CC BY
34Issues of Analysis
Vol. 2(20), No. 1, 2013
R. F. Shamqyan
ON SOME EXTREMAL PROBLEMS IN CERTAIN HARMONIC FUNCTION SPACES OF SEVERAL VARIABLES RELATED TO MIXED NORM SPACES
Abstract. In this paper we provide some (not new) estimates on distances from our two previous papers together with some new estimates. Namely some estimates on distances in spaces of harmonic functions in the unit ball and the upper half space are provided. New estimates concerning mixed norm spaces and general weighted Bergman spaces are obtained and discussed.
Key words: distance estimates, harmonic function, unit ball, Bergman spaces.
2010 Mathematical Subject Classification: 42B30.
§ 1. Introduction and preliminaries
The main goal of this note to present several new results on distances in harmonic spaces though some isolated results from our previous papers (see[1], [2]) will be also given to make the picture more complete. This direction of investigation related to extremal problems in harmonic function spaces started in [1], then continued in [2], [3]. In this note in particular a new sharp theorem on distances in harmonic function spaces in the unit ball will be added. An analogue of this assertion in case of upper half space was formulated and proved before in [2] and for completeness of exposition we will add it also here (but without proof) Note also this result in the unit ball in case of p > 1 was also provided before in [2], in this note we extend it to all values of positive p. The main tool for us will be so-called Whitney type decomposition of the unit ball of Euclidean space and this technique was used before in case of upperhalfspace by us in [2]. We also add important remarks concerning this extremal problem (distance function) in mixed norm spaces and weighted spaces of harmonic
© Shamoyan R. F., 2013
functions at the end of this paper. Harmonic function spaces were studied by many authors during last several decades we mention, for example, [4] and [5] and various references there.
Our line of investigation can be also considered as a continuation of papers on distances in analytic function spaces [6] and [7]. All main results are contained in the second section of the paper. The first section is devoted to preliminaries and main definitions which are needed for formulations of main results. Almost all objects we define in this section and definitions can be found in [5], [1], [2], [3] and in [8]. The Whitney decomposition and properities and estimates we used for our proofs can be found in various places, but we refer the reader to [1], [9] for case of upperhalfspace and for case of unit ball of Euclidean space (see also [2]). This paper also contains important additions namely we add important remarks about general Bergman spaces with general w weights, these new weighted harmonic spaces appeared for the first time in [10]). These results are also new and we hope to return to these issues related to general harmonic hW and Hw Bergman-type spaces in unit ball and upper halfspace also later.
Let B be the open unit ball in Rn, S = dB is the unit sphere in Rn,
for x E Rn we have x = rx', where r = \x\ = ^™=1 xj and x' E S.
Normalized Lebesgue measure on B is denoted by dx = dx1... dxn = = rn-1 drdx' so that JB dx = 1. We denote the space of all harmonic functions in an open set by h(fi). In this paper letter C designates a positive constant which can change its value even in the same chain of inequalities.
For 0 < p < to, 0 < r < 1 and f E h(B) we set
1/p
MP(f,r)= ( I \f(rx')\pdxr
with the usual modification to cover the case p = to.
For 0 < p < to and a > —1 we consider weighted harmonic Bergman spaces Aa = Aa(B) defined by
AS = {f E h(B) : \\f fAP = /\f(x)\p(1 — \x\2)adx < to I a J B
For p = to this definition is modified in a standard manner:
Aa = Aa (B) = {f E h(B) : \\f l|A~ =supx€B \f (x)\(1 — \x\2 T < to} ,
a > 0.
These spaces are complete metric spaces for 0 < p < to, they are Banach spaces for p > 1 ([5]). These spaces serve as particular cases of more general scales of mixed norm spaces Fp,q and Ba,q (see [2]) in unit ball of Rn. Note analytic area Nevanlinna type spaces of mixed norm type were studied previously in recent works of the author in [11], [12].
Next we need certain facts on spherical harmonics and the Poisson kernel, see [5] for a detailed exposition. Let Yjk be the spherical harmonics of order k, 1 < j < dk, on S. Next,
where bk = (bk,... ,b<kk), Yk = (Y1(k),..., Yjk)) and bk ■ Yk is interpreted
in the scalar product sense: bk ■ Yk = Y^<= 1 bkY(k.
We denote the Poisson kernel for the unit ball as usual by P(x, y'), it is given by
where un is the volume of the unit ball in Rn. We are going to use also a Bergman kernel for A^ spaces, (see [5]) this is the following function
(k)
are zonal harmonics of order k. Note that the spherical harmonics Yj , (k > 0, 1 < j < dk) form an orthonormal basis of L2(S,dx'). Every f E h(B) has an expansion
OO
f (x) = f (rx') = £rkbk ■ Yk(x'),
k=0
P (x,y') = Py. (x) = £ Yjk (y')Yjk\x;) =
k=0 j=1
x = rx' E B, y' E S,
rk pk zXk v), (1)
x = rx', y = py' E B.
This function is playing a very important role in many issues related with harmonic function spaces in unit ball (see [1], [2], [3], [5] and references there).
For details on this kernel we refer to [5], where the following theorem can be found.
Theorem 1. [5] Let p > 1 and 3 > 0. Then for every f E APp and x E B we have
f (x) = / [ Qp(x,y)f (py')(1 — p2)PPn-1py', y = py'.
J0 JSn-1
This theorem is a cornerstone for our approach to distance problems in the case of the unit ball. Everywhere below y = py” and x = rx” are in unit ball, sometimes this will be omitted by us and this will be clear from text. The following lemma gives estimates for this kernel, see [5], [4].
Lemma 1. 1. Let 3 > 0. Then, for x = rx', y = py' E B we have
I Qp(x,y) I< | px C|n+P ■
2. Let 3 > -1. Then
( C
Js _i 1 Qp(rx',y) 1 dx' < (1 - rp)1+P, 1 y 1 = P, 0 < r< 1.
3. Let 3 > n — 1, 0 < r < 1 and y' E Sn-1. Then
dx' C
-l Irx' — y' IP < (1 — r)p-n+1 .
The following simple lemma is classical see, for example, [5] and references there.
Lemma 2. [5] Let a > —1 and X > a +1. Then
H d < c(1—p)a+1-x, 0 < p<
The following lemma is purely technical, but it is vital for the proof of our main result.
Lemma 3. For 5 > -l, j > n + 5 and /3 > 0 we have
f \Q0(x,y)\*(l - \y\)sdy < C(l - \x\)s-Yx e B.
J B
Proof. Using Lemma l and Lemma l we obtain:
S: q (x,y) ^ (l -\ y \ )S dy < c IPx-tf d <
<C i(l - p) I dy dp <
< ci (l - p)s(l - rp)n-Y-1 dp <
J 0
< C(l - r)n+S-Y. □
Now we turn to the basic definitions for upperhalfspace, all of them are classical see [5] and references there or see [l], [2].
We first set R++1 = {(x,t) : x e Rn, t > 0} C Rn+1. We usually denote
points in R++1 by z = (x, t) or w = (y, s) where x,y e Rn and s,t > 0.
For 0 < p < to and a > -1 we consider spaces
<(RJ+1) = Aa = I f e h(R++1) : [ + \f (x,t)\ptadxdt < to} .
Also, for p = to and a > 0, we set
i“(RJ+1) = A™ = |f e h(R++1): sup \f(x,t)\ta < to
(x +)gR+-
j++l
(x,t) GR++ 1
These spaces have natural (quasi)-norms, for l < p < to they are Banach spaces and for 0 < p < l they are complete metric spaces (see [5]). They serve as particular cases of more general Fp,q and Ba,q mixed norm spaces (see, for example, [2] and references there).
Now we turn to formulate some known assertions in this case of up-perhalfspaces (see [5] and references there). We denote the Poisson kernel for R++1 by P (x, t), i.e.
P(x, t) = cn------- ---—1, x e Rn,t> 0.
( \ x \2 + t2) n+1
For an integer m > 0 we introduce a Bergman kernel Qm(z,w), where z = (x, t) e R++1 and w = (y, s) e R++1, by
(_2)m+1 Qm+1
Qm(z, w) =---------;---——— P(x — y,t + s).
, j m! dtm+1 v 7
The terminology is justified by the following result from [5] which is a complete analogue of integral representation of Bergman spaces in the unit ball we formulated above. Note it is well-known that these theorems on integral representations for ball and upperhalfspace have various applications in harmonic function theory.
Theorem 2. Let 0 < p < to and a > -l. If 0 < p < l and m > > a+n+1 - (n + l) or l < p < to and m > - l, then
f (z) = f f (w)Qm(z,w)sm dУds, f e Aa, z e R++1. (2)
J R++1
The following elementary estimate of this kernel is crucial and it is contained, for example, in [5]:
_n + m+1
\ Qm(z, w) \ < C \\x - y \2 + (s + t)^ 2 , (3)
z = (x,t),w = (y, s) e R++1.
It is well-known that the theory of Bergman spaces in unit ball and upperhalfspace are parallel to each other and it is natural to consider extremal problems in both spaces together. Various issues in these spaces can be solved using arguments related so-called properties of a certain family of special cubes which usually called Whitney cubes (see [8]) and references there. Here are two basic properties of Whitney type decomposition which we will need for upperhalfspace ^++1 (see, for example, [2]). There is a collection of closed cubes Ak in ^++1 with sides parallel to coordinate axes such that the following properties hold The union of these collection Ak of cubes gives all ^++1. The interior of Ak cubes are pairwise disjoint. Such type of family also exists in the unit ball (see, for example, [9] and references there). Related facts and estimates we need for these cubes and their centers can be found in [9] in unit ball and in
[2] in case of upperhalfspace ^++1.
§ 2. Sharp estimates for distances in harmonic Bergman function spaces of several variables in the unit ball and in R++1 and related theorems in mixed norm spaces and weighted Bergman spaces
In this section we investigate distance problems both in the case of the unit ball and in the case of the upper half space for harmonic functions of several variables. The method we use here originated in [13], see also
[1], [2], [3], [6], [7] for various modifications of this interesting approach, and applications of this method to various distance problems in analytic function spaces and harmonic function spaces in one and several variables. New assertions and important remarks concerning mixed norm harmonic function spaces and general harmonic weighted Bergman spaces will be added at the second part of this paper. The following lemma is elementary.
Lemma 4. Let 0 < p < to and a > -1. Then there is a C = CP;a,n such that for every f E (B) we have
ct + n
| f (x)| < C(1 - | x |)-— ||f A, x E B.
Proof. We use subharmonic behavior of | f |p, see for this for example [8], [5], to obtain
f (x)|p < (1 C n„ f 1 ,, f fe)|pdy <
(1 - m)n JB(x, iqM)
< C(,V |xl))„° f , |f(y)|p(1 -\y\)°dy <
(1 -m)n JB(x, iqM)
< c(1 - yxD-a-nIf fAP. □
This lemma shows that Aa is continuously embedded in At0)+n and
P
motivates the distance problem in unit ball which investigated in our Theorems below.
Lemma 5. Let 0 < p < to and a > -1. Then there is C = Cp,a,n such that for every f E Aa and every (x, t) E R++1 we have
ct + n+1
f (x,t)|< Cy ~IIf 1 A* . (4)
The above lemma states that A^ is continuously embedded in AO+
P
its proof is analogous to that of Lemma 2. This lemma motivates the distance problem in harmonic function spaces in upperhalfspaces which investigated in our Theorems below.
For e > 0, t > 0 and f E h(B) we set
The following assertion alone can be found among other things in [3] hovewer we decided to formulate it since the remaining part of it p < 1 case was open and we in this note close it in our next theorem, we also prove the first theorem to use it for further results and observations at the end of this note.
Theorem 3. Let p > 1, a > -1, X = , m E N0. Set, for
f E Aa+n (B): P
Then there is a m0 depending from a,p, n so that for all /3 > m0
In the following new sharp theorem on distances we cover the remaining case of p < 1. Note this result is new. Nevertheless the proof of this theorem is based on estimates connected with Whitney-type decomposition of the unit ball which we mentioned above(see [9]), and it is very close and parallel to the case of upperhalfspace case (see the last theorem of this paper below) and arguments in proofs are similar. The theorem in upperhalf space is formulated at the end of this paper and the proof of it is given in our previous work (see [2]). Hence here we omit details of proof, refereing the reader to [2].
Theorem 4. Let p < 1, a > -1, X = , m E N0. Set, for
f E AO+n (B): ^
U€,t(f) = U€;t = {x E B : \f (x)\(1 - \x\)t > e}.
P
P
tl (f) ~ t2 (f).
P
P
s2 (f) = inf<| e> 0: I ( U Qf3 (x,y)(1 -\v\y X dy\ X
x(1 — \x\)“dx < oo|.
Then there is an m0 depending on a, p, n so that for all 3 > m0
s1 (f) - s2 (f).
We now provide now a complete proof of first theorem we formulated above.
Proof. We begin with inequality t1 (f) > t2(f). Assume t1(f) < t2(f). Then there are 0 < e1 < e and f1 E Aa such that \\f — f1||^>° < e1 and
f \qp(x,v)\(1 -\v\)p-td^ (1 -\x\)adx = +°.
'U,t(f) J
Since (1 — \x\)t\f1 (x)\ > (1 — \x\)t\f (x)\ — (1 — \x\)t\f(x) — f1(x)\ for every x E B we conclude that (1 — \x\)t \f1(x) \ > (1 — \x\)t \f (x)\ — e1 and therefore
(e — e1)Xut,t(f)(x)(1 — \x\)-t < \f1 (x)\, x E B.
Hence
= f ( f \Qp(x,y)\(1 — \y\)P-td^ (1 — \x\)adx =
Jb \Jue,t(f) J
B X(—yy Q (x,y)\(1 — \y\)"d^ (1 — \x\)adx <
< °€’€1 f ( f \f1(y)Q(x,y)\(1 — \y\f dy1 (1 — \x\Tdx = M,
Jb \Jb /
and we are going to prove that M is finite, arriving at a contradiction. Let q be the exponent conjugate to p. We have, using Lemma 1,
1 (x) = (L \f1(y)\(1 — \y\)\Qp (x,y)\dy) =
\ p
\f1(y)\(1 — \y\f \Qp (x,y)\^(n +/3-e)\Qp (x,y)\^(n+e)dy <
p
< f if1(y)\p(1 — Mr3\Qn(x,y)\n±np^dy( [ \Q„(x,y)\n+qedvУ’/, < Jb \Jb J
, rt n+pQ-pe
< C(1 — \x\)-”' \f1(y)\p(1 — \y\)p3\Q3(xM-^+r-dy
Jb
for every e > 0. Choosing e > 0 such that a — pe > —1 we have, by Fubini’s theorem and Lemma 1:
M < C f \f1 (y)\p(1 — \y\)p3 !(1 — \xY-p,\Q3(x,y)\dxdy <
Jb Jb
< C f \f1 (y)\p(1 — \y\)ady < °.
Jb
In order to prove the remaining estimate t1 (f) < Ct2 (f) we fix e > 0 such that the integral appearing in the definition of t2(f) is finite and use Theorem on integral representation in unit ball formualted above, with 3 > max(t — 1, 0):
f (x) = f Q3 (x,y)f (y)(1 — \y\2 )3 dy+
e,t (f )
+ / Q3 (x,y)f (y)(1 — \y\2 )3 dy = f1(x) + f2(x).
JUeAf)
Since, by Lemma 1, \f1 (x)\ < 23 fB \Q@(x,y)\(1 — \w\)3-tdy < C(1 — \x\)-t we have \f1||a°° < Ce. Thus it remains to show that f2 E Aa and this follows from
If A < \\f IIA» / ( / \Qp (x,y)\(1 — \y\2 )3-t d^ X
“ * Jb \JUe,t(f) J
x(1 — \x\)“dx < o. □
The above theorem has a counterpart in the R++1 setting. As a preparation for this result we need the following analogue of Lemma 1.
Lemma 6. For S > —1,j>n +1 + S and m E N0 we have
m+1
\Qm(z,w)\n+m+i s5dyds < Ct5 ^+n+1, t> 0.
+
Proof. Using Fubini’s theorem and estimate (3) we obtain I (t) = \Qm (z,w)\n+m +1 ss dyds <
;m\
1++1
<crAS... dy ... Ids=
J0 \JRn [\y\2 + (s + t)2]7,
r OO
= C ss (s + t)n-Y ds = CtS-Y+n+1. □
0
For e > 0, X > 0 and f £ ^(R++1) we set:
V.Af) = {(^.t) £ R++1 : \f(x,t)\tX > e}.
Theorem 5. Let p > 1, a > —1, X = a+n+1, m £ N0 and Set, for f £ jq+n+1 (R++1) :
P
s1 (f) = distj^oo+ +1
a + n+1 P
P
s2(f) = in^ e > 0:1 ( I Qm(z,w)sm-Xdyds] tadxdt < to
;m\
1++1
Then there is an m0 depending on a,p, n so that for all m > m0 we have
s1 (f) - s2(f )-
The proof of this theorem closely parallels the proof of the previous one, in fact, the role of Lemma 1 is taken by Lemma 2 and the role of Theorem on integral representation in unit ball formulated above is taken by Theorem on integral representation in upperhalspace which was also given above. We leave details to the reader.
The following theorem covers the remaining case of p < 1. A sharp theorem for this values of p as above in parallel case of same values of p, but in unit ball is proved via direct application of Whitney decomposition of upperhalfspace (see [2]) we mentioned above and estimates for it which were also mentioned above by us (see also[9]). We remark here that this theorem below is not new and we add this assertion for completness of our exposition in these issues connected with extremal problems. The proof
of theorem will be omitted. We refer the reader to our previous papers
[1], [2] where the proof of this result can be seen.
Theorem 6. Let p < 1, a > -1, X = a+n+1, m E No and Set, for f E (R++1): "
S1 (f) = dist^oc, +, (f,K),
p
s2 (f) = inf ^ e > 0: J I J Qm(z,w)sm~xdyds] tadxdt < to
Then there is an m0 depending from a,p,n so that for all m > m0 we have
Si(f) - S2(f).
The proof closely parallels the proof we have in the unit brll case for same values of p (see [2]). At the end of this paper we add important remarks on distance theorems in so-called mixed norm and general weighted spaces of harmonic functions.
By S we define a class of all positive measurable functions v on (0, 1) for which there are positive numbers mv ,Mv and qv so that mv, qv E (0,1) and so that mv < v(jy) < Mv for all r and X,r E (0,1), X E (qv, 1). These spaces of special functions were mentioned in [5] and analytic Bergman spaces with these general weights were considered and studied in unit disk, polydisk and later in unit ball, see [10]. This S class includes various unusal weight-functions, for example, like ta (ln(cfor any positive numbers a and [3.
We define as in [10] the following general spaces of harmonic functions of Bergman-type in unit ball and upperhalfspace. Let hv (B) for all positive values of p be the space of all harmonic F functions in the unit ball with the following finite quazinorm.
f \F(x)|pv(1 — \x\)dx.
J B
These are Banach spaces for all p so that p > 1 and quazinormed spaces for all p < 1 (see [10]). We modify this quazinorm in a standard manner to define also h^ (B) as a space of F harmonic functions with finite quazinorm
sup \F(x) \v(1 — \x\).
xEB
P
P
Some results of this note can be extended directly to these general spaces since the Bergman representation formula for these general classes is also valid (see[10]) and it is a core of our approach in all proofs [1], [2],
[3]. We formulate a result from [10] it says that for all p > 1 and p = to and all a > s(v,p) = a"v+1, where av = l<0<grqv the intergal representation for all functions from hp via Qa(x,y) kernel is valid. The following theorem in particular can be found in [10]. We formulate it since it is not available in literature openly.
Theorem 7. [10] Let p > 1 or p = to and a > s(v,p) > 0. Then for every f E hp and x E B we have
f(x) = / [ Qa(x,y)f(py')(1 — p2)“pn-1 dPdy', y = py'.
Jo JSn-1
The complete analogue of this representation is valid for all p > 1 for spaces of Bergman type in upper halfspace Rn+1, which we defined above, but with general v weights from S class [10] These are spaces of F harmonic functions in upperhalfplane Hp(R++1) with finite quazinorm
/ \F (x”,xn+1)\p v(xn+1 )dxdxn+1
JR++1
for all positive p [10] with obvious modification for p = to case. And again for p > 1 these are Banach spaces, for all other positive p they are quazinormed spaces. Here again v is a positive slowly varying function on (0, to) from S class (see [10]).
In this case another integral representation for upperhalfspace R++1 with another kernel, which we also mentioned above is true. Note for other values of positive p, namely for p < 1 there is also another(more complicated) kind of integral representations [10] both unit ball and upperhalfspace. The following result for general weighted Bergman spaces also can be found in [10].
Theorem 8. Let 1 < p < to and m > m0 for certain fixed m0 depending from p, w, m0 = s(v,p) — 1 then the following representation is valid
f (z) = / f (w)Qm(z, w)smdyds, f E Hp, z E R++1. (5)
J R++1
Finally we remark results of these note partially can be extended to so-called mixed norm spaces of harmonic functions Fp,q and Bp,q in unit ball and their direct analogues in upperhalf space R++1 at the same time. For definition of these four scales of harmonic mixed norm spaces we refer the reader to [2], [1], [3]. Note Carleson-type embedding theorems for these spaces were studied in our previous paper [2]. The core of the proof of a distance theorem for these classes is the same and at the end of proof an appropriate (known) embeddings connecting classical Bergman Aa spaces with Fp,q and Bp,q classes in unit ball or R++1 should be used [2] We provide two natural examples in R++1 from a group of not sharp results for these classes in ball and upperhalfspace to readers noting we didn’t get yet any sharp result yet in this direction. Note the problem is motivated by an embedding Fp,q C A^ and also by another embedding Bp,q C A^ for q < p, which we actually already showed partially above (see also [1],
[2], [3]) where A = a+n+1, a > -1,p,q E (0, to).
Theorem 9. Let p,q E (0, oo),q < p, a > -1, A = a+'np+1, m E N0. Set, for f E Aa+n+1 (R++1) :
P
S1(f)=distAo (f,Bpa’q),
a + n+1 P
s2(f) = inf |e> 0 : J ^ J Qm(z,w)sm-Xdyd^ tadxdt < to
Then there is an m0 depending from p, q, n, a so that for all m > m0,
S2(f) < CS1 (f).
Theorem 10. Let q,p E (0, to), q < p, a > -1, A = a+'np+1, m E N0. Set, for f E AO+n+i (R++1): ^
P
S1 (f) = distjAoo+ +i (f,Fp,q),
a + n+1 P
P
s2 (f) = inf ^ e > 0: j ( J Qm(z,w)sm~xdyds\ tadxdt < to
Then there is an m0 depending from p, q, n, a so that for all m > m0,
s2(f) < Cs1 (f).
Proofs repeat arguments we provided above in combination with (known) embeddings between mixed norm and Bergman spaces [2], namely we use the embedding X C Aa where by X we denote one of these mixed norm spaces and which we should use at very last step of proof and we omit easy details here.
A note from the author. This mainly expository paper was prepared by Romi Shamoyan alone, however it contains also several old results from previous work with Dr. Milos Arsenovic. Dr. Milos Arsenovic declined my suggestion to join to this project (which was obviously needed from my point of view) in April 2012 when this work was in preparation, so the paper was posted in www.arxiv by Romi Shamoyan as the only author.
References
[1] Arsenovic M., Shamoyan R. F. On some extremal problems in spaces of harmonic functions // Romai Journal. 2011. Vol. 7. No. 1. P. 13-34.
[2] Arsenovic M., Shamoyan R. F. Embeddings, traces and multipliers in harmonic function spaces // Kragujevac Journal of Mathematics. 2013. Vol. 37. No. 1. P. 45-64. arXiv:1108.5343 [math.FA]
[3] Arsenovic M., Shamoyan R. F. Sharp theorems on multipliers and distances in harmonic function spaces. arXiv:1106.5481 [math.CV]
[4] Jevtic M., Pavlovic M. Harmonic Bergman functions on the unit ball in Rn // Acta Math. Hungar. 1999 Vol. 85 (1—2). P. 81-96.
[5] Djrbashian M., Shamoian F. Topics in the theory of Aa classes // Teubner Texte zur Mathematik. 1988. V. 105.
[6] Shamoyan R. F., Mihic O. On new estimates for distances in analytic function spaces in the unit disc, polydisc and unit ball // Bol. de la Asoc. Matematica Venezolana. 2010. V. 42, No. 2. P. 89-103.
[7] Shamoyan R. F., Mihic O. On new estimates for distances in analytic function spaces in higher dimension // Siberian Electronic Mathematical Reports. 2009. Vol. 6. P. 514-517.
[8] Stein E. M., Weiss G. Introduction to Fourier analysis on Euclidean spa,ces. Princeton: Princeton University Press, 1971.
[9] Avetisyan K. L. Weighted spaces of harmonic and analytic functions// Armenian Journal of Mathematics. PhD Thesis. Erevan: Erevan State University, 2009. Vol. 2. No. 4.
[10] Zakaryan M. Weighted spaces of harmonic functions in unit ball and upperhalfspaces // PhD Thesis. Erevan: Erevan State University, 1996.
[11] Shamoyan R. F. On characterizations of main parts of some mero-morphic classes of area Nevanlinna // Tr. Petrozavodsk. Gos. Univ. Ser. Mathematics. 2011. Vol. 18. P. 172-254.
[12] Shamoyan R. F., Haiying L. On decriptions of closed ideals of analytic area Nevanlinna type classes in a circular ring on a complex plane C // Issues of Analysis. 2012. Vol. 1 (19). No. 1. P. 24-31.
[13] Zhao R. Distance from Bloch functions to some Mobius invariant spa,ces // Ann. Acad. Sci. Fenn. 2008. Vol. 33. P. 303-313.
The work is received on February 21, 2013.
Bryansk State Technical University,
Department of Mathematics 241050, Bryansk.
E-mail: [email protected]
