ON MULTIPLIERS OF HOLOMORPHIC FP,Q α TYPE SPACES ON THE UNIT POLYDISC Текст научной статьи по специальности «Математика»

УДК 517.5

On Multipliers of Holomorphic Fp q Type Spaces on the Unit Polydisc

Romi F. Shamoyan*

Bryansk University, av. 50 October, 7, 241035, Bryansk,

Russia

MiloS ArsenoviC

Faculty of Mathematics, University of Belgrade, Studentski Trg., 16, 11000 Belgrade,

Serbia

Received 03.04.2012, received in revised form 17.05.2012, accepted 17.06.2012 We describe certain new spaces of coefficient multipliers of analytic Lizorkin-Triebel FP'q type spaces in the unit polydisc with some restriction on parameters. This extends some previously known assertions on coefficient multipliers in classical Bergman Apa spaces in the unit disk.

Keywords: multipliers, polydisc, analytic functions, analytic Lizorkin- Triebel spaces.

1. Introduction and Preliminary Results

The goal of this paper is to continue the investigation of spaces of coefficient multipliers of analytic FP'q Lizorkin-Triebel type spaces in the unit polydisc started in [1,2]. These Fpq type spaces serve as a very natural extension of the classical Hardy and Bergman spaces in the unit polydisc simultaneously. Spaces of Hardy and Bergman type in higher dimension were studied intensively by many authors during past several decades, see for example [3-8] and references therein. The investigation in this new direction of FP'q type classes in the unit polydisc was started in recent papers of the second author, see [1,2,9]. We note that complete analogues of these classes in the unit ball were studied also in recent papers of Ortega and Fabrega see [6-8] and references therein. See also [5,10,11] for some other properties of these new analytic FP'q type classes and related spaces in the unit disk or subframe. Below we list notation and definitions which are needed and in the next section we formulate and prove main results of this note.

Let D = {z e C : |z| < 1} be the unit disc in C, T = dD = {z e C : |z| = 1}, Dn is the unit polydisc in Cn, Tn c dDn is the distinguished boundary of Dn, Z+ = {n e Z : n > 0}, Z+ is the set of all positive multyindex and I = [0,1).

We use the following notation: for z = (zi,..., zn) e Cn and k = (ki,..., kn) e Z+ we set zk = zk1 •• • zt and for z = (zi,.. .,zn) e Dn and y e R we set (1 -|z |)Y = (1 -|zi |)Y •• • (1 — |zn|)Y and (1 — z)Y = (1 — z1)Y • • • (1 — zn)Y. Next, for z e Rn and w e Cn we set wz = (w1z1,..., wnzn). Also, for k e Z+ and a e R we set k + a = (k1 + a,..., kn + a). For z = (z1,..., zn) e Cn we set z = (z1,.. .,z„). For k e (0, +^)n we set r(k) = r(k1) •• • r(kn).

The Lebesgue measure on Cn = R2n is denoted by dV(z), normalized Lebesgue measure on Tn is denoted by = ... d£n and dR = dR1... dRn is the Lebesgue measure on [0, +^)n.

*shamoyan.r@gmail.com tarcenovis@matf.bg.ac.rs © Siberian Federal University. All rights reserved

The space of all functions holomorphic in Dn is denoted by H(Dn). Every f G H(Dn) admits an expansion f (z) = ^ akzk. For [ > —1 the operator of fractional differentiation is defined

ke z+

by

f (z)=V r(k + 3,+ 1) , akzk, z G Dn. (1)

n r([ + i)r(k + 1) k ' ^

For f G H(Dn), 0 < p < to and r G In we set

Mp(f,r)=(( |f (r£)|p dA/P , (2)

VI" /

with the usual modification to include the case p = to. For 0 < p < to we have analytic Hardy classes in the polydisc:

Hp(Dn) = {f G H(Dn) : ||f \\hp = sup Mp(f,r) < to}. (3)

rein

For n =1 these spaces are well studied. The topic of multipliers of Hardy spaces in polydisc is relatively new, see for example [3,12]. These spaces are Banach spaces for 1 < p and complete metric spaces for all other positive values of p. Also, for 0 <p < to, 0 < q < to and a > 0 we have mixed (quasi) norm spaces, defined below.

Apa'q (Dn) = { f G H (Dn) : f H«™ = h Mq (f, R)(1 - R)aq- 1 dR < ^ .

(4)

These spaces are Banach spaces for cases when 1 < min (p, q), and they are complete metric spaces for all other values of parameters. We refer the reader for these classes in the unit ball and the unit disk to [3,6-8] and references therein. Multipliers between spaces on the unit disc were studied in detail in [13].

As is customary, we denote positive constants by C, sometimes we indicate dependence of a constant on a parameter by using a subscript, for example Cq. We define now the main objects of this paper.

For 0 <p,q < to and a > 0 we consider Lizorkin-Triebel spaces Fp'q(Dn) = Fp'q consisting of all f G H(Dn) such that

llfIIF- =/ ( I If(R£)lq(1 — R)aq- 1 dR]P/qd£< to. (5)

a Jfn J

It is not difficult to check that those spaces are complete metric spaces, if min(p, q) > 1 they are Banach spaces.

We note that this scale of spaces includes, for p = q, weighted Bergman spaces Ava = Fp'f1,

see [4] for a detailed account of these spaces. On the other hand, for q = 2 these spaces coincide

with Hardy-Sobolev spaces namely Hp = F^ , for this well known fact see [3,11] and references

2

therein.

Finally, for a > 0 and [ > 0 we set

A~f(Dn) = {f G H(Dn) : |f ||A~,~ = sup (Mx(Daf, r))(1 — r)^ < to}. (6)

This space is a Banach space. For all positive values of p and s we introduce the following two new spaces.

The limit space case FpTO's(Dn) is defined as a space of all analytic functions f in the polydisc such that the function = supreI„ |f (r£)|(1 — r)s, £ G Tn is in Lp(Tn, d£).

Finally, the limit space case (Dn) is the space of all analytic functions f in the polydisc such that sup Mp(f, r)(1 — r)s < to.

rei"

Obviously the limit space is embedded in the last space we defined. This simple

observation will be used at the end of this note. These last two spaces are Banach spaces for all p > l and they are complete metric spaces for all other positive values of p.

The following definition of coefficient multipliers is well known in the unit disk. We provide a natural extension to the polydisc setup.

Definition 1. Let X and Y be quasi normed subspaces of H(Dn). A sequence c = {ck}keZ+ is

said to be a coefficient multiplier from X to Y if for any function f (z) = ^ akzk in X the

kez+

function h = Mcf defined by h(z) = ^ ckakzk is in Y. The set of all multipliers from X to Y

kez+

is denoted by MT (X, Y).

The problem of characterizing the space of multipliers (pointwise multipliers and coefficient multipliers) between various spaces of analytic functions is a classical problem in complex function theory, there is vast literature on this subject, see [4,13-15], and references therein.

In this paper we are looking for some extensions of already known theorems, namely we are interested in spaces of multipliers acting into analytic Lizorkin-Triebel FP'q type spaces in the unit polydisc and from these spaces into certain well studied classes like mixed norm spaces, Bergman spaces and Hardy spaces. We note that the analogue of this problem of description of spaces of multipliers in Rn for various functional spaces in Rn was considered previously by various authors in recent decades, for these we refer the reader to [16].

2. On Coefficient Multipliers of Analytic Lizorkin-Triebel Type Spaces in Polydisc

We start this section with several lemmas which will play an important role in the proofs of all our results. We note that these assertions can serve as direct natural extensions of previously known one dimensional results to the case of several complex variables. The following lemma in dimension one is well-known, see [1,4,17] and references therein. This is a several variables version of the so called Littlewood-Paley formula, see [17], which allows to pass from integration on the unit circle to integration over unit disk. We omit a simple proof which follows from a straightforward technical computation based on orthonormality of the trigonometric system, similarly to the classical planar case.

Lemma 1 ( [2]). For f, g G H(Dn), r G In and a > 0 we have

r f Da+1g(R£)fK)x

J0 J T"

(7)

The first part of the following lemma was stated in [2]. This lemma provides direct connection between mixed norm spaces and standard Bergman classes in higher dimensions. A detailed proof of the first part can be found in the cited paper of the second author and the proof of the second part is just a modification of that proof. We again omit details refereing the reader to [2].

f (rt)g(rt)dt =(2a)nH rj

2a

j=1

*n (rj2 - R2)aRi ••• j=i

' 1

T

0

n

Lemma 2. Let 0 < max(p, q) ^ s < to and a > 0. Then we have

f |f(w)|s(1 — |w|)s(a+ p)-2dV(w)) 1/S < C|fHf-«, f G Fp-q(Dn), J Dn /

f f |f (w)|s(1 — |w|)s(a+1 )-2dV(w))1/s < C||f ||A,„, f G A^'q(Dn). \JB" /

(8)

(9)

The following lemma is crucial for all proofs of necessity of multiplier conditions in our main results. It provides explicit estimates of the denominator of the Bergman kernel in various (quasi) norms in polydisc which appear in this note. Again these estimates are known and can be found in [1,2,9]. Let us note that the Bergman kernel in polydisc is a product of n one dimensional Bergman kernels. This fact often allows one to reduce calculations in several variables to the already classical one dimensional case, see for example [4] and references therein.

Lemma 3 ( [1]). Let 0 < p, q < to and set

/r(z)

1

(1 - Rz)W

ß> -1, R e

z e Dn

Then we have the following (quasi) norm estimates:

C

||fR||№(Bn) < (1 — R)^_i/p+i ,

If H ' C

ii/rii

AS'q (Dn)

Fa ' q (Dn)

<

<

(1 - R)ß-a-l/p+l>

C

(1 - R)ß-a-1/p+1 '

ß > - - 1

P

ß > a - 1 + -P

ß > a-1 + 1 P

(10)

(11) (12) (13)

Note the last estimate of last lemma is true also for two limit spaces and we

defined above

The following lemma follows from Lemma 2 but we include it here to stress its importance for later proofs.

Lemma 4 ( [4]). If 0 < v < 1, q> 1 — 2 and t > 0 then we have, for every f G H (Dn):

i |/(w)|t(1 - |w|)q dV (w) < c(f |/(w)|vt(1 - |w|)2v-2+qv dV (w))

Dn Dn

1/v

(14)

The proof of this lemma shows that if we replace here (1 — |w|)q by (1 — |w|r)q, where r G In, then the integrand on the right hand side changes to (1 — |w|r)qv(1 — |w|)2v-2|f (w)|vt

and conditions on parameters in this estimate will be ^ < v < 1. The proof of this last statement

is a modification of the well-known proof of Lemma 4, therefore we omit easy details. This remark will be nevertheless used by us in the proof of the main result of this note.

The first assertions we formulate provide conditions which are necessary, but not in general sufficient, in cases when all indexes are different in pairs of mixed norm spaces we consider in this note. By this we mean spaces of coefficient multipliers from Fp'q spaces to A^s spaces in the polydisc, and conversely. Even these assertions are more general than those that are present in the literature, see [4], a large survey article [3] by V. Shvedenko and more recent work [14,18,19] by O. Blasco, J. L. Arregui and M. Pavlovic. After that we formulate a theorem containing a sharp result which is one of the main results of this paper. This last result also extends

known assertions on coefficient multipliers of Bergman spaces in the unit disk in two directions at the same time: it deals with mixed norm F^'9 spaces and deals with polydisc instead of the disk. Since the proof of our main theorem contains in some sense proofs of all preliminary easy observations which we put below before formulation of main theorem we provide only complete sketches of proofs of these observations.

Let g G H(Dn), g(z) = J2 ckzk. If c = {ck}keZ+ is a coefficient multiplier from AP'9 to F|'s

kez+

where t ^ s, then the following condition holds:

sup Mt(Dmg, r)(1 - rf+m-a-1+1 < TO, (15)

rei"

where m is any number satisfying m>a — p +---1.

P

Moreover the last condition is a necessary condition for the following cases when we consider multipliers from F^'9 to A^'s or from F^'9 to F^'s for t ^ s.

However, we add additional conditions when we look at multipliers into Lizorkin-Triebel type spaces. These conditions appear due to the first embedding in the following chain F^'s ^ A^'s ^ of embeddings. The first embeddings holds for t ^ s and follows directly from Minkowski inequality. The second embedding is a well known classical fact and it holds for all strictly positive s, t and p, see, for example, [4,6,7].

In all of the above cases of necessity of condition (15) proof follows directly from Lemma 3 and the above embeddings in combination with the closed graph theorem which is always used in such situations, see the proof of theorem below as a typical example. We remark in addition that in the case of multipliers into Lizorkin-Triebel spaces we should use both embeddings, in the other case only the second one. We omit details referring readers to the proof of the first part of Theorem l.We will see this necessary condition on a multiplier which we just mentioned holds also when s < t,moreover we will show that it is sharp in this case.

If t = to and a = y+r, then the condition (15) is again necessary when considering multipliers from FY1'9 into We again omit a proof which is a modification of arguments of the proof

of the first part of Theorem 1.

Theorem 1. Let g G H (Dn), g(z) = ^keZ+ ck zk. Assume t < 1, 2 < s < t < to, 0 <

112 2

max(p, q) ^ s< to, p +— <a +— <-, m G N and m>--1.

t p t t

1o. {ck}kez+ G MT(F^9(Dn),4's(Bn)) if and only if

sup Mt(Dmg,r)(1 — r)m+1-P < to. (16)

rei"

2o. {ck}kez+ G MT(AS'9(Dn), A^s(D")) if and only if

sup Mt(Dmg,r)(1 — r)m+1-P < to. (17)

rei"

We remark that the above mentioned embedding A^'9 ^ F,P'9, p ^ q (which follows from Minkowski's inequality) allows us to deduce sufficiency of condition (17) from part 1o of the above theorem under additional condition p > q. However, this additional condition can be dropped with help of the second part of Lemma 2. Hence we will omit the complete proof of second part providing only the complete sketches of it.

Proof. 1o. Assume {ck}kez+ G MT(F^9(Dn), A^s(B")). An application of the closed graph

theorem gives ||Mcf ||At,s < C||/||Fp,q. Let w G Dn and set fw(z) = -, gw = Mcfw. Then

$ a 1 — wz

we have

Dmgw = DmMc/w = McDm/w = CMc--

(1 — wz)m+T

which, together with the third estimate from Lemma 3, gives

1

(1 - wz)m+i

C ( A

pgq ^ (1 - |w|)m-«-1/P+1 ■ (18)

Now Lemma 2, where /, p, q, s and a are replaced respectively by Dmgw, t, s, t and combined with the above estimate, gives

I |Dmgw (z)|4(1 — |z|)^-1dV (z)V<--—C-—T. (19)

JDJ V n V 117 V V (1 — |w|)m-a-1/p+1 V '

Setting |w| = r and using estimate

Mt(^r)(1 — r)" < c(f |^(z)|4(1 — |z|)i^-1dV(z^ / (20)

VD" J

we derive (16). The above argument provides also a complete proof of necessity in part 2o of the theorem. Indeed, the only change is that we use the second estimate in Lemma 3 instead of the third one. Moreover, similar arguments can be used to prove various necessity assertions we stated before this theorem.

Conversely, assume (16) holds and choose / G FP'9(Dn). Set h = Mc/ and choose n G Tn, r G In. Then we have, using Lemma 1:

h(r2n) = /T_ / (rnt)g(rt)dt = Dmg(R£)/(rne) 0 1 ■ ■ ■ Jo "f[( j - R2)m-1Ri...

The last expression can be transformed by a change of variables R = rp and we obtain an estimate

|h(r2n)| <C / |(Dmg)(rp£)/(rpne)| n(1 — p2)m-1dpde =

n

=c / |(Dmg)(rpe)/(rpne)^(1 — p2)m-1dpde.

JD" j=1

Now we apply Lemma 4 to an analytic function Dmg(rz)/(rnz) and obtain

~ n

|h(r2n)|4 < C / |(Dmg)(rpe)|i|/(rpn?)|^(1 — p2)i(m+1)-2dpde. (21)

JD" „-_1

Next we integrate over n G Tn. This yields, taking into account (16):

f |h(r2n)|4dn W / f |(Dmg)(rpe)|i|/(rpne)|4(1 — p2)i(m+1)-2dpdedn:

J T" JT" J I" J T"

=c f i i(Dmg)(rpe)iiMii(/,rp)(i - p2)i(m+i)-2dpde <

J Tn J In

^c f i Mtt(/,rp)(i - rp)-i(m+i-a+ß-i/p)(i - p2)i(m+i)-2dpde < JTn Jin

<C / I Mtt(/, rp)(1 - rp)i(a-ß+i/p)-2dpd£, t(m +1) > 2

Jjn J In

<C / |/(rw)|4(1 - r|w|)t(a-ß+i/p)-2dV(w).

J Dn

Now we apply Lemma 4 with v = s/t and with (1 — r|w|)q instead of (1 — |w|)q where q = t( p — fi + a) — 2, see comments after Lemma 4. This gives

M."(h, r2) 4 C / |/ (rw)|"(1 — r|w|)"(a-/+1/p)-2"/'(1 — |w|)2"/,-2dV(w). (22)

J Dn

Next we integrate both sides over r e In. This gives

llfrllAt,. = f M."(h, R)(1 — 4

J/.

/• r (1 — |w|)2s/t-2

4 C (1 — R)^-1 / |/ (Rw)|"--( — | " a . 0 . dV (w)dR 4

f f f (1 — | W| )2S/1-2 4 Ci(1—R) /.l|f(rR£)|" (1—Rw|)-.(a-g+./P)+2,/,drdRdi 4

C C (1 — | W| )2s/,-2

= C J J, M;(/'rR)(1 — R)S"-1 (1 — rU-^./^/. drdR 4 4 C J,. " — r>2"/'-2M"(/.r^r. (1 — rR)"(;iRlg-"-/P)+2"/, dRdr 4

4 C / M"(/,r)(1 — r)"(a+1/p)-2dr. ,.

At the last the assumption t(a + 1/p) < 2 allowed us to use the following well known estimate:

[. iT—R^dR 4 C I1 — rr-\ 0 4 r< 1

valid for a > —1 and A > a +1. Finally, Lemma 2, specifically embedding (8), allows us to conclude that ||h||At,s 4 C||/||Fp.q.

The proof of sufficiency in the AO/9 case goes along the same lines, the only difference being use of embedding (9) instead of (8) at the last step. □

The following theorem is parallel to the previous one, it gives characterization of multipliers into F/'" spaces. The proof goes along similar lines as the proof of the above theorem, therefore we omit it. Note here the necessity follows directly from the same arguments as in previous theorem ,but it is based on first estimate in second lemma.The sufficiency part also repeats the proof of previous theorem till the end, at last step we use the mentioned embedding between F£" and A^"

Theorem 2. Let g e H(Dn), g(z) = cfcAssume t 4 1, ^ < s 4 t < m, 0 < max(p, q) 4

fcez+ 2

112 2

s < m, fi + - < a + - < -, m e N ,m>--1.

t p t t

1o. |cfc}fc£Z+ e Mt(Fp'9(Dn), F//'"(Dn)) if and only if

sup M.(Dmg,r)(1 — r)m+1-P +/-a < m. (23)

re/.

2o. |cfc}fcez+ e MT(Aa'9(Dn), F/,'"(Dn)) if and only if

sup M.(Dmg,r)(1 — r)m+1-P +/-a < m. (24)

re/n

We remark partially results of this note were announced before in [2] without proofs. We conclude this paper with remarks on limit case spaces FP'TO'a, which are defined analogously to the spaces, the only difference being replacement of inner (quasi) norm Lq by norm. We also constantly use in arguments an obvious fact that weighted Hardy spaces AP'TO contain FP'TO'a classes in the polydisc.

Namely, we look for estimates on the rate of growth of a function g which represents a coefficient multiplier into the above described space. Again proofs are parallel to the one presented in the proof of Theorem 1.

We again assume that g = ^fcez+ cfczk is analytic in polydisc, by an estimate of the rate of growth of g we mean the following:

sup Mp(Dmg,r)(1 — r)T < TO. (25)

The condition (25) is often a necessary condition for {ck}keZ+ to be a multiplier and in that case t depends on parameters involved and can be explicitly specified for each pair of spaces separately using Lemma 3. A typical example is the following

Proposition 1. Assume 0 < v ^ p and m > p — 1 —s. If a sequence {ck }k£Z+ is a multiplier from

to FP'TO's, then the function g = J2 cfczk satisfies condition (25) with t = m + 1 + s — 1/p.

fcez+

The same condition on g function, but for different t is necessary for g to be a multiplier from mixed norm spaces we studied in this paper to FP'TO's or from Fto FP'TO'v. In the case of multipliers from FP'TO'v into classes the necessary condition is the same (for different

t) but with to instead of p. We leave simple proofs of these facts based on detailed outlines provided above to interested readers.

References

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Мультипликаторы голоморфных пространств типа Fa,q со смешанной нормой на единичном поликруге

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Мы описываем некоторые новые пространства коэффициентов мультипликаторов аналитических пространств типа ЕР'4 Лизоркина-Трибеля в единичном поликруге с некоторыми ограничениями на параметры. Это обобщает некоторые предыдущие известные утверждения на коэффициенты мульпликаторов в классических Аа пространствах Бергмана в единичном круге.

Ключевые слова: мультипликаторы, поликруг, аналитические функции аналитические пространства Лизоркина-Трибеля.