Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 1, February 2024
Page(s) 1 - 6
DOI https://doi.org/10.1051/wujns/2024291001
Published online 15 March 2024

© Wuhan University 2023

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

In this paper, a unit disc is denoted by D. The transformation of D is defined by:

Note that

is the hyperbolic metric for any , where is the pseudohyperbolic distance for any . For , an analytic function belongs to the space if

(1)

where dA(z) is an area measure on D normalized, which makes .

Equipped with the norm , the space is Banach. Generally, is the Bloch space if . If , coincides with , analytic function of bounded mean oscillation. is just the Dirichlet space. Regarding the theory of spaces, readers may refer to Refs. [1-4].

An analytic function belongs to the Bloch space, denoted by , if

The space consists of all bounded analytic functions on D with:

For a subarc , acts as the midpoint of and denotes the Carleson box:

for and for . A positive measure is a -Carleson measure if

where denotes the arc length of . An analytic function if and only if the positive measure is a p-Carleson measure.

The sequence space consists of all complex numbers , so that is a p-Carleson measure, where denotes the unit point-mass measure at and .

A sequence is called an interpolating sequence for if, for each bounded sequence of complex values, there exists an such that for all n.

A sequence in D is separated if .

Usually is an interpolating sequence for if and only if in D is separated and is a Carleson measure for . See Ref.[5] for interpolating sequence in . Readers can refer to Refs.[6,7] about the Hardy and Bergman space theory. See Ref.[8] for more interpolating sequences. Sundberg solved the interpolating question for BMOA in Ref.[9]. A necessary and sufficient condition is obtained for the interpolating sequence in the Bloch space [10]. Pascuas characterized the interpolating sequence in the Bloch space by the p-Carleson measure, as details in Ref. [11]. Ref. [12] gave a characterization of the interpolating sequence for and . The main result is listed as follows:

Theorem 1[12] Let . A sequence of points in the unit disc is an interpolating sequence for if and only if in D is separated and is a p-Carleson measure.

In the following analysis, (for two functions and ) if there is a constant C such that . (that is, is comparable with ) whenever .

1 Interpolating Sequence in

If the sequence is an interpolating sequence for the Bloch space, then

(2)

The aforementioned condition represents the separation criterion. It is subsequently reformulated into the following condition

(3)

where is the constant from the separation condition (2), coinciding with the one used in Ref.[13]. Readers can refer to Refs.[8] and [13] about the separation condition (3). The condition (2) holds, then is separated.

To a point z, a region is defined by

(4)

where denotes the radial projection of and . If the two regions , intersect and , then can be obtained and is outside of . The constants () and are chosen so that and . Those implications of the separation condition can be found in Ref. [13].

This paper chooses a constant so that . The constant will be needed to define the support of the function in the following Lemma 1. The paper constructs a function living essentially in a region , with reasonable estimates of how it behaves for all points.

Lemma 1   Let and . For any given point , there exists a function so that:

satisfies =1, and for points in , the value is estimated by:

Here is defined by and

For points outside of , the bound of is obtained by:

(5)

Further,

(6)

Proof   In this proof, a technique that is borrowed from Ref.[8] has been adapted to enhance its efficiency. The relation defining is as follows

where lives in the annulus

(7)

and is further restricted to a cone with the vertex in and a fixed small aperture. For all others , is taken to be zero. There is the following equation:

where is chosen so that . Observing the defining of , then we get:

The function is to be estimated. Let us suppose that is in . Let:

and

The contributions on and are considered. For any , there is by . Then:

For , after a calculation,

Since

and

then

For a point outside of , holds when belongs to the support of . Thus,

for sufficiently large s can be obtained so that the condition (5) holds. Finally, we can get (6) by the direct calculation.

An operator is defined as follows:

(8)

where is a measurable function on D. The following Lemma 2 is Lemma 3.1.2 in Ref.[2].

Lemma 2   Let . If is a p-Carleson measure on D, then

is also a p-Carleson measure on D.

A finite number of points can be added to an interpolating sequence, rendering it interpolating. This fact is employed in Ref. [13] to derive the implications of the separation condition.

Lemma 3   Let be a sequence in D so that . When (3) holds and , it can be founded by , and so that approximates it in the sense

The coefficients as well as are bounded by . Corresponding to , is the function in Lemma 1.

Proof   Without loss of generality, it is supposed =1. The points are ordered in sequence by their distance to the boundary. To a point , an increasing chain of regions is chosen at each step, selecting the smallest region strictly containing all the previous ones. is defined in Lemma 1, where is given by

The coefficients corresponding are already defined. An induction is assumed that . The coefficient corresponding to is determined by the equation:

Note that

Since is defined in the same way as , can be obtained. The above estimate gives . The asserted properties of is checked. A point is fixed, and a notation is kept as in the construction above. Here:

(9)

The first term of the right of (9) should be considered first. Since

then

By repeatedly applying the separation condition, there comes:

Then

Now, let us look to the second term of the right of (9). If is not in , by Lemma 1 the estimate is made as follows:

If is in , is founded in the chain not contained in so that Since

then . This gives the estimate

The above estimates show that

Note: the last term is finite by the p-Carleson measure condition. The left expression can be made smaller than a by removing finitely many points from the sequence.

Now, let us prove . If z is not contained in , then is easy to get. If z is included in some region , it is assumed that is the smallest, and is bounded by applying the estimates on .

This paper points out that the following result is essential Theorem 1. Here, a new method is used to construct the function sequence such that .

Theorem 2   A sequence is an interpolating sequence for if and only if the condition (3) holds and is a p-Carleson measure. Furthermore, there is an analytic function such that , for any sequence .

Proof   Note the condition (3) holds. By repeatedly applying Lemma 3 we can find a function with satisfying . Since each comes from a as in Lemma 1, is given by and

As a consequence of the separation condition, the support of is disjoint. Also, if the support of intersects a Carleson box , then is contained in . Just for this condition, a relation can be obtained:

Then , by Lemma 2.

Conversely, is a p-Carleson measure by Theorem 1. Since is also an interpolating sequence for Bloch space, the condition (3) holds by (2).

Remark 1   The analytic function in Theorem 1 is not unique. It is assumed that such that for any sequence , causing the following relation,

Since the inner function

by Theorem 5.2.1 in Ref.[1], then by Lemma 1 of Ref.[14]. is defined. Then and .

2 Interpolating Sequence in

For ,,an analytic function belongs to the space if

(10)

We know in this paper that the space is a subspace of by Corollary 2.8 of Ref.[15]. An analytic function if and only if the positive measure is an s-Carleson measure. belongs to a general of function space . See Ref.[15].

A sequence in the unit disc is an interpolating sequence for if each bounded sequence of complex values; there exists an so that for all n. Yuan and Tong gave a characterization of the interpolating sequences for in Ref. [16].

Lemma 4[16] Let Then is an interpolating sequence for if and only if is a separated sequence and is an s-Carleson measure.

The paper obtains a result about the space .

Theorem 3   Let Let be a sequence on D. Then is an interpolating sequence for if and only if the condition (3) holds and is an s-Carleson measure. Futhermore, there is an analytic function such that for any sequence .

The proof of Theorem 3 can be omitted, as it bears similarity to the proof of Theorem 2.

We know in this paper that contains only constant functions if . Qian and Ye gave the following result in Ref. [17].

Lemma 5 [17] Let , max Suppose is a sequence in D. Then is an interpolating sequence for if and only if is a separated sequence and is an s-Carleson measure.

We have a similar result for , where , We omitted the detail analysis.

References

  1. Xiao J. Holomorphic Q Classes[M]. Berlin: Springer-Verlag , 2001. [CrossRef] [Google Scholar]
  2. Xiao J. Geometric Qp Functions[M]. Basel-Boston-Berlin: Birkhäuser Verlag, 2006. [Google Scholar]
  3. Xiao J E. The Qp Carleson measure problem[J]. Advances in Mathematics, 2008, 217(5): 2075-2088. [CrossRef] [MathSciNet] [Google Scholar]
  4. Pau J, Peláez J Á. Multipliers of möbius invariant Qs spaces[J]. Mathematische Zeitschrift, 2009, 261(3): 545-555. [CrossRef] [MathSciNet] [Google Scholar]
  5. Carleson L. An interpolation problem for bounded analytic functions[J]. American Journal of Mathematics, 1958, 80(4): 921. [CrossRef] [MathSciNet] [Google Scholar]
  6. Duren P L. Extension of a theorem of Carleson[J]. Bulletin of the American Mathematical Society, 1969, 75(1): 143-146. [CrossRef] [MathSciNet] [Google Scholar]
  7. Hedenmalm H, Korenblum B, Zhu K H. Theory of Bergman Spaces[M]. New York: Springer-Verlag, 2000. [CrossRef] [Google Scholar]
  8. Böe B. Interpolating sequences for Besov spaces[J]. Journal of Functional Analysis, 2002, 192(2): 319-341. [Google Scholar]
  9. Carl S. Values of BMOA functions on interpolating sequences[J]. Michigan Mathematical Journal, 1984, 31(1): 21-30. [MathSciNet] [Google Scholar]
  10. Bøe B, Nicolau A. Interpolation by functions in the Bloch space[J]. Journal D'Analyse Mathematique, 2004, 94(1): 171-194. [CrossRef] [Google Scholar]
  11. Pascuas D. A note on interpolation by Bloch functions[J]. Proceedings of the American Mathematical Society, 2007, 135(7): 2127-2130. [CrossRef] [MathSciNet] [Google Scholar]
  12. Nicolau A, Xiao J. Bounded functions in Möbius invariant Dirichlet spaces[J]. Journal of Functional Analysis, 1997, 150(2): 383-425. [Google Scholar]
  13. Marshall D E, Sundberg C. Interpolating sequences for the multipliers of the Dirichlet space[EB/OL]. [2010-10-15]. http://www.math.washington.edu/~marshall/preprints/preprints.html. [Google Scholar]
  14. Peláez J Á. Inner functions as improving multipliers[J]. Journal of Functional Analysis, 2008, 255(6): 1403-1418. [Google Scholar]
  15. Zhao R. On a General Family of Function Spaces[D]. Fenn: University of Joensuu, 1996. [Google Scholar]
  16. Yuan C, Tong C Z. On analytic campanato and F(p, q, s) spaces[J]. Complex Analysis and Operator Theory, 2018, 12(8): 1845-1875. [CrossRef] [MathSciNet] [Google Scholar]
  17. Qian R S, Ye F Q. Interpolating sequences for some subsets of analytic Besov type spaces[J]. Journal of Mathematical Analysis and Applications, 2022, 507(2): 125838. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.