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<112π∫02π|f(reit)|dt<∞},
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:

f∈H1;

supn(1/an)∑j=0n(1/(j+1))∥sj(f)∥1<∞;

supn∥Pnf∥1<∞.

Here, Pnf=(1/an)∑j=0n(1/(j+1))sj(f), where an=∑j=0n(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<112π∫02π|f(reit)|pdt)1/p<∞.
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=1πrdrdθ.
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|f′(z)|pdλ(z))1/p<∞,
wheredλ(z)=dA(z)(1-|z|2)2
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<+∞,
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∫01(1-r2)p(t)|f′(reit)|p(t)rdrdtπ<+∞,
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 p̲:=inft∈𝕋p(t), p¯:=supt∈𝕋p(t). If p̲>0, then it is obvious thatHp¯⊂Hp(⋅)⊂Hp̲,Bp¯⊂Bp(⋅)⊂Bp̲.

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 Ã is defined byÃ(λ):=〈Ak̂λ,k̂λ〉(λ∈Ω),
where k̂λ:=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 {an}n≥0 of complex numbers an, let D{an} denote the associate diagonal operator acting in the Hardy space H2 by the formulaD{an}zn=anzn,n=0,1,2,….
It is known that the reproducing kernel of the Hardy space H2 has the form kλ(z)=1/(1-λ¯z)(λ∈𝔻). Then, it is easy to show that (see [11])D̃{an}(λ)=(1-|λ|2)∑k=0∞an|λ|2k(λ∈D),
that is, the Berezin symbol of the diagonal operator D{an} on the Hardy space H2 is a radial function.

Note that the inequality |f̂(n)|≤const, n≥0, is the necessary condition for the function f(z)=∑n=0∞f̂(n)zn to be in the spaces Hp(1≤p≤∞). Also note that if f∈Bp, then f̂(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=0∞f̂(n)zn∈Hol(𝔻) be a function with the bounded sequence {f̂(n)}n≥0 of Taylor coefficients f̂(n)=f(0)(n)/n!(n=0,1,2,…). Then, the following are true:

f∈Hp(·) if and only if
sup0<r<112π∫02π|D̃{f̂(n)eint}(r)|p(t)dt(1-r)p(t)<+∞;

if, in addition, f̂(n)=O(n-1) as n→∞, then f∈Bp(·) if and only if
∫02π∫01|D̃{(n+1)f̂(n+1)eint}(r)|p(t)(1+r)p(t)-2(1-r)2rdrdt<+∞.

Proof.

Indeed, by using the concept of Berezin symbols and formula (2.2), let us rewrite the function f(z)=∑n=0∞f̂(n)zn∈Hol(𝔻) as follows:
f(z)=f(reit)=∑n=0∞f̂(n)(reit)n=∑n=0∞f̂(n)eintrn=(1-r)∑n=0∞f̂(n)eintrn1-r=D̃{f̂(n)eint}(r)1-r,
thus
f(z)=D̃{f̂(n)eint}(r)1-r
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 f̂(n)=O(n-1) that the diagonal operator D{nf̂(n)} is bounded in H2 (and hence D{(n+1)f̂(n+1)eint} is bounded for every fixed t∈[0,2π)). Then, we have
f′(z)=(∑n=0∞f̂(n)zn)′=∑n=1∞nf̂(n)zn-1=∑n=0∞(n+1)f̂(n+1)zn=∑n=0∞(n+1)f̂(n+1)eintrn=(1-r)∑n=0∞(n+1)f̂(n+1)eintrn1-r=D̃{(n+1)f̂(n+1)eint}(r)1-r,
thus
f′(z)=D̃{(n+1)f̂(n+1)eint}(r)1-r.
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<+∞
if and only if
∫02π∫01|D̃{(n+1)f̂(n+1)eint}(r)|p(t)(1-r2)p(t)(1-r)p(t)r(1-r2)2drdt<+∞,
that is,
∫02π∫01|D̃{(n+1)f̂(n+1)eint}(r)|p(t)(1+r)p(t)-2(1-r)2rdrdt<+∞,
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?

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