CC BY
7
Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 1(21). C. 48-63. ISSN 2079-6641
DOI: 10.18454/2079-6641-2018-21-1-48-63
MSC 32A07, 432A10, 32A07
ON SOME NEW ESTIMATES RELATED WITH BERGMAN BALL AND POISSON INTEGRAL IN TUBULAR DOMAIN AND UNIT BALL
R. F. Shamoyan1, O. R. Mihic2
1 Department of Mathematics, Bryansk State Technical University, Bryansk 241050, Russia
2 Department of Mathematics, Fakultet organizacionih nauka, University of Belgrade, Jove Ilica 154, Belgrade, Serbia
E-mail: [email protected],[email protected]
We introduce new Herz type analytic spaces based on Bergman balls in tubular domains over symmetric cones and in products of such type domains. We provide for these Herz type spaces new maximal and embedding theorems extending known results in the unit disk. In addition we define new Poisson-type integral in the unit ball and extend a known classical maximal theorem related with it. Related results for such type integrals will be given.
Key words: tubular domains over symmetric cones, Herz type spaces, Bergman type integral operators, maximal theorems, embedding theorems, Poisson-type integral, unit ball.
© Shamoyan R. F. , Mihic O. R., 2018
Introduction and preliminaries
Let D be bounded domain. We denote by H(D) the space of all analytic functions in D. The analytic Hardy space is as usual
HD (D) = {f e H (D) : lim ^ | f (I - ev^ )| pd a ) < , 0 < p <
where V| is outer unit normal for tangent plane (dD), (see [6]) and where da is a Lebesgues measure on r.
Let HD be Hardy space in D and let also dD e C2. Let 0 < p < f e H^. Then we have
f sup |f (z)|pda) < Cp,a IIf ||Hp,
Jr zeAa (I)
where r = dD.
We refer to [6] for definition of Aa) region.
This maximal theorem was proved by E. Stein and L. Hormander, (see [6]). Related results for spaces of harmonic functions can be found in paper of [5]. For similar results for plurisubharmonic functions in bounded strongly pseudoconvex domains in Cn see [6]. We refer for such type results also to [7] and references there.
For such type results in harmonic functions paces in R++1 and Rn we refer the reader to [8], [9].
We in this paper find complete analogues of this theorem but in Bergman spaces in some unbounded tube domain in Cn using rather transparent arguments and ideas related with lattices.
In this paper we also introduce new Herz type analytic spaces in tubular domains over symmetric cones and in products of such type domains. We provide for these Herz type spaces new maximal and sharp embedding theorems extending known classical results in the unit disk. Also, we define Poisson-type integral in the unit ball and extend some known classical maximal theorems related with it.
We introduce some basic definitions, notations for tube (see [10], [12]). Let TQ = V + iQ be the tube domain over an irreducible symmetric cone Q in the complexification VC of an n-dimensional Euclidean space V. Following the notation of [2] we denote the rank of the cone Q 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 defined for n > 3 by
A« = {y e Rn : y2-----y2 > 0,yi > 0}.
It is equivalent to the forward light cone given by {y = (y1,y2,y') e Rn : y1y2 — |y'|2 > 0}. Light cones have rank 2. The determinant function in this case is given by the Lorentz
form A(y) = y2-----y;;, (see, for example, [1], [2], [3]).
h(TQ) denotes the space of all holomorphic functions on TQ. First we define some known function spaces on TQ, (see [1], [2], [3], [13]). For t e R+ and the associated determinant function A(x) we set
A~(Tq) = | F e h(TQ) : ||F||a~ = sup |F(x + iy)|AT(y) < , (1)
[ x+iyeTQ J
(see [2] and references there). It can be checked that this is a Banach space.
For 1 < p,q < v e R and v > — 1 we denote by Ay(TQ) the mixed-norm weighted Bergman space consisting of analytic functions F in TQ that
lF IIapq =
i/ y/p \1/q
jlj |F (x + Av (y) ^
\fl \v 1 )
<
This is a Banach space. Replacing above simply 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, for example, [1], [2], [3], [13]). It is known the Ap'q(TQ)
n
space is nontrivial if and only if v > — 1, (see, for example, [1], [2], [3], [13]) and we
r
will assume this everywhere below. When p = q we write (see [1], [2], [3])
APp,q(To)= AV (Tq).
This is the classical weighted Bergman space with usual modification when p =
The (weighted) Bergman projection Pv is the orthogonal projection from the Hilbert space Lp (Tq) onto its closed subspace Ap(Tq) and it is given by the following integral formula (see [1], [2], [3]).
Ppf (z)= CpJ Bp (z, w) f(w)AV-n (v)dudv, (2)
Tq
n
-(v +-) _
where Bv(z, w) = CvA r ((z — w)/i) is the weighted Bergman reproducing kernel, for Av(Tq), (see [1], [2], [3]). Below and here we use constantly the following notations w = u + iv e Tq and z = x + iy e Tq. We denote dvv(w) = Av(v)dudv.
Let us first recall the following known basic integrability properties for the determinant function, which appeared already above in definitions. Below we denote by As the generalized power function, (see [1], [2], [3]). Lemma 1. 1) The integral
Ja (y) = J
Rn
A
-a I x + iy
dx
n
converges if and only if a > 2— 1. In that case
r
Ja (y)= Ca A—a+n/r (y),
a e R, y e Q.
2)Let a e Cr and y e Q. For any multi-indices s and p and t e Q the function y m- Ap (y +1)As(y) belongs to L1(Q, ) if and only if ^s > g0 and ^(s + p) < — g^. In that case we have
J Ap (y + t)As(y) ^$7-7= CP,sAs+P (t). Q ( )
We refer to Corollary 2.18 and Corollary 2.19 of [13] for the proof of the above lemma or [2]. As a corollary of one dimensional version of second estimate and first estimate (see, for example, [13]) we obtain the following vital Forelly-Rudin estimate (3) which we will use in proofs of our main results.
J A? (y)|Ba+. (z, w)|dv(z) < CA-a (v), (3)
Tl
n
(Forelly-Rudin estimate in tube), ? > -1, a > — 1, z = x+iy, w = u + z'v, r > 0, z,w e Tl,
r
(see [12], [13]).
We denote by B(z,r) the Bergman ball in Tl, (see [12], [13]). Finally for completeness we provide very vital Whitney decomposition of tubular domain over symmetric cones based on Bergman balls. It was used during many proofs of various assertions (see, for example, [2], [3]).
Lemma 2. Given 8 e (0,1] there exist a sequence of points {zj} in Tl called 8-
lattice such that calling {Bj} and {#}} the Bergman balls with center zj and radius 8
8 and ^ respectively, then
1) the balls (Bj) are pairwise disjoint;
2) the balls (Bj) cover Tl with finite overlapping;
3) f As(y)dV(z) x f As(y)dV(z) = QA2n+s(/mzj), s > n - 1, J = |B8(zj)|x A2n (Imzj),
Bj (zj ,8) Bj (zj ,8) r
j = 1,..., m, J x A2n (Im w), w e B8 (zj).
We call by {zj} r-lattice of Tl below everywhere. This is a vital notation for this note.
We denote m cartesian products of tubes by T^, the space of all analytic function on this new product domain which are analytic by each variable separately will be denoted by H(T^). In this paper we will be interested on properties of certain analytic subspaces of H(T^). By m here and everywhere below we denote a natural number bigger than 1.
m
Let further dv(z1,...,zm) = n dv(zj) = dv(Y), zj e Tl, j = 1,...,m be the normalized
j=1
Lebesgues measure on product domain. We provide now some facts on function spaces on product of tube domains.
We denote as usual by dvy(z) = 8y(z)dv(z) = Ay(Im z)dv(z), y > -1, the weighted Lebesgues measure on T domain and similarly on products of such domains using products of A functions in a standard way for all y > -1. Using dva and At on product domain we can define A7^) and Aa,p(T£) = Ap) for 1 < p < a > -1. For example we have
!m
f e H(Tl) : sup | f )| n A(1m zj)t <«
zj eTl j=1
t > 0, tt = (z1,...,zm), zj e Tl, j = 1,..., m. These are Banach spaces.
In our last section we extend the notion of classical Poisson integral in a simple natural way in the unit ball and extend some well known estimates related with it,
in particular an extension of a known maximal theorem will be provided in the unit ball. Such type results probably can be proved in context of more general pseudoconvex domains with smooth boundary.
Similar maximal theorems we proved in section 2 in harmonic function spaces in the unit ball and R++1, were proved by T. Flett, (see [5]). They have many applications (see [17]). Various nice results related to various maximal theorems and Poisson integrals in various domains in Cn and their applications can be seen in [17].
We denote in this paper, as usual, by C,C1,C2,...,Ca various positive constants.
Maximal and embedding theorems in Herz spaces
The intention of this section is to provide new maximal and embedding theorems for Herz type spaces in tubular domains over symmetric cones. We alert the reader some arguments are sketchy since they can be easily recovered by readers based on simpler cases and remarks we make. All results of this section are known in particular case of simplest one domain namely the unit disk.
This topic is well-developed in the unit disk and other simple domains like unit ball and polydisk (see, for example [17], [18]). In [10], [12] this type embedding theorems can be seen in context of Bergman type harmonic spaces and tubular domains over symmetric cones TQ. In this section we add some new results in this direction in same tubular domains over symmetric cones but in new Herz-type analytic spaces.
Maximal theorems are vital classical topic in complex function theory in the unit disk. We provide such type theorems in Bergman type (Herz) spaces in the unit disk and then using same arguments in the different domains, namely, in tubular domain over symmetric cones in Cn which where under attention in recent decades (see [3], [12], [13]).
Our maximal theorems then will be used to get some new embedding theorems in tube domains.
Some related new results in analytic Herz type spaces in product domains will be also provided in this section. We start with the case of unit disk, then pass easily same arguments to more complicated domains as tubular domain in Cn.
Let U be the unit disk. Let H(U) be the class of all analytic functions in U. Let
Dyf (z) = £ (k + 1)yakzk, y e R k>0
be the fractional derivative of f e H(U). We denote by dm2 the normalized Lebesgues measure on U. We note the following simple arguments are valid in U. Let f e H(U), a > —1, then
f (z) = c(P) f f n ^wlf dm2(w), P > Po, Ju (1 — wz)p+2
p0 is large enough, z e U, (see [18]). This representation also is valid in tube, for Bergman spaces and A™(TQ) spaces, (see [1], [3], [12] and references there).
We will need Forelly-Rudin estimate in the unit disk
f (1 -|z|)a lu |1 - Wz|ß
dm2(z) < C(1 - W^-15-2, ß >-1, ß > a + 2, w e U.
Then we have based on this estimates for Bergman disk D(z,R) /(/) = f sup (|DY/(z)|(1 - |z|)T)(1 - |v|)sdm2(v)
I(f) < ci f i f )(1_ ,'wf dm2(w)(1 -|v|)T(1 -|v|)sdm2(v) Juju |1 - wz|ß+2
< cj |f(w)|(1 -|w|)T+s-Ydm2(w), JU
where t > 0, s > -1, Y > 0.
We used simple properties of lattice in U (see [18]): (1 — |v|) x (1 — |z|) x (1 — |w|) and |1 — wwv| x |1 — wz|, v, z e D(w, r), WW e U, w e U.
So I(f) < c||f ||A1 , t + s — y = a > —1. So we have that I(f) < c||f ||Ai., a > —1.
The p > 1 case for Aa needs only small modification.
The only new ingredient is an estimate which follows directly from well known Forelly-Rudin and Holder's inequality for each e > 0, p > —1, 1 < p < <*>, t > 0
f(w)|(1 — , A' r |f(w)|p(1 — |w|)Pp
< c^ ^^^2-ip-dm2(w)(1 - |z|)-ep, z G U.
Iu |1 — wwz|T ^ J ~ Ju |1 — wWz|Tp+2—ep
This estimate, with same proof, also is valid in tube, see estimate (3) (and see, for example [1], [3], [12] and references there). As a result, we have for p > 1
/ sup |DYf(z)|p(1 — |z|)T(1 — |v|)sdm2(v) < c / |f(w)|p(1 — |W|)T+s—adm2(w).
U zeD(v,R) JU
The repetition of these arguments leads to the same estimate, but on product domains (polydisk)
/.../ sup ••• sup Da1...zm |f(Zl,..., zm)|pX
■>U ^Uz1eD(v1,R) zmeD(vm,R)
m
X n(1 — |zj|)Tj(1 — |Vj|)sjdm2(V1)••• dm2(Vm) < j=1
< C ... |/(^)|p n(1 -|wj |)Tj+Sj-ajdm2(Wj),
JU JU j=i
with the same restrictions of parameters. (This remark will be used by as below to get same type result in different unbounded tube domains.)
These arguments under one condition on Bergman kernel leads to new maximal and embedding theorems in very general tube domains over symmetric cones and for Herz type spaces on them even on product of such domains.
We assume for Bergman kernel Bv, v > Vo, the following condition is valid for Bergman ball B(z,R) in tube.
sup |Bv(w, ww) | < c|Bv(z, ww) |, w G Th,
weB(z,R)
where v > v0, v0 is large enough.
Note the reverse is valid for c=1 obviously and this is valid in ball and polydisk (see, for example, [18]).
As usual we shall denote by Dz the natural extension to the complex space Cn of the generalized wave operator Dx of the cone Q:
°z = At^ ,
1 d
i d z
which is the differential operator of degree r defined by the equality:
1 d
a{ 1 Tz) [ei{ZlZ= A(Z)ei(zlZ), Z ^
Repeating arguments of the unit disk case step by step and using preliminaries of previous section about tubular domains and properties of analytic functions on them, we have the following:
n
Theorem 1. (a maximal theorem) Let f e AP (T£), Pj >--1, j = 1,..., m, Pj =
n r
Tj + sj — Yj > — 1, p > 1. Then we have that
¡r,p = suP •"/ suP |DL..,zmf 1 Pdv(wm)x
T,s JTQ Z1eB(w1,r) JTQ ZmeB(w
m
A Ti f T___ _ USifT............N \ ^ „II rll
)
x n ATj'(Im zj)ASj(Im wj)dv(wm)... dv(wi) < c|| f ||Aa,Tm
(a ^
j=i
for all Yj > Yo, j = 1,...,m where Yo is large enough and the reverse is also true if Yj = 0, j = 1,...,m.
The same proof can be given for very similar another maximal theorem. Theorem 2. Let f e App (Tfi), aj > - — 1, j = 1,..., m, aj = Tj + sj — Yj > - — 1, p > 1. Then we have that
IIf\\%,P = f ...••• f sup sup P?,...^lpdv(wm)x ST,s JTq JTaz1eB(wm,r) zmeB(w1,r)
m
x nATj(Im zj)ASj(Im wj)dv(w1)... dv(wm) < c\\f ||ap (Tm)
t- , ay Q '
J=1
for all Yj > Yo, j = 1,...,m where Yo is large enough and the reverse is also true if Yj = 0, j = 1,...,m.
Remark 1. Note, T. Flett proved some very similar to our maximal theorem results in Bergman harmonic spaces in R++1 in [5].
Note for Hp Hardy space in the unit disk we have similar type classical result
f sup | f (z)\pd% < C || f llPHP, 0 < p < ~ JT 7p r„ (%)
zerp £)
where 0 < p < <x>, rp(^) is a classical Lusin cone, and where T = {|z| = 1}, see [17].
The short proof in the unit disk was provided above. The T case is the same, properties of r-lattices for TQ must be used (see [12] and our first section). Note, for Yj = 0, j = 1,...,m this result is sharp.
The crucial ingredient is the next lemma (see [10], [12]).
Lemma 3. Let 1 < p < <*>, v > - — 1, f e AV, then for l > lo, m > 0, where lo is large enough, we have that
□lf (z) = c / Bv+i(z, w)Dmf (w)Am(lm w)dv(w), z e ; JTa
We have to use also short comments in tube we provided in the proof of the unit disk case by us. The rest is the repetition of arguments we gave above.
This theorem provided a way for various embedding theorems for Herz-type spaces. Our proof use also some arguments of [10], [12]. Namelly, combining results from [10], [12] and our maximal theorem we have: Theorem 3.
1) Let y > To, 7o is large enough. Let p > 1, let m = m1 x ■ ■ ■ x Mm be positive Borel measure on T^. Let p < q, if
(4 I f (^ )lqd m (^)) i < cii f lis-
then Mj(B(ak, r)) < cAq(Im ) for some qj = qj(p, y, T,s), aj = Tj + sj — y > - — 1, j = 1,..., m, Tj > 0, Sj > — 1, Tj = Sj — n, j = 1,..., m, where } is r-lattice. For y = 0 this condition on measure is sufficient for this embedding.
2) Let y > To, To is large enough. Let p > 1, let m be positive Borel measure on T^. Let
q < P, if
(J If(z)IqdM(z)) q < c||fllsrj
then M 5 (z)) e Li (Tfl) for some q, a = t + s — y > n — 1, t > o, s > —1, t = s — n, Aa+n (Imz) r r
s = pp-, v e (o, 1), if Pv is bounded on Lp, v > vo, vo is large enough. For y = o this p q
condition on measure is sufficient for this embedding.
Sketch of proof of Theorem . Indeed, in [10] we can see the complete description of M Borel positive measures defined in T^, so that the following embeddings are valid
(4 If(z)|qdM(z)) 1 < Cllf lAi(Tfl),
where q > p, a > - — 1, and r
(jTa I f (z)|qd M (z)) 1 < C1ii f lAi (Tn),
where q < p, a > - — 1.
r
Note, also, the same result with same proof is valid for q > p case for Bergman Ap(Tm) spaces on product T^ domains. It remains to combine this result with our maximal theorems to get easily what we need.
We define Sq'p(d/) replacing dv by d/ in quazinorms, where d/j, j = 1,...,m is a Borel measure on T .
Remark 2. For y = 0 similar result is valid for embedding of type || f |U,p, <
ST,s \d/i )
c|\f Wap(t%), p > q > 1.
Methods used in this paper also allows to find for 0 < p, q < <x>, a > —1 necessary conditions on measures (/1,...,/m) on TQ, so that the following embedding is valid for Herz type spaces
sup ••• sup |f (vW)|qHdM(wj) < Wl|P
m
(wj) < Ci || f ||aP
JTa JTaZieB(wi,r) Zm^B(Wm,r) j=l "
To obtain necessary condition for embeddings
11 f h^idji) < C| f hi{Tm)
or
11 f ^(dfl) < Cl11 f hi(TK)>
for all p, q g (0, <*>) and all aj > —1, Yj, Tj > 0, j = 1,..., m and for positive Borel measures ij on Ta, j = 1,...,m we have to use an elementary estimate
sup |$(z,w)| > |$(w,w)l
zGB(w,r)
for every measurable function and every w, w g Ta, and every Bergman ball B(w, r) c Ta, r > 0, w g Ta and standard arguments, see [10], based on a estimates from below of Bergman kernel on Bergman ball and Forelly-Rudin type estimate for Bergman kernel. For same type condition for j Borel measure on Ta embeddings || f HAqidu) < c|| f 11 So,p,
0 < p, q < or embeddings || f HAq(dj) < C1| f H^o.p we must use same type arguments
T,f
with condition discussed them partially below. Let now
Pm
P2
Ak (Tm) = {f e H (Tg) : ( I ... ^ | f (wi,..., Wm) |P1 d va, (wi) )Pl ... ] dVam (wm) ) < «}
1 < pj < <*>, j = l,..., m, vaj (w) = Aaj (Im w)d v (w), aj > -l, j = l,..., m.
These spaces are direct extensions of Bergman Apa(T ffl function classes. In our embeddings relating S0p(dj ) and Apa (T^) the Bergman space can be replaced by A0 (T^) easily also.
And we obtain again a necessary condition on j = (j1,...,jm) similarly. (Carleson type condition on measure). We here omit easy details.
Next in our maximal theorem for s0,P classes probably can be extended to S°',p mixed norm spaces where
S°'p = {f g H(1^) :
I sup •••(/" sup (|f(z1,...,Zm)l)P1 dVp1 (z1 M dVam(Zm) I <
'Ta. z1GB(w1,r) YTa.zmGB{w m,r) )
We will study this problem in our next paper of this topic.
Based on discussion above we get the following estimates which at the same time provide proofs of last assertions on embeddings. We have for standard test function (see [10]):
m 5a (z ■)
fz(w) = n-=j-, W G , j = 1,...,m,
j=i AP (zjj) j '
where 5(z) = A(1m z), z G T^, z g T^, z = (zi,...,Zm), for some a and p > Po,Po is large enough,
m /2n \ 1
II /?Ha№) ^ c n 5T (zj), zj G , j = 1,..., m, t = a - p + y + yj p < 0, 0 < p <
with some restrictions on p and a and more generally
II■fzIAp(Tm) = ( (XQ " ■ L 1 ("W)'P1 dVai(W1 )) P1 ■ ■ ■ dVam(Wm)
P2 X
\ Pm
n I 4 (w ^ (T-) < ^ n5Tj (Zj)
for some Tj, j = 1,..., m. Also, then we have
and also
(m \ / m
n 5 V (Zj )J (n Mj (B5 (Zj ))) , 5 > 0,
IIf-IU №) > M n5V(Zj) n Mj (B5(Zj)), V=1 / j=i
for some parameters V, V and zj G T^, j = 1,..., m.
First two estimates are based directly on Forelly-Rudin estimate in tube and the last two estimates are based on standard arguments related with estimates from below of Bergman kernel on Bergman ball, (see [10], [12], for example for similar type argument).
Remark 3. Similar type results (maximal and embedding theorems) by similar methods can be shown in the polydisk, in the unit ball and in pseudconvex domains with smooth boundary in Cm. This will be done in out next papers.
A maximal theorem in the unit ball related with the Poisson type integral
In this section we extend the notion of classical Poisson integral in a simple natural way in the unit ball and extend some well known estimates related with it, in particular an extension of a known maximal theorem will be provided in the unit ball. Such type results probably can be proved in context of more general pseudoconvex domains with smooth boundary.
Poisson type integrals on product domains considered recently also in papers [4] and [14].
In this short section we also in particular provide an extension of a known result concerning estimate from below of Poisson kernel (see, for example, [18]).
Let = {z e Cn : |z| < 1} be the unit ball and let Sn = {z e Cn : |z| = 1} be the unit sphere.
Let % e Sn, r > 0, Qr(%) = {z G Bn, d(z, %) < r}, d(z, w) = |1- < z, w > |1, z, w G Bn. We recall Qr(%) is Carleson tube at %. Let Ma(f)(%) = supzeDa(%) |f (z)|, where
a
Da(§) = {z G B„ : |1- < z, § > | < -(1 - |z|2), a > 1}
where § g sn. Let
r (1 -|z|)ndM(%) _ P[M](z)^S„ |1- %z|2n , z G B-
(m *)(% )=su>p ^br L ,*)d M (%),
where Q(%, *) = {n G Sn : d(%, n) < *}, * > 0, % e Sn, z G Bn.
Theorem A. (a maximal theorem in the ball, see [11], [18]) Let a > 1, then
7M(%) = (MaP[M])(%) < (%), % G Sn,
and ||//(%)||lp(sb) < c||f ||lp, p > 1 for a ^ positive complex finite Borel measure on Bn.
This maximal theorem result has many applications (see, for example, [11], [18]). We now show an extension of this, that the following result is valid also. Let further
& %)= (1 -N)n 7* ( z ^) nm=111 - z.% |aj ,
where % g Sn, zj G Bn, |zj| = |z|, aj > 0, j = 1,...,m. Theorem 4. Let
Pa= I jlj, 'jG B, j = -m, jf aj = 2n
Then we have that
K%)= sup PaM(?) < C1MM(%), r e (0,1),
zj GDt (% ),|zj|=r
where ^ is a positive Borel measure, and
||K% (f) 1 LP(SB,dff(%)) < c21 f ¡LP(SB,dff(%)), P > 1
Proof.
Let z e Da(%). Let z = (|z11^1,..., |zm|^m), |zj| = |z,|, 1 < i < j < m, r = |zj|, t = 8a(1 - r). Let V0 = {n G Sn : |1-< n, % > | < t}, 1 < k < N, 2Nt > 2, Vk = {n G Sn : 2k-1t < |1- < n, % > | < 2kt}.
Note that we have obviously
I Pa(zn)dM(n)= f Pa&n)dM(n)+ f f Pa(zn)dM(n).
v/SB JV0 k=1 Vk
It is enough to show that
f Pp(z,nW(n) < c(a,n)(M/(%)), % g Sn, zj GDa(%), j = 1,...,m,
■J Sn
for some constant c(a,n). We have by definition that Vk C Q(%,V2t), 1 < k < N and hence
/(Vk) < /(Q(%, Vfr)) < ((M/)(%))o(Q(%, y/fr)) < c(2kt)nM/i(%), 0 < k < N.
Pa(z, n) < 2n(1 — r)—n. We have hence, since /(V0) < ctnM/(%)
J Pa(z,n)d/(n) < c (Y^) (M/(%)) < Can(M/)(%). Then if n G Sn we have
|1— < z, % > | < ca (1 — ft2) < ¡fa |1— < z, n > I
d(z, %) < fP/d(z, n).
Hence we have from definition of d
d(%,n) < d(%,z) + d(z,n) < c'Pd(z,n), z = zj, j = 1,...,m.
Hence
J (Ppt,n))d(/(n)) < (S(pn))(M/)(%),
where 1 < k < N, the rest is clear.
The last estimate follows from the fact that
|1— < %,n > I < c(a,n)11— < zj,n > I for every zj, j = 1,...,m, 1 < k < N, n G Vk. And hence we have that
p c'(p, n)tn c"(p, n)
Pp(z n) < |1— < %, n > I2n < 4k«t« .
The second assertion of theorem follows from the first part of our theorem and the well known maximal theorem, (see [11], [18]). Theorem is proved. □
(1 — |z|2)n |1— < z, % >
on Bn unit ball (see [11], [17], [18]). Let P[f](z) = fSnP(z,%)f (%)da(%), f g L1(Sn,da), where a is a normalized measure on Sn.
It is natural to study the following extension of P[f], P[f](z), zj G Bn, j = 1,...,m, where
P[f](z)= / P(z, %)f (%)da(%),
J Sn
Let further, as usual, P(z, %) = -— g ^ l2n, z G Bn, % G Sn be the Poisson kernel
where
P(Z, ^ ) = Paß (1, ^ ) = n
_ n=i(l |zj |2)aj
«,ß^ j |1- < Zj, % > |j
aj,> 0, j = 1,...,m, aj = n, A/ = 2n and even more generally.
G(zi,..., zm,..., zj,..., zm, §i,..., n^, %), p=P^,
i=1 j , j
zj,eBn, i,j = 1,...,k and try to expand classical known assertions.
Very similar procedure of extension of Bergman projection was provided and used intensively in connection with trace problem (see [15], [16]).
We found the following results for Poisson P kernel as modification of known proofs for P kernel (see [18] for m = 1 case).
Proposition 1. Let ^, ¿Uj be positive Borel measures on Bn, and let p e (0, i = 1,..., m. Then we have that 1)
sup i P(w,z)du(w) > c M)), % e Sn, r e (0,1),
and even generally we have for rj e (0,1), j = 1,..., m. 2)
£2 ■ 1
sup / ...( |G(z,w)|P1 d^i(ziH P1 d^2(z2)••• djUm(zm) ) >
wjeBK,j=1...m \^BK \^BK /
> M. (Qri fe ))■■■ W-tom fe,)), 0 < p. < „, j = 1 m,
r«0 «0 '1 . . . / m
for some Og, a0 > 0, j = 1,..., m, where ^, ¿Uj, j = 1,..., m are positive Borel measures on Bn for some constants c1, c where e Sn, j = 1,..., m.
Proofs are heavily based on arguments of proofs of m = 1 case and they will be given elsewhere.
Remark 4. These results of Proposition extend some known assertions from [11] and [18].
References
[1] Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B., "Analytic Besov spaces and Hardy type inequalities in tube domains over symmetric cones", J. Reine Angew. Math., 647 (2010), 25-56.
[2] Bekolle D., Bonami A., Garrigos G., Nana C., Peloso M., Ricci F., Lecture notes on Bergman projectors in tube domain over cones, an analytic and geometric viewpoint, Proceeding of the International Workshop on Classical Analysis, Yaounde, 2001.
[3] Bekolle D., Bonami A., Garrigos G., Ricci F., "Littlewood - Paley decomposition and Besov spaces related to symmetric cones and Bergman Projections in Tube Domains", Proc. Lond. Math. Soc, 89:3 (2004), 317-360.
[4] Chyzhykov I., Zolota O., "Growth of the Poisson - Stieltjes integral in the polydisk", Zh. Mat. Fiz. Anal. Geom., 7(2) (2011), 141-157.
[5] Flett T., "Inequalities for the pth mean values of harmonic and subharmonic functions with p < 1", Proc. London Math. Soc., 20 (1970), 249-275.
[6] Henkin G. M., Chirka E.M., "Boundary properties of holomorphic functions of several complex variables", Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat., 4 (1975), 13-142.
[7] Krotov V. G., "Estimates of maximal operators connected with the boundary behaviour and their applications", Trudy Mat. Inst. Steklov, 190 (1989), 117-138.
[8] Nagel A., Rudin W., Shapiro J. H., "Tangential Boundary Behavior of Function in Dirichlet-Type Spaces", Annals of Mathematics Second Series, 1l6(2) (1982), 331-360.
[9] Nagel A., Stein M., "On certain maximal functions and approach regions", Advances in mathematics, 54 (1984), 83-106.
[10] Nana C., Sehba B., "Carleson Embeddings and Two Operators on Bergman Spaces of Tube Domains over Symmetric Cones", Integral Equations and Operator Theory, 83 (2015), 151-178.
[11] Rudin W., Function Theory in the Unit Ball of Cn, Springer, New York, 2008.
[12] Sehba B. F., "Bergman type operators in tube domains over symmetric cones", Proc. Edin-burg. Math. Society, 52 (2009), 529-544.
[13] Sehba B. F., "Hankel operators on Bergman Spaces of tube domains over symmetric cones", Integral Equations Operator Theory, 62:2 (2008), 233-245.
[14] Shamoyan R., "A note on Poisson type integrals in pseudoconvex and convex domains of finite type and some related results", Acta Universitatis Apulensis, 40 (2017), 19-27.
[15] Shamoyan R., Mihic O., "In search of traces of some holomorphic functions in polyballs", Revista Notas de MatemAtica, 4 (2008), 1-23..
[16] Shamoyan R., Mihic O., "On traces of holomorphic functions on the unit polyball", Appl. Anal. Discrete Math, 3 (2009), 198-211.
[17] Shvedenko S.V., "Hardy classes and related spaces of analytic functions in the unit disc, polydisc and ball, in Russian", Itogi nauki i tekhniki, VINITI, 1985, 3-124.
[18] Zhu K., Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. V. 226, Springer, New York, 2005.
References (GOST)
[1] Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B. Analytic Besov spaces and Hardy type inequalities in tube domains over symmetric cones // J. Reine Angew. Math. 2010. vol 647. pp. 25-56.
[2] Bekolle D., Bonami A., Garrigos G., Nana C., Peloso M., Ricci F. Lecture notes on Bergman projectors in tube domain over cones, an analytic and geometric viewpoint. Proceeding of the International Workshop on Classical Analysis. Yaounde, 2001
[3] Bekolle D., Bonami A., Garrigos G., Ricci F. Littlewood - Paley decomposition and Besov spaces related to symmetric cones and Bergman Projections in Tube Domains // Proc. Lond. Math. Soc. 2004. vol. 89. no. 3. pp. 317-360
[4] Chyzhykov I., Zolota O. Growth of the Poisson - Stieltjes integral in the polydisk // Zh. Mat. Fiz. Anal. Geom. 2011. vol. 7(2). pp. 141-157
[5] Flett T. Inequalities for the pth mean values of harmonic and subharmonic functions with P < 1 // Proc. London Math. Soc. 1978. vol. 20. pp. 249-275
[6] Henkin G.M., Chirka E. M. Boundary properties of holomorphic functions of several complex variables // Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. 1975. vol. 4. pp. 13-142.
[7] Krotov V. G. Estimates of maximal operators connected with the boundary behaviour and their applications // Trudy Mat. Inst. Steklov. 1989. vol. 190. pp. 117-138
[8] Nagel A., Rudin W. , Shapiro J. H. Tangential Boundary Behavior of Function in Dirichlet-Type Spaces // Annals of Mathematics Second Series. 1982. vol. 116(2). pp. 331-360
[9] Nagel A., Stein M. On certain maximal functions and approach regions // Advances in mathematics. 1984. vol. 54. pp. 83-106
[10] Nana C., Sehba B. Carleson Embeddings and Two Operators on Bergman Spaces of Tube Domains over Symmetric Cones // Integral Equations and Operator Theory. 2015. vol. 83. pp. 151-178
[11] Rudin W. Function Theory in the Unit Ball of Cn. New York: Springer, 2008.
[12] Sehba B. F. Bergman type operators in tube domains over symmetric cones // Proc. Edin-burg. Math. Society. 2009. vol. 52. pp. 529-544.
[13] Sehba B. F. Hankel operators on Bergman Spaces of tube domains over symmetric cones // Integral Equations Operator Theory. 2008. vol. 62. no. 2. pp. 233-245.
[14] Shamoyan R. A note on Poisson type integrals in pseudoconvex and convex domains of finite type and some related results // Acta Universitatis Apulensis. 2017. vol. 40. pp. 19-27.
[15] Shamoyan R., Mihic O. In search of traces of some holomorphic functions in polyballs // Revista Notas de Matematica. 2008. vol. 4. pp. 1-23.
[16] Shamoyan R., Mihic O. On traces of holomorphic functions on the unit polyball // Appl. Anal. Discrete Math. 2009. vol. 3. pp. 198-211.
[17] Shvedenko S. V. Hardy classes and related spaces of analytic functions in the unit disc, polydisc and ball
Itogi nauki i tekhniki, 1985. vol. 23. pp. 3-124.
[18] Zhu K. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics. vol. 226. New York: Springer, 2005.
Для цитирования: Shamoyan R. F., Mihic O. R. On some new estimates related with Bergman ball and Poisson integral in tubular domain and unit ball // Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 1(21). C. 48-63. DOI: 10.18454/2079-6641-2018-21-1-48-63
For citation: Shamoyan R. F., Mihic O. R. On some new estimates related with Bergman ball and Poisson integral in tubular domain and unit ball, Vestnik KRAUNC. Fiz.-mat. nauki. 2018, 21: 1, 48-63. DOI: 10.18454/2079-6641-2018-21-1-48-63
Поступила в редакцию / Original article submitted: 24.12.2017
В окончательном варианте / Revision submitted: 09.03.2018
Vestnik KRAUNC. Fiz.-Mat. Nauki. 2018. no.1(21). pp. 48-63. ISSN 2079-6641
DOI: 10.18454/2079-6641-2018-21-1-48-63
УДК 517.53+517.55
0 НЕКОТОРЫХ НОВЫХ ОЦЕНКАХ, СВЯЗАННЫХ
С ТЕОРЕМАМИ ОБ ОГРАНИЧЕННОСТИ
ПРОЕКТОРОВ ТИПА БЕРГМАНА И ИНТЕГРАЛОМ ПУАССОНА В ТРУБЧАТОЙ ОБЛАСТИ И ЕДИНИЧНОМ ШАРЕ
Р. Ф. Шамоян1, О. Р. Михич2
1 Брянский государственный технический университет, 241050, г. Брянск, Россия
2 Отдел математики, Университет Белграда, 154, Белград, Сербия
E-mail: [email protected],[email protected]
Введены новые аналитические пространства типа Герца, основанные на шарах Бергмана в трубчатых областях над симметричными конусами. Мы предлагаем для этих пространств типа Герца новые максимальные теоремы и теоремы вложения, расширяющие известные результаты в единичном круге. Кроме того, мы определяем новый интеграл типа Пуассона в единичном шаре и распространяем известную классическую максимальную теорему, связанную с ним. Соответствующие результаты для интегралов такого типа также будут приведены
Ключевые слова: трубчатые области над симметричными конусами, пространства типа Герца, интегральные операторы типа Бергмана, максимальные теоремы, теоремы вложения, интеграл типа Пуассона, единичный шар
(с) Шамоян Р.Ф., Михич О. Р., 2018
