Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 796798 | 5 pages | https://doi.org/10.1155/2012/796798

A Characterization of Some Function Classes

Academic Editor: Nicolae Popa
Received12 Feb 2009
Accepted02 Dec 2011
Published03 Jan 2012

Abstract

We give in terms of Berezin symbols a characterization of Hardy and Besov classes with a variable exponent.

1. Introduction and Notations

In his book [1, page 96], Pavlović proved the following characterization of functions belonging to the classical Hardy space: 𝐻1=𝐻1(𝔻)∶=𝑓∈Hol(𝔻)∶‖𝑓‖1=sup0<𝑟<112𝜋02𝜋||𝑓𝑟𝑒𝑖𝑡||𝑑𝑡<∞,(1.1) where 𝔻∶={𝑧∈ℂ∶|𝑧|<1} is the unit disc of the complex plain ℂ.

Theorem A. For ğ‘Ž function 𝑓 analytic in 𝔻, the following assertions are equivalent:(a)𝑓∈𝐻1; (b)sup𝑛(1/ğ‘Žğ‘›)∑𝑛𝑗=0(1/(𝑗+1))‖𝑠𝑗(𝑓)‖1<∞; (c)sup𝑛‖𝑃𝑛𝑓‖1<∞.
Here, 𝑃𝑛𝑓=(1/ğ‘Žğ‘›)∑𝑛𝑗=0(1/(𝑗+1))𝑠𝑗(𝑓), where ğ‘Žğ‘›=∑𝑛𝑗=0(1/(𝑗+1))(𝑛=0,1,2,…) and 𝑠𝑗(𝑓) are the partial sums of the Taylor series of 𝑓.

Recently, Popa [2] gave some generalization of this result by proving a similar characterization of upper triangular trace class matrices.

In the present paper, we give in terms of the so-called Berezin symbols a new characterization of analytic functions belonging to the Hardy class 𝐻𝑝(⋅) and Besov class 𝐵𝑝(⋅) with a variable exponent. Our results are new even for the usual Hardy and Besov spaces 𝐻𝑝 and 𝐵𝑝.

Recall that the Hardy space 𝐻𝑝=𝐻𝑝(𝔻)(1≤𝑝<∞) is the collection of holomorphic functions in 𝔻 which satisfy the inequality‖𝑓‖𝐻𝑝∶=sup0<𝑟<112𝜋02𝜋||𝑓𝑟𝑒𝑖𝑡||𝑝𝑑𝑡1/𝑝<∞.(1.2) Let 𝑑𝐴(𝑧) be the area measure on 𝔻 normalized so that the area of 𝔻 is 1. In rectangular and polar coordinates,1𝑑𝐴(𝑧)=𝜋1𝑑𝑥𝑑𝑦=𝜋𝑟𝑑𝑟𝑑𝜃.(1.3) For 1<𝑝<+∞, the Besov space 𝐵𝑝=𝐵𝑝(𝔻) is defined to be the space of analytic functions 𝑓 in such that‖𝑓‖𝐵𝑝∶=𝔻1−|𝑧|2𝑝||ğ‘“î…ž||(𝑧)𝑝𝑑𝜆(𝑧)1/𝑝<∞,(1.4) where𝑑𝜆(𝑧)=𝑑𝐴(𝑧)1−|𝑧|22(1.5) is the Möbius invariant measure on 𝔻. We refer to Duren [3] and Zhu [4] for the theory of these spaces.

Let 𝕋=𝜕𝔻, and let 𝑝=𝑝(𝑡), 𝑡∈𝕋, be a bounded, positive, measurable function defined on it. Following by Kokilashvili and Paatashvili [5, 6], we say that the analytic in the disc 𝔻 function 𝑓 belongs to the Hardy class 𝐻𝑝(⋅) ifsup0<𝑟<102𝜋||𝑓𝑟𝑒𝑖𝜃||𝑝(𝜃)𝑑𝜃=𝐶<+∞,(1.6) where 𝑝(𝜃)=𝑝(𝑒𝑖𝜃), 𝜃∈[0,2𝜋).

For 𝑝(𝜃)=𝑝=const>0, the 𝐻𝑝(⋅) class coincides with the classical Hardy class 𝐻𝑝.

Analogously, we say that the analytic in 𝔻 function 𝑓 belongs to the Besov class 𝐵𝑝(⋅) with a variable exponent if02𝜋101−𝑟2𝑝(𝑡)||ğ‘“î…žî€·ğ‘Ÿğ‘’ğ‘–ğ‘¡î€¸||𝑝(𝑡)𝑟𝑑𝑟𝑑𝑡𝜋<+∞,(1.7) where 𝑝(𝑡)=𝑝(𝑒𝑖𝑡), 𝑡∈[0,2𝜋).

For 𝑝(𝑡)=𝑝=const>0, the 𝐵𝑝(⋅) class coincides with the Besov class 𝐵𝑝.

Suppose that 𝑝∶=inf𝑡∈𝕋𝑝(𝑡), 𝑝∶=sup𝑡∈𝕋𝑝(𝑡). If 𝑝>0, then it is obvious that𝐻𝑝⊂𝐻𝑝(⋅)⊂𝐻𝑝,𝐵𝑝⊂𝐵𝑝(⋅)⊂𝐵𝑝.(1.8)

Recall that for any bounded linear operator 𝐴 acting in the functional Hilbert space 𝐻=𝐻(Ω) over some set Ω with reproducing kernel 𝑘𝜆(𝑧), its Berezin symbol 𝐴 is defined by𝐴̂𝑘𝐴(𝜆)∶=𝜆,̂𝑘𝜆(𝜆∈Ω),(1.9) where ̂𝑘𝜆∶=𝑘𝜆/‖𝑘𝜆‖ is the normalized reproducing kernel of 𝐻. (We mention [4, 7–11] as references for the Berezin symbols.)

2. On the Membership of Functions in Hardy and Besov Classes with a Variable Exponent

In this section, we characterize the function classes 𝐻𝑝(⋅) and 𝐵𝑝(⋅) in terms of Berezin symbols.

For any bounded sequence {ğ‘Žğ‘›}𝑛≥0 of complex numbers ğ‘Žğ‘›, let 𝐷{ğ‘Žğ‘›} denote the associate diagonal operator acting in the Hardy space 𝐻2 by the formula𝐷{ğ‘Žğ‘›}𝑧𝑛=ğ‘Žğ‘›ğ‘§ğ‘›,𝑛=0,1,2,….(2.1) It is known that the reproducing kernel of the Hardy space 𝐻2 has the form 𝑘𝜆(𝑧)=1/(1−𝜆𝑧)(𝜆∈𝔻). Then, it is easy to show that (see [11])𝐷{ğ‘Žğ‘›}(||𝜆||𝜆)=1−2î‚âˆžî“ğ‘˜=0ğ‘Žğ‘›||𝜆||2𝑘(𝜆∈𝔻),(2.2) that is, the Berezin symbol of the diagonal operator 𝐷{ğ‘Žğ‘›} on the Hardy space 𝐻2 is a radial function.

Note that the inequality |𝑓(𝑛)|≤const, 𝑛≥0, is the necessary condition for the function ∑𝑓(𝑧)=âˆžğ‘›=0𝑓(𝑛)𝑧𝑛 to be in the spaces 𝐻𝑝(1â‰¤ğ‘â‰¤âˆž). Also note that if 𝑓∈𝐵𝑝, then 𝑓(𝑛)=𝑂(𝑛−1/𝑝)(𝑝≥1) (see, for instance, Duren [3] and Zhu [4]).

Our main result is the following theorem.

Theorem 2.1. Let ∑𝑓(𝑧)=âˆžğ‘›=0𝑓(𝑛)𝑧𝑛∈Hol(𝔻) be a function with the bounded sequence {𝑓(𝑛)}𝑛≥0 of Taylor coefficients 𝑓(𝑛)=𝑓(𝑛)(0)/𝑛!(𝑛=0,1,2,…). Then, the following are true:(a)𝑓∈𝐻𝑝(⋅) if and only if sup0<𝑟<112𝜋02𝜋|||𝐷{𝑓(𝑛)𝑒int}√𝑟|||𝑝(𝑡)𝑑𝑡(1−𝑟)𝑝(𝑡)<+∞;(2.3)(b)if, in addition, 𝑓(𝑛)=𝑂(𝑛−1) as ğ‘›â†’âˆž, then 𝑓∈𝐵𝑝(⋅) if and only if 02𝜋10|||𝐷{(𝑛+1)𝑓(𝑛+1)𝑒int}√𝑟|||𝑝(𝑡)(1+𝑟)𝑝(𝑡)−2(1−𝑟)2𝑟𝑑𝑟𝑑𝑡<+∞.(2.4)

Proof. Indeed, by using the concept of Berezin symbols and formula (2.2), let us rewrite the function ∑𝑓(𝑧)=âˆžğ‘›=0𝑓(𝑛)𝑧𝑛∈Hol(𝔻) as follows: 𝑓(𝑧)=𝑓𝑟𝑒𝑖𝑡=âˆžî“ğ‘›=0𝑓(𝑛)𝑟𝑒𝑖𝑡𝑛=âˆžî“ğ‘›=0𝑓(𝑛)𝑒int𝑟𝑛=∑(1−𝑟)âˆžğ‘›=0𝑓(𝑛)𝑒int𝑟𝑛=𝐷1−𝑟{𝑓(𝑛)𝑒int}√𝑟,1−𝑟(2.5) thus 𝑓𝐷(𝑧)={𝑓(𝑛)𝑒int}√𝑟1−𝑟(2.6) for every 𝑧=𝑟𝑒𝑖𝑡∈𝔻, where, as usual, 𝑟=|𝑧| and 𝑡=arg(𝑧). Now, assertion (a) is immediate from the definition of considering space and formula (2.6).
Let us prove (b). Indeed, it follows from the condition 𝑓(𝑛)=𝑂(𝑛−1) that the diagonal operator 𝐷{𝑛𝑓(𝑛)} is bounded in 𝐻2 (and hence 𝐷{(𝑛+1)𝑓(𝑛+1)𝑒int} is bounded for every fixed 𝑡∈[0,2𝜋)). Then, we have ğ‘“î…žîƒ©(𝑧)=âˆžî“ğ‘›=0𝑓(𝑛)ğ‘§ğ‘›îƒªî…ž=âˆžî“ğ‘›=1𝑛𝑓(𝑛)𝑧𝑛−1=âˆžî“ğ‘›=0(𝑛+1)𝑓(𝑛+1)𝑧𝑛=âˆžî“ğ‘›=0(𝑛+1)𝑓(𝑛+1)𝑒int𝑟𝑛=∑(1−𝑟)âˆžğ‘›=0(𝑛+1)𝑓(𝑛+1)𝑒int𝑟𝑛=𝐷1−𝑟{(𝑛+1)𝑓(𝑛+1)𝑒int}√𝑟,1−𝑟(2.7) thus ğ‘“î…žî‚ğ·(𝑧)={(𝑛+1)𝑓(𝑛+1)𝑒int}√𝑟1−𝑟.(2.8) Therefore, by using formula (2.8), we have that 𝔻1−|𝑧|2𝑝(𝑡)||ğ‘“î…ž(||𝑧)𝑝(𝑡)𝑑𝐴(𝑧)1−|𝑧|22<+∞(2.9) if and only if 02𝜋10|||𝐷{(𝑛+1)𝑓(𝑛+1)𝑒int}√𝑟|||𝑝(𝑡)1−𝑟2𝑝(𝑡)(1−𝑟)𝑝(𝑡)𝑟1−𝑟22𝑑𝑟𝑑𝑡<+∞,(2.10) that is, 02𝜋10|||𝐷{(𝑛+1)𝑓(𝑛+1)𝑒int}√𝑟|||𝑝(𝑡)(1+𝑟)𝑝(𝑡)−2(1−𝑟)2𝑟𝑑𝑟𝑑𝑡<+∞,(2.11) as desired. The theorem is proved.

We remark that, in case of classical Hardy space, Theorem 2.1 shed some light on the following old problem for the Hardy space functions (see Privalov [12] and Duren [3]): how an 𝐻𝑝 function can be recognized by the behavior of its Taylor coefficients?

References

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Copyright © 2012 M. T. Karaev. 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.


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