+𝑓₂(𝑧), (50) where 𝐶(𝛽) is a well-known Bergman representation constant (see [31]).
For 𝑡<0, 𝑓₁(𝑧)≤𝐶_{𝐔⧵Ω𝜀,−𝑡}𝑓(𝑤)(1−|𝑤|)^𝛽|1−𝑤𝑧|^𝛽+2𝑑𝑚₂(𝑤)≤𝐶𝜀_𝐔(1−|𝑤|)^𝛽+𝑡|1−𝑤𝑧|^𝛽+2𝑑𝑚₂(𝑤)1≤𝐶𝜀(1−|𝑧|)^−𝑡. (51) So sup_𝑧∈𝐔|𝑓₁(𝑧)|(1−|𝑧|)^−𝑡<𝐶𝜀.
For 𝑠<0, 𝑡<0, we have _𝐔𝑓₂(𝑧)^𝑞(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧)≤𝐶_{𝐔Ω𝜀,−𝑡}(1−|𝑤|)^𝛽+𝑡|1−𝑤𝑧|^𝛽+2𝑑𝑚₂(𝑤)^𝑞(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧)≤𝐶. (52) So we finally have dist_𝐴^∞_−𝑡𝑓,𝐴^𝑞_{−𝑠𝑞−1}≤𝐶𝑓−𝑓_{2𝐴∞−𝑡}=𝐶𝑓_{1𝐴∞−𝑡}≤𝐶𝜀. (53) It remains to prove that 𝑙₂≤𝑙₁. Let us assume that 𝑙₁<𝑙₂. Then we can find two numbers 𝜀,𝜀₁ such that 𝜀>𝜀₁>0, and a function 𝑓_𝜀1∈𝐴^𝑞_{−𝑠𝑞−1}, ‖𝑓−𝑓_𝜀1‖_{𝐴∞−𝑡}≤𝜀₁, and ∫_𝐔(∫_{Ω𝜀,−𝑡}((1−|𝑤|)^𝛽+𝑡/|1−𝑧𝑤|^𝛽+2)𝑑𝑚₂(𝑤))^𝑞(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧)=∞. Hence as above we easily get from ‖𝑓−𝑓_𝜀1‖_{𝐴∞−𝑡}≤𝜀₁ that (𝜀−𝜀₁)𝜒_{Ω𝜀,−𝑡(𝑓)}(𝑧)(1−|𝑧|)^𝑡≤𝐶|𝑓_𝜀1(𝑧)|, and hence 𝑀=_𝐔𝐔𝜒_{Ω𝜀,−𝑡(𝑓)}(𝑧)(1−|𝑤|)^𝛽+𝑡|1−𝑤𝑧|^𝛽+2𝑑𝑚₂(𝑤)^𝑞×(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧)≤𝐶_𝐔𝐔𝑓_𝜀1(𝑤)(1−|𝑤|)^𝛽|1−𝑤𝑧|^𝛽+2𝑑𝑚₂(𝑤)^𝑞×(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧). (54) Since for 𝑞≤1, (see [31]) _𝐔𝑓_𝜀1(𝑤)(1−|𝑤|)^𝛼|1−𝑤𝑧|^𝑡𝑑𝑚₂(𝑤)^𝑞≤𝐶_𝐔𝑓_𝜀1(𝑤)^𝑞(1−|𝑤|)^{𝛼𝑞+𝑞−2}𝑑𝑚₂(𝑤)|1−𝑤𝑧|^𝑡𝑞, (55) where 𝛼>(1−𝑞)/𝑞, 𝑡>0, 𝑓_𝜀1∈𝐻(𝐔), 𝑧∈𝐔, and _𝐔(1−|𝑧|)^{−𝑠𝑞−1}|1−𝑤𝑧|^𝑞(𝛽+2)𝑑𝑚₂(𝑧)≤𝐶(1−|𝑤|)^{𝑞(𝛽+2)+𝑠𝑞−1}, (56) where 𝑠<0, 𝛽>((1−𝑠𝑞)/𝑞)−2, 𝑤∈𝐔. we get 𝑀≤𝐶_𝐔𝑓_𝜀1(𝑧)^𝑞(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧), (57) So, we arrive at a contradiction. The theorem is proved.

The following theorem is a version of Theorem 4 for the case 𝑞>1.

Theorem 5. Let 𝑞>1, 𝑠<0, 𝑡≤𝑠−(1/𝑞), 𝛽>(−1−𝑠𝑞)/𝑞, and 𝛽>−1−𝑡. Let 𝑓∈𝐴^∞_−𝑡. Then the following are equivalent: (a)𝑙₁=𝑑𝑖𝑠𝑡_𝐴^∞_−𝑡(𝑓,𝐴^𝑞_{−𝑠𝑞−1}); (b)𝑙₂̂=inf{𝜀>0∶∫_𝐔(∫_{Ω𝜀,−𝑡(𝑓)}((1−|𝑤|)^𝛽+𝑡/|1−𝑧𝑤|^2+𝛽)𝑑𝑚₂(𝑤))^𝑞(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧)<∞}.

The proof of Theorem 5 is the same actually as the proof of Theorem 4. The only difference is the boundedness of Bergman type projection operator but with the positive Bergman kernel. This fact will be heavily used by us below.

Indeed the close inspection of the proof of Theorem 4 shows that the proof of Theorem 5 is the same as the proof of Theorem 4, but here we will use (58) (see below) instead of (55). For 𝜀>0, 𝑞>1, 𝛽>0, 𝛼>−1/𝑞, _𝐔𝑓(𝑧)(1−|𝑧|)^𝛼|1−𝑤𝑧|^𝛽+2𝑑𝑚₂(𝑧)^𝑞≤𝐶_𝐔𝑓(𝑧)^𝑞(1−|𝑧|)^𝛼𝑞|1−𝑤𝑧|^{𝛽𝑞−𝜀𝑞+2}𝑑𝑚₂(𝑧)(1−|𝑤|)^−𝜀𝑞,𝑤∈𝐔, (58) which follows immediately from Hölder’s inequality and (56).

Remark 6. Analytic 𝐴^∞_𝛼 spaces in unit disk are well-known in literature as so-called growth spaces (see, e.g., [31]). It can be shown easily that these spaces are Banach spaces. These spaces are playing a vital role in this paper and are embedded in Bergman spaces 𝐴^𝑝_𝛽, for large enough 𝛽 (the same is valid in any bounded domain with 𝐶² boundary). Hence the representation (4) is valid also in the unit disk with 𝛽 large enough index depending on 𝛼 for all functions from such classes. The mentioned embedding is well-known and almost obvious and we leave the proof of it to interested readers. This fact is crucial for the proofs of Theorems 7 and 8 below and along with well-known Forelli-Rudin estimates, for to each domain we defined, (see [31]), actually serves as base of both proofs.

We formulate a general theorem for Siegel domains of second type in ℂ^𝑛. Then we use the same ideas to formulate the same result in minimal bounded homogeneous domains in ℂ^𝑛 using very recent advances of Yamaji (see [20, 22] and references there).

We formulate finally also some new one side estimates for distances based on projection theorems for symmetric domain of tube type, Lie ball, without proofs, since arguments are similar to unit disk case. Our results for Siegel domains of second type and minimal homogeneous domains are sharp.

Theorem 7. Let 𝐷 be Siegel domain of second type. Let 𝑟⁰_𝑗𝑛=max^𝑖+222𝑑−𝑞_𝑖𝑛,−1−^𝑖22𝑑−𝑞_𝑖,𝑗=1,…,𝑙. (59) Let 𝑓∈𝐴^∞_1+𝑟 and 𝑁_̃𝜀,𝑟𝑓=𝑧∈𝐷∶𝑓(𝑧)𝑏^1+𝑟(𝑧,𝑧)>𝜀̃, (60) where 𝜀̃ is a positive number. Then the following two quantities are equivalent 𝑙₁≍𝑙₂, where 𝑙₁=𝑑𝑖𝑠𝑡_𝐴^∞_1+𝑟𝑓,𝐴^1,𝑟𝑙₂=inf𝜀̃>0∶_{𝐷𝑁̃𝜀,𝑟(𝑓)}𝑏̃^{−𝑘+1+𝑟}(𝜏,𝜏)𝑏̃(𝜏,𝑧)^𝑘+1𝑑𝜈(𝜏)×𝑏̃^−𝑟(𝑧,𝑧)𝑑𝜈(𝑧)<∞, (61)(62) for all 𝑟 and 𝑘 so that, 𝑟=(𝑟₁,…,𝑟_𝑙), 𝑘=(𝑘₁,…,𝑘_𝑙), 𝑟_𝑗∈(𝑟^𝑗₀,∞), and 𝑘_𝑗∈(𝑘₀,∞), 𝑗=1,…,𝑙 and for certain fixed vector 𝑘₀=(𝑘¹₀,…,𝑘^𝑙₀), depending on 𝑟_𝑗 and on parameters of the Siegel 𝐷 domains 𝑑_𝑖, 𝑞_𝑖, and 𝑛_𝑖.

Proof of Theorem 7. We will follow the proof of Theorem 4. First we show that 𝑙₁≤𝐶𝑙₂. Let 𝑘∈ℝ^𝑙, 𝑟∈ℝ^𝑙, 𝑝∈ℝ₊, 𝑟_𝑗≥0, 𝑗=1,…,𝑙. Then 𝑓(𝑧)=𝐶_𝜀∫_𝐷𝑓(𝑢)𝑏^1+𝑘(𝑧,𝑢)𝑏^−𝑘(𝑢)𝑑𝑢, 𝑓∈𝐴^∞_1+𝑟, 𝑧∈𝐷, 𝑘_𝑖>𝑚_𝑖 for the same large enough 𝑚_𝑖 set depending on parameters of Siegel domain, for all 𝑓∈𝐴^1,𝑟(𝐷).
Hence we have that 𝑓=𝑓₁+𝑓₂𝑓₁(𝑧)≤𝐶_{𝐷⧵𝑁̃𝜀,𝑟}𝑓(𝑢)𝑏^1+𝑘(𝑧,𝑢)𝑏^−𝑘(𝑢,𝑢)𝑑𝑢≤𝐶𝜀_𝐷𝑏^1+𝑘(𝑧,𝑤)𝑏^{−𝑘+1+𝑟}()(𝑢)()𝑢,𝑢𝑑𝜈≤𝐶𝜀𝑏𝑧,𝑧^1+𝑟,𝑧∈𝐷. (63) Also, we have that _𝐷𝑓₂(𝑧)𝑏^−𝑟()(𝑧)≤𝑧,𝑧𝑑𝜈_{𝐷𝑁̃𝜀,𝑟}𝑏^{−𝑘+1+𝑟}(𝜏,𝜏)|𝑏(𝜏,𝑧)|^𝑘+1𝑑𝜈(𝜏)𝑏^−𝑟(𝑧,𝑧)𝑑𝜈(𝑧)≤𝐶. (64) Note then we used that for 𝑘_𝑗−1−𝑟_𝑗>𝑛_𝑖+222𝑑−𝑞_𝑖𝑘_𝑗−1−𝑟_𝑗−𝑛_𝑖22𝑑−𝑞_𝑖<𝑘_𝑗,𝑗=1,…,𝑙,_𝐷𝑏^1+𝑘(𝑧,𝑢)𝑏^{−𝑘+1+𝑟}(𝑢,𝑢)𝑑𝜈(𝑢)≤𝐶𝑏(𝑧,𝑧)^1+𝑟,𝑧∈𝐷. (65)
So dist_𝐴^∞_1+𝑟(𝑓,𝐴^1,𝑟)≤𝐶‖𝑓−𝑓₂‖_𝐴^∞_1+𝑟=𝐶‖𝑓₁‖_𝐴^∞_1+𝑟≤𝐶𝜀.
We show the reverse implication now following again unit disk case arguments (see Theorems 4 and 5). We have the following. Let us assume that 𝑙₁<𝑙₂. Then we can find two numbers 𝜀, 𝜀₁ such that 𝜀>𝜀₁>0 and a function 𝑓_𝜀1∈𝐴^1,𝑟, ‖𝑓−𝑓_𝜀1‖_𝐴1,𝑟≤𝜀₁ and hence 𝜀−𝜀₁𝜒_{𝑁𝜀(𝑓)}(𝑧)𝑏(𝑧,𝑧)^1+𝑟≤𝐶𝑓_𝜀1(𝑧) (66) and hence _𝐷𝐷𝜒_{𝑁𝜀(𝑓)}(𝑧)𝑏(𝑧,𝑧)^{1+𝑟−𝑘}|𝑏(𝜏,𝑧)|^𝑘+1𝑑𝜈(𝑧)×𝑏^−𝑟(𝜏,𝜏)𝑑𝜈(𝜏)≤𝐶_𝐷𝐷𝑓_𝜀1(𝑧)𝑏(𝑧,𝑧)^−𝑘|𝑏(𝜏,𝑧)|^𝑘+1𝑑𝜈(𝑧)×𝑏^−𝑟(𝜏,𝜏)𝑑𝜈(𝜏)≤𝐶_𝐷𝑓_𝜀1(𝑧)𝑏(𝑧,𝑧)^−𝑟𝑑𝜈(𝑧). (67) We used that (see [5, 8]) _𝐷|𝑏(𝜏,𝑧)|^𝑘+1𝑏^−𝑟(𝜏,𝜏)𝑑𝜈(𝜏)≤𝐶(𝑏(𝑧,𝑧))^𝑘−𝑟𝑟,𝑧∈𝐷_𝑗>𝑛_𝑖+222𝑑−𝑞_𝑖𝑘,𝑖=1,…,𝑙,_𝑗>𝑟_𝑗−𝑛_𝑖22𝑑−𝑞_𝑖,𝑖=1,…,𝑙. (68) These estimates give a lower estimate for 𝑘₀=(𝑘¹₀,…,𝑘^𝑙₀).
Theorem 7 is proved.

For 𝑝>1 and 𝛼>−1, denote by 𝐴^𝑝_𝛼(𝐷) the space of all functions 𝑓 holomorphic in 𝐷 and satisfying the condition _𝐷𝑓(𝑤)^𝑝dist(𝑧,𝜕𝐷)^𝛼𝑑𝜈(𝑤)<+∞, (69) where 𝑑𝜈 is the Lebesgue measure in 𝐷 domain.

Let 𝐾_𝛼 denote reproducing weighted Bergman kernel for weighted Bergman space 𝐴²_𝛼 (see [11, 12]) 𝐾_𝛼(𝑧,𝑤)=𝐾(𝑧,𝑤)^1+𝛼, (70) where 𝐾(𝑧,𝑤) is a Bergman kernel for 𝐴²₀. Based on preliminaries of previous section on Bergman type spaces in minimal homogeneous domains and arguments of proof of Theorems 4 and 5 and comments related to them, we can formulate the following result.

Note that it is known that there exists a constant 𝛼_min such that 𝐴²_𝛼 is nontrivial if and only if 𝛼>𝛼_min, (see [11, 12]). From now, we consider nontrivial weighted Bergman spaces.

Theorem 8. Assume that 𝛼>𝛼_min and let 𝒰 be bounded minimal homogeneous domains with 𝐶² boundary. Let 𝐺_𝜀,𝛼(𝑓)={𝑧∈𝒰∶|𝑓(𝑧)||𝐾_𝛼(𝑧,𝑧)|^−1/2>𝜀}. Then for 𝑓∈𝐴^∞_𝛼 we have that 𝑙₁≍𝑙₂, where 𝑙₁=𝑑𝑖𝑠𝑡_𝐴^∞_𝛼𝑓,𝐴²_𝛼,𝑙₂=inf𝜀>0∶_{𝒰𝐺𝜀,𝛼}𝐾(𝑧,𝑧)^(1−𝛼)/2𝐾_𝛼(𝑧,𝑎)𝑑𝜈(𝑧)²×𝐾(𝑎,𝑎)^−𝛼𝑑𝜈(𝑎)<∞, (71) for all 𝛼>𝛼₀ for some fixed 𝛼₀.

We add only the full sketch of proof of this theorem since it is quite similar to proofs of previous theorems. We consider the operator 𝑃⁺_𝒰 defined by 𝑃⁺_𝒰𝑔(𝑧)=_𝒰𝐾_𝒰(𝑧,𝑤)𝑔(𝑤)𝑑𝜈(𝑤),𝑔∈𝐿²(𝒰,𝑑𝜈). (72)𝑃⁺_𝒰 is operator 𝒰 on 𝐿²(𝒰,𝑑𝜈). The fact that Bergman type projection 𝑃⁺_𝒰 with positive Bergman kernel is a bounded operator in 𝐿²_𝛼 spaces (weighted Lebegue spaces) can be seen in recent paper [11]. It is well-known also, since 𝐴²_𝛼 is a Hilbert space that the Bergman reproducing formula for all functions from these spaces is valid and since our domain is bounded, this space for large enough 𝛼 contains 𝐴^∞_𝛽 for any 𝛽. So the Bergman representation formula for each function taken from 𝐴^∞_𝛽 is also valid with large enough 𝛼. These two facts along with Forelly-Rudin type estimates for these domains (see [12]) complete the proof.

All other results of this paper are one side estimates for distance function and are fully based on results on boundedness of Bergman type projection with positive Bergman kernel as we have seen in unit disk case in Bergman type spaces, but over specific domain which we mentioned above in previous section as separate assertions.

Based on preliminaries of previous section on Bergman type spaces in Lie ball and arguments of proof of Theorems 4 and 5 and comments related to them, we can formulate the following result.

Theorem 9. Let 𝐷 be Lie ball with 𝐶² boundary. Let 𝑝∈((2𝑛−2)/𝑛,(2𝑛−2)/𝑛−2), 𝑛>2, 𝑓∈𝐴^∞_𝑛/𝑝. Let 𝐷_𝑡={𝑧∶|𝑓(𝑧)|𝑑𝑖𝑠𝑡(𝑧,𝜕𝐷)^𝑡≥𝜀}, 𝑡=𝑛/𝑝. Then 𝑑𝑖𝑠𝑡_𝐴^∞_𝑛/𝑝𝑓,𝐴^𝑝₀≥𝐶inf𝜀>0∶_𝐷𝐷𝜒_𝐷𝑡(𝑧)𝑑𝑖𝑠𝑡(𝑧,𝜕𝐷)^𝑡×𝐵_𝐷(𝑧,𝑤)𝑑𝜈(𝑧)^𝑝𝑑𝜈(𝑤)<∞. (73)

We omit details of the proof referring to unit disk case.

Based on preliminaries of previous section on Bergman type spaces in bounded symmetric domains of tube type and arguments of proof of Theorems 4 and 5 and comments related to them, we can formulate the following result.

Theorem 10. Let 𝐷 be bounded symmetric domain of tube type with 𝐶² boundary. Let 𝑝∈(1,∞), 𝑛>2, 𝑓∈𝐴^∞_𝑛/𝑝. Let 𝐷¹_𝑡={𝑧∶|𝑓(𝑧)|𝑑𝑖𝑠𝑡(𝑧,𝜕𝐷)^𝑡≥𝜀}, 𝑡=𝑛/𝑝. Then 𝑑𝑖𝑠𝑡_𝐴^∞_𝑛/𝑝𝑓,𝐴^𝑝₀≥𝐶inf𝜀>0∶_𝐷𝐷𝜒_𝐷1𝑡(𝑧)𝑑𝑖𝑠𝑡(𝑧,𝜕𝐷)^𝑡𝐵^𝐷_𝜈×(𝑧,𝑤)𝑑𝜈(𝑧)^𝑝𝑑𝜈(𝑤)<∞. (74)

Remark 11. It is known that (see [30]) if 𝛼>−1 and 𝑝∈(0,∞) and if 𝐷 is general bounded domain with 𝐶² boundary then dist(𝑧,𝜕𝐷)^{(𝑛+𝛼+1)/𝑝}𝑓(𝑧)≤𝐶𝑓_𝐴^𝑝_𝛼. (75) This allows posing a dist problem for weighted Bergman spaces in such type domains and based on preliminaries of previous section to formulate Theorem 9 even for weighted Bergman spaces. We leave this to readers.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.