ON SOME NEW ESTIMATES FOR INTEGRALS OF THE SQUARE FUNCTION AND ANALYTIC BERGMAN TYPE CLASSES IN SOME DOMAINS IN CN Текст научной статьи по специальности «Математика»

Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 31. № 2. C. 32-55. ISSN 2079-6641

MSC 32A37, 30H20 Research Article

On some new estimates for integrals of the square function and analytic Bergman type classes in some domains in Cn

R.F. Shamoyan1, E.B. Tomashevskaya2

1 Department of Mathematical Analysis, Bryansk State University named after Academician I. G. Petrovsky, Bryansk, 241036, Bryansk, y,. Bezhitskaya, 14, Russia

2 Department of Mathematics, Bryansk State Technical University, Bryansk 241050, Russia

E-mail: [email protected], [email protected]

The purpose of the note is to obtain equivalent quasinorm, sharp estimates for the quasinorm of Hardy's and new Bergman's analytic classes of in the polydisk. We extend some classical onedimensional assertions to the case of several complex variables. Our results more precisely provide direct new extention of some known one variable theorems concerning area integral to the case of simplest product domains namely the unit polydisk in Cn. Let further D be a bounded or unbounded domain in Cn. For example, tubular domain over symmetic cone or bounded pseudoconvex domain with smooth boundary. Our results can be probably extended to the case of products of such type complicated domains, namely even to D x ■ ■ ■ x D. This can be probably done based on some approaches we suggested and used in this paper. On the other hand our results in simpler case namely in the unit polydisk may also have various interesting applications in complex function theory in the unit polydisk. We finnaly provide similar type sharp. results in some new Bergman spaces in bounded strongly pseudoconvex domains

Keywords: polydisk, integral operators, analytic functions, analytic spaces, Hardy class, new Bergman space, pseudoconvex domains.

DOI: 10.26117/2079-6641-2020-31-2-32-55

Original article submitted: 30.04.2020 Revision submitted: 11.06.2020

For citation. Shamoyan R. F., Tomashevskaya E. B. On some new estimates for integrals of the square function and analytic Bergman type classes in some domains in Cn. Vestnik KRAUNC. Fiz.-mat. nauki. 2020, 31: 2,32-55. DOI: 10.26117/2079-6641-2020-31-2-32-55

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Shamoyan R. F., Tomashevskaya E. B., 2020

Introduction

In Euclidean space Rn the problem of finding equivalent quasinorm and exact estimates for the norms of certain function spaces has been solved by many authors (see, for

Funding. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors

example, [19]). Recently, a large number of papers have appeared, where a similar problem was considered for analytic functions in the disk. Different equivalent quasinorms for analytic new Bergman type classes in disk U = {z: |z| < 1} of complex plane C were derived in [10], [11],[20], [21]. The papers [1],[3] gives several characteristics of the analytic classes in bidisk. Finally, at [1], [14] an equivalent quasinorm is indicated for space Lizorkin-Triebel in the polydisk. In the first paragraph well-known estimates for the quasi-norm Hardy classes will be generalized to the case of the polydisk. In the second paragraph, equivalent quasinorms for the new Bergman type classes introduced in [14] are derived. It should be noted that the new functional spaces (see [7]) and their properties modified by us for the case of the polydisk play an essential role in the note.

We finnaly at the end of the paper provide new similar type sharp results in new Bergman type analytic function spaces in bounded pseudoconvex domains with smooth boundary. Note here we again choose the context of the unit disk, then show in details how to pass the same proof to these general domains.

New estimates of the quasinorm of Hardy classes in the polydisk

In this section we will extend some classical onedimensional assertions to Hardy spaces of several variables. We will present the following notation. Let Un - be the unit

polydisk of Cn, Un = {ze Cn,|zj| < 1, j = 1,...,n}, Tn = {ze Cn, |zj| = 1, j = 1,...,n} -mn(§) and m2n(®) be the normalized Lebesgue measures on Tn and Un accordingly.

Let further Ta(§) = Ta(§1,...,§„) = r^(§1) x ••• x ra„(§„), where a, > 0,§,- e T,i = 1,..., n,

and let also

ra; (%*■) = z e U : 1 - % iz < ad 1 - |z|2 , * = 1,-, n;

1 - Zn % n

/Un(%1,...,%n,ti,...,tn) = ]ze Un : 1 -Z1% 1

< t1,...,

< tn >;

tj > 0, j = 1,...,n.

We omit aj below sometimes dealing with ra(§). lUn = nUn,7t« = n Tn. For a measurable f function in Un we put

(

Aq(/) (%) =

= sup

1

\Ta (%) A. (/))(% )= sup

zeUn

1

| / (Z1,..., zn)|qdm2n(z)

(1 - |z|)2

\

1/q

, q <

/

,...,zn)|,z e ra(%)% e Tn.

q

Cq(/))(% )) ' =

... sup

1

| / (z1,...,zn)|q

t; |1t (%n, tn)| / (1 -|zn|r' t! |1t (%1, t1)| / (1 - |z11)

Iu (%n,tn) Iu (%1,t1)

dm2n(z),

§ e Tn,§ = (§1,...,§„),tj > 0, j = 1,...,n.

Also let H(Un) and Hp(Un), 0 < p < ^ are the space of all holomorphic functions in Un and Hardy class in the polydisk accordingly.

Hp(Un) = {f e H (Un) : sup| | f (r£ )| ) < , 0 < p < <*>.

r

These are Banach spaces for all p > 1, and quazinormed spaces for other values of parameters.

Remark 1. The values given above in Rn+1 were first introduced in [7] for the so-called new functional spaces in Rn+1.

A well-known statement of the theory of Hardy Hp spaces states that if a function f belongs to the Hardy class Hp, 0 < p < then the quantity S(f) is finite.

(

\

p/2

S(/) = / f |D/(z)| (1 - |z|)2k-2dm2(z)

\r(l)

) <

/

Dk is a well-known differential operator of an analytic f function in the unit disk on the complex plane C (see, for example, [6], [8]) .

S operator is known as the Luzin area integral operator and the above statement about its boundedness in Hardy Hp classes, 0 < p < ^ in the disk U was established in [22]. In Theorem 1, relying on the multidimensional maximum theorem established in [22], two generalizations are given - direct polydisk analogues of this statement for p < 2.

In Theorem 2 for p > 2 this result will be generalized in two different ways at once. Let Da be the fractional derivative of the f function, a > 0, Da : H(Un) ^ H(Un),

(Daf) (zi,...,zn) = £ (k + 1)aaki...kBzi1...zi1, (zi,...,zn) e Un,

|k|>0

(k + 1) = n(kj + 1),(see [8],[5],[21]). j=1

We define fractional derivative also for negative indexes, namely we put

D-a (Da)^.f)= f, for a > 0.

Let further C1,C2,C(n) be various positive constants.

Everywhere below the notation A < B denotes that there is the constant C > 0 such that A < CB, the notation A = B denotes that there are the constants C1 > 0 and C2 > 0 such that C2B < A < C1B .

Theorem 1.

A) Let f e H(Un), Assume that

+ , F(zi,...,z„)= L (k2 + 1)... (k„+ 1)aki...kBzi1 ...z£ ki ...kB>0

is in c/ass Hp(Un), 0 < p < 2,

Sn(/) =

( (

J J ••• j |d/(Z1,...,z„)|2dm2(z) \r(|1) r(|B) /

(1)

\ p/2 \

1/p

/

oo

n

Then next estimates are correct

Ci I |D-ef | |Hp(UB) < S«(f) < C2|| f ||hp(u«), 0 < p <

2,

(2)

where e - arbitrary positive number and D(f) = D1(f). 5) Let 0 < p < 2, f e H(Un). Then next estimates are correct

C1 ||D-ef ||hp(u«) ^ Sn(f) ^ f 1 |HP(Un),

where e - arbitrary positive number and

S«(f) =

Df

dm2(zi)

T r(§B) T r(§i)

■ dm(§i))2 ...dm2(zn^ dm(§n)j .(3)

Remark 2. For n = 1 the condition (1) disappears and the statements of the theorem coincide and are well known (see [21],[22]).

Remark 3. It will be seen from the proof that by slightly modifying the reasoning in a similar way a somewhat more general result for Hardy-Sobolev classes can be proved. They are defined as follows:

HOP H f G H(Un) :

Da f

hp

< ^ï,

a > 0, P e (0, Proof of Theorem 1.

Lemma 1. Let f e H(Un), then next estimates are correct

sup

T« Vzef(S )

Daf(z)(i -|z|)a dm«(§) < Ci||f ||Hp,0 < P < +-,a > 0; (4)

J ■■■ J /|Df(zi, §2,..., §n)

T TU n-i

X

X

p-2

ziF(zi, §2,..., §n) (i - |zi |)dm2(zi)dmi (§2)... dmi (§«) < C2IIF||pp(UK), (5)

0 < p < <*>;

/• P-2 2

J ziF (zi,..., zn) Df(zi,..., zn) (i - |zi I)... (i -|zn|)dm2(zi,..., z«) <

Un

< C3

[■■■J J |f(§i,z2,...,zn)|P(i - M)... (i - |zn|)dm(§)dm2(z2),...,dm2(z«).

U U T

p

2

Here and hereafter F is defined by (1) 0 < p < <*>. The first inequality is proved in [15] and relies on the multivariable maximal theorem proved in [16], [22]. The second and third inequalities are not difficult to establish directly relying on one consequence of Green's formula (see [16], [22]).

W(ei6)d6 = J (log î^ AW(z)dxdy,

U

where A is the Laplace operator, W e C2(U U T), W(0) = 0.

p

Selecting the function (|zF(pz,z2,...,zn)|2 + e)2,0 < p < 1,e > 0 as W and applying this formula we get

|F(pe'®1,z2,...,Zn)|2 + e)2 d6i =

= log — A (|zF(pz,Z2,...,Zn) |2 + e)2 dm2(z) + 2ne2 ;

U

|z|

passing to the limit at e ^ 0 and given that AF = 4J^F we get

||f( p e'®1, Z2,..., zn)

Pd6i = ^

J |zF (pz, Z2,..., Zn)|p-2x

U

1

x|Df(pz,z2,...,zn)| logndm2(z).

| z|

Now the inequalities (2) and (3) are easily obtained by relying on this proportion. We can integrate both parts either by Tn—1 with measure dm(;2)...dm(;n) or by Un—1 with measure (1 — |z2|)...(1 — |zn|)dm2(z2)...dm2(zn) appropriately. From here we have for 0 < p < 2

r P-2 2

J F (zi,..., Zn) Df(zi,..., Zn) (1 -|zi|)... (1 -|zn|)dm2(z) <

-jn

f

< C1 sup

|Z1|,...,|ZK|

I |f (zi,..., Zn) |pdmn(<§ )(1 - |Z2|)p... (1 - |zn|)p If (1 - r)1-pdr

\TB

1

/

<

<C2||f||Hp,0 <p < 2. (6)

Let us first establish that condition (2) is necessary for f function to belong to the Hardy Hp class. Let 0 < p < 2, then we have

SP (f) < Ci I I sup

F(zi,...,Zn)

(2-p)P s 2

\r(Si) r(|B)

|Df |2dm2n(z) |F (z)|2-p

\

p/2

dmn(^ ).

/

n

Recall that F is defined in equality (1). Next we apply the Holder's inequality with 2

2—p ^ p

the exponent p' = 22p, 4 = p and (7') and we obtain

2-p

2

p/2

Sp(f) < C^ sup |f) x ij |F(z)|2-p|Df(z)|2(1 - |z|)dm2«(z)

Next we use the condition of the Theorem and estimates (4) and (6) respectively, we

get

(2-p)p p2 Sp(/) — C ||F ||Hp2(UB) 11 f 1 |H2 p(Ub) '

From here it is easy to get SP (/) |— C |/, 0 < p < 2. Let us now prove the left inequality. Let p — 1. Then from known estimate (see [8]) we have

/ |f (z)|(1 -|z|)adm2„(z)l < C3 J |f (z)|p(1 - |z|)ap+2p-2dm2„(z), (6')

WK J

p < 1, (a + 2)p > 1,f g H(Un). Hence we deduce

11f(n§1,...,r„§,)|pdm„(<§) <

< CW I I |Df (w)|p

1

1 - wz

p (1 - |w|)ap+2p-2dm2n(w) j dm„(<§) <

< C5 J |Df (w)|p(1 - |w|)p-1dm2„(w), (6'')

rj G (0, 1), j = 1,'.',p.

For 1 < p — 2 the estimate (6") can be obtained similarly using instead (6') estimate

WB

| f (z)|(1 -|z|) |1 - wz|ß+2

J ^ f(z)|p(1 -|z|)ap(1 -|w|)-ep ,,

dm2n(z) j < Q j U ()| (1 -(£)p+2 |) dm2n(z)

I Un |

where a > — 1, e > 0,^ > 0, which is easily obtained from the Holder's inequality. Next, we will need the following estimates, they are well-known at p = 1 (see [7]),

| f (z)|g(z)

|1 - z|

dm2„(z)

< C7|(Aq(f)(<§))(Cq(g)(§))dm„(«§)

(7)

|$(z)|(1 -|z|)adm2„(z) < C8

|*(z)|(1 -|z|)a-1dm2„(z)dm„(^ ).

(7')

u«

Estimates (7) and (7') are not difficult to obtain from one-dimensional version by sequential application over each variable. Based on (7') and (6") we deduce

D- f

P C P 1

HP < c9 J Df (w) (1 - |w|)p-1(1 - |w|)edm2n(w) <

U«

< C10

P \ 2/P

Df (w) (1 -|w|)P

\

P/2

\r(l1) r(&)

(1 - |w|)2

-dm2«(w)

x

/

X

\

V (2/P)'

(1 -|w|)-2+e (2/P)dm2«(w)

) < C„(S«(f))P.

\r(li) r(&) /

The first part of the theorem is established.

Remark 4.

Note that above we (right inequality) have modified the reasoning applicable to n = 1, previously by various authors (see, for example, [5], [6], [16], [22]).

An essential role in our reasoning is played by the maximum theorem in Un, established in [22], the integrals of the squares by Luzin and values Aq(f) and Cq(f) in Un chosen in a suitable way.

We establish the second statement of the theorem. Applying the estimate (7') for each variable separately with the exponent 2/p > 1 and repeating the above reasoning for each variable, we obtain

||D-ef(r£)|Pdm„(<§) < C12 J

Df (w) (1 — |w|)P+e

(1 - |w|)

(

< C13

U«-1 T

X

J |^./(w)|2dm2(w1)

\r(l1)

((1 -|W2|) ... (1 -|w«|))P+£ (1 -|W2|) ... (1 |w«|)

dm2„(w) <

\ P/2 /

dm(^1 )x

dm2n-2(w) < C14 (

(f )

Let us now prove the right inequality. For this we modify the arguments given in proving of the corresponding inequality A). Let

F =

dn 1W2 ... w«f

d W2... d w« '

Reasoning similarly we have

Si =

(

/

dnzi...z„/(z)

dzi ... dZn

\

p/2

dm2(zi)

dm(^) <

/

< Ci5 (/ sup |F(z)|pdm(^i)

1 J zier(li)

2-p 2

p/2

|F (z)|'

2 dn(zi. .. zn/)

dzi. . d zn

(i - |zi|)dm2(zi)

< Ci6

dn-i(z2. .. zn/ (z))

d z2. . d zn

P(i- p/2)

Hi (U)

d n-i(z2. . zn/(z))

d z2.. . d zn

<

P'( P/2) Hp (U)

d n-i(z2. .. zn/(z))

d z2 . . d zn

Hi; (u )

We integrate the last inequality by r(£2) and T by z2. We apply Minkowski's inequality, Fubini's theorem and repeat the reasoning for on the variable z2. Repeating this procedure n — 2 times will come to the desired result. Theorem 1 is fully proved. For the f function, f e H(Un) denote

2

2

G(/,a,p,y) = | J ... J |D/(z)

a+2

Tn Y(|i) r(|n)

\ P/2

. (1 — |z|)a/p+(a+2)k—2—Ydm2n(zH dmn(<§).

The following theorem 2 generalizes the known one-dimensional estimates from below for the Hardy class norm Hp(U) for p > 2 (in two directions at once). This theorem also establishes the e — accuracy of this estimate.

Theorem 2. Let p > 2, f e HP(Un), 0 < a < p. Then

G(f,a,p,0) < C2||f ||Hp (8)

This estimate is accurate in the following sense. For all e, e > 1 — 2/p there will always be a number, a > 0, such that the inequality is true

||f iiHp < Ci(G(f,a,p,e)),

where C;, C2 - there are some positive constants.

Proof of Theorem 2.

The theorem we prove in bidisk. The General case is exhausted similarly. The proof of the estimate (8) relies on the ratio (7) and (7'). Indeed, using duality arguments and the formula

\

g(w)dm2(w)

\r(§)

/

)dm(§) = J g(z)y Ar(§)(z1)^(§)dm(§)dm2(z)

U T

(A is a characteristic function of r(£)) for each variable separately we deduce G(/, a, p, o)= M2 — cj ||Dk/(z)|a+2(1 — |z|) x

U U

x JAp(|2)(z2) I yAp(|1)(z1)y(£1,£2)dm(£) I dm(£2)dm2(z1)dm2(z2),

t \r /

where ^(£1, £2) G Lq(T2),q = (f)',t = a(1 + fc) + 2k — 2.

Next given the estimate (7) with the exponent q = <*>, q' = 1 we deduce the inequality.

M2 < CW / sup sup

T T ^(§1) z2er(|2)

x sup sup

Dkf (z) (1 - |z|)2kx 1

z1GP(,1) z2GP(&)(1 -|z1|)(1

Ar(|2)(z2) I y Ap(|1)(z)^(§b §2)dm(§0 j x

Dkf (z) (1 - |z|)a(1/p+k)) dm(&),dm(^2) <

< C2

_ |z|)a(1/p+k)

(1 -|z|)

Dkf (z)

x sup sup

T T Uer(li) z2er(|2)

L^(TB)

Dkf (z) (1 - |z|)2M J>x

X

x (M(^)(£1,£2)) dm(£1)dm(£2), (8') where M(/) is a maximal function of Hardy-Littlewood.

M( f )(§1, = sup

1

sup

"T It I I "T

t1>0 U t2>0

1

fet |

(91,92)|dm(^1)

§2/7,

dm(92.) (9)

To estimate the first multiplier, it is sufficient to apply the Holder inequality with the exponent p/2 and two maximal theorems (in the polydisk), one of which was mentioned above (see the estimate (4)) and the latter theorem was established in [15]. The second estimate on the action of the Hardy-Littlewood operator is derived by applying a one-dimensional result (see [22]) by each variable.

Now let's estimate the second multiplier. We have

A = (|Dkf (z)|a(1 - |z|)ka+a/p) (§) < C1

1

sup

2 |1t(§2,t2)|

y (1 -|z2|)ka+a/p-1x

Iu (§2/2)

2

2

x sup

i

t! |1t (<§i, ti)|

|Dk/(z)|a(i - |zi|)ka+a/p-idm2(zi)dm2(z2).

(i0)

Iu (li/i)

We -will evaluate "the inner sup"(use the Holder inequality with the exponent (p/a)') for J |Dkf (z)|a dm(<i). |i—<lim l<ti

1

sup^) / / |Dkf(z)|a(i — |zi|)ka+a/p—idm2(zi) < i—tti |i— <finil<ti

< C2 sup(t— i) / M^(Dk/(zi))a/pii(p/a)' (i-|zi|)ka+a/p-idm|zi|<

ti 1J

i-ti

a/p

< C3

D(z2)/(<§i, |z2|&) dm(^i)

The second multiplier is evaluated similarly. So,

a < cii f iiHp .

(11)

Remark 4'. The above inequality (8) is well known for n = i (see [22]). To establish the second inequality, we use the known one-dimensional embedding (see [4]-[7],[21],[22]).

AS(U) c Hp(U),r < p,s — i/r = — ^,p > 2,r < a + 2, where Ar is analytic new Bergman's class in U.

(i2)

AS(U) = ^ / e H(U) : J |Dk/(z)|r(i -|z|)(k-s)r-idm2(z)

U

< ^

Using embedding (12) (by each variable) and estimate (7') with an exponent , we deduce

p

Hp <

<C

\p

|Dk/(z)|a+2(i - |z|)a/p+(a+2)k-2-edm2(z)

x

/

X

C(a+2)(i -|z|)Y , (i3)

where

i

(¥)''=fc+nb){(k-s)r-^(-f+c+H (14)

It remains to be noted that from the conditions of the Theorem and the obtained their relation is possible to choose a such that 7 > 0, and the latter is sufficient for the finiteness of the second multiplier in (13). Theorem is proved.

New equivalent quasinorm to the Bergman type analytic space in the polydisk

In this section we will find some equivalent norms (quazinorms) for new Bergman type classes of analytic functions. Let's introduce new Bergman type spaces in the polydisk as follows

H f e H(Un)

Lp,q

J (J |Rsf (r| )|q(1 - r)VdrJ dm„(«§ ) < -,

p, q e (0, -), Y >-1, l > 0}, (15)

T« \ 0

where - differential operator acting as a bounded operator from H(Un) to H(Un)

(Rf) (z) = £ (ki + ■ ■ ■ + kn + 1)sakzk, f (z) = £ . kl,...,kB>0 |k|>0

Below we establish the theorem for the quasinorm || ||psyl for p = q using some new space and thier relation to new functional spaces (see [4]-[7])but in Un.

Theorem 4.

1). Let Rsf (z,..., z) G L0;P+n ), 0 < p < -, r, s > 0, n G N, 7 = ^ - 1 and assame a// conditions imposed on 7, ^ in 2) are valid. Then we have

p —

Lp, p

j j |Dßf(zi,...,zn)|p(1 -|z|)^ +p(ß-n)-2dm2n(z)

\r« (I )

dmn(I ); (16)

2). Let 0 < p < -,ß > n, 7,s > 0,n e N, n > max ( 1 + 2+1, 3+2 - 1).

p

^p.p

J J |Dßf (z1,...,zn)|p(1 - |z|)r+p(ß-n)-1dm2n(z)

\r« (S )

where D : H(Un) ^ H(Un) is a differential operator

dmn (I ), (17)

Dß /) (z) = £

|k|> 0

r(k + ß + 1) r(k + 1)r(ß + 1)

,ß > 0;r(k + ß + 1) = nr(kj + ß + 1).

j=i

Mixed norm analogues of these results by similar methods can be also obtained (the case when p is not equal to q) but there our results are not sharp. Remark 5. The Theorem 3 was announced at [3].

Remark 6. For n = 1 operators and Ds, classes Lp'^ and Lp'J, relations (16) and (17) coincide and Theorem 3 is not difficult to deduce from the well-known embedding theorems and the Hardy-Littlewood theorem (see [22], [3], [21]) .

The proof of Theorem 3. First, we prove the second part of the theorem. Note that

G/) = J |D/r(z)|q(1 - |z|)a-2dm2n(z) = ra

Fa

rmz) R

(1 - fR)ß+1

dm„(i )

(1 -|z|)a-2|z|dm2n(z), (1V)

s > 0, a > 0,fR(z) = f (Rz),R GI, where a will be chosen below. Next we will use the following two relations

J/(r1?1,..., rni„)g(r1i1,..., r„F„)dmn(i)

J J/(rw)(g(rw)) flog p^l pdpdm„(p)

0

; (18)

where s > 0, /, g G H (Un), w = (pm,..., pnçn), P dp = n Pkdpk.

k=1

q

1

—s

1

IJ |)|q (1 -|w|)a+n-1d |w|dmn(^) <J |$(w)|q (1 -|w|)a/ndm2n(w), (19)

Tn 0 Un

where a > -1,q G (0, $ G H(Un).

The first estimate is easily deduced from the following equality

C I . . . I '. . . ' *h)/(*1'. . . ' kn) (r2k1 . . . rn2kn) =

k1>0 kn>0

1

log iWp) M2(kl+ +kn)MdM = //rsZ(™>)g(rw) flog-U dpdm„(p)

0

where p = |w|.

The second estimate we see in [1], [8]. Using successively (19), (6') and estimate (6"), we deduce the following inequalities

1|f(z)|(1 -W)'dm„(,)d|z| I <

0

|1 - w z|ß

<r f f If(z)|q(1 -|z|)q-2+(^)q , () (20) < QJJ-^-dm2«(z), (20)

where w e Un,z = (|z|^i,..., |z|^n),1 > 0,y > -1 for q < 1,y > -qj - 1 and

11 f(z) | (1 -| z | )Ydm.(9)d| z |I <

0

|1 - w z|ß

Y+1 — n

<r if | f (z)|q(1 -|z|^3+n-q(1 -|w|)e q

< -11 - wz|ßq-(2+e)q+2-dm2»(z), (21)

Tn In

where j8 > 0, y > -1 for q > 1, y > - q? - 1. Next from (17') given (18), (20), (21) we get

G f ) <

f (1 | ha-2 [} 1 |RsfR(w)|p (1 -|w|)td|w|dm„(^)dm2n(w) (21/)

< rW (1 -|Z|) J J-J-|-- w Rz|(ß+1) p- (21)

f(|1,...,|n) Tn 0 0

for p < 1, where t = Sn + p - 2 and

G (fR2 ) <

< r r (1 - |z|)a-2 fj... 1 |RsfR(w)|p (1 - |w|)t(1 - |z|)-epdmw(9)dm2n(w) (21//)

- 0 0 |1 -Rwz|(^+1)p+2-(2+e)p

for p > 1, where t = (n - 1) p. Let a = y + p (£ - n) + 1. Given that

(1 — |z|)a-2dm2n(z) < C n

„ J |1 — zR0W|(ß+1)p " 7=i |1 — RÄ|p(ß+1)—a:

Wk g U,<§k G T,R0 GI,I =(0,1), (see [8], [22]), and passing to the limit for R0 ^ 1 from (21') we get

G(■ /) < //7"^ß-if diw|dm„(,),t = SP + P — 2.

T« /K 11 W' 1

To obtain the first half of the statement of the second part of the Theorem for p < 1 it remains to integrate both parts of the last inequality in T n and to take into account the following known estimate (see [8])

n h ,v) dmn(^) < CI! (1 ,1 hv-1' V > 1 (22)

Tn vfe=1 |1 - §fcWfc| / k=1 (1 -|wk|)

To obtain the first half of the statement of the second part of the theorem for p > 1 from (21"), it is necessary to conduct similar argument by putting a = y + ^p + 2 - sn. And given the (22) and conditions of the Theorem it is not difficult to deduce that

J G(f)dmn(S) < C||f ^.

yn

We now will prove the inverse estimates. Thus we will complete the proof of the second part of the second half of Theorem 3 completely. Based on formula (7') it is easy to see that it is enough to set the estimate

p

LS,Y < C

Un

Note that from (18) it follows that

j |D/(z) |P (1 — |z|)Y+p(ß—s/n)dm2„(z).

1

|DaRs/(z)| < C(Y,n)j y |Dß/(rw)

0 Tn

x

x

RS+JDa D—ß

1

1 — rw ^

(1 — Iwl)—1d |w|dp, (22')

where z = r2^,w = (|w|^1,..., |w|$n),t - relatively large.

Let p < 1. We will need an analogue of the estimate (6') for the subframe of Un. We have

1

1 V

h f (z)|(1 -|z|)ad |z|dm„(p )| <

1

< C(p)/ J |f (z)|p(1 -|z|)ap+(n+1)(p-1)d|z|dm„(p), (23)

Tn 0

a > "p

(1p-p^ - 1. It is easy to deduce, given that

M (f,n,...,r„) < C J!(1 -rk)1-1/pMp(f,n,...,r„),p < 1,

k=1

and from the elementary inequality 1

JG(p)(1 -p/dp I < c|(G(p))p(1 -p)tp+p-1dp, 00 where p < 1, t > -1, where G is an increasing function. Next we note that

a-1+1

Rs+t

1 - rw ^

p

dm„(^) <

n 1

<C1 jc n(1 -MM)p<*~-i+1)-1,zrn«n); (24)

£ a,-=s+t j>0

where a and t - are sufficiently large positive numbers.

Estimates (24) are not difficult to deduce directly from Newton's binomial formula and (22). Given (23) and (24) we get

Un

|DaRsf (z)|p (1 -|z|)Y+apdm2„(z) <

1

< Q J J |Dßf (w) [ (1 - |w|)(t-1)p+(«+1)(p-1)x

0 Tn

1 1

x £ 1 1_(1 - |z|)Y+apd|w|d|z|dmn(^)_<

j0 J J (1 -|w||z1|)p(a1+a-1+1)-1... (1 -|w||z„|)p(an+a-1+1)-1 < _, j 0 0 £ aj=s+t

1

< C2 y J |Dßf (w)|p (1 -|w|)Yn+1pn-ps+n-1dm„(p)d|w| <

0 Tn

< r3

Un

D1 f (w) p (1 - |w|)Y+p(1 -s/n)dm2„(w),

1

where in the last inequality we used the estimate (19).

For p > 1 instead of estimating (23), we should use the analogue (6") for subostov (subpolydisk) (which easily follows from the Holder inequality)

J — (/ / f^ ™ m) p<

\0 1 1 /

< C4 / l |Rsf(m)|p (1 -W)"-1'p «M^-P -1 zn I ))-1+1/n+e p d , m | , " 0 /, |1 - wz I(Y-1) p+2+ep ,

where e is any number, e > 0 and to carry out similar above to the reasoning. The second part of the theorem is proved.

To prove the first part of the theorem, it is necessary to use the already established second part of Theorem 3 and the following Lemma A. Lemma A.

1). The following estimate is valid

Si < CSi, (25) (1)

where

Si =//1 f (z)|q(1 - | z |)admn(p) |z | d | z | 0

and

S1 = [ | f (w) |q J! (1 - | wk |) ^ -1dm2n(w), ^ k—1

U« —

here z — (| z 191,..., | z |9n),f G H(Un),0 < q < a > -1. 2). Let S1 < <*>, then if

S2 — J | f(z,..., z) | q(1 - | z | )a+n-1dm2(z) < q G (0, ~).

U

Then the inverse of (25) is also true. The proof of Lemma A. The first inequality is actually established in [8]. (It is a theorem on mapping from polydisk to diagonal in Ap classes ). It is necessary to use the dyadic division of the parallelepiped

In — J [1 - 2-k1;1 - 2-k1 -1) x---x [1 - 2-k«; 1 - 2-k«-1)

k1 ,...,kn>0

and to limit the summation for n-dimensional sums on the diagonal (see [8]). We prove the second assertion of the Lemma. In [8] it is fixed that for the function f(z1,...,zn) :

f (z,..., z)(1 -|z|) Ydm2(z) (1 - zzO^ ... (1 - zzn)^ the value S1 is finite if only S2 < and the inequality S1 < constS2 is also true, so

r, x n( A/-/(z,...,z)(1 -|z|)Ydm2(z) . _

f(zi,...,z„) = C(y,n) ^—-—-m,zj e U,j = 0,...,m;

U (1 - zzi) n ... (1 - zzn) n

1

J | f (zi,..., zn)|q(1 -|z|) -1dm2n(z) < cj | f(z,..., z)|q(1 -|z|)a+n-1dm2(z),

Un U

0 < q < —, a > -1, n > 1.

It remains to use once again the dyadic partition of the subostov ( polydisk ). Un = {z = ..., r£n), ^ g T, r g I} and the summation of the n- dimensional sum along the diagonal to establish the estimate (see [1], [3], [8]).

1

J | f (z,..., z)|q(1 - |z|)a+n-1dm2 (z) < C| J | f(z1,..., zn)|q(1 -|z|)admn(^)d|z|,

Un 0 Tn

z = (|z|^1,..., |z|^n),0 < q < —,a > -1 (see [1], [3], [8]). At the same time it is necessary to take into account also that the function defined on subostov Un can be extended uniqily to arbitrary point of the unit polydisk Un (see [8], [18]) . The Lemma A is proved.

On some new equivalent norms for analytic Bergman type spaces in bounded strongly pseudoconvex domains

In this final section we prove similar type result for Bergman type spaces in bounded pseudoconvex domains.

Let D be a bounded strictly pseudoconvex domain in Cn with smooth boundary, let d (z) = dist (z, d D).

Then there is a neighborhood U of D and p g C—(U) such that D = {z g U : p (z) > 0}, | vP(z)| > c > 0 for z g dD,0 < p(z) < 1 for z g D and —p is strictly plurisubharmonic in a neighborhood U0 of dD. Note that d(z) x p(z),z g D. Then there is an r0 > 0 such that the domains Dr = {z g D: p(z) > r} are also smoothly bounded strictly pseudoconvex domains for all 0 > r > r0. Let do> be the normalized surface measure on dDr and dv the Lebesgue measure on D. The following mixed norm spaces were investigated in [20]. For 0 < p < —, 0 < q < —,8 > 0 and k = 0,1,2,... set

p.,«=(e (r8 L,|D"f|PdCTrfPdr)1/q,0 < q <—

|a |<k

and weighted Hardy space (A0,— = Hp)

\ 1/q

= sup £ rM |Daf |, 0 < q <

r8

0<r<r0 "'dDr J

where Da is a derivative off (see [20]) The corresponding spaces Ap,q = A8P,q(D) = {f g H(D) : ||f||pq,8;k<—} are complete quasi normed spaces, for p,q > 1 they are Banach spaces. We mostly deal with the case k = 0, when we write simply Ap,q and ||f||pq,8. We also consider this spaces for p = — and k = 0, the corresponding space is denoted by A—,p = A—,p(D) and consists of all f g H(D) such that

rr \ 1/p

,,p,8 = ( jT(sup |f |)pr8p—^rj < Also, for 8 > -1, the space A^ = (D) consist of all f e H(D) such that

||f||aj = sup|f(z)|p(z)8 <

8 zeD

and the weighted Bergman space Ap = A^(D) = Ap+^D) consists of all f e H(D) such that

IIf llAp = iL If(z)lPP8(z)dv(z^1/P <

We denote by Kfi the weighted Bergman kernel on D (see [4], [3]).

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

f (z) = Cfi L f (£ )K (z, £ )Pt (£ )dv(£), f e A1 (D), z e D, (*)

where K(z, £) is a kernel of type t, that is a smooth function on D x D such that |K(z,£)| < C|4>(z,£)|-(n+1+t), where 4>(z,£) is so called Henkin-Ramirez function for D. Note that (*) holds for functions in any space X that embeds into A1.

The following is the main result of this section.

Theorem 4. Let p,q e (0,<x>),y> —1, fi > —1. Then for f e H(D) we have

q/p p / \ q/p

|f |p | efi+( r+1)q/pde x f\f |f (z)|pdY(z)dv(z) | rfidr.

Remark. This theorem almost with similar proof probably can be extended also to Ap,~and A~,p,p e (0,~), 8 > 0.

Note in the Unit disk we see this result in [14].

We will need the following simple lemmas for the proof of our main result. Lemma D. (see [14]). Let 0 < p < ^ and fi > 0. Let (an) be a sequence of nonnegative

real numbers such that £ 2—nfiap < <*>. Then there is a constant C > 0, depending only

n=1

on p and fi, such that

£ 2—nfiap < C(a0 + £ 2—nfi |an — an—11p).

n=1 n=1

Lemma B. (see [18], [20]). 1. Let G e L1(De), w e c"(d).

£

J G(z)d£ x | J G(z)w(z)d£. D 0 d d£

2. Let

hP (r) = J | f | pda.

Then function is non icreasing, and

(t50)hp (2t0) < cj 2t0hp

(r)r5 dr,

t0

0 > 0,0 < p < —, 5 > 0. 3. Let 0 < p < —,

r

n

«n = J J I ft (z)|p (z)w(z)dv(z), w e C—(D).

0 d d£

rn = 2-n,n = 0,1,...,t > 0.

Then the assertion of Lemma D is valid.

For any Lebesgue measurable function f in U, we define

1/p

MP(r,f)= (/ If^)IPdm(^^

where 0 < p < —.

If 0 < p < —, 0 < q < —, and a > -1, let

q,a = /(1 - r2)a Mp(r, f )qdr, I

where I = [0,1). The mixed norm space Hp,q,a = Hp,q,a(U) is then defined to be the space of function f holomorphic in U, (f e H(U)) such that ||f ||p, q,a < —.

The result of this section in the unit disk can be expanded to this classes in the polydisk from the unit disk (see below).

First we have a new characterization of the mixed norm spaces Hp,q,a(U).

Let 0 <p,q < — and -1 < 0,7< —. A function f eH(U) belongs to Hp,q,0 +p/q(r+1)(U) if and only if

( \

/ If(z)|p(1 -|z|)Ydm2(z)

0 \z|<r J

q/p

(1 - r)ßdr < -

Note that, it is our theorem in the unit disk. We follow arguments from [14] first assume that ||f ||p,qj3 +q/p(y+1) < —. For 0 < t < 1 define ft(z) = f(tz),z e U. Let r„ = 1 -2-n,n = 0,1,...'. Then

- r)ß

\

q/p

| ft (z)| p(1 -|z|) Ydm2(z)

Vzl

<r

dr =

(

I / (1 - r)ß

n=1

rB- 1

\

q/p

| ft (z)| p(1 — |z|) Ydm2(z)

\z|<r

dr < C1 £ 2—+1Uq/p,

/

n=1

where

An = J |ft(z)|p(1 — |z|)Ydm2(z).

|z|<rB

Now by using Lemma D we find that

— r)ß

\

q/p

| ft (z)| p(1 — |z|) Ydm2(z)

\|z|<r

dr

/

(

< C2 I 2—+1)

n=1

\

q/p

| ft (z)|p(1 — |z|) Ydm2(z)

\rK— 1<|z|<rB

rn

<

/

< C3 £ Mp(rn,ft)q2—n(Y+1)p/q <

n=1

'"n+1

< C4 U Mp(rn,ft)q(1 — r)ß+q/p(Y+1)dr < C5 j (1 — r)ß+q/p(Y+1)Mp(rn,ft)qdr.

Letting t ^ 1, we get

— r)ß

\

q/p

| f (z)| p(1 — |z|) Ydm2(z)

Vz|

<r

dr < ^f ||q,q,ß +q/p(Y+1).

/

Conversely

1

n

1

1

1

p,q,ß +q/p(Y+1)

= y"(1 - r)ß+q/p(Y+1)Mp(r, f )qdr = 0

£ / (1 - r)ß+q/p(Y+1)Mp(r,f)qdr < C7 £ 2-n(ß+q/p(Y+1)+1)Mp(r„,f)q <

n=1

< Cg £

n=1 /

\

q/p

| f (z)|p(1 -|z|)Y dm2(z)

\rB<|z|<rB+1

2-n(ß + 1) <

< C9 £

n=1

/

\ q/P On+2

J I f (z)| p(1 -|z|)Ydm2(z)

\rB <|z|<rB+1 y

rB+2 /

J (1 - r)ßdr <

rB+1

< C10 £ /(1 - r)ß

B=1rn+1

\

q/p

| f (z)I p(1 -|z|)Y dm2(z)

№

<r

dr <

/

q/p

i '

< C11J(1 - r)ß J |f (z)|p(1 -|z|)Ydm2(z)

dr.

0 \|z|<r y

Note we used Lemma D in our arguments above. The carefull analysis of the unit disk proof we just provided shows that the repetition of arguments provided in the unit disk and applications of two lemmas, Lemma D and Lemma B, easily lead us also to the proof of the main theorem of this final section.

We leave these easy details to interested readers. For any Lebesgue measurable function f in Un, we define

Mp(r,f)= f| |f (r£)|pdm„(£)

Vb

where 0 < p < — and = (ri ,..., rn£n).

If 0 < p < —,0 < q < —, and a = (a1,...,an),«/ > -1, j = 0,...,n, let

II f llp,q,a = / (n (1 - r2 )ajMp(r, f)qj dr,

where In = [0,1)n and dr = dr1... drn. The mixed norm space Hp,q,a = Hp,q,a(Un) is then defined to be the space of function f holomorphic in Un, (f e H(Un)) such that

11 f 11 p,q,a < —.

The result of this section in the unit disk can be expanded to these classes in the polydisk from the unit disk.

It will be nice to obtain some direct analogues of our results of first sections in more general bounded pseudoconvex domains or in unbounded tube domains over symmetric cones.

1

q

References

[1] Shamoyan R. F., "O prostranstvakh golomorfnykh v polikruge funktsiy tipa Lizorkina-Tribelya (On spaces of holomorphic functions in polydisk Lizorkin-Tribel type )", Izv. NAN Arm, 2002, №3, 57-78.

[2] Shamoyan R. F., "On BMOA-type characteristics for one class of holomorphic functions in a disk", Siberian Math. J., 44:3 (2003), 539-560.

[3] Shamoyan R. F., "On the quasinorm of holomorphic functions from classes Lizorkin-Tribel in subostov", Symposium "Fourier Series and their applications" (in Russian), 2002, 5455.

[4] Cohn W., "A factorization theorem for the derivative of a function in Hp", Proc. AMS., 127:2 (1999), 507-517.

[5] Cohn W., "Bergman projections and operators on Hardy spaces", Funct. Anal. J., 144 (1997), 1-19.

[6] Cohn W., Verbitsky I., "Factorization of Tent spaces and Hankel operators", Journal of Functional Analysis, 175 (2000), 308-329.

[7] Coifman R., Meyer Y., Stein E., "Some new functional spaces and their application to harmonic analysis", Journal of Functional Analysis, 62:2 (1985), 304-335.

[8] Djrbashian M.M., Shamoian F. A., Topics in the theory of Aa spaces, Teubner-Verl., Leipzig, 1988.

[9] Dorronsoro J., "Mean Oscillation and Besov Spaces", Canadian Mathematical Bulletin, 28:4 (1985), 474-480.

[10] Dyakonov K. M., "Besov spaces and outer functions", Michigan Math. J., 45:1 (1998), 143-157.

[11] Dyakonov K. M., "Equivalent norms on Lipschitz-type spaces of holomorphic functions", Acta Math, 178:143-167 (1997).

[12] Dynkin E. M. Free interpolation sets for Holder classes, Math. USSR-Sb., 37:1 (1980), 97-117.

[13] Fergusson S., Sadosky C., "Hankel operators and bounded mean oscillation on the polydisk", Math. Anal. and Applic J., 2002, 241-267.

[14] Jevtic M., Pavlovic M., Shamoyan R. F., "A note on diagonal mapping theorem in spaces of analytic functions in the unit polydisk", Publ. Math. Debrecen, 74/1-2 (2009), 1-14.

[15] Guliyev V. S., Lizorkin P. I."Classes of holomorphic and harmonic functions in a polycircle in connection with their boundary values"", Research on the theory of differentiable functions of many variables and its applications, Proc. Steklov Inst. Math., 204, 1994, 117-135.

[16] Gvaradze M., "Mnozhiteli odnogo klassa analiticheskikh funktsiy, opredelennykh na polidiske", Tr. Tbil. Mat. In-ta (In Russian), 66 (1980), 15-21.

[17] Koosis J. B., Bounded analytic functions, Academic Press, Orlando, Fla, 1981, 356 p pp.

[18] Krantz S., Ma D., "Bloch functions on strongly pseudoconvex donains", Indiana University Mat. Journal, 37:1 (1988), 145-163.

[19] Marcinkciewicz J. Riewicz, Zygmund A., "A theorem of Lusin", Duke Math. J., 4 (1938), 473-485.

[20] Ortega J.M., Fabrega J., "Mixed-norm spaces and interpolation", Studia Math, 109:3 (1994), 233-254.

[21] Rudin W., Function theory in the polydisk, Benjamin, New York, 1969.

[22] Tribel Kh., Theoriya funktsionalnykh prostranstv (The theory of functional spaces)., Moscow, 1986, 448 pp.

[23] Yoneda R., "Characterizations of Bloch space and Besov spaces by oscillations", Hokkaido Math. J, 29:2 (2000), 409-451.

[24] Zhu K., Operator theory in function spaces, Springer, New York, 1990, 280 pp.

[25] Zygmund A., Trigonometric series. V.I, II, Cambridge University Press, 1959.

References (GOST)

[1] Shamoyan R. F. O prostranstvakh golomorfnykh v polikruge funktsiy tipa Lizorkina-Tribelya (On spaces of holomorphic functions in polydisk Lizorkin-Tribel type ) // Izv. NAN Arm., 2002. no. 3, pp. 57-78.

[2] Shamoyan R. F. On BMOA-type characteristics for one class of holomorphic functions in a disk // Siberian Math. J. 2003. vol. 44. no. 3. pp. 539-560.

[3] Shamoyan R. F. On the quasinorm of holomorphic functions from classes Lizorkin-Tribel in subostov // Symposium "Fourier Series and their applications" (in Russian), 2002. pp. 54-55.

[4] Cohn W. A factorization theorem for the derivative of a function in // Proc. AMS. 1999. vol. 127. no. 2. pp. 507-517.

[5] Cohn W. Bergman projections and operators on Hardy spaces // Funct. Anal. J. 1997. vol. 144. pp. 1-19.

[6] Cohn W., Verbitsky I. Factorization of Tent spaces and Hankel operators // Journal of Functional Analysis. 2000. vol. 175. pp. 308-329.

[7] Coifman R., Meyer Y., Stein E. Some new functional spaces and their application to harmonic analysis // Journal of Functional Analysis. 1985. vol. 62. no. 2. pp. 304-335.

[8] Djrbashian M. M., Shamoian F. A. Topics in the theory of A^ spaces. Leipzig. Teubner-Verl., 1988.

[9] Dorronsoro J. Mean Oscillation and Besov Spaces // Canadian Mathematical Bulletin. 1985. vol. 28. no. 4. pp. 474-480.

[10] Dyakonov K.M. Besov spaces and outer functions // Michigan Math. J. 1998. vol. 45. no. 1. pp. 143-157.

[11] Dyakonov K. M. Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math. 1997. vol. 178. pp. 143-167.

[12] Dynkin E.M. Free interpolation sets for Holder classes // Math. USSR-Sb. 1980. vol. 37. no. 1. pp. 97-117.

[13] Fergusson S., Sadosky C. Hankel operators and bounded mean oscillation on the polydisk // Math. Anal. and Applic J. 2002. pp. 241-267.

[14] Jevtic M., Pavlovic M., Shamoyan R. F. A note on diagonal mapping theorem in spaces of analytic functions in the unit polydisk // Publ. Math. Debrecen. 2009. vol. 74/1-2. pp. 1-14.

[15] Guliyev V. S., Lizorkin P. I. Classes of holomorphic and harmonic functions in a polycircle in connection with their boundary values". Research on the theory of differentiable functions of many variables and its applications // Proc. Steklov Inst. Math., 1994. vol. 204. pp. 117-135.

[16] Gvaradze M. Mnozhiteli odnogo klassa analiticheskikh funktsiy, opredelennykh na polidiske // Tr. Tbil. Mat. In-ta (In Russian). 1980. vol. 66. pp. 15 - 21.

[17] Koosis J. B. Bounded analytic functions. Orlando, Fla: Academic Press, 1981. 356 p.

[18] Krantz S., Ma D. Bloch functions on strongly pseudoconvex donains // Indiana University Mat. Journal. 1988. vol. 37. no 1. pp. 145-163.

[19] Marcinkciewicz J. Riewicz, Zygmund A. A theorem of Lusin // Duke Math. J. 1938. vol. 4. pp. 473-485.

[20] Ortega J.M., Fabrega J. Mixed-norm spaces and interpolation // Studia Math. 1994. vol. 109. no. 3. pp. 233-254.

[21] Rudin W. Function theory in the polydisk. Benjamin. New York. 1969.

[22] Tribel Kh., Theoriya funktsionalnykh prostranstv (The theory of functional spaces). Moscow. 1986. 448 p.

[23] Yoneda R., Characterizations of Bloch space and Besov spaces by oscillations. Hokkaido Math. J. 2000. vol. 29. no. 2. pp. 409-451.

[24] Zhu K. Operator theory in function spaces. New York: Springer. 1990, 280 p.

[25] Zygmund A. Trigonometric series Vols. I, II (2nd ed.). Cambridge University Press, 1959.

Vestnik KRAUNC. Fiz.-Mat. Nauki. 2020. vol. 31. no. 2. pp. 32-55. ISSN 2079-6641

УДК 517.55+517.33

Научная статья

О некоторых новых оценках интегралов функции площадей и аналитических классов типа Бергмана в некоторых областях в Сп

Р. Ф. Шамоян1, Е. Б. Томашевская2

1 Брянский государственный университет имени академика И. Г. Петровского, 241036, г. Брянск, Россия

2 Брянский государственный технический университет, 241050, г. Брянск, Россия E-mail: [email protected], [email protected]

В работе приведены новые эквивалентные квазинормы для некоторых новых пространств типа Бергмана в полидиске и в ограниченных псевдовыпуклых областях. Подобные оценки установлены также для классов типа Харди в полидиске. Эти результаты обобщают некоторые известные одномерные неравенства для пространств типа Харди и классов типа Бергмана в единичном круге. на случай полидиска и ограниченной псевдовыпуклой области. Оценки такого типа могут иметь также различные приложения. Пусть D ограниченная или неограниченная область в Cn (ограниченная псевдовыпуклая или неограниченная трубчатая область над симметрическим конусом). Подходы, примененные в данной работе при доказательстве утверждений в полидиске могут быть, по-видимому, также использованы для доказательства подобных приведенных в данной работе оценок, но в полиобластях D х ■ ■ ■ х D существенно более общего типа, чем единичный полидиск.

Ключевые слова: интегральные операторы, аналитические функции, псевдовыпуклые области, полидиск, классы типа Бергмана, классы Харди.

Для цитирования. Шамоян Р. Ф., Томашевская Е. Б. О некоторых новых оценках интегралов функции площадей и аналитических классов типа Бергмана в некоторых областях в Сп // Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 31. № 2. С. 32-55. 001: 10.26117/20796641-2020-31-2-32-55

Конкурирующие интересы. Авторы заявляют, что конфликтов интересов в отношении авторства и публикации нет.

Авторский вклад и ответственность. Все авторы участвовали в написании статьи. Авторы несут полную ответственность за предоставление окончательной версии статьи в печать. Окончательная версия рукописи была одобрена всеми авторами.

DOI: 10.26117/2079-6641-2020-31-2-32-55

Поступила в редакцию: 30.04.2020

В окончательном варианте: 11.06.2020

@ Шамоян Р. Ф., Томашевская Е. Б., 2020

Финансирование. Исследование выполнялось без финансирования