A note on a distance function in Bergman type analytic function spaces of several variables Текст научной статьи по специальности «Математика»

УДК 517.55

A Note on a Distance Function in Bergman Type Analytic Function Spaces of Several Variables

Romi F. Shamoyan* Sergey M. Kurilenko^

Laboratory of complex and functional analysis Bryansk State University Bezhitskaya, 14, Bryansk, 241036

Russia

Received 10.08.2014, received in revised form 10.10.2014, accepted 20.11.2014 New sharp estimate concerning distance function in certain Bergman-type spaces of analytic functions on tube domains over symmetric cones is obtained. This is the first result of this type for tube domains over symmetric cones. New similar results in analytic mixed norm spaces on products of tube domains over symmetric cones will also be provided.

Keywords: distance estimates, tube domains, Bergman spaces.

1. Introduction and preliminaries

In this note we obtain a sharp distance estimate in spaces of analytic functions in tube domains over symmetric cones.

This line of investigation can be considered as continuation of previous papers (see, for example, [1,2] and [3] and references there).

This new results are contained in the second and third section of this note. We remark that for the first time in literature we consider this type extremal problem related with distance estimates in spaces of analytic functions on tube domains over symmetric cones.

The first section contains known required preliminaries on analysis on symmetric cones.

In one dimensional tubular domain which is upperhalfspace C+ (see, for example, [4]) our theorem is not new and it was obtained recently in [5].

Moreover arguments we provided below in proof are similar to those we have in one dimension and the base of proof is again the so-called Bergman reproducing formula, but in tubular domain over symmetric cone (see, for example, [4] for this integral representation).

We shortly remind the history of this problem.

Recently various papers appeared where arguments which can be seen in [6] were extended in various directions (see, for example, [1-3]).

In particular in mentioned papers various new results on distances for analytic function spaces in higher dimension (unit ball and polydisk ) were obtained. Namely new results for large scales of analytic mixed norm spaces in higher dimension were proved.

* rsham@mail.ru tSergKurilenko@gmail.com © Siberian Federal University. All rights reserved

Later several new sharp results for harmonic functions of several variables in the unit ball and upperhalfplane of Euclidean space were also obtained (see, for example, [1] and references there).

We mention separately [5] and [7] where the case of higher dimension was considered in special cases of analytic spaces on subframe and new similar results in the context of bounded strictly pseudoconvex domains with smooth boundary were also provided.

The classical Bergman representation formula in various forms and in various domains serves as a base in all these papers in proofs of main results.

We would like to note also recently Wen Xu (see [8]) repeating arguments of Ruhan Zhao in the unit ball obtained results on distances from Bloch functions to some Mobius invariant function spaces in one and higher dimension in a relatively direct way.

Probably for the first time in literature these extremal problems connected with distances in analytic spaces appeared before in [9] and in [10], where this problem was formulated probably for the first time and certain concrete cases connected with spaces of bounded analytic functions in the unit disk were considered.

These results much later were mentioned in [11]. Some other results on distance problems in BMOA spaces can be found also in [12].

Various other extremal problems in analytic function spaces also were considered before in various papers (see, for example, [13-16]).

In those papers other results related to this topic and some applications of certain extremal problems can be found also.

The motivation of this problem related with distance function is to find a concrete formula which will help to calculate this function more concretely via the well-known Bergman kernel.

The goal of this note to develop further some ideas from our recent mentioned papers and to present a new sharp theorem in tubular domain over symmetric cones.

For formulation of our results we will need various standard definitions from the theory of tubular domains over symmetric cones (see, for example, [4,17-19]).

Let Tq = V + ill be the tube domain over an irreducible symmetric cone l in the complexi-fication VC of an n-dimensional Euclidean space V. Let H(Tq) be the space of all holomorphic functions on Tq. Following the notation of [17] and [4] we denote the rank of the cone l by r and by A the determinant function on V.

Letting V = rn, we have as an example of a symmetric cone on rn the Lorentz cone An which is a rank 2 cone defined for n ^ 3 by

An = {y e rn : y\-----y2n > 0, yi > 0}.

The determinant function in this case is given by the Lorentz form

A (y) = y\-----yn

(see, for example, [4]).

Let us introduce some convenient notations regarding multi-indexes.

If t = (ti,... ,tr), then t* = (tr,... ,ti) and, for a e r, t + a = (ti + a,... ,tr + a). Also, if t,k e rr, then t < k means tj < kj for all 1 ^ j ^ r.

We are going to use the following multi-index

go =((j - 1)d) , where (r - 1)d = - - 1.

\ 2 J i^j^r 2 r

For t e r+ and the associated determinant function A(x) we set

A?(Tn) = \ F gH(Tq): ||FIU- = sup |F(x + iy)|AT(y) < ,

x+iyeTn

This is a Banach space. We denote by As the generalized power function (see for this function, for example, [4,17]).

n

For 1 ^ p,q < and v e r, and v >--1 we denote by Ap,q(Tq) the mixed-norm

r

weighted Bergman space consisting of analytic functions F in Tq such that

Il F ||Ap„ = i f^V IF (x + iyWdxJ V (y)

q'p dy Y,q

A(y)n/r

< <x>.

This is a Banach space. For various properties of such functions we refer the reader to [4]. Replacing above A by L we will get as usual the corresponding larger space of all measurable functions in tube over symmetric cone with the same quazinorm (see [17,18]).

n

It is known the AP'q(Tq) space is nontrivial if and only if v >--1, (see, for example, [4,19]).

r

When p = q we write (see, for example, [4])

AP'q (Tq) = AP (Tq).

This is the classical weighted Bergman space with usual modification when p = to.

The (weighted) Bergman projection Pv is the orthogonal projection from the Hilbert space L"l(Tq) onto its closed subspace A^(Tq) and it is given by the following integral formula (see [4]).

Pvf(z) = Cv\ Bv(z,w)f(w)dVv(w), (1)

■JTn

where

Bv(z,w) = A-(v+ r )((z - w)/i) is the Bergman reproducing kernel for

A2 (Tq)

(see [4,17]).

Here we used the usual notation

dVv (w) = Av-r (v)dudv,

(see [4]).

Below and here we use constantly the following notations w = u + iv e Tq and also z = x + iy e Tq.

Hence for any analytic function from A^(Tq) the following integral formula is valid (see also [4])

f(z) = Cv ( Bv(z,w)f(w)dVv(w). (2)

In this case sometimes below we say simply that the f function allows Bergman representation via Bergman kernel with v index.

Note these assertions have direct copies in simpler cases of analytic function spaces in unit disk, polydisk, unit ball, upperhalfspace C+ and in spaces of harmonic functions in the unit ball

or upperhalfspace of Euclidean space Rn. These classical facts are well-known and can be found, for example, in [20] and in some items from references there.

Above and throughout the paper we write C (sometimes with lower or upper indexes) to denote positive constants which might be different each time we see them (and even in a chain of inequalities), but are independent of the functions or variables being discussed.

In this paper we will also need a pointwise estimate for the Bergman projection of functions in Lp,q(Tq), defined by integral formula (1), when this projection makes sense. Note such estimates in simpler cases of unit disk, unit ball and polydisk are well-known, (see, for example, [20]). Let us first recall the following known basic integrability properties for the determinant function, which appeared already above in definitions.

By A^ (x) we denote determinant function (see [4]).

Lemma 1. Let a £ cr and y £ l.

1. The integral

' x + iy

My) = /

J Rn

A-

dx

n

converges if and only if Rea > gg +—. In that case

Ja(y) = Ca 1A—a+n/r (y)|-

2. For any multi-indices s and ft from Cr and t £ l the function

y ^ A^(y + t)As(y)

dy

belongs to L 1 (l, An/r ( )) if and only if Res > go and Re(s + ft) < —gg. In that case we have

Ap (y + t)As(y)A„dy(y) = Cp,sAs+p (t).

We refer to Corollary 2.18 and Corollary 2.19 of [19] for the proof of the above lemma or [4] We will need only "one-dimensional" version of this estimate.

Indeed as a corollary of one dimensional versions of these estimates (see, for example, [18] Theorem 3.9) we obtain the following vital estimate (3) which we will use in proof of our main result. Note also it is a complete analogue of the well-known Forelli-Rudin estimate for Bergman kernel in analytic spaces in the unit ball.

f A^(y)|Ba+^+n(z,w)ldV(z) < CA-», (3)

■JTn

n

ft > —1, a>--1, z = x + iy, w = u + iv (see [18]).

r

n

Let t be the set of all triples (p, q, v) such that 1 < p,q < <x, v >--1.

r

The following vital pointwise estimate can be found, for example, in [4].

Lemma 2. Suppose (p, q, v) £ t. Then we

|Pvf (z)| < Cpiqir,v,nA-q-r (Imz)||f||Ap,q. Proof. This is a consequence of the lemma 1 and Holder's inequality (see [4]).

2. New estimates for distances in analytic function spaces in tube domains over symmetric cones.

In this paper we restrict ourselves to Q irreducible symmetric cone in the Euclidean vector

space Rn of dimension n,endowed with an inner product for which the cone Q is self dual. We

denote by Tq = Rn + iQ the corresponding tube domain in Cn.

This section is devoted to formulation and proof of the main result of this paper. As previously

in case of analytic functions in unit disk, polydisk, unit ball, and upperhalfspace C+ and in case

of spaces of harmonic functions in Euclidean space (see, for example, [1-3,5-7]) the role of the

Bergman representation formula is crucial in these issues and our proof is heavily based on it. A

variant of Bergman representation formula is available also in Bergman-type analytic function

spaces in tubular domains over symmetric cones and this known fact (see [4,17-19]), which is

crucial also in various problems in analytic function spaces in tubular domains (see [4] and various

references there) is also used in our proof below.

The following result can be found in [18] (section 4).

n 11 n

For all 1 < p < to and 1 < q < to and for all — < p1, where--\— = 1 and--1 < v and

r pi p r

n

for all functions f from AP'q and for all--1 < a the Bergman representation formula with a

r

index or with the Bergman kernel Ba(z, w) is valid.

We remark this result is a particular case of a more general assertion for analytic mixed norm AP'q classes (see, for example, [18]), which (after some analysis of our proof below) means that our main result admits also some extensions, even to mixed norm spaces which we defined above. This will be discussed in our next paper which is in preparation.

We will also need for our proofs the following important fact on integral representation (see,

nn

for example, [21]). Let v >--1, a >--1, then for all functions from the Bergman

integral representation with Bergman kernel Ba+V (z,w) (with a + v index) is valid. We also note that by lemma 2 we have

|f (x + iy)IAr + q (y) < Cpqrv llf IIAp'q, x + iy G Tq, (p,q,v) G t. Hence we have a continuous embedding Ap ^ A°S + v for (p,p,v) G t and hence we have a

rp P

problem to find sharp estimates

distA5p + p (f,Ap)

where f G A. n | v .

rp + p

This problem is solved in our next theorem below, which is the main result of this note. Let us set, for f e H(Tq), s e R and e > 0 :

Ve,s(f) = {x + iy G Tq : If(x + iy)IA"(y) > e}

Let also w = u + iv G Tq, z = x + iy G Tq. We denote by N1 and by N2 two sets-the first one is Ve,s(f), the other one is the set of all those points, which are in tubular domain Tq, but not in N1.

in \ n 1 / n \

Theorem 1. Let 1 < p < to, v > p (--1) , ¡3 > t \---1, t = - (v \— ) . Set, for

\r J r p V rJ

f G A^S

+ P :

h(f )=distA~ + v (f, AP),

i2(/)=f e> 0:/ (/ A*-n (y)dxdy< J .

I JTn\Jv€,t(f) ap+ r ((z - w)/t) J J

Then there is a positive large enough number ft0 depending on parameters involved, so that

for all ft > ft0 we have Z1(/) x Z2(/).

n n

Proof. Let v >--1, t>--1, then as we indicated above for all functions from AS the

integral representations of Bergman with Bergman kernel

B(t+v)(z,w)

is valid.

We denote below the double integral which appeared in formulation of our theorem by G(/) and we will show first that 11(/) < CZ2(/). We assume now that Z2(/) is finite.

We use the Bergman representation formula which we provided above, namely (2), and using conditions on parameters we have the following equalities.

First obviously we have by remark from which we started this proof that for large enough ft

/(z) = cJ Bp(z,w)/(w)dVp(w) = /i(z) + /2(z),

JTn

/i(z) = Cp f Bp(z,w)/(w)dVp(w), J N2

/2(z) = Cp [ Bp(z,w)/(w)dVp(w). J N1

Then we estimate both functions separately using lemmas provided above. Using definitions of N1 and N2 and using estimate (3) we will have

/1 e AS? + P

rp 1 p

and

/2 e Ap.

We will prove both estimates below. First we easily note the last inclusion follows directly from the fact that Z2 is finite. Indeed we have

i |/2(z)|pAv-r (y)dxdy < Ci (Cp i |Bp (z,w)||/(w)|dVp (w))pAv-r (y))dxdy. ■JTn JTn JN1

Turning to the first estimate it can be easily seen that the norm of /i can be estimated from above by Ce, for some positive constant C, since we have immediately

r

|/i(z)|Arp+p(y) < C(A^ + p(y))(cJ |Bp(z,w)||/(w)|dVp(w))

N2

and hence

sup |/l(w)|At(v) < Ce.

N2

Note the last estimate follows directly from definition of N2 set and the following inequality which follows from (3) (see also [18]).

f A-t(y)B(z,w)^Vp(z) < CA-t(v), JTn

z = x + iy, w = u + iv, for all 3, so that 3 > 3o, for some large enough fixed 3o, which depends

1A ( n \ (n

on n, r, v, ... . .

p r r

immediately one part of our theorem. Indeed, we have obviously

T-tt/, UJ — (x-ptty, iui an ou bnab i-J ^ yu, iui ouinG laigc cnuugn IIACU yu, win^n Licpciiuo

r, v, p and for t = ^p^ (v + and v > p — (see [18] Theorem 3.9). This gives

h < C2llf — f2||Ar = CsIfilAr < C4e

It remains to prove that l2 < l1. Let us assume l1 <l2. Then there are two numbers e and e1, both positive such that there exists fei, so that this function is in Ap and e > e1 and also the following conditions holds.

II f — fei IUr < ei

and G(f) = to, where G is a double integral in formulation of theorem in l2 (see for similar arguments, in one dimension, for example, [5]). Next from

If — fei Ikr < ei

we have the following two estimates, the second one is a direct corollary of first one. First we have for z = x \ iy

(e — ei)Tve>t (z)A-t(y) < Cf (z)I,

where TVe t(z) is a characteristic function of V = Ve,t(f) set we defined above.

And from last estimate we have directly multiplying both sides by Bergman kernel Bp (z,w) and integrating by tube Tq both sides with measure dVp.

G(f ) < ci (Lf ))pAv - n (y)dydx,

JTo

'To

where

L = Lf ,z)

and

(w)IIBß (z, w)ldVß (w)

Lf ,z)= f | fei (w)IIBß (z,w)ldVß (w). JTo

'To

n n n f 1 1 \

Put 3 +— = k1 + k2, where k1 = 3---M, k2 = M + 2—1 —\--.

r r r p p i

By classical Holder inequality with p and p1, p-1 + p-1 = 1 we have

Lp < CI1I2,

where

Ii(f)= i |fi(z)|p|As((z — w)/i)A(ß-n)p(y)dxdy,

J To

'To

p

where f1 = fei and

I2pi = |Av((z — w)/i)ldxdy, JTo

n n s = ^p — 2--ßp + P~,

n

v = —2--^pi.

r

n 1

Note from here 12 can be calculated directly using (3). We have 12 < CA-Mp(v), ^ >---. It

rpi pi

remains to integrate both sides of the upper estimate for L by Tq . Choosing then parameter, so that the estimate (3) can be used again after changing the order of integration and making some minor final calculations with indexes we will get what we need. Note here we have to use also the fact

(U

v > p--1

V r

which was given in formulation of our theorem. Indeed we will have finally,

Tn \ JTn

but we also have

/ (z)|Bß(z,w)|dVß(z)J A-r (v)dV(w) < C/ ||Ap. ) < C/ ,

/ei G AP.

This will give a contradiction with the equality G(f) = to. We proved the estimate which we wanted to show from the start. The proof of our theorem is now complete.

Note finally the main theorem can be easily reformulated in terms of analytic Besov spaces. For this we have to use known embeddings connecting analytic Besov and Bergman spaces (see, for example [19,21]).

We define now the same analytic function spaces we considered in this paper in product domains (products of tubular domains over symmetric cones Th x Th ). We define first the space of all analytic function spaces in T™ as H(TQ) is the space of all analytic f (zi,..., zm) analytic by each variable separately.

Let also

ap(Tm) = 4 / g h(Tm) :

'Tn

'Tn

|/(zi,..., zm)|p^ A(Im Zj))v-ndujdvj < to j=i

where 1 ^ p < to, v >--1,

r

Ar(Tm)H / G h(Tm) : sup |/(zi,...:

zeTn

|II A4(Im Zj ) < to

j=i

t G R.

Spaces of functions on product domains (not only analytic) were under attention during last several decades (see, for example [22-24] and various references there).

We note Ap and are both Banach spaces for values of parameters we defined them. Repeating arguments we provided during the proof of main theorem of this paper we arrive at following extension of that theorem to the case of product domains:

(n \ n 1 ( n \

Theorem 2. Let m > 1, 1 < p< to, v > p--1 , ft0 > t +---1, t = - v +— ;

Vr/ r pVr/

f G A^T^). Then for all ft > ft0

distA- (Tm)(/,AV (Tm))

p

m

m

z

m

inf ^ £ > 0:/ ... I I I j-j ) duj dvi | x

Im z

j=1'

>To JTo\JVsAf) n"=1 Aß+ n)

xü [A"-n (lmyj)]dxjdyj < ,

j=i

where

m

VE,t(f ) ={ z G Tm : If (zi, ...,zm)ll[ At(Im zj) > £ ¡> .

j=i

We omit the proof since it is based mainly on simple procedure of adding variables.

And we provide one more extension to so called mixed norm spaces on product domains T^.

3. Extremal problems in mixed norm spaces in product of tubular domains over symmetric cones

We define mixed norm spaces on product of tubular domains

Ap (Tm) = APl'-'Pm (Tm) =

Q I ' Vi Vm \ q L >

= \f G H(Tm):^JT ...^T If(zi ,...,zm)IPiAVl-S (Imzi )dv(zi^^

... AVm-n (Imzm))1/Pm < to} n

where pi > 1, i = 1,... ,m, vj >--1, j = 1,... ,m. These are Banach spaces.

r

In Rn such spaces studied in [25]. See also [24,26] for similar analytic spaces.

We will need for our theorem the following lemmas. Proofs are the same as in [24] for unit

disk case, we omit details refering to [24]. n

Lemma 3. Let a.j >--1, j = 1,..., m, 1 ^ pi < to, i = 1,..., m. Then

r

sup If(z)| [A21 +a)/p(Imz) < cIIfIU(Tm);

Lemma 4. Let vj > v0, j = 1,... ,m for some fixed large enough v0. Then we have the following embeddings

A^om) c avo (Tm) c A^(Tm)

n

where v0 = v0(a1, ..., am,p), 1 ^ pj < to, j = 1,. .. ,m, t = v0 +--.

r

Based on these lemma 3 and 4 we formulate the following theorem for Aa spaces.

nn Theorem 3. Let 1 < pj < to, aj >--1, j = 1,...,m, f G Af(Tm), tj = vj + -,

j = 1,... ,m, vj > v0, 3 > 3o for some fixed large enough 3o and v0. Then we have the estimate

P [ f fif Yj=1 Ap-t-S (Imzj) \

distAT{тm)(f, APa) > inf< £ > 0: - [ j n, zj-u,j, dujdvj) x

m

zerm

m

x n ^ - r (Im yj)dxjdyj < TO

j=i

This work was supported by the Russian Foundation for Basic Research (grant 13-353

01-97508) and by the Ministry of Education and Science of the Russian Federation (grant

1.1704.2014K).

References

[1] M.Arsenovic, R.Shamoyan, On some extremal problems in spaces of harmonic functions, ROMAI J., 7(2011), 13-34.

[2] R.Shamoyan, O.Mihic, On new estimates for distances in analytic function spaces in the unit disc, polydisc and unit ball, Bollet. de la Asoc. Matematica Venezolana, 42(2010), no. 2, 89-103.

[3] R.Shamoyan, O.Mihic, On new estimates for distances in analytic function spaces in higher dimension, Siberian Electronic Mathematical Reports, 6(2009), 514-517.

[4] D.Bekolle, A.Bonami, G.Garrigos, C.Nana, M.Peloso, F. Ricci, Lecture notes on Bergman projectors in tube domain ver cones, an analytic and geometric viewpoint, Proceeding of the International Workshop on Classical Analysis, Yaounde, 2001.

[5] R.Shamoyan, M.Arsenovic, Some remarks on extremal problems in weighted Bergman spaces of analytic functions, Communication of the Korean Math. Society, 27(2012), no. 4, 753-762.

[6] R.Zhao, Distance from Bloch functions to some Mobius invariant spaces, Ann. Acad. Sci. Fenn., 33(2008), 303-313.

[7] R.Shamoyan, M.Radnia, Some new estimates for distances in analytic function spaces of several complex variables and double Bergman representation formula, J. of Nonlinear Sci. and Appl., 3(2010), no. 1, 48-54.

[8] W.Xu, Distances from Bloch functions to some Mobious invariant function spaces, J. of Function Spaces and Appl., 7(2009), 91-104.

[9] J.M.Anderson, J.Clunie, Ch.Pommerenke, On Bloch functions and normal functions, J. Reine. Angew. Math., 270(1974), 12-37.

[10] J.M.Anderson, Bloch functions — the basic theory, Operators and Function Theory, Lancaster, Reidel, Dordrecht, 153(1985), 1-17.

[11] J.Xiao, Geometric Qp functions, Frontiers in Mathematics, Birkhauser-Verlag, 2006.

[12] P.Ghatage, D.Zheng, Analytic functions of bounded mean oscillation and the Bloch space, Integ. Equat. Oper. Theory, 17(1993), 501-515.

[13] L.Ahlfors, Bounded analytic functions, Duke Math. J., 14(1947), 1-14.

[14] W.Rudin, Analytic functions of Hardy class, Trans. Amer. Math. Soc., 78(1955), 46-66.

15] D.Khavinson, M. Stessin, Certain linear extremal problems in Bergman spaces of analytic functions, Indiana Univ. Math. J., 46(1997), no. 3, 933-974.

16] S.Khavinson, On an extremal problem in the theory of analytic function, Russ. Math. Survey, 32(1949), no. 4, 158-159.

17] J.Faraut, A.Koranyi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994.

18] B.F.Sehba, Bergman type operators in tube domains over symmetric cones, Proc. Edinburg. Math. Society, 52(2009), no. 2, 529-544.

19] D.Debertol, Besov spaces and boundedness of weighted Bergman projections over symmetric tube domains, Dottorato di Ricerca in Matematica, Universita di Genova, Politecnico di Torino, 2003, 21-72.

20] P.Duren, A.Schuster, Bergman spaces, Mathematical Surveys and Monographs, v. 100, AMS, RI, 2004, 251-256.

21] D.Bekolle, A.Bonami, G.Garrigos, F.Ricci, B.Sehba, Analytic Besov spaces and Hardy type inequalities in tube domains over symmetric cones, J. Reine. Angew. Math., 647(2010), 2556.

22] R.F.Shamoyan, E.V.Povprits, Multifunctional analytic spaces on products of Bounded strictly pseudoconvex domains and embedding theorems, Kragujevac J. of Math., 37(2013), no. 2, 221-244.

23] P.Jakobczak, The boundary regularity of the solution of the df equation in products of strictly pseudoconvex domains, Pacific J. of Math., 121(1986), no. 2, 371-386.

24] F.A.Shamoyan, O.V.Yaroslavtseva, Continuous projections, duality, and the diagonal mapping in weighted spaces of holomorphic functions with mixed norm, Zap. Nauchn. Sem. POMI, 247(1997), 268-275, (in Russian).

25] A.Benedek, R.Panzone, The space Lp, with mixed norm, Duke Math. J., 28(1961), no. 3, 301-324.

26] S.M.Nikolskiy, Approximation of functions of several variables and embedding theorems, Springer-Verlag, Berlin, 1975.

О функции дистанции в пространствах типа Бергмана аналитических функций нескольких переменных

Роми Ф. Шамоян Сергей М. Куриленко

Получена новая точная оценка, относящаяся к функции 'расстояния в некоторых пространствах аналитических функций типа Бергмана в трубчатых областях над симметрическими конусами. Это первые результаты такого рода в трубчатых областях над симметрическими конусами.

Ключевые слова: оценки расстояний, трубчатая область.