A Characterization of Some Function Classes

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=sup_0<𝑟<112𝜋^02𝜋||𝑓𝑟𝑒^𝑖𝑡||𝑑𝑡<∞,(1.1) where 𝔻∶={𝑧∈ℂ∶|𝑧|<1} is the unit disc of the complex plain ℂ.

Theorem A. For 𝑎 function 𝑓 analytic in 𝔻, the following assertions are equivalent:(a)𝑓∈𝐻¹; (b)sup_𝑛(1/𝑎_𝑛)∑^𝑛𝑗=0(1/(𝑗+1))‖𝑠_𝑗(𝑓)‖₁<∞; (c)sup_𝑛‖𝑃_𝑛𝑓‖₁<∞.
Here, 𝑃_𝑛𝑓=(1/𝑎_𝑛)∑^𝑛𝑗=0(1/(𝑗+1))𝑠_𝑗(𝑓), where 𝑎_𝑛=∑^𝑛𝑗=0(1/(𝑗+1))(𝑛=0,1,2,…) and 𝑠_𝑗(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 𝐻^𝑝(⋅) and Besov class 𝐵_𝑝(⋅) with a variable exponent. Our results are new even for the usual Hardy and Besov spaces 𝐻^𝑝 and 𝐵_𝑝.

Recall that the Hardy space 𝐻^𝑝=𝐻^𝑝(𝔻)(1≤𝑝<∞) is the collection of holomorphic functions in 𝔻 which satisfy the inequality‖𝑓‖_𝐻^𝑝∶=sup_0<𝑟<112𝜋^02𝜋||𝑓𝑟𝑒^𝑖𝑡||^𝑝𝑑𝑡^1/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,𝑑𝐴(𝑧)=1𝜋𝑑𝑥𝑑𝑦=𝜋1𝑟𝑑𝑟𝑑𝜃.(1.3) For 1<𝑝<+∞, the Besov space 𝐵_𝑝=𝐵_𝑝(𝔻) is defined to be the space of analytic functions 𝑓 in such that‖𝑓‖_𝐵𝑝∶=_𝔻1−|𝑧|²^𝑝||𝑓^(𝑧)||^𝑝𝑑𝜆(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 𝐻^𝑝(⋅) ifsup_0<𝑟<1^02𝜋||𝑓𝑟𝑒^𝑖𝜃||^𝑝(𝜃)𝑑𝜃=𝐶<+∞,(1.6) where 𝑝(𝜃)=𝑝(𝑒^𝑖𝜃), 𝜃∈[0,2𝜋).

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

Analogously, we say that the analytic in 𝔻 function 𝑓 belongs to the Besov class 𝐵_𝑝(⋅) with a variable exponent if^02𝜋¹⁰1−𝑟²^𝑝(𝑡)||𝑓^𝑟𝑒^𝑖𝑡||^𝑝(𝑡)𝑟𝑑𝑟𝑑𝑡𝜋<+∞,(1.7) where 𝑝(𝑡)=𝑝(𝑒^𝑖𝑡), 𝑡∈[0,2𝜋).

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

Suppose that 𝑝∶=inf_𝑡∈𝕋𝑝(𝑡), 𝑝∶=sup_𝑡∈𝕋𝑝(𝑡). If 𝑝>0, then it is obvious that𝐻𝐵^𝑝_𝑝⊂𝐻^𝑝(⋅)⊂𝐻^𝑝,⊂𝐵_𝑝(⋅)⊂𝐵_𝑝.(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 𝑘_𝜆(𝑧), its Berezin symbol 𝐴 is defined by𝐴(𝜆)∶=𝐴𝑘̂_𝜆,𝑘̂_𝜆(𝜆∈Ω),(1.9) where 𝑘̂_𝜆∶=𝑘_𝜆/‖𝑘_𝜆‖ 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 𝐻^𝑝(⋅) and 𝐵_𝑝(⋅) in terms of Berezin symbols.

For any bounded sequence {𝑎_𝑛}_𝑛≥0 of complex numbers 𝑎_𝑛, let 𝐷_{𝑎𝑛} denote the associate diagonal operator acting in the Hardy space 𝐻² by the formula𝐷_{𝑎𝑛}𝑧^𝑛=𝑎_𝑛𝑧^𝑛,𝑛=0,1,2,….(2.1) It is known that the reproducing kernel of the Hardy space 𝐻² has the form 𝑘_𝜆(𝑧)=1/(1−𝜆𝑧)(𝜆∈𝔻). Then, it is easy to show that (see [11])𝐷_{𝑎𝑛}(𝜆)=1−||𝜆||²^∞_𝑘=0𝑎_𝑛||𝜆||^2𝑘(𝜆∈𝔻),(2.2) that is, the Berezin symbol of the diagonal operator 𝐷_{𝑎𝑛} 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)=∑^∞𝑛=0𝑓(𝑛)𝑧^𝑛 to be in the spaces 𝐻^𝑝(1≤𝑝≤∞). Also note that if 𝑓∈𝐵_𝑝, 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)=∑^∞𝑛=0𝑓(𝑛)𝑧^𝑛∈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∈𝐻^𝑝(⋅) if and only if sup_0<𝑟<112𝜋^02𝜋|||𝐷_{𝑓_{(𝑛)𝑒int}}√𝑟|||^𝑝(𝑡)𝑑𝑡(1−𝑟)^𝑝(𝑡)<+∞;(2.3)(b)if, in addition, 𝑓(𝑛)=𝑂(𝑛⁻¹) as 𝑛→∞, then 𝑓∈𝐵_𝑝(⋅) 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)=∑^∞𝑛=0𝑓(𝑛)𝑧^𝑛∈Hol(𝔻) as follows: 𝑓(𝑧)=𝑓𝑟𝑒^𝑖𝑡=^∞_𝑛=0𝑓(𝑛)𝑟𝑒^𝑖𝑡^𝑛=^∞_𝑛=0𝑓(𝑛)𝑒^int𝑟^𝑛=(1−𝑟)∑^∞𝑛=0𝑓(𝑛)𝑒^int𝑟^𝑛1−𝑟=𝐷^{𝑓_{(𝑛)𝑒int}}√𝑟1−𝑟,(2.5) thus 𝑓(𝑧)=𝐷^{𝑓_{(𝑛)𝑒int}}√𝑟1−𝑟(2.6) for every 𝑧=𝑟𝑒^𝑖𝑡∈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_{𝑛𝑓_(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 𝑓^(𝑧)=^∞_𝑛=0𝑓(𝑛)𝑧^𝑛^=^∞_𝑛=1𝑛𝑓(𝑛)𝑧^𝑛−1=^∞_𝑛=0(𝑛+1)𝑓(𝑛+1)𝑧^𝑛=^∞_𝑛=0(𝑛+1)𝑓(𝑛+1)𝑒^int𝑟^𝑛=(1−𝑟)∑^∞𝑛=0(𝑛+1)𝑓(𝑛+1)𝑒^int𝑟^𝑛=𝐷1−𝑟_{(𝑛+1)𝑓_{(𝑛+1)𝑒int}}√𝑟1−𝑟,(2.7) thus 𝑓^(𝑧)=𝐷^{(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)||𝑓^(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 𝐻^𝑝 function can be recognized by the behavior of its Taylor coefficients?