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Sharp inclusions and lacunary series in mixed-norm spaces on the polydisc K.L. Avetisyan

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Faculty of Physics, Yerevan State University, Alex Manoogian st. 1, Yerevan 0025, Armenia Version of record first published: 20 Sep 2011.

To cite this article: K.L. Avetisyan (2013): Sharp inclusions and lacunary series in mixed-norm spaces on the polydisc, Complex Variables and Elliptic Equations: An International Journal, 58:2, 185-195 To link to this article: http://dx.doi.org/10.1080/17476933.2011.561839

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Complex Variables and Elliptic Equations, 2013 Vol. 58, No. 2, 185–195, http://dx.doi.org/10.1080/17476933.2011.561839

Sharp inclusions and lacunary series in mixed-norm spaces on the polydisc K.L. Avetisyan* Faculty of Physics, Yerevan State University, Alex Manoogian st. 1, Yerevan 0025, Armenia Communicated by R.P. Gilbert

Downloaded by [K.L. Avetisyan] at 03:17 26 March 2013

(Received 14 July 2009; final version received 5 February 2011) We establish the sharpness and strictness of continuous inclusions in mixed-norm spaces of n-harmonic functions on the unit polydisc of Cn. To this end, we modify a wellknown counterexample of Hardy and Littlewood and give a characterization of lacunary series with Hadamard gaps in mixed-norm and weighted Hardy spaces. Keywords: mixed-norm; polydisc; Hadamard gaps; lacunary series AMS Subject Classifications: 32A37; 32A05

1. Introduction Let Un ¼ {z ¼ (z1, . . . , zn) 2 Cn : jzjj 5 1, 1 j n} be the unit polydisc in Cn and U1 ¼ D the unit disc, and let Tn ¼ {w ¼ (w1, . . . , wn) 2 Cn : jwjj ¼ 1, 1 j n} be the n-dimensional torus, the distinguished boundary of Un. We will deal with n-harmonic functions on the polydisc Un, i.e. functions harmonic in each variable zj separately. Denote by h(Un) and H(Un) the sets of n-harmonic and holomorphic functions in Un, respectively. The pth integral mean of a measurable function f in Un is denoted as usual by Mp ð f; rÞ ¼ f ðrÞ p n , r ¼ ðr1 , . . . , rn Þ 2 ½0, 1Þn , 0 5 p 1, L ðT ;dmn Þ

where dmn is the n-dimensional Lebesgue measure on Tn. The collection of n-harmonic (holomorphic) functions f, for which k f khp ¼ supr 2 ð0,1Þn Mp ð f; rÞ 5 þ1, is the usual Hardy space hp (respectively Hp). The quasi-normed space h(p, q, )(0 5 p, q 1, ¼ (1, . . . , n)) is the set of those functions f n-harmonic in the polydisc Un, for which the quasi-norm 8 !1=q Z n n > Y Y > > > ð1 rj Þj q1 Mqp ð f; rÞ drj , 0 5 q 5 1, < ð0,1Þn j¼1 j¼1 k f kp,q, ¼ n Y > > > ð1 rj Þj Mp ð f; rÞ, q ¼ 1, sup > : r 2 ð0,1Þn j¼1

*Email: [email protected] ß 2013 Taylor & Francis

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is finite. If for each j, 1 j n, (1 r) Mp(u; r) ¼ o(1) as rj ! 1, then we say that nharmonic function u belongs to the little space h0(p, 1, ). For the subspaces consisting of holomorphic functions let Hð p, q, Þ ¼ HðUn Þ \ ð p, q, Þ,

H0 ð p, 1, Þ ¼ HðUn Þ \ h0 ð p, 1, Þ:

For p ¼ q 5 1 the spaces H(p, q, ), h(p, q, ) coincide with the wellknown weighted Bergman spaces, while for q ¼ 1 they are known as weighted Hardy or growth spaces. A lot of work is devoted to the mixed-norm and Bergman spaces consisting of holomorphic or pluriharmonic functions. We refer the reader to [1–4] for n-harmonic mixed-norm spaces on the polydisc. In [1], among others, the following theorem is proved.


THEOREM A Let 0 5 p, q 1, ¼ (1, . . . , n), ¼ (1, . . . , n), j, j 2 R, 1 j n. Then the following inclusions are continuous: ðiÞ hð p, q, Þ hð p, q, Þ,

j j ð1 j nÞ,

ðiiÞ hð p, q, Þ hð p0 , q, Þ, ðiiiÞ hð p, q, Þ hð p, q0 , Þ,

0 5 p0 5 p 1, 0 5 q 5 q0 1,

ðivÞ hð p, q, Þ hð p0 , q, Þ,

j j þ 1=p 1=p0 , p p0 1,

ðvÞ hð p, q, Þ hð1, q0 , Þ, j 4 j þ 1=p, 0 5 q0 1, ðviÞ hð p, q, Þ hð p, q0 , Þ, j 4 j , 0 5 q0 1, ðviiÞ Hp Hð p0 , q, 1=p 1=p0 Þ, 0 5 p 5 p0 1, p q 1, ðviiiÞ hp hð p0 , q, 1=p 1=p0 Þ, 1 5 p 5 p0 1, p q 1, ðixÞ hp hð p0 , q, Þ,

j 4 1=p 1=p0 , 0 5 p 5 p0 1,

ðxÞ If uð p, q, Þ, 0 5 q 5 1,

then

u 2 h0 ð p, 1, Þ:

It is natural to ask whether these inclusions are strict and sharp. The main purpose of this article is to prove the strictness and sharpness of the inclusions (i)–(x) in an appropriate sense. THEOREM 1 Let 0 5 p, q 1, j 4 0, 1 j n. Then all the inclusions (i)–(x) are strict and best possible in a certain sense.

2. Notation and preliminaries We will use the conventional multi-index notations: r ¼ (r11, . . . , rnn), ¼ 11 nn , dr ¼ dr1 drn for 2 Cn, r 2 [0, 1)n, ¼ (1, . . . , n). Let Nn , Znþ denote the sets of all n-tuples of positive integers and nonnegative integers, respectively. Throughout this article, the letters C(, , . . .), C, etc., stand for positive different constants depending only on the parameters indicated. For A, B 4 0 the notation A B denotes the two-sided estimate c1A B c2A with some inessential positive constants c1 and c2 independent of the variable involved. Define the following test function: cj n Y b e , z 2 Un , Fb,c ðzÞ :¼ 1 zj j log 1 z j j¼1

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where b ¼ (b1, . . . , bn), c ¼ (c1, . . . , cn), bj, cj 2 R. The following lemmas can be proved by a direct estimation, for the proof see [4, Section 2.3] or [5]. LEMMA 1

Suppose that n ¼ 1, b, c 2 R, 0 5 p 1, 0 5 q 5 1, 4 0. Then

(a) Fb,c is in H( p, q, ) if and only if b 5 þ 1p , c 2 R or b ¼ þ 1p , c 4 1q. (b) Fb,c is in H( p, 1, ) if and only if b 5 þ 1p , c 2 R or b ¼ þ 1p , c 0. (c) Fb,c is in H0( p, 1, ) if and only if b 5 þ 1p , c 2 R or b ¼ þ 1p , c 4 0. LEMMA 2

Suppose 4 0, p 4 0, ak 0, Ik ¼ f j 2 N; 2k j 5 2kþ1}, k ¼ 1, 2, . . . . Then Z

1

ð1 rÞ

1

0

1 X

!p ak r

k

k¼1

1 X 1 dr 2k k¼0

X

!p ,

aj

j 2 Ik


where the involved constants C ¼ C(p, ) depend only on p and . LEMMA 3

Let p 4 0, ak 0, N 2 N. Then ! !p ! N N N X X X p p p1 p1 ak ak maxf1, N g ak : minf1, N g k¼1

k¼1

k¼1

Lemma 2 is due to Mateljevic´ and Pavlovic´ [6], while Lemma 3 is an easy consequence of Ho¨lder’s inequality.

3. Lacunary series in H(p, q, a) This section can be viewed as a continuation of [2] where lacunary series in growth spaces H(p, 1, ) are studied. Lacunary series in classical function spaces such as Bloch, Bergman, Besov, Dirichlet, Q-type spaces, have been extensively studied recently [7–21]. Recall that a sequence fmk g1 k¼0 of positive integers is said to be for all lacunary (or Hadamard) if there exists a constant 4 1 such that mmkþ1 k k ¼ 0, 1, 2, . . . . A corresponding power series is called a lacunary series. For the polydisc we will consider the lacunary series of the form X m1,k n,kn ak1 ...kn z1 1 zm , z 2 Un : ð1Þ f ðzÞ ¼ n k 2 Znþ

The following theorem is an extension of classical Paley–Kahane–Khintchine inequalities to the polydisc. THEOREM B. ([2]) Let fmj, kj g1 kj ¼0 , j ¼ 1, 2, . . . , n be arbitrary lacunary sequences and f be a holomorphic function in Un given by a convergent lacunary series (1). Then for any p, 0 5 p 5 1, f is in Hardy space Hp ifPand only if {ak} 2 ‘2. Moreover, the corresponding norms are equivalent: k f kHp ð k 2 Znþ jak1...kn j2 Þ1=2 , where the involved constants are independent of f. Let R be the Hadamard operator of fractional integro-differentiation of order ¼ (1, . . . , n), j 2 R, X R f ðzÞ ¼ ð1 þ k1 Þ1 ð1 þ kn Þn ak1...kn zk11 zknn : k 2 Znþ

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The following two theorems characterize lacunary series in weighted Hardy spaces H( p, 1, ) and are essentially proved in [2]. 1 THEOREM 2 Let mj,kj kj ¼0 , j ¼ 1, 2, . . . , n be arbitrary lacunary sequences, ¼ (1, . . . , n), j 4 0, ¼ (1, . . . , n), j 2 R, and f be a holomorphic function in Un given by a convergent lacunary series (1). Then the following statements are equivalent: ðaÞ R f 2 Hð1, 1, Þ; ðbÞ R f 2 Hð p, 1, Þ

ðcÞ R f 2 Hð p, 1, Þ

for all p 2 ð0, 1Þ;

jak j

ðdÞ sup k 2 Znþ


for some p 2 ð0, 1Þ;

1 1 m1,k 1

n n mn,k n

5 þ1:

Also, corresponding norms are equivalent: kR f k1,1, kR f kp,1, sup

k 2 Znþ

jak j 1 1 m1,k 1

n n mn,k n

:

The next assertion is a ‘little oh’ version of Theorem 2. 1 THEOREM 3 Let mj,kj kj ¼0 , j ¼ 1, 2, . . . , n be arbitrary lacunary sequences, ¼ (1, . . . , n), j 4 0, ¼ (1, . . . , n), j 2 R, and f be a holomorphic function in Un given by a convergent lacunary series (1). Then the following statements are equivalent: ðaÞ R f 2 H0 ð1, 1, Þ; ðbÞ R f 2 H0 ð p, 1, Þ

for some p 2 ð0, 1Þ;

ðcÞ R f 2 H0 ð p, 1, Þ

for all p 2 ð0, 1Þ;

ak

ðdÞ lim

kj !1 m1 1 1,k1

Proof of Theorems 2 and 3 R f ðzÞ ¼

n n mn,k n

¼0

for each 1 j n:

The series expansion of R f is lacunary, too,

X

mk

kn ð1 þ mk1 Þ1 ð1 þ mkn Þn ak z1 1 zm n :

k 2 Znþ

So, it suffices to apply Theorems 3 and 4 in [2] to the function R f.

g

Remark 1 It is easily seen that Theorems 2 and 3 cover all (weighted) Bloch and little Bloch spaces and generalize and improve the corresponding results in [7,8,11,16,17,21,22]. Versions of Theorems 2 and 3 for the unit ball in Cn are given in [15]. The main result of this section is the following theorem extending Theorem 2 to all q 2 (0, 1).

189


1 THEOREM 4 Let mj,kj kj ¼0 , j ¼ 1, 2, . . . , n be arbitrary lacunary sequences, 0 5 q 5 1, ¼ (1, . . . , n), j 4 0, and f be a holomorphic function in Un given by a convergent lacunary series (1). Then the following statements are equivalent: ðaÞ f 2 Hð1, q, Þ; ðbÞ f 2 Hð p, q, Þ for some p 2 ð0, 1Þ; ðcÞ f 2 Hð p, q, Þ for all p 2 ð0, 1Þ; X ak1 ...kn q 5 þ1: ðdÞ mk11 q mknn q k 2 Zn þ

Also, corresponding norms are equivalent:


0 k f k1,q, k f kp,q, @

X

k 2 Znþ

Proof

11=q jak1 ...kn jq A : mk11 q mknn q

We may assume that n ¼ 2. Let f ðz1 , z2 Þ ¼

P1

m

j,k¼0

ajk z1 j zn2k .

The implication (a) ) (b) is obvious because of the elementary inclusion H(1, q, ) H(p, q, ). The implication (b) ) (c) follows from Theorem B which asserts that Mp( f; r1, r2) Ms( f; r1, r2) for any s, 0 5 s 5 1. For proving the implication (c) ) (d), assume that f 2 H(2, q, ). Then, by Theorem B, we have k f kq2,q,

Z 1Z

1 q1

Z

ð1 rÞ

¼ 0

0

T

Z 1Z

ð1 rÞ 0

Z

q1

0

!q=2

m jajk j2 r1 j rn2k

dr1 dr2

j,k¼0

1

¼C

j,k¼0

1 X

1

C

!q=2 X 2 1 m m n n j j k k ajk r1 1 r2 2 dm2 ðÞ dr1 dr2 2

ð1 r2 Þ 0

2 q1

Z

1 1 q1

ð1 r1 Þ 0

1 X

!q=2 m Gj ðr2 Þr1 j

dr1 dr2 ,

j¼0

P 2 nk where Gj ðr2 Þ :¼ 1 k¼0 jajk j r2 . Applying Lemmas 2 and 3, and then Fubini’s theorem, we obtain k f kqp,q, C

Z

1

ð1 r2 Þ2 q1

0

Z

1

ð1 r2 Þ2 q1

C 0

Z C 0

ð1 r2 Þ2 q1

X

1 @ m1 q 2 m m¼0

j

1q=2 Gj ðr2 ÞA

dr2

2 Im

q=2 1

dr2 1 q Gj ðr2 Þ m mj 2 Im j

1 X X m¼0

1

0

1 X

1 q=2 X 1 dr2 1 q ðGj ðr2 Þ mj j¼0

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K.L. Avetisyan


!q=2 Z1 1 1 X X 1 n 2 q1 ¼C ð1 r2 Þ jajk j2 r2k dr2 mj1 q 0 j¼0 k¼0 !q=2 1 1 X X 1 X 1 2 C jajk j mj1 q m ¼ 0 2m2 q n 2 Im j¼0 k ! 1 1 1 X 1 X jajk jq X X 1 X jajk jq C ¼C 1 q 2 q 1 q 2 q , mj m ¼ 0 nk 2 Im nk m nk j¼0 j¼0 k¼0 j where C ¼ C(p, q, 1, 2, 1, 2). Proceeding to the implication (d) ) (a), we write q X Z 1Z 1 1 mj mj nk nk q1 q k f k1,q, ¼ ð1 rÞ sup ajk r1 1 r2 2 dr1 dr2 0 0 2 T2 j,k¼0 ! q Z 1Z 1 1 X m C ð1 rÞq1 jajk j r1 j rn2k dr1 dr2 : 0

0

j,k¼0

Estimating as above by using Lemmas 2 and 3 leads to k f kq1,q, C

1 X 1 X jajk jq 1 q 2 q , m nk j¼0 k¼0 j

g

as desired. This completes the proof of Theorem 4.

The following is a generalization and an immediate consequence of Theorem 4. COROLLARY 1 Let fmj,kj g1 kj ¼0 , j ¼ 1, 2, . . . , n, be arbitrary lacunary sequences, 0 5 q 5 1, ¼ (1, . . . , n), j 4 0, ¼ (1, . . . , n), j 2 R, and f be a holomorphic function in Un given by a convergent lacunary series (1). Then the following statements are equivalent: ðaÞ R f 2 Hð1, q, Þ; ðbÞ R f 2 Hð p, q, Þ for some p 2 ð0, 1Þ; ðcÞ R f 2 Hð p, q, Þ for all p 2 ð0, 1Þ; X jak1 ...kn jq ðdÞ 5 þ1: ð1 1 Þq mkðnn n Þq k 2 Znþ mk1 Also, corresponding norms are equivalent: 0 kR f k1,q, kR f kp,q, @

X

k 2 Znþ

ak ...k q 1

mkð11 1 Þq

n

mkðnn n Þq

11=q A

:

Remark 2 In [16,21], versions of Theorem 4 are proved for weighted Bergman spaces in the unit disc, ball and polydisc. In [17], the equivalence of (b) and (d) in Corollary 1 is proved for ordinary derivatives of functions holomorphic in the unit disc.

191


Remark 3 Substituting (j 4 j) in place of , we see that Corollary 1 covers all Besov spaces and generalizes the previous similar results in [7,8,21,23].

4. Pointwise estimates in H( p, q, a) We now turn to some pointwise estimates for lacunary series. It is well known that arbitrary function f 2 H(p, q, ) satisfies the pointwise estimate


j f ðzÞj Cð p, q, , nÞ

k f kp,q, ð1 jzjÞþ1=p

,

z 2 Un ,

ð2Þ

where the exponent þ 1/p in (2) is best possible for general functions. Indeed, the inclusion H( p, q, ) H(1, 1, þ 1/p ") is false for any small " 4 0. The function Fþ1/p,2/q is in H(p, q, ), by Lemma 1, but Fþ1/p,2/q 2= H(1, 1, þ 1/p "). The following theorem shows that lacunary series in H( p, q, ) grow more slowly near the distinguished boundary than general functions of H( p, q, ). THEOREM 5 Let 0 5 p, q 1, fmj,kj g1 kj ¼0 , j ¼ 1, 2, . . . , n be arbitrary lacunary sequences, ¼ (1, . . . , n), j 4 0, and f be a function of H( p, q, ) given by a convergent lacunary series (1). Then j f ðzÞj Cð, p, q, , nÞ

k f kp,q, , ð1 jzjÞ

z 2 Un ,

ð3Þ

where the exponents j cannot be decreased. Proof By the inclusion (iii) of Theorem A, H( p, q, ) H( p, 1, ). Since the function f 2 H( p, 1, ) is given by a convergent lacunary series, we obtain by Theorem 2 that ð1 jzjÞ j f ðzÞj k f k1,1, k f kp,1, Ck f kp,q, ,

z 2 Un ,

as desired. Now we will show that no one of the exponents j may be decreased in (3). We assume that there exists some 1, 0 5 1 5 1, such that for every lacunary series f 2 H(p, q, ) there exists a constant C 4 0 such that j f ðzÞj

Ck f kp,q, 1

ð1 jz1 jÞ ð1 jz2 jÞ2 . . . ð1 jzn jÞn

,

z 2 Un ,

that is f 2 H(1, 1, (1, 2, . . . , n)). Then choosing a multiindex ¼ ( 1, . . . , n) such that 1 5 1 5 1 and 0 5 j 5 j for all 2 j n, define the example X k kn f0 ðzÞ ¼ 2k1 1 2kn n z21 1 z2n , z 2 Un : k 2 Znþ

By Theorems 2 and 4, f0 2 H( p, q, ), but, on the other hand, f0 2= H(1, 1, g (1, 2, . . . , n)). This contradiction completes the proof of the theorem. Although we cannot decrease the exponents j in (3), however we can improve the estimates (3) in the following sense.

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THEOREM 6 Let fmj,kj g1 kj ¼0 , j ¼ 1, 2, . . . , n be arbitrary lacunary sequences, 0 5 p 1, 0 5 q 5 1, ¼ (1, . . . , n), j 4 0, and f be a function of H( p, q, ) given by a convergent lacunary series (1). Then for each 1 j n we have 1 f ðzÞ ¼ o ð4Þ as jzj j ! 1 : ð1 jzjÞ Proof By (x) of Theorem A, f 2 H0( p, 1, ). Then Theorem 3 with j ¼ 0, asserts that f 2 H0( p, 1, ) is equivalent to f 2 H0(1, 1, ) for lacunary power series f, and the relations (4) follow. Of course, the exponents j in (4) cannot be decreased because of Theorem 5. g


5. A Hardy–Littlewood-type counterexample Hardy and Littlewood [24, p. 416] defined the following important function f ðzÞ :¼

eim=2 , ð1 zÞ1=p

p¼

1 , mþ1

m 2 N, z 2 D,

ð5Þ

as an example of holomorphic function in D whose real part is in Hardy space hp(D), e 0 5 p 5 1, but f 2= Hp(D); moreover, Mpp ð f; rÞ log 1r for all 0 r 5 1. Later, Duren and Shields [25, p. 257] applied the example (5) to prove the falsity of the inclusion hp hð1, 1, 1=p 1Þ,

0 5 p 5 1,

ð6Þ

on the unit disc D. For a polydisc version of (6), see [26, p. 140]. Now we are able to improve the result of Duren and Shields. THEOREM 7 Let 0 5 p 5 1, p 5 p0 1, j 4 1p p10 for all 2 j n. Then the inclusion !! 1 1 p , 2 , . . . , n h h p0 , 1, ð7Þ p p0 1 , m ¼ 1, 2, . . . . Hence, the inclusion is false at least for p ¼ mþ1 !! 1 1 , 2 , . . . , n hp h p0 , q, p p0

ð8Þ

1 ðm ¼ 1, 2, . . .Þ and each q, 0 5 q 1. is false for p ¼ mþ1

Proof In view of the inclusion (iii) in Theorem A, it suffices to prove only the falsity of (7). Define the functions eiðmþ1Þ=2 1 log , z 2 Un , 1=p 1 z1 ð1 z1 Þ uðz1 , . . . , zn Þ :¼ Re gðz1 , . . . , zn Þ, z 2 Un , gðz1 , . . . , zn Þ :¼

which are modifications of (5). It is easily seen by Lemma 1 that j gðzÞj ¼ jF1=p,1 ðz1 Þj

and g 2= H p0 , 1,

1 1 , 2 , . . . , n p p0

!! :


193

Then u 2= h p0 , 1, ð1p p10 , 2 , . . . , n Þ since the operator of pluriharmonic conjugation is bounded in mixed-norm spaces h(p, q, ) on the polydisc (see, e.g. [3]). On the other hand, assuming zj ¼ rj eij , we get eiðmþ1Þ=2 1 i1 log uðe , . . . , ein Þ ¼ Re 1 ei1 ð1 ei1 Þ1=p 1 1 ðm þ 1Þ 1 ðm þ 1Þ i1 i1 ¼ log j1 e argð1 e cos j þ sin Þ : mþ1 2 2 2 sin 1 2 Consequently, Cp Cm i1 , uðe , . . . , ein Þ m ¼ j1 j j1 j1=p1

ei 2 T n ,


so u 2 hp(Un). Thus, the falsity of the inclusion (7) is proved.

g

Remark 4 Note that the example of Hardy (5) is not sufficient for and Littlewood proving the falsity of the inclusion hp h p0 , 1, 1p p10 . In fact, we have proved that inclusion (ix) in Theorem A is sharp in the sense that no other choice for the components j in (ix) is permitted. Remark 5 The question of the falsity of the inclusions (7) and (8) for values of 1 p 2 (0, 1) other than p ¼ mþ1 ðm ¼ 1, 2, . . .Þ remains as an open question.

6. Proof of Theorem 1 (i) The inclusion (i) is strict if j 4 j for anyone j, say 1 4 1. Indeed, according to Lemma 1 the holomorphic function Fþ1/p,0 belongs to h( p, q, ), but not to h( p, q, ), 0 5 p 1, 0 5 q 5 1. Also, the holomorphic function Fþ1/p,0 belongs to h( p, 1, ), but not to h( p, 1, ), 0 5 p 1. (ii) The strictness of the inclusion (ii) is proved by the examples Fþ1/p,0 for 0 5 q 5 1 and Fþ1=p0 ,0 for q ¼ 1. (iii) The strictness of the inclusion (iii) is proved by the examples Fþ1/p,0 for q0 ¼ 1, and Fþ1/p,1/q for 0 5 q 5 q0 5 1. (iv) The sharpness of the inclusion (iv) in a strong form is proved in [1, p. 733]. Namely, the condition j j þ 1/p 1/p0 (1 j n) is necessary and sufficient for the inclusion (iv). The strictness of the inclusion (iv) is proved by the example X k kn k1 kn 2k1 1 2kn n z21 1 z2n , z 2 Un , f1 ðzÞ ¼ k 2 Znþ

which is in H(p, q, ), but not in H(p0, q, þ 1/p 1/p0), by Theorems 4 and 2. (v)–(vi) The inclusions (v) and (vi) are strict because of the example f1 or Fþ1/p,0. On the other hand, the inclusions (v) and (vi) are sharp for q0 5 q in the sense that no other choice for the components j is permitted. The function Fþ1=p,1=q0 gives a suitable example.

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K.L. Avetisyan

(vii)–(ix) The strictness of the inclusions (vii)–(ix) can be proved by the example P k kn f2 ðzÞ ¼ k 2 Znþ z21 1 z2n , z 2 Un :


(x)

The inclusions (vii) and (viii) are sharp in the sense that the condition p q is essential, that is for p 4 q the inclusions (vii) and (viii) are false. A corresponding example can be provided by the function F1/p,, where 1/p 5 5 1/q. Indeed, F1/p, is in Hp but not in H(p0, q, 1/p 1/p0), by Lemma 1. On the other hand, the inclusions (viii) and (ix) are sharp in the sense that the parameter p in (viii) cannot be decreased, and no other choice for the components j in (ix) is permitted, by Theorem 6. The strictness of the inclusion (x) follows from the example Fþ1/p,1/q. Indeed, by Lemma 1, Fþ1/p,1/q 2 H0( p, 1, ), but Fþ1/p,1/q 2= H( p, q, ). The sharpness of the inclusion (x) is understood in the sense that none of the components j can be decreased. Namely, the inclusion h p, q, ð1 , 2 Þ h0 p, 1, ð1 ", 2 Þ is false for any 0 5 p 1, 0 5 q 5 1, 0 5 " 5 . The function Fþ1/p"/ g 2,0 gives a corresponding example. This article represents a part of the author’s Dc.Sc. thesis [4] written at Yerevan State University.

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