Some remarks on distances in spaces of analytic functions in bounded domains with c 2 boundary and admissible domains Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК

Том 15 Выпуск 3 (2014)

УДК 511.6

НЕКОТОРЫЕ ЗАМЕЧАНИЯ О ДИСТАНЦИЯХ В ПРОСТРАНСТВАХ АНАЛИТИЧЕСКИХ ФУНКЦИЙ В ОГРАНИЧЕННЫХ ОБЛАСТЯХ

_ _ О р. _ __ __

С границей из с2 и в допустимых ОБЛАСТЯХ12

Р. Ф. Шамоян, С. М. Куриленко (г. Брянск)

Аннотация

Воспроизводящая формула Бергмана и различные оценки для проектора Бергмана с положительным воспроизводящим ядром, а также точные оценки типа Форелли-Рудина для ядра Бергмана играют важную роль в некоторых новых экстремальных задачах, связанных с так называемой функцией дистанции в пространствах аналитических функций в различных областях в Сп.

В этой работе, опираясь на известные теоремы вложения для пространств аналитических функций в ограниченных областях с границей из С2 и в допустимых областях, мы получили новые результаты связанные с экстремальной задачей для пространств типа Бергмана аналитических функций.

Также мы приводим некоторые утверждения для пространств Неван-линны и БЫСЛ, для аналитических пространств Бесова в областях с границей из С2 и в допустимых областях.

Отметим также, что проблемам, связанные с регулярностью проектора Бергмана, которую мы часто используем в доказательствах, уделяется большое внимание. Многие оценки для воспроизводящих операторов и их ядер привлекают внимание математиков уже более 40 лет.

Структура этой работы такая же, как и в предыдущих работах по данной теме: сначала мы излагаем некоторые полученные ранее факты, связанные с проектором Бергмана, а потом, опираясь на них, доказываем оценки для функции дистанции.

Основываясь на известных результатах для классических пространств аналитических функций в различных областях в Сп, мы получаем несколько новых утверждений для функции дистанции на произведении строгоп-севдовыпуклых областей с гладкой границей, в ограниченных областях с границей из С2 и в допустимых областях.

хРабота поддержана грантом РФФИ (грант 13-01-97508)

2Классификация по МБС 2010: первичная 42Б15, вторичная 42Б30.

Кроме того, нами получены некоторые точные результаты на произведении строгопсевдовыпуклых областей с гладкой границей в Cn, расширяющие наши результаты для строгопсевдовыпуклых областей.

Ключевые слова: оценки дистанций, аналитические функции, ограниченные области, допустимые области, псевдовыпуклые области.

Библиография: 22 наименования.

UDC 511.6

SOME REMARKS ON DISTANCES IN SPACES OF ANALYTIC FUNCTIONS IN BOUNDED DOMAINS WITH C2 BOUNDARY AND ADMISSIBLE DOMAINS3 4

R. Shamoyan, S. Kurilenko (Bryansk)

Abstract

The so-called Bergman representation formula (reproducing formula) and various estimates for Bergman projection with positive reproducing kernel and sharp Forelli-Rudin type estimates of Bergman kernel are playing a crucial role in certain new extremal problems related to so-called distance function in analytic function spaces in various domains in Cn.

In this paper based on some recent embedding theorems for analytic spaces in bounded domains with C2 boundary and admissible domains new results for Bergman-type analytic function spaces related with this extremal problem will be provided. Some(not sharp) assertions for BMOA and Nevanlinna spaces, for analytic Besov spaces in any D bounded domain with C2 boundary or admissible domains in Cn will be also provided.

We remark for readers in addition the problems related to regularity of Bergman projection which we use always in proof in various types of domains with various types of boundaries (or properties of boundaries) are currently and in the past already are under intensive attention. Many estimates for reproducing operators and there kernels and Lp boundedness of Bergman projections have been also the object of considerable interest for more than 40 years. These tools serve as the core of all our proofs. When the boundary of the domain D is sufficiently smooth decisive results were obtained in various settings. Our intention in this paper is the same as a previous our papers on this topic, namely we collect some facts from earlier investigation concerning Bergman projection and use them for our purposes in estimates of distY(f, X) function (distance function).

3This work was supported by the Russian Foundation for Basic Research (grant 13-01-97508).

4Mathematics Subject Classification 2010 Primary 42B15, Secondary 42B30.

Based on our previous work and recent results on embeddings in classical analytic spaces in domains of various type in Cn we provide several new general assertions for distance function in products of strictly pseudoconvex domains with smooth boundary, general bounded domains with C2 boundary and in admissible domains in various spaces of analytic functions of several complex variables.These are first results of this type for bounded domains with C2 boundary and admissible domains. In addition to our results we add some new sharp results for special kind of domains so called products of strictly pseudoconex domains with smooth boundary in Cn extendind our sharp results in strictly pseudoconex domains.

Keywords: Distance estimates, analytic function, bounded domains, admissible domains, pseudoconvex domains.

Bibliography: 22 titles.

1. Introduction

The so-called Bergman representation formula (reproducing formula) and various estimates for Bergman projection with positive reproducing kernel and sharp Forelli-Rudin type estimates of Bergman kernel are playing a crucial role in certain new extremal problems related to so-called distance function in analytic function spaces in various domains in Cn.( see, for example, [15], [16], [17], [18], [9] and various references there).

In this paper based on some recent embedding theorems for analytic spaces in bounded domains with C2 boundary and admissible domains (see, for example,

[1], [2], [3] )new results for Bergman-type analytic function spaces related with this extremal problem will be provided.Some(not sharp) assertions for BMOA and Nevanlinna spaces, for analytic Besov spaces in any D bounded domain with C2 boundary or admissible domains in Cn will be also provided.

We remark for readers in addition the problems related to regularity of Bergman projection which we use always in proof in various types of domains with various types of boundaries (or properties of boundaries) are currently and in the past already were under intensive attention. (see, for example, [7], [5] and references there).

Many estimates for reproducing operators and there kernels and Lp boundedness of Bergman projections have been also the object of considerable interest for more than 40 years. These tools serve as the core of all our proofs. When the boundary of the domain D is sufficiently smooth decisive results were obtained in various settings (see [7], [5] and references there).

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

Following our previous paper [16], [15], [9] we easily see to obtain a sharp result for distance function we need only several tools and the scheme here is the following.

First we need an embedding of our quazinormed analytic space (in any domain) into another one (X C Y), this immediately pose a problem of disty (f, X) = inf || f—g\\y

gEX

for all f G 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 sharp estimates of Bergman kernel. These three tools were used in general Siegel domain of second type, polydisk and unit ball in [16], [15], [17], [18](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 analytic spaces in pseudoconvex domains, in spaces in any bounded domain D with C2 boundary and in so-called admissible domains also.

The plan of this note is the following. We provide first two typical results on this topic with complete proofs taken from one of our previous recent paper but in more general setting of products of strictly pseudoconvex domains with smooth boundary and then in next sections discuss its various modification and extensions in more general settings based on same ideas in a sketchy form.

We denote as c or as C with various indexes various positive constants below. We denote by A a characteristic function of a set everywhere below.

2. A distance theorem for analytic Bergman spaces in products of bounded strictly pseudoconvex domains with smooth boundary

Functional spaces and their properties on m products of strictly pseudoconvex domains with smooth boundary were studied in various papers (see, for example, [22],[21] and references there). The goal of this section is to provide new sharp distance theorem in m- products of bounded strictly pseudoconvex domains with a smooth boundary in Cn but with a complete proof only for m = 1 case.

Proofs of m = 1 and general case are very similar and we choose it to shorten the exposition. Moreover proofs related with this particular case can be seen in[9]. The reason we do this to heavily shorten this and last sections where series of other similar type results with similar proofs based also on recent work from [1],

[2], [3] in more general settings namely in bounded domains with C2 boundary and admissible domains will be formulated by us. So we first consider Bergman type spaces in D C Cn, where D is smoothly bounded relatively compact strictly pseudoconvex domain, providing sharp results in this case in this section. In one of the next sections of this paper this results will be partially generalized, but proofs mostly will be omitted there since it is based on this proof. The theory of bounded strictly pseudoconvex domains in Cn with smooth boundary and function spaces of several complex variables on them is a subject of very active study (see, for example, [9], [13], [11], [14] and various references there).

Our proofs in this section are heavily based on estimates from [14], where even more general situation was considered when our domains are embedded in so-called Stein manifolds. For definition of Bergman spaces and other objects we refer the reader to [14].

We define Bergman spaces as usual as follows.

A*(D) = jf gH(D): J |f(z)|p8d(z)a-1dV(z) <roj 0<p<ro, a > 0,

where H(D) is a space of all analytic functions in D and dV(z) (or dv(z)) is a Lebegues measure in D and 8D (z) is a distance from z to boundary of D [14].

Since |f (z)|p is subharmonic (even plurisubharmonic) for a holomorphic f in D, we have Ap(d) C A°°(D) for 0 < p < ro, sp > n and t = s. Also, Ap(D) C A1(D) for 0 < p S 1 and Ap(D) C A^(D) for p > 1 and t sufficiently large. Therefore we have an integral representation.

f(z) = i f(£)K(z^^dV(0, (1)

Jd

where K(z, £) is a kernel of type n +1 +1 ,[14] that is a measurable function on D x D such that |K(z,£)| S C|<l(z,^)|-(n+1+i), where $(z,£) is so called Henkin-Ramirez function for D. From now on we work with a fixed Henkin-Ramirez function $ and a fixed kernel K of type n + t + 1. We are going to use the following results from

[14].

Lemma 1. ([14], Corollary 5.3.) If r > 0, 0 < p S 1, s > —1, p(s + n + 1) > n and f G H(D), then we have

[ |f (0||$(z,5)ri‘K)dvkAp S C J |f (0n$(;,0r<ip(‘+n+1Mn+1)(0dv(?).

Lemma 2. ([14], Corollary 3.9.) Assume K (z,£) is a kernel of type ft, and a>0 satisfies a + n — ft <0. Then we have

f K(z,^)iCT-1(z)dV(z) S CiCT+n-^(0-Jd

Note we use same 8 function as a function of z1,... ,zm variables in product domain Dm as products of m such functions, this will be clear from context below. And we use Vm as a Lebegues measure on Dm A natural problem is to estimate distAoo(D)(f, Af(D)) where 0 < q < ro, sq > n and f G A^°(D). We give sharp estimates below, treating cases 0 < q S 1 and q > 1 separately. Note the same problem is valid for same spaces but on product domains.

Note moreover all we discussed is also valid in m-products of D domains and analytic spaces on them. This includes the last embedding, the integral representation

and the estimates of Kernel. (m products of kernels). This can be obtained by using "one dimensional" result m -times by each variable separately. Below we formulate a general theorem in it is "m- product version" . For m = 1 case this theorem can be seen before in [9]. To define all objects in product spaces and analytic spaces on products of pseudoconvex domains themselves we use standard procedures which we see in polydsk (see, for example, [16]). Note also various problems of analytic function theory on products of pseudoconvex domains were under attention, see, for example, [21] and [22] and references there.

Theorem 1. Let m G N, 0 < q S 1, sq > n, f G A°(Dm) and t > s is sufficiently large. Then w1 = where

^1 = distAjo(Dm)AS w2 = inf |e > 0 : |Km(z1,...,zm,£)|8i-s(£)dVm(£)^| 8sq-n-1(z)dVm(z) <ro

1 Dm \ JQ£,s

where K(z, £) is the above kernel of type n +1 + 1, Km(z1,..., zm, £) is a m-product of such kernels and

n£,s = {z G Dm : |f (z)|8s(z) ^ e}.

Proof. We assume m = 1.The proof of general case is the same. Let us prove that w1 S ^2. We fix e > 0 such that the above integral is finite and use (1):

f (z) = i f (£ )K (z,£)dV (£) + i f (£ )K (z,£)dV (0 = f1(z) + f2(z).

Jd\q£,s

^£,S

We estimate f1 :

|f 1 (z)| S Ce I |K(z,e)|8i-‘dV(e) S Ce I S Ce8-‘(z),

Jd Jd |$(z,e)|n+t+1

where the last estimate is contained in [14] (see p. 375). Next,

Hf2|IV = i |f2(z)|q8sq-n-1(z)dV(z) S s Jd

S C i ( i |f(e)|K(z,e)8*(e)dV(e) ) 8sq-n-1(z)dV(z) S C'||f||«oo. Jd yJqz,s J

Now we have

distAj°(D) (f, AS(D)) S ||f — f2|Aj°(D) = |f1|Aj°(D) S Ce.

Now assume that ^ < w2. Then there are e > e1 > 0 and /ei G Aq (D) such that 11/ — fei IUj° « ei and

I = i ( i |K(z,Ç)|£i-s(£)dV(Ç)) ¿sq-n-1(z)dV(z) = «x

«/ D ^ J ^£,S J

As in the case of the upper half-plane one uses ||/ — /ei ||aj° « e1 to obtain

(e — e1)Xn£,s(z)^-S(z) « CWei (z)|.

Now the following chain of estimates leads to a contradiction:

\ q

ri—s / ^ \ xsq—n— 1/

xn£,s(e)^i-s(e)K (z,e)dV (e) ¿sq-n-1(z)dV (z)

ID \JD )

S cj (J\fn(5)|5‘(5)K(z,e)dV(e)],5’q-"-1(z)dV(z)

S CI (jD 1/1 (e)|,5‘(e)|¥cz,in++r)55sq-n-1(z)dVW

sc / f |f.. (e } dV WdV (e)

JWd |$(z,e)|q(n+i+1)

sc i |fei (e )|q ^sq-n-1(e )dV (e) <ro, (2)

JD

where we used Lemma 1 and Lemma 2 with ft = q(n + t + 1), a = sq — n. □

Next theorem deals with the case 1 < q < ro.

Theorem 2. Let m G N, q > 1, sq > n, t > s, t > and f G A~(Dm).

Then w1 = w2 where

^1 = distAj® (Dm)(f, AS

^2 = inf < e > 0 : I I I |Km(z,£)|£i-s(£)dVm(£ ) | ¿sq-n-1(z)dVm(z ) < œ > .

lDm \ Jn£,s

Proof. We again restric ourselves to m = 1 case. See also [9]. The proof of

general case is the same. A carefull inspection of the proof of the previous theorem

shows that it extends to this q > 1 case also, provided one can prove the estimate:

j = X (// K)|i‘K)/f (o ) q^-1«^, (z )

« c f |/eiK)|qisq-n-1(Ç)dv(Ç) <œ

D

where q > 1. Using Holder’s inequality and Lemma 2, with a = 1 and ft = n + 1+ pe, we obtain

I (z)

(7 / K)|i‘K)K (z,e )dv (eA ’

I-

Jd

CI

Jd

f /(e)ia6‘"(e)dV(e), ( r dV(e) y/p

D \<l(z,e)\ra+1+iq-£q Ud \i(z,e)in+1+pV

; r /(e)\q(e)dv(e) r„(z),

JD \$(z,e )\n+1+iq-£q

a

and this gives

< C /* r / (e)\q(e)^-q£+sq-ra-1(z) ^ JdJd \<l(z,e)\ra+1+iq-£q

^ C i /(e)\q¿sq-n-1(e)dv(e) <»,

II

Jd Jd

/ /

Jd

dV(z) dV(e)

where we again used Lemma 2, with ft = n + 1 + tq — eq and a = q(s — e) — n > 0. □

Remark 1. We note that some results of this section on distances can be extended to so-called minimal bounded homogeneous domains. ficCn and A2a Hilbert spaces on them. Indeed, our proofs are based on Bergman representation formula, and Forelli-Rudin type estimates for integrals

where K(z,w) is a Bergman reproducing kernel for the weighted Bergman space A0,(n). All relevant estimates for this case can be found in recent papers [19] and

3. Extremal problems in analytic spaces in bounded domains with C2 boundary.

The intention of this section which is heavily based on previous one to consider the same extremal problem but in more general setting. All results of this section are based on series of recent theorems on embeddings of analytic function spaces in bounded domains with C2 boundary( see [1], [2], [3], [4] and references there). Note our general results provide only one side estimates for dist function. Note also we will not provide proofs but only short comment after each assertion. Since all proofs are modifications of proofs of theorems of previous section. Let D c Cn be a bounded domain with C2 boundary and let H(D) = {/ holomorphic in D}. For

[20].

z G D let Sd (z) denote the distance from z to dD. We define for every a > 0, p < to the measure dva = (Sd(z)a-1)dV where dV (or dV^) denotes the volume element and the weighted Bergman space

A£(D) = {/ G H(D) : ||/||p,„ = ^ \/(w)\pdV;(w) < w

A-CT(D) = {/ G H(D) : ||/||-CT = sup(^D(w)CT\/(w)\) < to : w G D} , a > 0.

Ap = Hp for all positive p. If D has C2-boundary then the following inequality holds (see [1], [2])

\/(z)\ ^ (Sd(z)-"bll/|U V/ G A*(D);Vz G D

and

A*(D) C A-^(D); see [1].

Also we have H^(D) C A-CT(D), a > 0, see [1].

In summary we have if D has C2 boundary then we have the following chains of embeddings

n + a

H~(D) C A-(J(D) c Aa(D) C A-~+~(D) c Ns(D)

for all 0 < p, a < w, 0 < a < a; ft > 0; (see [1]) where

N(D) = {/ G H(D) : ^(log+ \/(w)\)dVs(w) < to} , ft > 0

is a Bergman-Nevanlinna class in D (see for all this [1] and references there). Each embedding as we noticed above pose a problem of finding distY(/,X), / G Y\X, XCY.

In [2] the authors showed for Hardy spaces Hp(D) C AS(D); for 0 < p < q < to; ft > 0 with ^ = n+s; for all bounded domains in Cn with C2 boundary. This poses a problem distAq(/, Hp); / G AS for any bounded domain in Cn with C2 boundary. We formulate a general result for A^ space.

Let

B(Aa, /) = inf { e > 0 : sup (z)(Sd(z)) + +s

|z|<1

<

where s ^ n±a, t = — n±a and

pp

nr,£ = {z G D : \/(z)\SD(z) ^ e}; t =--------------; a > 0.

p

Theorem 3. Let D be any bounded domain in Cn. Let also D has C2 boundary.

n + a

Then for all /, / G A p

B(Aa,/) ^ c(distA_^(D)(/,Aa(D))); 0 < p < to; a>°.

The proof follows directly from arguments of proof of theorem 1 and will be omitted. Note similarly we can get one side estimates for first and third embeddings in chain of embeddings we provided above Similar results are valid for pairs (B, Bp), where B is a Bloch space,Bp is an analytic Besov space. Since Bp C B as we mentioned above for all p G (0, to). Similarly using other embeddings we mentioned above we can formulate a result like this using the embedding H^ C A-CT(D), where a G (0, a) on estimates of distA-<x (/, H^(D)) or distA—(/, Hp(D)); 0 < / < n(p — 1)on bounded D domain with C2 boundary in Cn.

The proof of theorem 3 can be easily recovered from arguments we provided above in theorem 1 (see, for example, for one-dimensional case [16] and references there).

The following theorem is based on already mentioned embedding for any bounded D domain with C2 boundary.

Let = {z G H : \/(z)\S(z)CT ^ e}, then we have

Theorem 4. Let D be bounded domain with C2 boundary. Let / G A-CT (D). Then for a G (0, a)

distA--(d) (/, H~(D)) ^ inf{e > 0 : sup[Xn£(j(z)]S(z)CT < to}

z€D

The proof follows easily from discussion we had above based on arguments of proof of theorem 1 we omit details. Finally based on embeddings (see [1], [2], [3]) H ^ C Hq, and Ap C Hq, p<q<TO, a>0, n++a = ^ similar results based on proof of theorem 1 can be obtained for any bounded domain with C2 boundary. We leave this to interested readers.

4. On extremal problems in analytic spaces in admissible domains and in bounded strictly pseudoconvex domains with smooth boundary

The main goal of this section to turn to recent results from [12], [11], [6] on

general admissible domains to get new estimates for dist function in this setting. All

results are one side estimates and we don’t discuss proofs since all of them are based on a proof of theorems 1, 2 from section 1. We say a smoothly bounded domain in Cn is an admissible domain (see [7]) if D is one of the following.

1) a strictly pseudoconvex domain

2) a pseudoconvex domain of finite type in C2

3) a convex domain of finite type in Cn

Such type domains were under intensive study [13], [11], [12], [6]. It is natural to give estimates for dist function also in this case in this more general setting. But first we generalize all results of first section (in strongly pseudoconvex domains with smooth boundary)to so -called mixed norm spaces. Note we again omit details of proofs since it is a modification of proof of theorem 1.

Let D = {z : p(z) <0} be a bounded strictly pseudoconvex domain of CN with C° boundary. We assume that the strictly plurisubharmonic function p is of class C° in a neighbourhood of D, that — 1 ^ p(z) <0, z G D, |dp| ^ c0 > 0 for |p| ^ r0. Let A?^(D) = {/ holomorphic in D such that [[/||p,q,<s,fc < to} where

p,qAfc = | ! (/ \D/\pdar) (r^q 1 )dr

|a|<fc V-'dDr /

here we denote by Dr = {z G Cn : p(z) < — r}, dDr it is boundary dar the normalized surface measure on dDr and by dz the normalized volume element on D (see [5], [10]).

For p = q, k = 0:

** = V iJ/(z)lP(—p(z))'-1dv(z)

and hence we have a standard Bergman class A?(D) see [5], [10].

A?,, = j / G H (D) : ^ |3“/ IV1 (z)dv(z)) < to

q > 0, s G N, 0 < p < to.

Theorem 5. Let m > 0, 0 < q ^ 1; sq > n, q ^ p, / G A°(D), t > t0, where

t0 is a large enough positive number. Then for any strictly bounded pseudoconvex

domain D with smooth boundary we have w2 ^ c^1 where

^1 distAj° (D)(/, Asq-n (D)) distAf° (/, Asq-n+mq,m))

and

^2 = inf (e>0: f (l |K (z,Z )|^i-,(C )dV (Z A ¿,9-n-1dV (z) <to

[ ./D \./n£,s y

Theorem 6. Let D be any strictly bounded pseudoconvex domain with smooth

boundary . Let m > 0, q > 1; sq > n, q ^ p, q ^ to, t > t0, where t0 is large enough

positive number ,/ G A°(D). Then w2 ^ cw1 where

^1 distAj° (/, Asqi-n+mq,m);

and

^2 = inf (e>0: I (i \K (z,Z )\^i-s(C )dV (Z )^j ¿sq-n-1dV (z) <to [ Jd \JnE,s J

For proofs of these assertions we refer the reader to our paper [6] and embeddings from [10], [5]. Combining these two we’ll get proofs immediately.

Note for some values of parameters a sharp results for these pair of spaces were obtained previously by us see [16]. We now define several new analytic spaces in D domain. We list then various (known) embeddings which lead to distance estimates. Then provide new one side estimates.

For a bounded strictly pseudoconvex domain D we have also BMOA(D) C H2(D), where we define for any admissible domain Hp(D) and BMOA(D) in the following standard way (see [8], [12])

Hp(D) = if G H(D) : sup ( f \f\pda£) " < to

^ 0<£<£0 \ JODs /

where again De = {z G Cn : p(z) < — e}, e > 0; and D = {z G Cn : p(z) < 0}, 0 < p ^ to, da£ is surface measure on SDe. For any admissible domain we define also

BMOA(D) = If G H :

2

BMOA

= sup / \f (w) — f (z)\ xP(z,w)da(w); z G D > < to,

JdD J

where P(z,w) = (S(z,z)-1)\S(z,w)\2; z G D, w G 5D is a Poisson-Szego Kernel, S(z,w) - Szego Kernel. see [5], [13], [8].

Let p > n, D is any bounded domain with C2 boundary in Cn. Then the Besov space is a space with norm see [11], [10]

B'(D> = ( L iV”+If<2)iP[i<z)]p(“+1)dA(z}) p < to, (3)

d H f

ID

where

■(z)

|V"+1 (f (z ))| =

|a|^ra+l

and dÀ(z) = K(z,z)dV(z) - biholomorphically invariant measure and K(z, w) is the Bergman kernel, as above dV is a volume measure of D [10], [11].

For p = to, Bœ = B/(D) this is a classical Bloch space (see [5], [13], [8]).

We add more embeddings from [13], [8], [11], [5] for admissible D domains.

We say f G VMOA if f G BMOA and

lim I |f (w) - f (z)|2p(z,w)d^(w) = 0.

z^dD Jbd

We have Bp(D) C VMOA(D) for 1 < p < to. Also we have

n > 0

b ^ sup

z€D

c

|V”+1(/ (z))|(i(z)”+‘)(log )

and H^ C B/, H^ C Bp; where

|f ||bz = sup \Vkf (z)\(ifc(z)) < to, 1 ^ p ^ to, k > 0.

z€D

The log Besov space is the following class of analytic functions [13], [8], [11].

LBp(D) = {f G H(D)

|V”+1f (z)|p(i(2)‘”+1)p)(log ) 1 dA(z)

' D ^(z)

< to}

p G (0, to). We have LBp C Bp(D) C Bq(D) C B/(D): 0 < p < q < to (see [11]).

We also have for bounded strictly pseudoconvex domains BMOA C Lp(dD) if H(D) : 1 ^ p < to (see [12]). We define Nevanlinna classes for all bounded domains with C2 boundary in Cn (see [13], [8], [11]).

Let De = {p < (—e)}; D be bounded domain with C2 boundary; D = {p < 0}; dp = 0 on dD. De are subdomain of D 0 < e < e0 with C1 boundary a, are surface measures on dD and dDe respectively.

For 0 < p < to Hardy class is as above

Hp(D) = {f G H(D) : sup / |f |pda£ < to

0<£<£0 \ JODs /

if we put log+ |f | = max(log |f |,0) instead of |f |p then we set analytic Nevanlinna class N (D). Note N (D) D Hp(D) for all 1 ^ p ^ to. Also it is known for all bounded D domains with C2 boundary (see [7], [8], [9]).

n

sup |f(z)|dzst(z,<9D)n ^ ci||f ||hp(d);

z£D

n + 1

sup |f (z)|dist(z, 3D)~np~ ^ C2|f ||ap(d);

z€D

p G (0, to)

We also have an embedding of atomic Hardy spaces into standard Hardy classes for admissible domains Hp(D) C Hp(D); 0 < p ^ 1, where

m œ

HA (D) = {£ ÀjAj : Aj is holomorphic p-atom £ |Àj |p < to} j=i j=i

(see about these spaces and the definition of p-atom (holomorphic) in [9] for all admissible domains).

Based on several from these embeddings we formulate another (not sharp) three theorems on distances below. Short proofs of all of them are fully based on arguments of proof of theorem 1 and we leave them to interested readers.

In all assertions below our domain is admissible,in an assertion related with analytic Nevanlinna spaces it can be also simply bounded with C2 boundary. A sharp version of our first assertion is probably valid based on projection theorems for Besov spaces from [11].

Theorem 7. Letp > n; f g Bl. Let S/ = {z g D : \Vn+1f (z)\(£(z))n+1 ^ e}, n G N.

Let D1 = distB1 (f, Bp) and also let

D2 = inf{e > 0 : / XSs/(z)dA(z) < to}.

Jd ,

Then ¿D1 ^ D2.

Note similar assertion is valid if we replace Bp(D) by LBp(D) or any analytic function space X, so that X C Bl(D) for any admissible D domain.

Next we formulate an assertion which in particular concerns analytic Nevanlinna spaces in bounded domains in Cn with C2 boundary.

Theorem 8. 1) Let p > n and let also f G Hp. Then we have K1 ^ cK2, where K1 = distHp (f, Bp) and

PT

K2 = inf{e > 0 : / XKs,f(p)p2(n+1)p-(n+1)dp < to}

J° '

where

K£;/ = {p G (0,e°) : i \f(Z)\pdap(Z) ^ e}

JdDp

2) Let f G H 1(D). Then we have s1 ^ c(s2) where

s1 = distHi (f, H 1at) s2 = inf{e > 0 : sup AKsf(r) < to}

°<t<£o ,

and where

K£,/ = {p G (0,e°) :/ \f(Z)\d^p(Z) ^ e}

JdDp

3) Let p > 1, let also f G N(D), then we have T1 ^ cr2; where

T1 = distN(d) (f, Hp(D))

and where

Me,/ = {p G (0,e°) : i (log+\f(Z)\)pd^p(Z) ^ e}

JdDp

T2 = inf{e > 0 : sup (AMs,f(r)) < to}

°<T<£0 ,

Finally we formulate another assertion which concerns BMOA type spaces in admissible domains in Cn.

Theorem 9. Let D be any admissible domain in Cn 1 ) Let f G H1 then we have K1 ^ cK2, where

Ki = distH 1 (f, BMOA)

K2 = inf(e > 0 : sup ÀKsf(r) < to}

0<r<£0 ’

where

K£,/ = (P G (0,e0) : f |f(Z)|d^p(C) ^ e}

JODp

2) Let f G A.“/p then we have for p G (0, to) R1 ^ cR2, where

R1 = distp-/p(f,Hp)

R2 = inf(e > 0 : sup f(r) < to}

0<r<£0 ’

5. Conclusion

Conditions of similar type for distance function based on embeddings Ap^_si) C

Hpi with some restriction on p, s1, s2 can be obtained similarly. These are conditions

of type sup ÀMsf(r) < to where M£,/ should be appropriately defined. We leave 0<r<£0 ’

this to readers. Proofs of all these assertions are modifications of proof of theorem 1 and we omit details.

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Bryansk State University Поступило 13.01.2014