Boundedness and compactness of an integral-type operator from ...
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Author's personal copy Applied Mathematics and Computation 218 (2012) 5414–5421
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Boundedness and compactness of an integral-type operator from Bloch-type spaces with normal weights to F(p, q, s) space Stevo Stevic´ Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
a r t i c l e
i n f o
a b s t r a c t The boundedness and compactness of the integral-type operator
Keywords: Integral-type operator Bloch-type space F(p, q, s) space Normal weight Unit ball Boundedness Compactness
Lgu ðf ÞðzÞ ¼
Z
1
Rf ðuðtzÞÞgðtzÞ 0
dt ; t
z 2 B;
where g is a holomorphic function on the open unit ball B in Cn such that g(0) = 0 and u is a holomorphic self-map of B, from Bloch-type spaces Bl and Bl;0 , where l is a normal weight, to F(p, q, s) space on B are characterized, solving a recent open problem. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Let B be the open unit ball in Cn ; S ¼ @B its boundary, D the open unit disk in C; HðBÞ the class of all holomorphic functions on B; SðBÞ the class of all holomorphic self-maps of B and H1 ðBÞ the space of all bounded holomorphic functions P k . By with the supremum norm kf k1 ¼ supz2B jf ðzÞj. Let z = (z1, . . . , zn) and w = (w1, . . . , wn) be points in Cn and hz; wi ¼ nk¼1 zk w BX we denote the closed unit ball in a Banach space X. P The radial derivative of an f 2 HðBÞ with the Taylor expansion f ðzÞ ¼ jbjP0 ab zb is defined by
Rf ðzÞ ¼
X
j b j ab zb ;
jbjP0
where b = (b1, b2, . . . , bn) is a multi-index and zb ¼ zb11 zbnn . It is easy to see that Rf ðzÞ ¼ hrf ðzÞ; zi. Let l be a positive continuous function on B (weight). The Bloch-type space Bl consists of all f 2 HðBÞ such that
bl ðf Þ ¼ sup lðzÞjRf ðzÞj < 1: z2B
The little Bloch-type space Bl;0 consists of all f 2 HðBÞ such that
lim lðzÞjRf ðzÞj ¼ 0:
jzj!1
With the norm kf kBl ¼ jf ð0Þj þ bl ðf Þ the Bloch-type space becomes a Banach space and Bl;0 is its subspace. A positive continuous function m on the interval [0, 1) is called normal [14] if there are d 2 [0, 1) and a and b, 0 < a < b such that
mðrÞ mðrÞ is decreasing on ½d; 1Þ and lim a ¼ 0; r!1 ð1 rÞ ð1 rÞa mðrÞ mðrÞ is increasing on ½d; 1Þ and lim ¼ 1: r!1 ð1 rÞb ð1 rÞb E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.028
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If we say that a function m : B ! ½0; 1Þ is normal we also assume that it is radial, i.e. m(z) = m(jzj), z 2 B. Let 0 < p, s < 1, n 1 < q < 1. A function f 2 HðBÞ is said to belong to Fðp; q; sÞ ¼ Fðp; q; sÞðBÞ space if
kf kpFðp;q;sÞ :¼ jf ð0Þjp þ sup a2B
Z
q s jRf ðzÞjp 1 jzj2 h ðz; aÞ dv ðzÞ < 1;
ð1Þ
B
where
hðz; aÞ ¼ ln
1 ; jua ðzÞj
is the Green’s function for B with logarithmic singularity at a, ua is the involutive Möbius transformation of B interchanging the points 0 and a [12], and dv(z) is the Lebesgue volume measure. If q + s 6 1, then F(p, q, s) consists of constant functions. Assume that g 2 HðBÞ; gð0Þ ¼ 0 and u 2 SðBÞ. Operator Lgu with symbols u and g is defined as follows (see [20,22,25])
Lgu f ðzÞ ¼
Z
1
Rf ðuðtzÞÞgðtzÞ
0
dt ; t
z 2 B;
f 2 HðBÞ:
ð2Þ
For related one-dimensional operators, see, e.g. [7–9] and [23]. Some related integral-type operators in Cn , are treated, e.g., in [2–6,10,11,15–18,21,24,26–28,30,33–39]. For other product-type operators see, for example, [31,32] and the references therein. In [29], among others, we studied integral-type operator (2), between F(p, q, s) space and Bloch-type spaces, where the boundedness and compactness of the operator from F(p, q, s) space to Bloch-type spaces are completely characterized. However the boundedness and compactness of the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ, were completely characterized for the R1 case when the weight function l satisfies the condition 0 ldtðtÞ < 1, while the case
Z
1
0
dt
lðtÞ
¼1
ð3Þ
was treated only for l(z) = (1 jzj2)a, a > 1. In this paper we characterize the boundedness and compactness of the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ, when l is a normal weight, solving an open problem in [29]. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation a b means that there is a C > 0 such that a 6 Cb. If both a b and b a hold, then one says that a b. 2. Auxiliary results Several auxiliary results which will be used in the proofs of the main results in the paper are listed below. Lemma 1 [25]. Assume that g 2 HðBÞ; gð0Þ ¼ 0 and u 2 SðBÞ. Then for every f ; g 2 HðBÞ it holds
R½Lgu ðf ÞðzÞ ¼ Rf ðuðzÞÞgðzÞ: Let Ps(z) be a homogeneous polynomial of degree s. The next Aleksandrov’s lemma [1] is well-known. Lemma 2. Let q 2 N n f1g. Then there are s 2 (0, 1) and k0 2 N, depending only on n, and homogeneous polynomials ðlÞ Pqk ðzÞ ; l 2 f1; . . . ; k0 g, such that k2N
ðlÞ
max max jPqk ðfÞj 6 1 and
16l6k0
f2S
ðlÞ
max min jPqk ðfÞj P s:
16l6k0
f2S
ð4Þ
The proof of the next lema is standard (for the original idea see [13] and for a detailed proof of a similar result see, for example [17, Lemma 3]). Lemma 3. Assume that g 2 HðBÞ; gð0Þ ¼ 0; u 2 SðBÞ; 0 < p; s < 1; n 1 < q < 1, and l is a normal weight. Then the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ is compact if and only if for any bounded sequence ðfk Þk2N in Bl ðor Bl;0 Þ converging to zero uniformly on compact subsets of B,
lim kLgu ðfk ÞkFðp;q;sÞ ¼ 0:
k!1
The following point evaluation estimate was proved in Lemma 3.1 in [34]. Lemma 4. Assume that f 2 HðBÞ and l is a normal weight. Then
Z jf ðzÞj 6 Ckf kBl 1 þ
0
for some C > 0.
jzj
ds ; lðsÞ
ð5Þ
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The following lemma was essentially proved in [19] (see Theorems 1, 2, 3, 4). Since the proof is a slight modification of the proofs in [19] it will be omitted. Lemma 5. Let l be a normal weight and f 2 HðDÞ such that f ðzÞ ¼ statements hold true:
P1
nk k¼1 ak z ,
where nkþ1 =nk ¼ q > 1; k 2 N. Then the following
(a) f 2 Bl if and only if lim supk!1 nk l 1 n1 jak j < 1. k (b) f 2 Bl;0 if and only if limk!1 nk l 1 n1 jak j ¼ 0. k
The following lemma is the main tool in solving the open problem. Lemma 6. Assume that l is a normal weight. Then there are N 2 N; r 0 2 ð0; 1Þ and functions f1 ; . . . ; fN 2 Bl such that
jRf1 ðzÞj þ þ jRfN ðzÞj P
1 lðzÞ
ð6Þ
for r0 6 jzj < 1. ðlÞ
Proof. Let q, s, k0, and the polynomials ðP qk ðzÞÞk2N ; l ¼ 1; . . . ; k0 , be as in Lemma 2. The following gap series are considered
Gl ðzÞ ¼
1 X j¼1
ðlÞ
P j ðzÞ q ; l 1 q1j qj
for z 2 B;
l = 1, . . . , k0, for a large natural number q, which will be specified later. Applying Lemma 5 to the slice functions j
ðlÞ 1 X Pqj ðfÞwq ; ðGl Þf ðwÞ ¼ Gl ðwfÞ ¼ 1 j j¼1 l 1 qj q
w 2 D;
which for a fixed f 2 @B are lacunary series in D obviously belonging to HðDÞ, with ðlÞ
Pqj ðfÞ ; aj ¼ l 1 q1j qj
j 2 N;
and nj = qj, and using the first inequality in (4), we obtain ðGl Þf 2 Bl ðDÞ, for each f 2 @B. From this and since for every f 2 HðBÞ the following formula holds
lðrÞRf ðrfÞ ¼ reih lðrÞffe0 ih ðreih Þ; where w = reih, it follows that Gl 2 Bl ðBÞ, for each l 2 {1, . . . , k0}. Set
fl ðzÞ ¼ RGl ðzÞ ¼
1 X j¼1
ðlÞ
P j ðzÞ q ; l 1 q1j
for z 2 B;
l = 1, . . . , k0. Since nm+1 = qnm and by the normality of l, it is easy to see that, for large enough q we have
l 1 q1m
6 qb qa 6 1 l 1 qmþ1
ð7Þ
for every m 2 N. We prove that
lðjzjÞ
k0 X
jfl ðzÞj P C 1 ;
ð8Þ
l¼1
for some C1 > 0, when 1 qm 6 jzj 6 1 qðmþ1=2Þ ;
m 2 N.
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By Lemma 2 and some simple inequalities for any z 2 B, we have that k0 X l¼1
ðlÞ ðlÞ k0 k0 m1 X X Pqj ðzÞ X jPqm ðzÞj P 1 1 l 1 qj l¼1 l 1 qm l¼1 j¼1
k0 X X 1 jfl ðzÞj ¼ l¼1 j¼1
m
m 1 X jzjq k0 Ps l 1 q1m j¼1
j 1 X jzjq k0 l 1 q1j j¼mþ1
ðlÞ k0 X Pqj ðzÞ X 1 l 1 q1j l¼1 j¼mþ1
ðlÞ P qj ðzÞ l 1 q1j
j
jzjq ¼ I 1 I2 I 3 : l 1 q1j
From the inequality
1=2 m m mþ1=2 q ð1 qm Þq 6 jzjq 6 ð1 qðmþ1=2Þ Þq ; we have that, for q large enough,
q1=2 m 1 1 6 jzjq 6 3 2
ð9Þ
and hence
I1 P
s 1 : l 1 q1m
3
On the other hand, from (7) we have
1
I 2 6 k0
l 1
1 qm
m1 X j¼1
1 k0 < ; qaðmjÞ l 1 1m ðqa 1Þ q
for large enough q. Using (7) we get mþ1 1 X jzjq I 3 6 k0 l 1 q1m j¼mþ1
l 1 q1m
j
jzjðq q 1 l 1 qj
mþ1 Þ
mþ1 m 1 i X mþ2 mþ1 jzjq ðjzjq Þq qb 6 k0 qb qb jzjðq q Þ ¼ k0 m 2 l 1 q1m i¼0 l 1 q1m 1 qb jzjq ðq qÞ
1=2
k0 qb 2q 6 : 3=2 1=2 l 1 q1m 1 qb 2ðq q Þ The preceding estimates for I1, I2 and I3 imply that, for q large enough, m 2 N and (in the ranges of z specified) k0 X l¼1
1
jfl ðzÞj P l 1 q1m
s k0 a 3
q 1
1=2
k0 qb 2q 1 qb 2ðq
3=2 q1=2 Þ
! :
Hence for sufficiently small e, from
1 qmþ1=2
6 1 jzj 6
1 qm
and the normality of l we obtain k0 X
jfl ðzÞj P
l¼1
C
lð1 q1m Þ
P
C
lðjzjÞ
:
Similarly, for
Glþk0 ðzÞ ¼
1 X j¼1
ðlÞ
P qj ðzÞ ; nj l 1 n1
for z 2 B;
j
1
l = 1, . . . , k0, where q is a large natural number of the form s2, and nj ¼ qjþ2 , for flþk0 ðzÞ ¼ RGlþk0 ðzÞ, we obtain
lðjzjÞ
k0 X
jflþk0 ðzÞj P C 2
l¼1
for some C2 > 0, when 1 q(m+1/2) 6 jzj 6 1 q(m+1), m 2 N.
ð10Þ
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From (8), (10), and for a fixed but sufficiently large q, inequality (6) follows on the annulus 1 q1 < jzj < 1, with N = 2k0, and for the functions Gl ðzÞ=ð2C 1 Þ; Glþk0 ðzÞ=ð2C 2 Þ; l ¼ 1; . . . ; k0 , as desired. h 3. Boundedness and compactness of Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ In this section we prove the main results in this paper, that is, we characterize the boundedness and compactness of the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ. Theorem 1. Assume that g 2 HðBÞ; gð0Þ ¼ 0; u 2 SðBÞ; 0 < p; s < 1; n 1 < q < 1; q þ s > 1 and l is a normal weight satisfying condition (3). Then the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ is bounded if and only if
M1 :¼ sup
Z
a2B
B
q juðzÞjp jgðzÞjp s 1 jzj2 h ðz; aÞdv ðzÞ < 1: p l ðjuðzÞjÞ
ð11Þ
Moreover, if the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ is bounded, then the next asymptotic relation holds
kLgu kBl ðor
Bl;0 Þ!Fðp;q;sÞ
M11=p :
ð12Þ
Proof. First assume that Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ is bounded. Then by Lemma 6, it follows that there are N 2 N; r0 2 ð0; 1Þ and functions f1 ; . . . ; fN 2 Bl such that (6) holds for r0 < jzj < 1. From this and by some elementary inequalities we get N X
kLgu ðfj ÞkpFðp;q;sÞ ¼
j¼1
N X
sup a2B
j¼1
¼
N X
sup
j¼1
a2B
P C sup
a2B
B
Z
q s jRLgu ðfj ÞðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ q s jRfj ðuðzÞÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ
B
q jgðzÞjp 1 jzj2
Z
a2B
P C sup
Z
juðzÞj>r0
Z
lp ðjuðzÞjÞ
s
h ðz; aÞdv ðzÞ
q juðzÞjp jgðzÞjp 1 jzj2
lp ðjuðzÞjÞ
juðzÞj>r0
s
h ðz; aÞdv ðzÞ:
ð13Þ
On the other hand, for the test functions
~f ðzÞ ¼ z 2 Bl ; j j
j ¼ 1; . . . ; n;
we get
sup a2B
Z B
q s juj ðzÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ 6 kLgu ð~f j ÞkpFðp;q;sÞ < 1
for each j 2 {1, . . . , n}, from which it follows that
sup a2B
Z B
n q X s juðzÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ 6 CkLgu kpBl !Fðp;q;sÞ k~f j kpBl < 1:
ð14Þ
j¼1
We also have that
sup a2B
Z juðzÞj6r 0
q juðzÞjp jgðzÞjp 1 jzj2
l uðzÞjÞ p ðj
s
h ðz; aÞdv ðzÞ 6 max
06t6r0
1 sup lp ðtÞ a2B
Z
q s juðzÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ:
ð15Þ
B
From (13)–(15) we easily get
M1 6 CkLgu kpBl !Fðp;q;sÞ :
ð16Þ
If the operator Lgu : Bl;0 ! Fðp; q; sÞ is bounded, we use the functions
ðfj Þr ðzÞ ¼ fj ðrzÞ 2 Bl;0 ; were r P r0 > 0, and obtain
j ¼ 1; . . . ; N;
Author's personal copy S. Stevic´ / Applied Mathematics and Computation 218 (2012) 5414–5421
CkLgu kpBl;0 !Fðp;q;sÞ
N X
kfj kpBl P
j¼1
N X
kLgu ððfj Þr ÞkpFðp;q;sÞ P C sup a2B
j¼1
PC
q jgðzÞjp 1 jzj2
Z r 0 <jr uðzÞjq
q s jRfk ðuðzÞÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ
6 CM 1 sup jrfk ðzÞjp þ C e sup kfk kpBl : jzj6q
ð31Þ
k2N
Letting k ? 1 in (31) and using the boundedness of the sequence ðfk Þk2N we have that
lim sup kLgu ðfk ÞkpFðp;q;sÞ 6 CLp e; k!1
from which by Lemma 3 and since e is an arbitrary positive number it follows that the operator Lgu : Bl ! Fðp; q; sÞ is compact. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
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Author's personal copy Applied Mathematics and Computation 218 (2012) 5414–5421
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Boundedness and compactness of an integral-type operator from Bloch-type spaces with normal weights to F(p, q, s) space Stevo Stevic´ Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
a r t i c l e
i n f o
a b s t r a c t The boundedness and compactness of the integral-type operator
Keywords: Integral-type operator Bloch-type space F(p, q, s) space Normal weight Unit ball Boundedness Compactness
Lgu ðf ÞðzÞ ¼
Z
1
Rf ðuðtzÞÞgðtzÞ 0
dt ; t
z 2 B;
where g is a holomorphic function on the open unit ball B in Cn such that g(0) = 0 and u is a holomorphic self-map of B, from Bloch-type spaces Bl and Bl;0 , where l is a normal weight, to F(p, q, s) space on B are characterized, solving a recent open problem. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Let B be the open unit ball in Cn ; S ¼ @B its boundary, D the open unit disk in C; HðBÞ the class of all holomorphic functions on B; SðBÞ the class of all holomorphic self-maps of B and H1 ðBÞ the space of all bounded holomorphic functions P k . By with the supremum norm kf k1 ¼ supz2B jf ðzÞj. Let z = (z1, . . . , zn) and w = (w1, . . . , wn) be points in Cn and hz; wi ¼ nk¼1 zk w BX we denote the closed unit ball in a Banach space X. P The radial derivative of an f 2 HðBÞ with the Taylor expansion f ðzÞ ¼ jbjP0 ab zb is defined by
Rf ðzÞ ¼
X
j b j ab zb ;
jbjP0
where b = (b1, b2, . . . , bn) is a multi-index and zb ¼ zb11 zbnn . It is easy to see that Rf ðzÞ ¼ hrf ðzÞ; zi. Let l be a positive continuous function on B (weight). The Bloch-type space Bl consists of all f 2 HðBÞ such that
bl ðf Þ ¼ sup lðzÞjRf ðzÞj < 1: z2B
The little Bloch-type space Bl;0 consists of all f 2 HðBÞ such that
lim lðzÞjRf ðzÞj ¼ 0:
jzj!1
With the norm kf kBl ¼ jf ð0Þj þ bl ðf Þ the Bloch-type space becomes a Banach space and Bl;0 is its subspace. A positive continuous function m on the interval [0, 1) is called normal [14] if there are d 2 [0, 1) and a and b, 0 < a < b such that
mðrÞ mðrÞ is decreasing on ½d; 1Þ and lim a ¼ 0; r!1 ð1 rÞ ð1 rÞa mðrÞ mðrÞ is increasing on ½d; 1Þ and lim ¼ 1: r!1 ð1 rÞb ð1 rÞb E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.028
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If we say that a function m : B ! ½0; 1Þ is normal we also assume that it is radial, i.e. m(z) = m(jzj), z 2 B. Let 0 < p, s < 1, n 1 < q < 1. A function f 2 HðBÞ is said to belong to Fðp; q; sÞ ¼ Fðp; q; sÞðBÞ space if
kf kpFðp;q;sÞ :¼ jf ð0Þjp þ sup a2B
Z
q s jRf ðzÞjp 1 jzj2 h ðz; aÞ dv ðzÞ < 1;
ð1Þ
B
where
hðz; aÞ ¼ ln
1 ; jua ðzÞj
is the Green’s function for B with logarithmic singularity at a, ua is the involutive Möbius transformation of B interchanging the points 0 and a [12], and dv(z) is the Lebesgue volume measure. If q + s 6 1, then F(p, q, s) consists of constant functions. Assume that g 2 HðBÞ; gð0Þ ¼ 0 and u 2 SðBÞ. Operator Lgu with symbols u and g is defined as follows (see [20,22,25])
Lgu f ðzÞ ¼
Z
1
Rf ðuðtzÞÞgðtzÞ
0
dt ; t
z 2 B;
f 2 HðBÞ:
ð2Þ
For related one-dimensional operators, see, e.g. [7–9] and [23]. Some related integral-type operators in Cn , are treated, e.g., in [2–6,10,11,15–18,21,24,26–28,30,33–39]. For other product-type operators see, for example, [31,32] and the references therein. In [29], among others, we studied integral-type operator (2), between F(p, q, s) space and Bloch-type spaces, where the boundedness and compactness of the operator from F(p, q, s) space to Bloch-type spaces are completely characterized. However the boundedness and compactness of the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ, were completely characterized for the R1 case when the weight function l satisfies the condition 0 ldtðtÞ < 1, while the case
Z
1
0
dt
lðtÞ
¼1
ð3Þ
was treated only for l(z) = (1 jzj2)a, a > 1. In this paper we characterize the boundedness and compactness of the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ, when l is a normal weight, solving an open problem in [29]. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation a b means that there is a C > 0 such that a 6 Cb. If both a b and b a hold, then one says that a b. 2. Auxiliary results Several auxiliary results which will be used in the proofs of the main results in the paper are listed below. Lemma 1 [25]. Assume that g 2 HðBÞ; gð0Þ ¼ 0 and u 2 SðBÞ. Then for every f ; g 2 HðBÞ it holds
R½Lgu ðf ÞðzÞ ¼ Rf ðuðzÞÞgðzÞ: Let Ps(z) be a homogeneous polynomial of degree s. The next Aleksandrov’s lemma [1] is well-known. Lemma 2. Let q 2 N n f1g. Then there are s 2 (0, 1) and k0 2 N, depending only on n, and homogeneous polynomials ðlÞ Pqk ðzÞ ; l 2 f1; . . . ; k0 g, such that k2N
ðlÞ
max max jPqk ðfÞj 6 1 and
16l6k0
f2S
ðlÞ
max min jPqk ðfÞj P s:
16l6k0
f2S
ð4Þ
The proof of the next lema is standard (for the original idea see [13] and for a detailed proof of a similar result see, for example [17, Lemma 3]). Lemma 3. Assume that g 2 HðBÞ; gð0Þ ¼ 0; u 2 SðBÞ; 0 < p; s < 1; n 1 < q < 1, and l is a normal weight. Then the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ is compact if and only if for any bounded sequence ðfk Þk2N in Bl ðor Bl;0 Þ converging to zero uniformly on compact subsets of B,
lim kLgu ðfk ÞkFðp;q;sÞ ¼ 0:
k!1
The following point evaluation estimate was proved in Lemma 3.1 in [34]. Lemma 4. Assume that f 2 HðBÞ and l is a normal weight. Then
Z jf ðzÞj 6 Ckf kBl 1 þ
0
for some C > 0.
jzj
ds ; lðsÞ
ð5Þ
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The following lemma was essentially proved in [19] (see Theorems 1, 2, 3, 4). Since the proof is a slight modification of the proofs in [19] it will be omitted. Lemma 5. Let l be a normal weight and f 2 HðDÞ such that f ðzÞ ¼ statements hold true:
P1
nk k¼1 ak z ,
where nkþ1 =nk ¼ q > 1; k 2 N. Then the following
(a) f 2 Bl if and only if lim supk!1 nk l 1 n1 jak j < 1. k (b) f 2 Bl;0 if and only if limk!1 nk l 1 n1 jak j ¼ 0. k
The following lemma is the main tool in solving the open problem. Lemma 6. Assume that l is a normal weight. Then there are N 2 N; r 0 2 ð0; 1Þ and functions f1 ; . . . ; fN 2 Bl such that
jRf1 ðzÞj þ þ jRfN ðzÞj P
1 lðzÞ
ð6Þ
for r0 6 jzj < 1. ðlÞ
Proof. Let q, s, k0, and the polynomials ðP qk ðzÞÞk2N ; l ¼ 1; . . . ; k0 , be as in Lemma 2. The following gap series are considered
Gl ðzÞ ¼
1 X j¼1
ðlÞ
P j ðzÞ q ; l 1 q1j qj
for z 2 B;
l = 1, . . . , k0, for a large natural number q, which will be specified later. Applying Lemma 5 to the slice functions j
ðlÞ 1 X Pqj ðfÞwq ; ðGl Þf ðwÞ ¼ Gl ðwfÞ ¼ 1 j j¼1 l 1 qj q
w 2 D;
which for a fixed f 2 @B are lacunary series in D obviously belonging to HðDÞ, with ðlÞ
Pqj ðfÞ ; aj ¼ l 1 q1j qj
j 2 N;
and nj = qj, and using the first inequality in (4), we obtain ðGl Þf 2 Bl ðDÞ, for each f 2 @B. From this and since for every f 2 HðBÞ the following formula holds
lðrÞRf ðrfÞ ¼ reih lðrÞffe0 ih ðreih Þ; where w = reih, it follows that Gl 2 Bl ðBÞ, for each l 2 {1, . . . , k0}. Set
fl ðzÞ ¼ RGl ðzÞ ¼
1 X j¼1
ðlÞ
P j ðzÞ q ; l 1 q1j
for z 2 B;
l = 1, . . . , k0. Since nm+1 = qnm and by the normality of l, it is easy to see that, for large enough q we have
l 1 q1m
6 qb qa 6 1 l 1 qmþ1
ð7Þ
for every m 2 N. We prove that
lðjzjÞ
k0 X
jfl ðzÞj P C 1 ;
ð8Þ
l¼1
for some C1 > 0, when 1 qm 6 jzj 6 1 qðmþ1=2Þ ;
m 2 N.
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By Lemma 2 and some simple inequalities for any z 2 B, we have that k0 X l¼1
ðlÞ ðlÞ k0 k0 m1 X X Pqj ðzÞ X jPqm ðzÞj P 1 1 l 1 qj l¼1 l 1 qm l¼1 j¼1
k0 X X 1 jfl ðzÞj ¼ l¼1 j¼1
m
m 1 X jzjq k0 Ps l 1 q1m j¼1
j 1 X jzjq k0 l 1 q1j j¼mþ1
ðlÞ k0 X Pqj ðzÞ X 1 l 1 q1j l¼1 j¼mþ1
ðlÞ P qj ðzÞ l 1 q1j
j
jzjq ¼ I 1 I2 I 3 : l 1 q1j
From the inequality
1=2 m m mþ1=2 q ð1 qm Þq 6 jzjq 6 ð1 qðmþ1=2Þ Þq ; we have that, for q large enough,
q1=2 m 1 1 6 jzjq 6 3 2
ð9Þ
and hence
I1 P
s 1 : l 1 q1m
3
On the other hand, from (7) we have
1
I 2 6 k0
l 1
1 qm
m1 X j¼1
1 k0 < ; qaðmjÞ l 1 1m ðqa 1Þ q
for large enough q. Using (7) we get mþ1 1 X jzjq I 3 6 k0 l 1 q1m j¼mþ1
l 1 q1m
j
jzjðq q 1 l 1 qj
mþ1 Þ
mþ1 m 1 i X mþ2 mþ1 jzjq ðjzjq Þq qb 6 k0 qb qb jzjðq q Þ ¼ k0 m 2 l 1 q1m i¼0 l 1 q1m 1 qb jzjq ðq qÞ
1=2
k0 qb 2q 6 : 3=2 1=2 l 1 q1m 1 qb 2ðq q Þ The preceding estimates for I1, I2 and I3 imply that, for q large enough, m 2 N and (in the ranges of z specified) k0 X l¼1
1
jfl ðzÞj P l 1 q1m
s k0 a 3
q 1
1=2
k0 qb 2q 1 qb 2ðq
3=2 q1=2 Þ
! :
Hence for sufficiently small e, from
1 qmþ1=2
6 1 jzj 6
1 qm
and the normality of l we obtain k0 X
jfl ðzÞj P
l¼1
C
lð1 q1m Þ
P
C
lðjzjÞ
:
Similarly, for
Glþk0 ðzÞ ¼
1 X j¼1
ðlÞ
P qj ðzÞ ; nj l 1 n1
for z 2 B;
j
1
l = 1, . . . , k0, where q is a large natural number of the form s2, and nj ¼ qjþ2 , for flþk0 ðzÞ ¼ RGlþk0 ðzÞ, we obtain
lðjzjÞ
k0 X
jflþk0 ðzÞj P C 2
l¼1
for some C2 > 0, when 1 q(m+1/2) 6 jzj 6 1 q(m+1), m 2 N.
ð10Þ
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From (8), (10), and for a fixed but sufficiently large q, inequality (6) follows on the annulus 1 q1 < jzj < 1, with N = 2k0, and for the functions Gl ðzÞ=ð2C 1 Þ; Glþk0 ðzÞ=ð2C 2 Þ; l ¼ 1; . . . ; k0 , as desired. h 3. Boundedness and compactness of Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ In this section we prove the main results in this paper, that is, we characterize the boundedness and compactness of the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ. Theorem 1. Assume that g 2 HðBÞ; gð0Þ ¼ 0; u 2 SðBÞ; 0 < p; s < 1; n 1 < q < 1; q þ s > 1 and l is a normal weight satisfying condition (3). Then the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ is bounded if and only if
M1 :¼ sup
Z
a2B
B
q juðzÞjp jgðzÞjp s 1 jzj2 h ðz; aÞdv ðzÞ < 1: p l ðjuðzÞjÞ
ð11Þ
Moreover, if the operator Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ is bounded, then the next asymptotic relation holds
kLgu kBl ðor
Bl;0 Þ!Fðp;q;sÞ
M11=p :
ð12Þ
Proof. First assume that Lgu : Bl ðor Bl;0 Þ ! Fðp; q; sÞ is bounded. Then by Lemma 6, it follows that there are N 2 N; r0 2 ð0; 1Þ and functions f1 ; . . . ; fN 2 Bl such that (6) holds for r0 < jzj < 1. From this and by some elementary inequalities we get N X
kLgu ðfj ÞkpFðp;q;sÞ ¼
j¼1
N X
sup a2B
j¼1
¼
N X
sup
j¼1
a2B
P C sup
a2B
B
Z
q s jRLgu ðfj ÞðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ q s jRfj ðuðzÞÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ
B
q jgðzÞjp 1 jzj2
Z
a2B
P C sup
Z
juðzÞj>r0
Z
lp ðjuðzÞjÞ
s
h ðz; aÞdv ðzÞ
q juðzÞjp jgðzÞjp 1 jzj2
lp ðjuðzÞjÞ
juðzÞj>r0
s
h ðz; aÞdv ðzÞ:
ð13Þ
On the other hand, for the test functions
~f ðzÞ ¼ z 2 Bl ; j j
j ¼ 1; . . . ; n;
we get
sup a2B
Z B
q s juj ðzÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ 6 kLgu ð~f j ÞkpFðp;q;sÞ < 1
for each j 2 {1, . . . , n}, from which it follows that
sup a2B
Z B
n q X s juðzÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ 6 CkLgu kpBl !Fðp;q;sÞ k~f j kpBl < 1:
ð14Þ
j¼1
We also have that
sup a2B
Z juðzÞj6r 0
q juðzÞjp jgðzÞjp 1 jzj2
l uðzÞjÞ p ðj
s
h ðz; aÞdv ðzÞ 6 max
06t6r0
1 sup lp ðtÞ a2B
Z
q s juðzÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ:
ð15Þ
B
From (13)–(15) we easily get
M1 6 CkLgu kpBl !Fðp;q;sÞ :
ð16Þ
If the operator Lgu : Bl;0 ! Fðp; q; sÞ is bounded, we use the functions
ðfj Þr ðzÞ ¼ fj ðrzÞ 2 Bl;0 ; were r P r0 > 0, and obtain
j ¼ 1; . . . ; N;
Author's personal copy S. Stevic´ / Applied Mathematics and Computation 218 (2012) 5414–5421
CkLgu kpBl;0 !Fðp;q;sÞ
N X
kfj kpBl P
j¼1
N X
kLgu ððfj Þr ÞkpFðp;q;sÞ P C sup a2B
j¼1
PC
q jgðzÞjp 1 jzj2
Z r 0 <jr uðzÞjq
q s jRfk ðuðzÞÞjp jgðzÞjp 1 jzj2 h ðz; aÞdv ðzÞ
6 CM 1 sup jrfk ðzÞjp þ C e sup kfk kpBl : jzj6q
ð31Þ
k2N
Letting k ? 1 in (31) and using the boundedness of the sequence ðfk Þk2N we have that
lim sup kLgu ðfk ÞkpFðp;q;sÞ 6 CLp e; k!1
from which by Lemma 3 and since e is an arbitrary positive number it follows that the operator Lgu : Bl ! Fðp; q; sÞ is compact. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
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