Báo cáo hóa học: " Research Article Integral-Type Operators from F p, q, s Spaces to Zygmund-Type Spaces on the Unit Ball Congli Yang1, 2" ppt

Lappan, “Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal,” in Complex Analysis and Its Applications, vol.. Stevi´c, “On an integral oper

Volume 2010, Article ID 789285, 14 pages

doi:10.1155/2010/789285

Research Article

Zygmund-Type Spaces on the Unit Ball

Congli Yang1, 2

1 Department of Mathematics and Computer Science, Guizhou Normal University,

550001 Gui Yang, China

2 Department of Physics and Mathematics, University of Eastern Finland, P.O Box 111,

80101 Joensuu, Finland

Correspondence should be addressed to Congli Yang,congli.yang@uef.fi

Received 7 May 2010; Revised 21 September 2010; Accepted 23 December 2010

Academic Editor: Siegfried Carl

Copyrightq 2010 Congli Yang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Let HB denote the space of all holomorphic functions on the unit ball B ⊂ C n This paper investigates the following integral-type operator with symbol g ∈ HB, Tgf z 

1

0f tzRgtzdt/t, f ∈ HB, z ∈ B, where Rgz n

j1z j ∂g/∂z j z is the radial derivative of g.

We characterize the boundedness and compactness of the integral-type operators Tgfrom general

function spaces F p, q, s to Zygmund-type spaces Z μ , where μ is normal function on 0, 1.

1 Introduction

holomorphic functions onB Let z  z1, , z n  and w  w1, , w n be points in Cnand

z, w  z1w1 · · ·  z n w n

Let

j1

z j ∂f

stand for the radial derivative of f ∈ HB For a ∈ B, let gz, a  log1/|ϕ a z|, where ϕ ais the M ¨obius transformation ofB satisfying ϕ a 0  a, ϕ a a  0, and ϕ a  ϕ−1

a For 0 < p, s <∞,

−n − 1 < q < ∞, we say f ∈ Fp, q, s provided that f ∈ HB and

fp

Fp,q,sf0p sup

a∈B



BRfz p

1− |z|2 q

The space Fp, q, s, introduced by Zhao in 1, is known as the general family of

function spaces For appropriate parameter values p,q, and s, Fp, q, s coincides with several

1 < s < ∞ see 2, where Bα , 0 < α < ∞, consists of those analytic functions f in D for

which

f

Bα  sup

z∈D



0 see 3, Fp, p − 2, 0 is the classical Besov space B p , and, in particular, F2, 1, 0 is just the Hardy space H2 The spaces

F 2, 0, s are Q sspaces, introduced by Aulaskari et al.4,5 Further, F2, 0, 1  BMOA, the analytic functions of bounded mean oscillation Note that Fp, q, s is the space of constant functions if q  s ≤ −1 More information on the spaces Fp, q, s can be found in 6,7 Recall that the Bloch-type spacesor α-Bloch space B α  Bα B, α > 0, consists of all

b α f

 sup

z∈B



1− |z|2 α

0B  Bα

lim

|z| → 1



1− |z|2 α

0 is

a closed subspace ofBα If α 1, we write B and B0forB1andB1

0, respectively

constants 0≤ δ < 1 and 0 < a < b such that

μ r

r→ 1

μ r

1 − r a  0,

μ r

r→ 1

μ r

1 − r b  ∞.

1.6

sup

z∈B



Write

f

Zf0  sup

z∈B



1− |z|2 R2f z.

1.8

With the norm Z,Z is a Banach space Z is called the Zygmund space see 8 Let Z0

lim

|z| → 1



sup

z∈B μ |z|R2f z < ∞.

1.10

fZ

μ f0  sup

z∈B μ |z|R2f z,

1.11

lim

|z| → 1 μ |z|R2f z  0, 1.12

say that f ∈ Z1−r: Z if and only if f ∈ AB, and there exists a constant C > 0 such that

operator is

Tgf z 

1

0

f tzRgtz dt

all 0 < p, s < ∞, −n − 1 < q < ∞ see 10 Recently, Li and Stevi´c discussed the boundedness

Zygmund spaces, see12,13

from general analytic spaces Fp, q, s to Zygmund-type spaces.

In what follows, we always suppose that 0 < p, s < ∞, −n − 1 < q < ∞, q  s > −1 Throughout this paper, constants are denoted by C; they are positive and may have different

values at different places

2 Some Auxiliary Results

In this section, we quote several auxiliary results which will be used in the proofs of our main

Lemma 2.1 If f ∈ Fp, q, s, then f ∈ B n1q/p and

f

Bn1q/p ≤f

Lemma 2.2 see 9 For 0 < α < ∞, if f ∈ B α , then for any z∈ B

f z ≤

⎧

⎪

⎪

⎪

⎪

Cf

Cf

Bαlog 2

1− |z|2, α  1,

Cf

Bα



1− |z|2 1−α

, α > 1.

2.2

Lemma 2.3 see 15 For every f, g ∈ HB, it holds RTgfz  fzRgz.

Lemma 2.4 see 10 Let p  n  1  q Suppose that for each w ∈ B, z-variable functions g w

satisfy |g w z| ≤ C/|1 − z, ω|, then



B

g ω zp

1− |z|2 q

Lemma 2.5 Assume that g ∈ HB, 0 < p, s < ∞, −n − 1 < q < ∞, and μ is a normal function on

0, 1, then Tg: Fp, q, s → Z μ or Z μ,0  is compact if and only if Tg: Fp, q, s → Z μ or Z μ,0

is bounded, and for any bounded sequence {f k}k∈N in F p, q, s which converges to zero uniformly on

compact subsets of B as k → ∞, one has lim k→ ∞ gf Z

μ  0.

Hence, we omit the details

Lemma 2.6 Let μ be a normal function A closed set K in Z μ,0 is compact if and only if it is bounded and satisfies

lim

|z| → 1sup

f ∈K μ |z|R2f z  0.

2.4

3 Main Results and Proofs

Now, we are ready to state and prove the main results in this section

Theorem 3.1 Let 0 < p, s < ∞, −n − 1 < q < ∞, and let μ be normal, g ∈ HB and n  1  q ≥ p,

then T g : Fp, q, s → Z μ is bounded if and only if

i for n  1  q > p,

z∈B μ |z|Rgz1 − |z|2 −n1q/p

z∈B μ |z|R2gz

1− |z|2 1−n1q/p

ii for n  1  q  p,

z∈B μ |z|Rgz1 − |z|2 −1

z∈B μ |z|R2gz log 2

Proof i First, for f, g ∈ HB, suppose that n  1  q > p and f ∈ Fp, q, s By Lemmas

2.1–2.3, we writeR2f  RRf We have that

Tgf

Zμ Tgf0  sup

z∈B μ |z|R2 Tgf

z

≤ sup

z∈B μ |z|RfzRgz  fzR2gz

≤fBn1q/psup

z∈B μ |z|Rgz1 − |z|2 −n1q/p

 Cf

Bn1q/psup

z∈B μ |z|R2gz

1− |z|2 1−n1q/p

≤ Cf

Fp,q,ssup

z∈B μ |z|Rgz1 − |z|2 −n1q/p

 Cf

Fp,q,ssup

z∈B μ |z|R2gz

1− |z|2 1−n1q/p

.

3.5

Hence,3.1 and 3.2 imply that Tg: Fp, q, s → Z μis bounded

f z ≡ 1 ∈ Fp, q, s, we see that g ∈ Z μ, that is,

sup

z∈B μ |z|R2gz < ∞.

3.6

For w∈ B, set

f w z 



1− |w|2 1n1q/p

1 − z, w2n1q/p −



1− |w|2

1 − z, w n1q/p , z ∈ B.

3.7

Then, w Fp,q,s ≤ C by 14 and f w w  0.

Hence,

∞ >Tgf w

Zμ ≥ sup

z∈B μ |z|R2 Tgf w

z

 sup

z∈B μ |z|Rf

w zRgz  f w zR2gz

≥ μ|w|Rf

w wRgw  f w wR2gw

 μ|w|Rf w wRgw

 μ |w|Rgw|w|2



1− |w|2 n1q/p

3.8

From3.8, we have

sup

|w|>1/2

μ |w|Rgw



1− |w|2 n1q/p < 4 sup

|w|>1/2

μ |w|Rgw|w|2



1− |w|2 n1q/p ≤ 4Tgf w

On the other hand, we have

sup

|w|≤1/2

μ |w|Rgw



1− |w|2 n1q/p < C sup

|w|≤1/2

Combing3.9 and 3.10, we get 3.1

In order to prove3.2, let w ∈ B and set

It is easy to see that h w w  1/1 − |w|2n1q/p−1,Rh w w ≈ |w|2/ 1 − |w|2n1q/p.

C Hence,

∞ >Tgh w

Zμ ≥ sup

z∈B μ |z|R2 Tgh w

z

 sup

z∈B μ |z|Rh

w zRgz  h w zR2gz

≥ μ|w|R2gw

1− |w|2 1−n1q/p

−μ |w|Rgw|w|2



1− |w|2 n1q/p

3.12

From3.1 and 3.12, we see that 3.2 holds

ii If n1q  p, then, by Lemmas2.1and2.2, we have Fp, q, s ⊆ B1, for f ∈ Fp, q, s,

we get

Tgf

Zμ Tgf0  sup

z∈B μ |z|R2 Tgf

z

≤ sup

z∈B μ |z|RfzRgz  fzR2gz

≤f

B 1sup

z∈B μ |z|Rgz1 − |z|2 −1

 Cf

B 1sup

z∈B μ |z|R2gz log 2

1− |z|2

≤ Cf

Fp,q,ssup

z∈B μ |z|Rgz1 − |z|2 −1

 Cf

Fp,q,ssup

z∈B μ |z|R2gz log 2

1− |z|2.

3.13

Applying3.3 and 3.4 in 3.13, for the case n  1  q  p, the boundedness of the operator

Tg: Fp, q, s → Z μfollows

f w z 



1− |w|2 2

1 − z, w2 −



1− |w|2

then w Fp,q,s ≤ C by 14

By the boundedness of Tg, it is easy to see that

μ |w|Rgw|w|2



By3.14 and 3.15, in the same way as proving 3.1, we get that 3.3 holds

∞ >Tgf w

Zμ ≥ sup

z∈B μ |z|R2 Tgf w

z

≥ sup

z∈B μ |z|Rf

w zRgz  f w zR2gz

≥ μ|w|R2gw log 2

1− |w|2 − μ |w|Rgw|w|2



3.17

From3.15 and 3.17, we see that 3.4 holds The proof of this theorem is completed

Theorem 3.2 Let 0 < p, s < ∞, −n − 1 < q < ∞, and let μ be normal, g ∈ HB and n  1  q ≥ p,

then the following statements are equivalent:

A Tg: Fp, q, s → Z μ is compact;

B Tg: Fp, q, s → Z μ,0 is compact;

C i for n  1  q > p,

lim

|z| → 1 μ |z|Rgz1 − |z|2 −n1q/p

lim

|z| → 1 μ |z|R2gz

1− |z|2 1−n1q/p

ii for n  1  q  p,

lim

|z| → 1 μ |z|Rgz1 − |z|2 −1

lim

|z| → 1 μ |z|R2gz log 2

Proof. B ⇒ A This implication is obvious.

A ⇒ C First, for the case n  1  q > p.

inB such that limk→ ∞|z k |  1 Denote f k z  f z k z, k ∈ N, and set

f k z 



1− |z k|2 1n1q/p

1 − z, z k2n1q/p −



1− |z k|2

1 − z, z kn1q/p , k ∈ N.

3.22

lim

k→ ∞Tgf k

Tgf k

z∈B μ |z|R2 Tgf k

z

 sup

z∈B μ |z|Rf

k zRgz  f k zR2gz

≥ μ|z k|Rfk z k Rgz k  fk z kR2gzk

 μ|z k|Rfk z kRgzk

 μ |z k |Rgz k |z k|2



1− |z k|2 n1q/p

3.24

From3.23 and 3.24, we obtain

lim

k→ ∞

μ |z k |Rgz k



1− |z k|2 n1q/p  limk→ ∞μ |z k |Rgz k |z k|

2



1− |z k|2 n1q/p  0, 3.25

Similarly, we take the test function

f k z 



1− |z k|2 2

1 − z, z k2 −



1− |z k|2

obtain that3.20 holds for the case n  1  q  p.

For proving3.19, we set

then k Fp,q,s ≤ C, and {h k}k∈Nconverges to 0 uniformly on any compact subsets ofB as

lim

k→ ∞Tgh k

Further, we have

Tgh k

z∈B μ |z|R2 Tgh k

z

 sup

z∈B μ |z|Rh

k zRgz  h k zR2gz

≥ μ|z k|Rh

k z k Rgz k   h k z kR2gzk

≥ μ|z k|R2gzk

1− |z k|2 1−n1q/p

−μ |z k|Rgzk|zk|2



1− |z k|2 n1q/p

3.29

From3.25, 3.28, and 3.29, it follows that

lim

k→ ∞μ |z k|R2gzk

1− |z k|2 1−n1q/p

which implies that3.19 holds

ii Second, for the case n  1  q  p, take the test function

f k z  log2/1 − z, zk

2 log

2/

Tgf k

Hence, we have that

Tgf k

z∈B μ |z|R2 Tgf k

z

 sup

z∈B μ |z|Rf

k zRgz  f k zR2gz

≥ μ|z k|R2gzk log 2

1− |z k|2 − 2μ |z k|Rgzk|zk|2



1− |z k|2 .

3.33

From3.20, 3.32, and 3.33, it follows that

lim

k→ ∞μ |z k|R2gzk log 2

which implies that3.21 holds

2.2, we have that

μ |z|R2 Tgf

z  μ|z|RfzRgz  fzR2gz

≤ CfFp,q,s μ |z|Rgz1 − |z|2 −n1q/p

 Cf

Fp,q,s μ |z|R2gz

1− |z|2 1−n1q/p

.

3.35

Note that3.18 and 3.19 imply that

lim

|z| → 1 μ |z|R2gz  0. 3.36

Further, they also imply that3.1 and 3.2 hold From this andTheorem 3.1, it follows that

lim

|z| → 1 sup

f F p,q,s≤1

μ |z|R2 Tgf

z  0.

3.37

Similarly, we obtain that3.37 holds for the case n  1  q  p by 3.20 and 3.21 Exploiting Lemma 2.6, the compactness of the operator Tg: Fp, q, s → Z μ,0follows The proof of this theorem is completed

Finally, we consider the case n  1  q < p.

Theorem 3.3 Let 0 < p, s < ∞, −n − 1 < q < ∞, and let μ be normal, g ∈ HB, n  1  q < p, then

the following statements are equivalent:

A Tg: Fp, q, s → Z μ is bounded;

B g ∈ Zμ and

sup

z∈B μ |z|Rgz1 − |z|2 −n1q/p

here

Theorem 3.4 Let 0 < p, s < ∞, −n − 1 < q < ∞, and let μ be normal, g ∈ HB and n  1  q < p,

then the following statements are equivalent:

A Tg: Fp, q, s → Z μ is compact;

B g ∈ Zμ and

lim

|z| → 1 μ |z|Rgz1 − |z|2 −n1q/p

Proof A ⇒ B We assume that Tg : Fp, q, s → Z μ is compact For f ≡ 1, we obtain that

g∈ Zμ Exploiting the test function in3.22, similarly to the proof ofTheorem 3.2, we obtain that3.39 holds As a consequence, it follows that

lim

|z| → 1 μ |z|Rgz  0. 3.40

B ⇒ A Assume that {f k}k∈N is a sequence in Fp, q, s such that sup k∈N k Fp,q,s ≤ L < ∞,

lim

k→ ∞sup

z∈B

From3.39, we have that for every ε > 0, there is a δ ∈ 0, 1, such that, for every δ < |z| < 1,

μ |z|Rgz



and from3.39 that

μ |z|R2 Tgf k

z  μ|z|Rf

k zRgz  f k zR2gz

≤ sup

|z|≤δ

μ |z|Rf k zRgz  sup

δ<|z|<1

μ |z|Rf k zRgz

g

Zμsup

z∈B

f k z

|z|≤δ Rf k z  f k

Bn1q/p sup

δ<|z|<1

μ |z|Rgz



1− |z|2 n1q/p

g

Zμsup

z∈B

f k z

≤ G μsup

|z|≤δ Rf k z  εL  gZμsup

z∈B

f k z.

3.44

positive number, we obtain

lim

k→ ∞Tgf k

proof of this theorem is completed

Theorem 3.5 Let 0 < p, s < ∞, −n − 1 < q < ∞, and let μ be normal, g ∈ HB and n  1  q < p,

then the following statements are equivalent:

A Tg: Fp, q, s → Z μ,0 is compact;

B g ∈ Zμ,0 and

lim

|z| → 1 μ |z|Rgz1 − |z|2 −n1q/p

Proof A ⇒ B For f ≡ 1, we obtain that g ∈ Z μ,0 In the same way as inTheorem 3.4, we get that3.46 holds

μ |z|R2 Tgf

z  μ|z|RfzRgz  fzR2gz

≤ Cf

Fp,q,s μ |z|Rgz1 − |z|2 −n1q/p

 Cf

Fp,q,s μ |z|R2gz.

3.47

This along with Theorem 3.2implies that Tg Fp,q,s ≤ 1} is bounded Taking the

Acknowledgments

The author wishes to thank Professor Rauno Aulaskari for his helpful suggestions This research was supported in part by the Academy of Finland 121281

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