On Distance Function in Some New Analytic Bergman Type Spaces in ℂ^𝑛

Abstract

We extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new analytic Bergman type spaces in higher dimensions in ℂ^𝑛. Also, we study the same problem in anisotropic mixed norm ℎ(𝑝,𝑞,𝑠) spaces consisting of 𝑛-harmonic functions on the unit polydisc of ℂ^𝑛.

1. Introduction and Preliminaries

The goal of this paper is to add several new results for distances in analytic Bergman type and Bergman type n-harmonic function spaces of functions of several variables.

It turns out that some of our distance theorem, which we proved before for the case of unit disk, are valid under certain natural conditions in various domains and various Bergman type analytic spaces. Namely, we look at analytic Bergman spaces in minimal ball and so-called analytic 𝐾-Bergman spaces and in analytic Bergman-Orlicz spaces. 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 turn also to polydisk and 𝑛-harmonic function spaces on them providing also a sharp result there. Note in less general case, namely, in analytic Bergman type spaces in polydisk, this result is already known (see [1, 2]).

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

Following our previous paper [1, 2], we easily see that, to obtain a sharp result for distance function, we only need several tools and the scheme here is as follows. 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, and then we need the Bergman reproducing formula for all 𝑓 functions 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.

First we provide a known result in the unit disk with complete proof taken from our previous paper ([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, and then we will see arguing similarly as in unit disk; we will complete the proof of more complicated cases easily. The complete formulation of our theorems will be given, but only 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.

We will need various definitions for formulations of main results. All facts about polydisk are taken from [5].

Let 𝑛∈ℕ and ℂ^𝑛={𝑧=(𝑧₁,…,𝑧_𝑛):𝑧_𝑘∈ℂ, 1≤𝑘≤𝑛} be the 𝑛-dimensional space of complex coordinates. We denote the unit polydisk by 𝐔^𝑛=𝑧∈ℂ^𝑛:z_𝑘<1,1≤𝑘≤𝑛 (1) and the distinguished boundary of 𝐔^𝑛 by 𝑇^𝑛=𝑧∈ℂ^𝑛:𝑧_𝑘=1,1≤𝑘≤𝑛. (2)

By 𝑚_2𝑛 we denote the volume measure on 𝐔^𝑛 and by 𝑚_𝑛 we denote the normalized Lebesgue measure on 𝑇^𝑛. Let 𝐻(𝐔^𝑛) be the space of all holomorphic functions on 𝐔^𝑛.

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.

We denote the interval (0,1) by 𝐼, accordingly (0,1)^𝑛 by 𝐼^𝑛.

The Hardy spaces, denoted by 𝐻^𝑝(𝐔^𝑛)(0<𝑝≤∞), are defined by 𝐻^𝑝(𝐔^𝑛)={𝑓∈𝐻(𝐔^𝑛):sup_0≤𝑟<1𝑀_𝑝(𝑓,𝑟)<∞}, where 𝑀^𝑝_𝑝𝑓,𝑟=_𝑇𝑛𝑓(𝑟𝜉)^𝑝𝑑𝑚_𝑛(𝜉),𝑀_∞𝑓,𝑟=max_{𝜉∈𝑇𝑛}𝑓(,𝐔𝑟𝜉)𝑟∈𝐼,𝑓∈𝐻^𝑛. (3) As usual, we denote by 𝛼^→ the vector (𝛼₁,…,𝛼_𝑛).

For 𝛼_𝑗>−1, 𝑗=1,…,𝑛,0<𝑝<∞, recall that the weighted Bergman space 𝐴^𝑝_→𝛼(𝐔^𝑛) consists of all holomorphic functions on the polydisk such that 𝑓^𝑝_{𝐴𝑝→𝛼}=_𝐔𝑛𝑓(𝑧)^𝑝𝑛_𝑖=11−𝑧_𝑖^2𝛼𝑖𝑑𝑚_2𝑛(𝑧)<∞. (4)

When 𝛼₁=⋯=𝛼_𝑛=𝛼, then we use notation 𝐴^𝑝_𝛼(𝐔^𝑛).

If 𝑢 is 𝑛-harmonic (harmonic by each variable), then as usual 𝑢𝑟₁𝑒^𝑖𝜑1,…,𝑟_𝑛𝑒^𝑖𝜑𝑛=^∞_{𝑘1,…,𝑘𝑛=−∞}𝐶_{𝑘1,…,𝑘𝑛}^𝑛_𝑗=1𝑟^|𝑘𝑗|_𝑗𝑒^{𝑖𝑘𝑗𝜑𝑗}. (5) Let further ℎ𝛼^𝑝→=𝑢is𝑛-harmonic:_{𝐔𝑛𝑛𝑘=1}1−𝑧_𝑘^𝛼𝑘𝑢𝑧₁,…,𝑧_𝑛^𝑝𝑑𝑚_2𝑛(𝑧)<∞,0<𝑝<∞,𝛼_𝑘>−1,𝑘=1,…,𝑛. (6)

Also, we denote by 𝐁_𝑛 the unit ball in ℂ^𝑛,𝐁_𝑛={𝑤∈ℂ^𝑛:|𝑤|<1}. 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, satisfying the condition _𝐁𝑛𝑓(𝑤)^𝑝1−|𝑤|^2𝛼𝑑𝜈(𝑤)<+∞, (7) where 𝑑𝜈 is the Lebesgue measure in ℂ^𝑛≡ℝ^2𝑛 (see [6]).

Further, for a complex number 𝛽 with Re𝛽>−1, put 𝑐_𝑛𝛽=Γ𝑛+1+𝛽𝜋^𝑛.⋅Γ1+𝛽 (8)

Let 𝐴^∞_𝛼(𝐁_𝑛)={𝑓∈𝐻(𝐁_𝑛):sup_{𝑧∈𝐁𝑛}|𝑓(𝑧)|(1−|𝑧|)^𝛼<∞},𝛼>0.

Remark 1. Analytic 𝐴^∞_𝛼 spaces are well known in literature as so-called growth spaces (see, e.g., [5]). 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. Hence the representation (10) with 𝛽 large enough index depending on 𝛼 for all functions from such classes is valid. The mentioned embedding is well known and almost obvious and we leave the proof of it to interested readers. This fact nevertheless is crucial for the proof of Theorems 18 and 20 below and actually serves as base of both proofs, along with well-known Forelli-Rudin estimates for unit ball (see [5]).

We have the following theorem.

TheoremA (see [6]). Assume that 1≤𝑝<+∞,𝛼>−1 and that the complex number 𝛽 satisfies the condition Re𝛽≥𝛼,𝑝=1,Re𝛽>𝛼+1𝑝−1,1<𝑝<∞. (9) 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, (10)(11) where ⟨⋅,⋅⟩ is the Hermitian inner product in ℂ^𝑛.

For 𝑛>1, the theorem was proved in [7] (when 𝛼=0) and in [8, 9] (when 𝛼>−1).

In monographs [5] one can find numerous applications of the formulas (10) and (11) in the complex analysis. In [6] the author generalized Theorem A to the space with anisotropic weight function such as assuming that 1≤𝑝<+∞,𝛼=(𝛼₁,…,𝛼_𝑛)∈ℝ^𝑛 satisfies the conditions 𝛼_𝑛𝛼>−1,_𝑛+𝛼_𝑛−1𝛼>−2,_𝑛+𝛼_𝑛−1+𝛼_𝑛−2⋮𝛼>−3,_𝑛+𝛼_𝑛−1+𝛼_𝑛−2+⋯+𝛼₂(),𝛼>−𝑛−1_𝑛+𝛼_𝑛−1+𝛼_𝑛−2+⋯+𝛼₂+𝛼₁>−𝑛, (12) and then introduced the spaces 𝐴^𝑝_𝛼(𝐁_𝑛) of functions 𝑓 holomorphic in 𝐁_𝑛 satisfying the condition _𝐁𝑛𝑓(𝑤)^𝑝𝑛_𝑖=11−𝑤₁²−𝑤₂²−⋯−𝑤_𝑖^2𝛼𝑖𝑑𝜈(𝑤)<+∞. (13)

For these anisotropic spaces, similarities of the integral representations (10) and (11) are obtained, but in the mentioned generalization special kernels 𝑆_{𝛽1𝛽2⋯𝛽𝑛}(𝑧,𝑤)≡𝑆_𝛽(𝑧,𝑤) (where 𝛽_𝑖,1≤𝑖≤𝑛, are associated with 𝛼_𝑖,1≤𝑖≤𝑛, and 𝑝 in a special way) appear instead of 𝑐_𝑛(𝛽)⋅(1−⟨𝑧,𝑤⟩)^{−(𝑛+1+𝛽)} (see [6], Theorem 4.7).

Suppose now that 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛)∈ℝ^𝑛. We put 𝑚_𝛼=_𝛼𝑖<0−𝛼_𝑖,𝑙_𝛼=_𝛼𝑖<0𝛼_𝑖. (14) We will write 𝛼≺(∗) only if the following conditions are satisfied: 𝛼_𝑛+𝛼_𝑛−1+⋯+𝛼_𝑖>−(𝑛+1−𝑖)(1≤𝑖≤𝑛). (15) Similarly, if 𝛽=(𝛽₁,𝛽₂,…,𝛽_𝑛)∈ℂ^𝑛, then we will write 𝛽≺(∗) if only Re𝛽≡(Re𝛽₁,Re𝛽₂,…,Re𝛽_𝑛)≺(∗).

We have the following lemmas.

Lemma 2 (see [6]). If 𝛼≺(∗), then 1−𝑤_𝐁𝑛^𝑛_𝑖=11²−𝑤₂²−⋯−𝑤_𝑖^2𝛼𝑖(𝑤)=𝜋𝑑𝜈_𝑛∏^𝑛_𝑖=1𝛼_𝑛+𝛼_𝑛−1+⋯+𝛼_𝑖+𝑛+1−𝑖<+∞. (16)

We need also heavily the well-known Forelli-Rudin type estimates for unit ball (see [5]).

Proposition 3. Suppose 𝑐>0 is real and 𝑡>−1. Then the integral 𝐽_𝑐,𝑡(𝑧)=_𝐁𝑛1−|𝑤|^2𝑡𝑑𝜈(𝑤)|1−⟨𝑧,𝑤⟩|^{𝑛+1+𝑡+𝑐},𝑧∈𝐁_𝑛, (17) has the following asymptotic property: 𝐽_𝑐,𝑡∼1−|𝑧|^2−𝑐as|𝑧|⟶1−. (18)

Definition 4. Assume that 0<𝑝<+∞, 𝛼=(𝛼₁,…,𝛼_𝑛)≺(∗). Denote by 𝐿^𝑝_𝛼(𝐁_𝑛) the space of all complex-valued functions 𝑓 in 𝐁_𝑛 with 𝑀^𝑝_𝛼𝑓≡_𝐁𝑛𝑓(𝑤)^𝑝×^𝑛_𝑖=11−𝑤₁²−𝑤₂²−⋯−𝑤_𝑖^2𝛼𝑖𝑑𝜈(𝑤)<∞. (19) We obviously have 𝑀^𝑝_𝛼𝑓+𝑔≤2^𝑝𝑀^𝑝_𝛼𝑓+𝑀^𝑝_𝛼𝑔,𝑀^𝑝_𝛼𝑐𝑓=𝑐^𝑝𝑀^𝑝_𝛼𝑓. (20)

Lemma 5 (see [6]). Assume that 0<𝑝<+∞, 𝛼≺(∗), and 𝑓∈𝐻(𝐁_𝑛). Then 𝑓(𝑧)^𝑝≤2^{2𝑛+𝑙𝛼}𝑐_𝑛⋅(1−|𝑧|)^{2𝑛+𝑙𝛼}⋅𝑀^𝑝_𝛼𝑓∀𝑧∈𝐁_𝑛, (21) where 𝑐_𝑛=𝜈(𝐁_𝑛) is the volume of the unit ball of ℂ^𝑛.

Note that this lemma leads to an embedding theorem immediately which allows us to pose a distance problem as unit disk case.

Corollary 6 (see [6]). 𝐻^𝑝_𝛼(𝐁_𝑛) is a closed subspace in 𝐿^𝑝_𝛼(𝐁_𝑛).

We will need some facts from theory of Bergman-Orlicz spaces (see [10, 11]).

Let now Ψ:[0,∞)→[0,∞) be a continuous nondecreasing function which vanishes and is continuous at 0. Given a probabilistic space (Ω,ℙ), we define the Orlicz class 𝐿^Ω(Ω,ℙ) as the set of all (equivalence classes of) measurable functions 𝑓 on Ω such that ∫_ΩΨ(|𝑓|/𝐶)𝑑ℙ<∞ for some 0<𝐶<∞. We used to define the Morse-Transue space 𝑀^Ψ(Ω,ℙ) by 𝑀^Ψ(Ω,ℙ)=𝑓:Ω⟶ℂmeasurable,_ΩΨ𝑓𝐶𝑑ℙ<∞forany𝐶>0, (22) and we also introduce the following set: ℒ^Ψ()=𝑓Ω,ℙ:Ω⟶ℂmeasurable,_ΩΨ𝑓𝑑ℙ<∞. (23)

These three sets are not vector spaces and do not coincide, but we have 𝑀^Ψ(Ω,ℙ)⊂ℒ^Ψ(Ω,ℙ)⊂𝐿^Ψ(Ω,ℙ). (24)

We also define the Luxembourg gauge on 𝐿^Ψ(Ω,ℙ) by 𝑓_Ψ=inf𝜆>0,_ΩΨ𝑓𝜆𝑑ℙ≤1. (25)

This functional is homogeneous and is 0 if and only if 𝑓=0 but it is not subadditive.

We say that the two functions Ψ₁ and Ψ₂ as above are equivalent if there exists some constant 𝐶 such that 𝐶Ψ₁(𝑐𝑥)≤Ψ₂(𝑥)≤𝐶⁻¹Ψ₁𝐶⁻¹𝑥, (26)

for any 𝑥 large enough. Two equivalent functions define the same Orlicz class with equivalent Luxembourg functionals.

In order to define a good topology on 𝐿^Ψ(Ω,ℙ) and to get properties convenient for our purpose, we will assume that Ψ satisfies the following definition.

Definition 7. Let 0<𝑝≤1. One says that Ψ:[0,∞)→[0,∞) is a growth function of order 𝑝 if it satisfies the two following conditions. (1)Ψ is of lower type 𝑝, that is, Ψ(𝑦𝑥)≤𝑦^𝑝Ψ(𝑥) for any 0<𝑦≤1 and at least for 𝑥 large enough.(2)𝑥↦Ψ(𝑥)/𝑥 is nonincreasing, at least for every 𝑥 large enough.We will say that Ψ is a growth function if it is a growth function of order 𝑝 for some 0<𝑝≤1.

Sometimes we will use lower 𝑝 index for our growth function below.

In particular, a growth function Ψ is equivalent to the function 𝑥↦∫^𝑥₀(Ψ(𝑠)/𝑠)𝑑𝑠 which is concave (see [12]). Now, for such a concave growth function of order 𝑝, 𝐿¹(Ω,ℙ)⊂𝑀^Ψ(Ω,ℙ)=ℒ^Ψ(Ω,ℙ)=𝐿^Ψ⊂𝐿^𝑝(),𝑓Ω,ℙ_Ψ≲min_ΩΨ𝑓𝑑ℙ,_ΩΨ𝑓𝑑ℙ^𝑝, (27) while 𝑓_Ψ∶=_ΩΨ𝑓𝑑ℙ≲max𝑓_Ψ,𝑓^𝑝_Ψ. (28)

Moreover, if we define 𝑑_Ψ(𝑓,𝑔)=‖𝑓−𝑔‖_Ψ (𝑑^Ψ(𝑓,𝑔)=‖𝑓−𝑔‖^Ψ), then 𝑑_Ψ and 𝑑^Ψ (that we simply denote by ‖⋅‖_Ψ and ‖⋅‖^Ψ) are two equivalent metrics on 𝐿^Ψ(Ω,ℙ) for which it is complete. Without loss of generality, we then assume that every growth function that we consider further is concave and diffeomorphic.

When we deal with spaces of holomorphic functions, it is very natural to require subharmonicity. Then we will assume that Ψ is such that Ψ(|𝑓|) is subharmonic when 𝑓 is holomorphic. We will refer to such a function as a subharmonic-preserving function.

Here are some examples of (concave) growth function that we may consider further.

For Ψ a subharmonic-preserving growth function and 𝛼>−1, the weighted Bergman-Orlicz space 𝐴^Ψ_𝛼 of the ball consists of those holomorphic functions on 𝐁_𝑛 which belongs to the Orlicz space 𝐿^Ψ(𝐁_𝑛,𝜈_𝛼), where, as usual, for any real number 𝛼,𝑑𝜈_𝛼(𝑧)=𝑐_𝛼(1−|𝑧|²)^𝛼𝑑𝜈(𝑧) for |𝑧|<1. Here, if 𝛼≤−1, 𝑐_𝛼=1 and if 𝛼>−1, 𝑐_𝛼=Γ(𝑛+1+𝛼)/Γ(𝑛+1)Γ(𝛼+1) is the normalizing constant so that 𝜈_𝛼 has unit total mass.

Further, we will denote by ‖⋅‖_𝛼,Ψ the corresponding Luxembourg (quasi)norm and by ‖⋅‖^Ψ_𝛼 the quantity ∫_𝐁𝑛Ψ(|𝑓|)𝑑𝜈_𝛼.𝐴^Ψ_𝛼 is metric space for the distance 𝑑_𝛼,Ψ or 𝑑^Ψ_𝛼 defined by, respectively, 𝑑_𝛼,Ψ(𝑓,𝑔)=‖𝑓−𝑔‖_𝛼,Ψ and 𝑑^Ψ_𝛼(𝑓,𝑔)=‖𝑓−𝑔‖^Ψ_𝛼. If Ψ(𝑡)=𝑡^𝑝, then we recover the usual weighted Bergman space 𝐴^𝑝_𝛼. We have the following inclusions: 𝐴¹_𝛼⊂𝐴^Ψ_𝛼⊂𝐴^𝑝_𝛼, (29) whenever Ψ is a growth function of order 𝑝.

Note that the second inclusion which we see before leads to another inclusion, namely, 𝐴^Ψ_𝛼, is embedded in 𝐴^∞_{(𝛼+𝑛+1)/𝑝}. This last inclusion will lead us to distance theorem. We showed a complete analogue of a theorem in the unit disk below.

It is important to mention that a linear operator 𝑇 from 𝐴^Ψ1_𝛼 to 𝑋 with 𝑋=𝐿^Ψ2(𝐁_𝑛,𝜈_𝛼) or 𝑋=𝐴^Ψ2_𝛼, where Ψ₁ and Ψ₂ are two growth functions, is continuous if and only if it maps a bounded set into a bounded set, or equivalently if and only if there exists a constant 𝐶>0 such that max𝑇𝑓_𝛼,Ψ2,𝑇𝑓^Ψ2_𝛼≤𝐶, (30) for any 𝑓∈𝐴^Ψ1_𝛼 such that min(‖𝑓‖_𝛼,Ψ1,‖𝑓‖^Ψ1_𝛼)≤1. If 𝑋=ℂ, then 𝑇 is bounded if and only if |𝑇(𝑓)|≤𝐶 for 𝑓 as previously.

The next proposition says that the point evaluation functionals are continuous on 𝐴^Ψ_𝛼.

Proposition 8 (see [11]). Let 𝛼>−1 and let Ψ be a subharmonic-preserving growth function. For any 𝛼∈𝐁_𝑛 and any 𝑓∈𝐴^Ψ_𝛼, one has 𝑓(𝑎)≤Ψ⁻¹21−|𝑎|^{(𝑛+𝛼+1)}𝑓_𝛼,Ψ. (31)

The proof is the same as that of [10, Proposition 1.9] and so is omitted (still use the hypothesis that Ψ(|𝑓|) is subharmonic). We easily deduce from this and the completeness of 𝐿^Ψ the following result.

Corollary 9 (see [11]). 𝐴^Ψ_𝛼, endowed with 𝑑_𝛼,Ψ or 𝑑^Ψ_𝛼, is a complete metric space.

Some facts from theory of 𝑛-harmonic function spaces will be needed by us. We will use the same notation for 𝑀_𝑝(𝑓,𝑟) for harmonic and analytic functions.

Definition 10. The quasinormed space 𝐿(𝑝,𝑞,𝛼) (0<𝑝,𝑞≤∞,𝛼=(𝛼₁,…,𝛼_𝑛), 𝛼_𝑗>−1, 𝑗=1,…,𝑛 is the set of those functions 𝑓(𝑧) measurable in the polydisc 𝐔^𝑛, for which the quasinorm 𝑓_{𝑝,𝑞,𝛼}=^{𝐼𝑛𝑛𝑗=1}1−𝑟_𝑗^{𝛼𝑗𝑞−1}𝑀^𝑞_𝑝𝑓,𝑟^𝑛_𝑗=1𝑑𝑟_𝑗^1/𝑞,0<𝑞<∞,esssup_{𝑟∈𝐼𝑛}^𝑛_𝑗=11−𝑟_𝑗^𝛼𝑗𝑀_𝑝𝑓,𝑟,𝑞=∞, (32) is finite. For the subspace of 𝐿(𝑝,𝑞,𝛼) consisting of 𝑛-harmonic functions let ℎ(𝑝,𝑞,𝛼)=ℎ(𝐔^𝑛)∩𝐿(𝑝,𝑞,𝛼) (see [13]).

For 𝑝=𝑞<∞ the spaces ℎ(𝑝,𝑞,𝛼) coincide with the well-known weighted Bergman spaces, while for 𝑞=∞ they are known as growth spaces.

Let now 1≤𝑝,𝑞≤∞,𝛼_𝑗>0. The following inclusion will be used by us (see [13]): ℎ𝑝,𝑞,𝛼⊂ℎ(∞,∞,𝛼). (33)

This inclusion as in unit disk case (see [2]) allows us to pose a distance problem in spaces of 𝑛-harmonic functions we consider in this paper (we also did the same in spaces of analytic functions in the unit polydisk (see [1])).

Now we define Poisson kernel 𝑃_𝛼 in the unit disk 𝑃_𝛼(𝑧)(2=Γ𝛼+1)Re(1−𝑧)^𝛼+1−1,𝑧∈𝐔,𝛼≥0. (34) Let 𝑃_𝛼(𝑧,𝜉)=𝑃_𝛼(𝑧𝜉). For polydisk we have 𝑃_𝛼(𝑧,𝜉)=^𝑛_𝑗=1𝑃_𝛼𝑗𝑧_𝑗,𝜉_𝑗, (35) where 𝛼=(𝛼₁,…,𝛼_𝑛),𝛼_𝑗≥0,𝜉∈𝐔^𝑛,𝑧∈𝐔^𝑛. 𝑃_𝛼 is 𝑛-harmonic by both variables 𝑧 and 𝜉 and obviously holds 𝑃_𝛼(𝑧,𝜉)=𝑃_𝛼(𝜉,𝑧)=𝑃_𝛼(𝑧,𝜉) (see [13]).

Reproducing integral formula Poisson-Bergman type will be presented in the following theorem.

Theorem B (see [13]). Let 𝛼_𝑗>0, 𝑢∈ℎ(𝑝,𝑞,𝛼). If 0<𝑝, 𝑞≤∞, 𝛽_𝑗>max{𝛼_𝑗+1/𝑝−1,𝛼_𝑗}, or 1≤𝑝≤∞, 0<𝑞≤1, 𝛽_𝑗≥𝛼_𝑗 (1≤𝑗≤𝑛)𝑢(𝑧)=1∏^𝑛_𝑗=1Γ𝛽_𝑗×_𝐔𝑛^𝑛_𝑗=11−𝜁^{2𝛽𝑗−1}𝑃_𝛽(𝑧,𝜁)𝑢(𝜁)𝑑𝑚_2𝑛(𝜁),𝑧∈𝐔^𝑛. (36)

Let 𝑆_𝛽,𝜆 be a linear operator Bergman type 𝑆_𝛽,𝜆𝑓(𝑧)=1−|𝑧|^2𝜆Γ×𝛽+𝜆_𝐔𝑛1−𝜉^2𝛽−1𝑃_𝛽+𝜆(𝑧,𝜉)𝑓(𝜉)𝑑𝑚_2𝑛(𝜉). (37)

Theorem C (see [13]). Let 1≤𝑝,𝑞≤∞,𝛼_𝑗,𝛽_𝑗,𝜆_𝑗∈ℝ,𝛽_𝑗>𝛼_𝑗>−𝜆_𝑗,1≤𝑗≤𝑛. Then the operator 𝑆_𝛽,𝜆 continuously maps the space 𝐿(𝑝,𝑞,𝛼) into itself. Moreover, then operator 𝑆_𝛽,𝜆 is bounded in 𝐿(𝑝,𝑞,𝛼) if and only if 𝛽_𝑗>𝛼_𝑗>−𝜆_𝑗,1≤𝑗≤𝑛.

We will also need basic facts from theory of Bergman spaces in minimal ball (see [14, 15]). For the convenience of readers we will give some details about them.

We consider the domain 𝔹_∗ in ℂ^𝑛,𝑛≥2, defined by 𝔹_∗=𝑧∈ℂ^𝑛:|𝑧|²+|𝑧•𝑧|<1, (38) where 𝑧•𝑤=^𝑛_𝑗=1𝑧_𝑗𝑤_𝑗, (39) for 𝑧 and 𝑤 in ℂ^𝑛. This is the unit ball of ℂ^𝑛 with respect to the norm 𝑁_∗(𝑧)∶=|𝑧|²+|𝑧•𝑧|,𝑧∈ℂ^𝑛. (40) The norm 𝑁∶=𝑁_∗/√2 was introduced by Hahn and Pflug in [16], where it was shown to be the smallest norm in ℂ^𝑛 that extends the Euclidean norm in ℝ^𝑛 under certain restrictions.

The domain 𝔹_∗ has since been studied by Hahn and Pflug [16], Mengotti and Youssfi [15], and Oeljeklaus and Youssfi [17]. In particular, it is well known that the automorphism group of 𝔹_∗ is compact and its identity component is Aut⁰_𝒪𝔹_∗=𝑆¹⋅𝑆𝑂(𝑛,ℝ), (41) where the 𝑆¹-action is diagonal and the 𝑆𝑂(𝑛,ℝ)-action is by matrix multiplication (see [17], e.g.). It is also well known that 𝔹_∗ is a nonhomogeneous domain. Its singular boundary consists of all points 𝑧 with 𝑧•𝑧=0, and the regular part of the boundary of 𝔹_∗ consists of strictly pseudoconvex points.

For each 𝑠>−1 we let 𝑣_𝑠 denote the measure on 𝔹_∗, defined by 𝑑𝑣_𝑠(𝑧)∶=1−𝑁²_∗(𝑧)^𝑠𝑑𝑣(𝑧), (42) where 𝑣 denotes the normalized Lebesgue measure on 𝔹_∗. For all 0<𝑝<∞ we define 𝒜^𝑝_𝑠𝔹_∗∶=ℋ𝔹_∗∩𝐿^𝑝𝔹_∗,|𝑧•𝑧|^(𝑝−2)/2𝑑𝑣_𝑠, (43) where ℋ(𝔹_∗) is the space of all holomorphic functions on 𝔹_∗. Naturally, the spaces 𝒜^𝑝_𝑠(𝔹_∗) are called weighted Bergman spaces of 𝔹_∗.

When 𝑝=2, there exists an orthogonal projection from 𝐿²(𝔹_∗,𝑑𝑣_𝑠) onto 𝒜²_𝑠(𝔹_∗). This will be called the weighted Bergman projection and is denoted by 𝐏_𝑠,𝔹∗.

It is well known that 𝐏_𝑠,𝔹∗ is an integral operator on 𝐿²(𝔹_∗,𝑑𝑣_𝑠), namely, 𝐏_𝑠,𝔹∗𝑓(𝑧)=_𝔹∗𝐊_𝑠,𝔹∗(𝑧,𝑤)𝑓(𝑤)𝑑𝑣_𝑠(𝑤), (44) where 𝐊_𝑠,𝔹∗()=𝐀()𝑧,𝑤𝑋,𝑌𝑛²+𝑛−𝑠𝑣_𝑠𝔹_∗𝑋²−𝑌^𝑛+1+𝑠 (45) is the weighted Bergman kernel. Here 𝑋=1−𝑧•𝑤,𝑌=(𝑧•𝑧)𝑤•𝑤, (46) and 𝐀(𝑋,𝑌) is the sum ^∞_𝑗=0𝑐_{𝑛,𝑠,𝑗}𝑋^{𝑛+𝑠−1−2𝑗}𝑌^𝑗×2()()𝑛+𝑠𝑋−𝑛−2𝑗+𝑠𝑛+1+2𝑠𝑋𝑛+𝑠+1²−𝑌. (47) As usual, 𝑐_{𝑛,𝑠,𝑗}==()⋯𝑛+𝑠+12𝑗+1𝑛+𝑠+1)(𝑛+𝑠𝑛+𝑠−2𝑗+12𝑗+1! (48) is the binomial coefficient and 𝑣_𝑠(𝔹_∗) is the weighted Lebesgue volume of 𝔹_∗. See [15, 17] for these formulas and more information about the weighted Bergman kernels.

Because of the infinite sum 𝐴(𝑋,𝑌) the formula for 𝐊_𝑠,𝔹∗ is not really a closed form. As such, it is inconvenient for us to do estimates for 𝐏_𝑠,𝔹∗ directly. Thus we employ a technique that was used in [15]. More specifically, we relate the domain 𝔹_∗ to the hypersurface 𝕄 of the Euclidean unit ball in ℂ^𝑛+1 defined by 𝕄=𝑧∈ℂ^𝑛+1⧵{0}:𝑧•𝑧=0,|𝑧|<1. (49) If Pr∶ℂ^𝑛+1→ℂ^𝑛 is defined by Pr𝑧₁,…,𝑧_𝑛,𝑧_𝑛+1=𝑧₁,…,𝑧_𝑛, (50) and 𝐅=Pr_∣𝕄, then 𝐅∶𝕄→𝔹_∗−{0}. Let ℍ=𝑧∈ℂ^𝑛+1⧵{0}:𝑧•𝑧=0. (51) It was proved in [17] that there is an 𝑆𝑂(𝑛+1,ℂ)-invariant holomorphic form 𝛼 on ℍ. Moreover, this form is unique up to a multiplicative constant. In fact, after appropriate normalization, the restriction to ℍ∩(ℂ⧵{0})^𝑛+1 of this form is given by 𝛼(𝑧)=^𝑛+1_𝑗=1(−1)^𝑗−1𝑧_𝑗𝑑𝑧₁∧⋯∧𝑑𝑧_𝑗∧⋯∧𝑑𝑧_𝑛+1. (52)

Our norm estimates for the weighted Bergman projection 𝐏_𝑠,𝔹∗ on 𝐿^𝑝 spaces will be based on the corresponding estimates on 𝕄. Therefore, we will consider the spaces 𝐿^𝑝_𝑠(𝕄) consisting of measurable complex-valued functions 𝑓 on 𝕄 such that 𝑓^𝑝_{𝐿𝑝𝑠(𝕄)}=_𝕄𝑓(𝑧)^𝑝1−|𝑧|^2𝑠𝛼(𝑧)∧𝛼(𝑧)𝐶<∞, (53) where 𝐶∶=(−1)^{𝑛(𝑛+1)/2}(2𝑖)^𝑛. (54) We use 𝐴^𝑝_𝑠(𝕄) to denote the subspace of all holomorphic functions in 𝐿^𝑝_𝑠(𝕄). The weighted Bergman projection 𝐏_𝑠,𝕄 is then the orthogonal projection from 𝐿²_𝑠(𝕄) onto 𝐴²_𝑠(𝕄). Again, it is well known that 𝐏_𝑠,𝕄 is an integral operator on 𝐿²_𝑠(𝕄) given by the formula 𝐏_𝑠,𝕄𝑓(𝑧)=_𝕄𝐊_𝑠,𝕄(𝑧,𝑤)𝑓(𝑤)1−|𝑤|^2𝑠𝛼(𝑧)∧𝛼(𝑧)𝐶, (55) where 𝐊_𝑠,𝕄 is the corresponding Bergman kernel.

The starting point for our analysis is the following closed form of 𝐊_𝑠,𝕄, which was obtained as Theorem 3.2 in [15].

Lemma 11 (see [14]). The weighted Bergman kernel 𝐊_𝑠,𝕄 of 𝐴²_𝑠(𝕄) is given by 𝐊_𝑠,𝕄()=𝐶(()𝑧,𝑤𝑛−1+𝑛+1+2𝑠𝑧•𝑤)(1−𝑧•𝑤)^𝑛+𝑠+1, (56) where 𝐶 is a certain constant that depends on 𝑛 and 𝑠.

Let 𝑓:𝔹_∗→ℂ be a measurable function. We define a function 𝐓𝑓 on 𝕄 by (𝑧)𝑧𝐓𝑓∶=^𝑛+12(𝑛+1)^21/𝑝(𝑧)=𝑧𝑓∘𝐅_𝑛+1𝑓𝑧₁,…,𝑧_𝑛2(𝑛+1)^21/𝑝. (57) The operator 𝐓 will also play a key role in our analysis. In particular, we need the following result which was obtained as Lemma 4.1 in [10].

Lemma 12 (see [14]). For each 𝑝⩾1 and 𝑠>−1 the linear operator 𝐓 is an isometry from 𝐿^𝑝(𝔹_∗,|𝑧•𝑧|^(𝑝−2)/2𝑑𝑣_𝑠) to 𝐿^𝑝_𝑠(𝕄). Moreover, we have 𝐏_𝑠,𝕄𝐓=𝐓𝐏_𝑠,𝔹∗ on 𝐿^𝑝(𝔹_∗,|𝑧•𝑧|^(𝑝−2)/2𝑑𝑣_𝑠).

Theorem B of [15] states that the operator with positive kernel 𝐏_𝑠,𝔹∗ maps 𝐿^𝑝(𝔹_∗,|𝑧•𝑧|^(𝑝−2)/2𝑑𝑣_𝑠) boundedly onto 𝒜^𝑝_𝑠(𝔹_∗) for all 𝑝>1. We need to derive an improved version of the classical Forelli-Rudin integral estimates in the case of the minimal ball.

Lemma 13 (see [14]). Suppose 𝑇>0, 𝐴>−1, and 𝑑 is a nonnegative integer. Then there exists a constant 𝐶>0 (depending on 𝑑, 𝑇, and 𝐴, but not on 𝑡 and 𝑠) such that _𝕄|𝑧•𝑤|^2𝑑|1−𝑧•𝑤|^{𝑛+𝑡+𝑠+1}1−|𝑤|^2𝑠𝛼(𝑤)∧𝛼(𝑤)⩽𝐶Γ(𝑠+1)Γ(𝑡)1−|𝑧|^2𝑡, (58) for all −1<𝑠<𝐴,0<𝑡<𝑇, and 𝑧∈𝕄.

For any 𝑠>−1 we consider the measure 𝑑𝜆_𝑠(𝑧)=Γ(𝑛+𝑠)()Γ()Γ()|𝑧|2𝜔𝜕𝕄𝑛−1𝑠+11−^2𝑠𝛼(𝑧)∧𝛼(𝑧), (59) on 𝕄, where 𝑤,𝜔 is the (2𝑛−1)-form on 𝜕𝕄 defined by 𝜔(𝑧)𝑉₁,…,𝑉_2𝑛−1=𝛼(𝑧)∧𝛼(𝑧)𝑧,𝑉₁,…,𝑉_2𝑛−1. (60)

Proposition 14 (see [14]). Suppose 1≤𝑝<∞ and −1<𝑠<∞. Then 𝑔(𝑧)^𝑝≤11−|𝑧|^{2𝑛+1+𝑠𝕄}𝑔(𝑤)^𝑝𝑑𝜆_𝑠(𝑤) (61) for all 𝑔∈𝒜^𝑝_𝑠(𝕄) and 𝑧∈𝕄.

This proposition leads directly to embeddings between 𝒜^∞_𝑠 and 𝒜^𝑝_𝑠1 for some 𝑠 and 𝑠₁ as we had above for unit disk case (see [2]) and we showed that this repeating arguments of unit disk case leads to new theorem we formulate below, in next section, for Bergman spaces in minimal ball.

2. On Distances in 𝐾-Bergman Spaces, Large Bergman-Orlicz Spaces in ℂ^𝑛, and Bergman Spaces in Minimal Ball

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 higher dimensions. Since all proofs are short, the repetitions of arguments which are needed in higher dimension will be omitted; only sketches will be given.

It is well known that 𝐴^𝑞_{−𝑠𝑞−1}(𝐔)⊂𝐴^∞_−𝑡(𝐔),𝑡=𝑠−1/𝑞,𝑡<0,𝑠<0 (see [18]).

It is easy to note, based on previous section results, that the complete analogues of the embedding we just provided are valid also in Bergman spaces in minimal ball for 𝑛-harmonic function spaces in polydisk and in 𝐾-Bergman spaces, and this is based completely on preliminaries we have in previous section. We leave this easy task to readers. This allows us to pose a distance 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. It is easy to note that each argument below is also valid not only in unit disk but based on preliminaries in the previous section in Bergman spaces in minimal ball and polydisk and 𝐾-Bergman spaces and Bergman-Orlicz spaces. So this theorem is very typical for us though it is known.

Theorem 15. 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−⟨𝑧,𝑤⟩|^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−⟨𝑤,𝑧⟩)^𝛽+2𝑑𝑚₂(𝑤)+_{Ω𝜀,−𝑡}𝑓(𝑤)(1−|𝑤|)^𝛽(1−⟨𝑤,𝑧⟩)^𝛽+2𝑑𝑚₂(𝑤)=𝑓₁(𝑧)+𝑓₂(𝑧), (62) where 𝐶(𝛽) is a well-known Bergman representation constant (see [5, 18]).
For 𝑡<0, 𝑓₁(𝑧)≤𝐶_{𝐔⧵Ω𝜀,−𝑡}𝑓(𝑤)(1−|𝑤|)^𝛽|1−⟨𝑤,𝑧⟩|^𝛽+2𝑑𝑚₂(𝑤)≤𝐶𝜀_𝐔(1−|𝑤|)^𝛽+𝑡|1−⟨𝑤,𝑧⟩|^𝛽+2𝑑𝑚₂(𝑤)1≤𝐶𝜀(1−|𝑧|)^−𝑡. (63) 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−⟨𝑤,𝑧⟩|^𝛽+2𝑑𝑚₂(𝑤)^𝑞×(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧)≤𝐶. (64) So we finally have dist_𝐴^∞_−𝑡𝑓,𝐴^𝑞_{−𝑠𝑞−1}≤𝐶𝑓−𝑓_{2𝐴∞−𝑡}=𝐶𝑓_{1𝐴∞−𝑡}≤𝐶𝜀. (65)
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−⟨𝑧,𝑤⟩|^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−⟨𝑤,𝑧⟩|^𝛽+2𝑑𝑚₂(𝑤)^𝑞×(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧)≤𝐶_𝐔𝐔𝑓_𝜀1(𝑤)(1−|𝑤|)^𝛽|1−⟨𝑤,𝑧⟩|^𝛽+2𝑑𝑚₂(𝑤)^𝑞×(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧). (66) Since for 𝑞≤1 (see [5, 18]) _𝐔𝑓_𝜀1(𝑧)(1−|𝑧|)^𝛼𝑑𝑚₂(𝑧)|1−⟨𝑤,𝑧⟩|^𝑡𝑞≤𝐶_𝐔𝑓_𝜀1(𝑧)^𝑞(1−|𝑧|)^{𝛼𝑞+𝑞−2}𝑑𝑚₂(𝑧)|1−⟨𝑤,𝑧⟩|^𝑡𝑞, (67) where 𝛼>(1−𝑞)/𝑞, 𝑡>0, 𝑓_𝜀1∈𝐻(𝐔), 𝑤∈𝐔, and _𝐔(1−|𝑧|)^{−𝑠𝑞−1}|1−⟨𝑤,𝑧⟩|^𝑞(𝛽+2)𝑑𝑚₂(𝑧)≤𝐶(1−|𝑤|)^{𝑞(𝛽+2)+𝑠𝑞−1}, (68) where 𝑠<0, 𝛽>((1−𝑠𝑞)/𝑞)−2, and 𝑤∈𝐔. We get 𝑀≤𝐶_𝐔𝑓_𝜀1(𝑧)^𝑞(1−|𝑧|)^{−𝑠𝑞−1}𝑑𝑚₂(𝑧). (69) So, we arrive at a contradiction. The theorem is proved.

The following theorem is a version of Theorem 15 for the case 𝑞>1.

Theorem 16. 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)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)<∞}.

The proof of Theorem 16 is the same actually as the proof of Theorem 15. The only difference is the boundedness of Bergman type projection operator but with positive Bergman kernel. This fact will be used by us below heavily.

Indeed the close inspection of the proof of Theorem 15 shows that the proof of Theorem 16 is the same as the proof of Theorem 15. but here we will use (70) (see below) instead of (67). For 𝜀>0,𝑞>1,𝛽>0,𝛼>−1/𝑞 (see [18]) _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, (70) which follows immediately from Hölder's inequality and (68) (see [18]).

We formulate one more theorem in this direction.

Theorem 17. Let 𝛽_𝑗>𝛽₀, 𝑗=1,…,𝑛, for large enough 𝛽₀, 𝑝>1, and 𝛼_𝑗>0, 𝑗=1,…,𝑛. Then for 𝑓∈ℎ(∞,∞,𝛼^→) we have dist_ℎ(∞,∞,^→_𝛼)𝑓,ℎ𝑝,𝑝,𝛼^→≍inf𝜀>0:_{𝐔𝑛Ω0→𝛼𝜀,𝑛𝑗=1}𝑃_𝛽𝑗𝑧_𝑗,𝑤_𝑗𝑓(𝑤×_𝑛𝑗=1)1−𝑤_𝑗^{𝛽𝑗+𝛼𝑗}𝑑𝑚_2𝑛(𝑤)^𝑝×^𝑛_𝑘=11−𝑧_𝑘^𝛼𝑘𝑑𝑚_2𝑛(𝑧)<∞, (71) where Ω⁰_̃𝑡𝜀,=𝑧∈𝐔^𝑛:𝑓(𝑧)^𝑛_𝑘=11−𝑧_𝑘^̃𝑡,̃≥𝜀𝑡≥0,𝜀>0. (72)

Note also that now even a more general theorem for all mixed norm spaces is valid, namely, for ℎ(𝑝,𝑞,𝛼) spaces based on facts we provided already above concerning projection theorems. We leave this to interested readers.

The scheme of proofs of all high dimensional theorems of this section is the same. The main tools are (see [1, 2]) the Bergman representation formula for larger space (for each 𝑓 function from 𝑌,𝑓∈𝑌,𝑋⊂𝑌). Then we use the fact that ‖|𝑃|𝑓‖_𝑋≤𝐶‖𝑓‖_𝑋,𝑓∈𝑋, where |𝑃| is a Bergman projection with positive Bergman kernel. Then we use Forelli-Rudin type estimate. As example we provide a complete proof of Theorem 17. For a parallel proof of other assertions we refer the reader to facts we provided above taken from mentioned papers and leave this procedure to readers.

Now we return to the proof of Theorem 17. We follow arguments of unit disk case.

Proof of Theorem 17. Let sup_{𝑟∈𝐼𝑛}(∏^𝑛_𝑗=1(1−𝑟_𝑗)^𝛼𝑗)𝑀_∞(𝑓,𝑟)<∞.
Then ∫_𝐔𝑛|𝑓(𝑧)|^𝑝(∏^𝑛_𝑗=1(1−|𝑧_𝑗|)^𝛽𝑗)𝑑𝑚_2𝑛(𝑧)<∞ for all 𝛽_𝑗>𝛽₀,𝑗=1,…,𝑛, where 𝛽₀ is large enough. This means that based on lemmas on 𝑛-harmonic functions from the previous section that for large enough 𝛽_𝑗,𝛽_𝑗>𝛽₀,𝑗=1,…,𝑛, 𝑓(𝑧)=1∏^𝑛_𝑗=1Γ𝛽_𝑗×_𝐔𝑛^𝑛_𝑗=11−𝜉_𝑗^𝛽𝑗−1×𝑃^→_𝛽(𝑧,𝜉)𝑓(𝜉)𝑑𝑚_2𝑛(𝜉),𝑧∈𝐔^𝑛,𝑃^→_𝛼(()𝑧,𝜉)≤𝐶𝛼,𝑛^𝑛_𝑗=111−𝜉_𝑗𝑧_𝑗^𝛼𝑗+1,𝑧,𝜉∈𝐔^𝑛,𝛼_𝑗≥0. (73) Then we have 𝛽_𝑗>𝛽₀,𝑗=1,…,𝑛: 𝑓=𝑓₁+𝑓₂ as in unit disk case, 𝑓₁(𝑧)≤𝐶₁𝜀1∏^𝑛_𝑗=11−𝑧_𝑗^𝛼,𝑓₂(𝑧)≤𝐶₂𝑓_{ℎ(∞,∞,𝛼)}. (74)
So finally we have dist_{ℎ(∞,∞,𝛼)}(𝑓₂,ℎ(𝑝,𝛼))≤𝐶‖𝑓−𝑓₂‖_{ℎ(∞,∞,𝛼)}=𝐶‖𝑓₁‖_{ℎ(∞,∞,𝛼)}.
To get the reverse we again repeat and follow arguments of unit disk case and use the fact from the previous section 𝑆_𝛽0𝑓(𝑧)=𝐶𝛽_0𝐔𝑛^𝑛_𝑗=11−𝜉_𝑗^2𝛽0−1𝑃_𝛽0(𝑧,𝜉)𝑓(𝜉)𝑑𝑚_2𝑛(𝜉) (75) maps for large enough 𝛽₀ and for all 𝛼_𝑗>0,1≤𝑝,𝑞≤∞,𝑗=1,…,𝑛,𝐿(𝑝,𝑞,𝛼) into 𝐿(𝑝,𝑞,𝛼).

The following theorem is based heavily on Remark 1, which we provided above, and Forelli-Rudin type estimates for unit ball (see [5] and Proposition 3) and some elementary arguments we see in proof of unit disk case.

We formulate complete analogue of Theorem 15 in 𝐾-Bergman spaces.

Theorem 18. Let 𝛽>𝛽₀ for large enough 𝛽₀,𝛼^→≺(∗),0<𝑝<∞,𝑙_𝛼=∑_𝑖>0𝛼_𝑖. Let 𝑔∈𝐻(𝐁_𝑛). If ‖|𝑃_𝛽|𝑔‖_{𝐴𝑝→𝛼}≤𝐶‖𝑔‖_𝐴^𝑝_→𝛼⇔_{𝐁𝑛𝐁𝑛}𝑔(𝑤)(1−|𝑤|)^𝛽|1−⟨𝑧,𝑤⟩|^𝑛+1+𝛽𝑑𝜈(𝑤)^𝑝⋅^𝑛_𝑖=11−^𝑖_𝑗=1𝑧_𝑗^2𝛼𝑖𝑑𝜈(𝑧)≤𝐶_𝐁𝑛𝑔(𝑤)^𝑝⋅^𝑛_𝑖=11−^𝑖_𝑗=1𝑤_𝑗^2𝛼𝑖𝑑𝜈(𝑤), (76) then for 𝑓∈𝐴^∞_{(2𝑛+𝑙𝛼)/𝑝}(𝐁_𝑛) we have 𝑑𝑖𝑠𝑡_𝐴^∞_{𝛼(2𝑛+𝑙)/𝑝(𝐁𝑛)}𝑓,𝐻^𝑝_{𝛼1,…,𝛼𝑛}𝐁_𝑛≍inf𝜀>0:_{𝐁𝑛Ω1𝜀,𝑡(𝑓)}(1−|𝑤|)^𝛽+𝑡𝑑𝜈(𝑤)|1−⟨𝑧,𝑤⟩|^{𝑛+1+𝛽𝑝}⋅^𝑛_𝑖=11−^𝑖_𝑗=1𝑤_𝑗^2𝛼𝑖𝑑𝜈(𝑧)<∞, (77) where Ω¹_̃𝑡𝜀,𝑓=𝑧∈𝐁_𝑛:𝑓(𝑧)(1−|𝑧|)^̃𝑡,̃̃>𝜀𝜀>0,𝑡>0,𝑡=2𝑛+𝑙_𝛼𝑝. (78)

Remark 19. Note that the condition in Theorem 18 is true when 𝛼_𝑗=0,𝛼_𝑛≠0 or 𝛼_𝑗=0,𝑗=1,…,𝑛. This is a classical fact of theory of Bergman spaces in the unit ball (see [5]).

The following theorem is based heavily on Remark 1, which we provided above, and Forelli-Rudin type estimates for unit ball (see [5] and Proposition 3) and some elementary arguments we see in proof of unit disk case.

We formulate complete analogue of Theorem 15 in Bergman-Orlicz spaces.

Theorem 20. Let 𝛼>−1,0<𝑝<∞. Then for 𝑓∈𝐴^∞_{(𝑛+1+𝛼)/𝑝}(𝐁_𝑛) we have dist_𝐴^∞_{(𝑛+1+𝛼)/𝑝(𝐁𝑛)}𝑓,𝐴^Ψ𝑝_𝛼𝐁_𝑛≍𝜀>0∶_𝐁𝑛Ψ_{𝑝Ω2𝜀,(𝑛+1+𝛼)/𝑝}(1−|𝑤|)^{𝛽−(𝑛+1+𝛼)/𝑝}𝑑𝜈(𝑤)|1−⟨𝑧,𝑤⟩|^𝑛+1+𝛽𝑑𝜈_𝛼<∞, (79) where Ω²_̃𝑡𝜀,=𝑧∈𝐁_𝑛∶𝑓(𝑧)(1−|𝑧|)^̃𝑡>𝜀,𝜀>0,𝑡̃>0, (80) if ‖|𝑃_𝛽|𝑔‖_𝐿^Ψ_𝑝𝛼≤𝐶‖𝑔‖_𝐿^Ψ_𝑝𝛼 for 𝑔∈𝐴^Ψ𝑝_𝛼(𝐁_𝑛)),𝛼>−1,𝛽>𝛽₀, for large enough 𝛽₀, where 𝑃_𝛽 is a weighted Bergman projection in Bergman-Orlicz spaces.

We formulate complete analogue of Theorem 15 in minimal ball.

Theorem 21. Let 𝛽>𝛽₀ for large enough 𝛽₀,1<𝑝<∞,−1<𝑠<∞. Then for any 𝑓∈𝐴^∞_{(𝑛+1+𝑠)/𝑝}(𝑀) for which Bergman representation (55) is valid 𝑑𝑖𝑠𝑡_𝐴^∞_{(𝑛+1+𝑠)/𝑝(𝑀)}𝑓,𝐴^𝑝_𝑠(𝑀)≍inf𝜀>0∶_{𝑀Ω3𝜀,(𝑛+1+𝑠)/𝑝}𝐾_𝛽,𝑀(𝑓(𝑤)×(|𝑤𝑧,𝑤)1−|)^𝛽+𝑡𝑑𝜆(𝑤)^𝑝𝑑𝜆_𝑠(𝑤)<∞, (81) where 𝑡=−(𝑛+1+𝑠)/𝑝 and Ω³_̃𝑡𝜀,=𝑧∈𝑀∶𝑓(𝑧)(1−|𝑧|)^̃𝑡≥𝜀,𝑡̃>0,𝜀>0. (82)

Conflict of Interests

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

Acknowledgment

This paper is supported by MNTR Serbia, Project 174017.