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: š»1=š»1ī‚»(š”»)āˆ¶=š‘“āˆˆHol(š”»)āˆ¶ā€–š‘“ā€–1=sup0<š‘Ÿ<11ī€œ2šœ‹02šœ‹||š‘“ī€·š‘Ÿš‘’š‘–š‘”ī€ø||ī‚¼š‘‘š‘”<āˆž,(1.1) where š”»āˆ¶={š‘§āˆˆā„‚āˆ¶|š‘§|<1} is the unit disc of the complex plain ā„‚.

Theorem A. For š‘Ž function š‘“ analytic in š”», the following assertions are equivalent:(a)š‘“āˆˆš»1; (b)supš‘›(1/š‘Žš‘›)āˆ‘š‘›š‘—=0(1/(š‘—+1))ā€–š‘ š‘—(š‘“)ā€–1<āˆž; (c)supš‘›ā€–š‘ƒš‘›š‘“ā€–1<āˆž.
Here, š‘ƒš‘›š‘“=(1/š‘Žš‘›)āˆ‘š‘›š‘—=0(1/(š‘—+1))š‘ š‘—(š‘“), where š‘Žš‘›=āˆ‘š‘›š‘—=0(1/(š‘—+1))(š‘›=0,1,2,ā€¦) and š‘ š‘—(š‘“) are the partial sums of the Taylor series of š‘“.

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 š»š‘(ā‹…) and Besov class šµš‘(ā‹…) with a variable exponent. Our results are new even for the usual Hardy and Besov spaces š»š‘ and šµš‘.

Recall that the Hardy space š»š‘=š»š‘(š”»)(1ā‰¤š‘<āˆž) is the collection of holomorphic functions in š”» which satisfy the inequalityā€–š‘“ā€–š»š‘ī‚µāˆ¶=sup0<š‘Ÿ<11ī€œ2šœ‹02šœ‹||š‘“ī€·š‘Ÿš‘’š‘–š‘”ī€ø||š‘ī‚¶š‘‘š‘”1/š‘<āˆž.(1.2) Let š‘‘š“(š‘§) be the area measure on š”» normalized so that the area of š”» is 1. In rectangular and polar coordinates,1š‘‘š“(š‘§)=šœ‹1š‘‘š‘„š‘‘š‘¦=šœ‹š‘Ÿš‘‘š‘Ÿš‘‘šœƒ.(1.3) For 1<š‘<+āˆž, the Besov space šµš‘=šµš‘(š”») is defined to be the space of analytic functions š‘“ in such thatā€–š‘“ā€–šµš‘ī‚µī€œāˆ¶=š”»ī€·1āˆ’|š‘§|2ī€øš‘||š‘“ī…ž||(š‘§)š‘ī‚¶š‘‘šœ†(š‘§)1/š‘<āˆž,(1.4) whereš‘‘šœ†(š‘§)=š‘‘š“(š‘§)ī€·1āˆ’|š‘§|2ī€ø2(1.5) is the Mƶbius invariant measure on š”». We refer to Duren [3] and Zhu [4] for the theory of these spaces.

Let š•‹=šœ•š”», and let š‘=š‘(š‘”), š‘”āˆˆš•‹, be a bounded, positive, measurable function defined on it. Following by Kokilashvili and Paatashvili [5, 6], we say that the analytic in the disc š”» function š‘“ belongs to the Hardy class š»š‘(ā‹…) ifsup0<š‘Ÿ<1ī€œ02šœ‹||š‘“ī€·š‘Ÿš‘’š‘–šœƒī€ø||š‘(šœƒ)š‘‘šœƒ=š¶<+āˆž,(1.6) where š‘(šœƒ)=š‘(š‘’š‘–šœƒ), šœƒāˆˆ[0,2šœ‹).

For š‘(šœƒ)=š‘=const>0, the š»š‘(ā‹…) class coincides with the classical Hardy class š»š‘.

Analogously, we say that the analytic in š”» function š‘“ belongs to the Besov class šµš‘(ā‹…) with a variable exponent ifī€œ02šœ‹ī€œ10ī€·1āˆ’š‘Ÿ2ī€øš‘(š‘”)||š‘“ī…žī€·š‘Ÿš‘’š‘–š‘”ī€ø||š‘(š‘”)š‘Ÿš‘‘š‘Ÿš‘‘š‘”šœ‹<+āˆž,(1.7) where š‘(š‘”)=š‘(š‘’š‘–š‘”), š‘”āˆˆ[0,2šœ‹).

For š‘(š‘”)=š‘=const>0, the šµš‘(ā‹…) class coincides with the Besov class šµš‘.

Suppose that š‘āˆ¶=infš‘”āˆˆš•‹š‘(š‘”), š‘āˆ¶=supš‘”āˆˆš•‹š‘(š‘”). If š‘>0, then it is obvious thatš»š‘āŠ‚š»š‘(ā‹…)āŠ‚š»š‘,šµš‘āŠ‚šµš‘(ā‹…)āŠ‚šµš‘.(1.8)

Recall that for any bounded linear operator š“ acting in the functional Hilbert space š»=š»(Ī©) over some set Ī© with reproducing kernel š‘˜šœ†(š‘§), its Berezin symbol ī‚š“ is defined byī‚ī«š“Ģ‚š‘˜š“(šœ†)āˆ¶=šœ†,Ģ‚š‘˜šœ†ī¬(šœ†āˆˆĪ©),(1.9) where Ģ‚š‘˜šœ†āˆ¶=š‘˜šœ†/ā€–š‘˜šœ†ā€– is the normalized reproducing kernel of š». (We mention [4, 7ā€“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 š»š‘(ā‹…) and šµš‘(ā‹…) in terms of Berezin symbols.

For any bounded sequence {š‘Žš‘›}š‘›ā‰„0 of complex numbers š‘Žš‘›, let š·{š‘Žš‘›} denote the associate diagonal operator acting in the Hardy space š»2 by the formulaš·{š‘Žš‘›}š‘§š‘›=š‘Žš‘›š‘§š‘›,š‘›=0,1,2,ā€¦.(2.1) It is known that the reproducing kernel of the Hardy space š»2 has the form š‘˜šœ†(š‘§)=1/(1āˆ’šœ†š‘§)(šœ†āˆˆš”»). Then, it is easy to show that (see [11])ī‚š·{š‘Žš‘›}(ī‚€||šœ†||šœ†)=1āˆ’2ī‚āˆžī“š‘˜=0š‘Žš‘›||šœ†||2š‘˜(šœ†āˆˆš”»),(2.2) that is, the Berezin symbol of the diagonal operator š·{š‘Žš‘›} on the Hardy space š»2 is a radial function.

Note that the inequality |īš‘“(š‘›)|ā‰¤const, š‘›ā‰„0, is the necessary condition for the function āˆ‘š‘“(š‘§)=āˆžš‘›=0īš‘“(š‘›)š‘§š‘› to be in the spaces š»š‘(1ā‰¤š‘ā‰¤āˆž). Also note that if š‘“āˆˆšµš‘, then īš‘“(š‘›)=š‘‚(š‘›āˆ’1/š‘)(š‘ā‰„1) (see, for instance, Duren [3] and Zhu [4]).

Our main result is the following theorem.

Theorem 2.1. Let āˆ‘š‘“(š‘§)=āˆžš‘›=0īš‘“(š‘›)š‘§š‘›āˆˆHol(š”») be a function with the bounded sequence {īš‘“(š‘›)}š‘›ā‰„0 of Taylor coefficients īš‘“(š‘›)=š‘“(š‘›)(0)/š‘›!(š‘›=0,1,2,ā€¦). Then, the following are true:(a)š‘“āˆˆš»š‘(ā‹…) if and only if sup0<š‘Ÿ<11ī€œ2šœ‹02šœ‹|||ī‚š·{īš‘“(š‘›)š‘’int}ī‚€āˆšš‘Ÿī‚|||š‘(š‘”)š‘‘š‘”(1āˆ’š‘Ÿ)š‘(š‘”)<+āˆž;(2.3)(b)if, in addition, īš‘“(š‘›)=š‘‚(š‘›āˆ’1) as š‘›ā†’āˆž, then š‘“āˆˆšµš‘(ā‹…) if and only if ī€œ02šœ‹ī€œ10|||ī‚š·{(š‘›+1)īš‘“(š‘›+1)š‘’int}ī‚€āˆšš‘Ÿī‚|||š‘(š‘”)(1+š‘Ÿ)š‘(š‘”)āˆ’2(1āˆ’š‘Ÿ)2š‘Ÿš‘‘š‘Ÿš‘‘š‘”<+āˆž.(2.4)

Proof. Indeed, by using the concept of Berezin symbols and formula (2.2), let us rewrite the function āˆ‘š‘“(š‘§)=āˆžš‘›=0īš‘“(š‘›)š‘§š‘›āˆˆHol(š”») as follows: ī€·š‘“(š‘§)=š‘“š‘Ÿš‘’š‘–š‘”ī€ø=āˆžī“š‘›=0īī€·š‘“(š‘›)š‘Ÿš‘’š‘–š‘”ī€øš‘›=āˆžī“š‘›=0īš‘“(š‘›)š‘’intš‘Ÿš‘›=āˆ‘(1āˆ’š‘Ÿ)āˆžš‘›=0īš‘“(š‘›)š‘’intš‘Ÿš‘›=ī‚š·1āˆ’š‘Ÿ{īš‘“(š‘›)š‘’int}ī‚€āˆšš‘Ÿī‚,1āˆ’š‘Ÿ(2.5) thus š‘“ī‚š·(š‘§)={īš‘“(š‘›)š‘’int}ī‚€āˆšš‘Ÿī‚1āˆ’š‘Ÿ(2.6) for every š‘§=š‘Ÿš‘’š‘–š‘”āˆˆš”», where, as usual, š‘Ÿ=|š‘§| and š‘”=arg(š‘§). 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 īš‘“(š‘›)=š‘‚(š‘›āˆ’1) that the diagonal operator š·{š‘›īš‘“(š‘›)} is bounded in š»2 (and hence š·{(š‘›+1)īš‘“(š‘›+1)š‘’int} is bounded for every fixed š‘”āˆˆ[0,2šœ‹)). Then, we have š‘“ī…žīƒ©(š‘§)=āˆžī“š‘›=0īš‘“(š‘›)š‘§š‘›īƒŖī…ž=āˆžī“š‘›=1š‘›īš‘“(š‘›)š‘§š‘›āˆ’1=āˆžī“š‘›=0ī(š‘›+1)š‘“(š‘›+1)š‘§š‘›=āˆžī“š‘›=0ī(š‘›+1)š‘“(š‘›+1)š‘’intš‘Ÿš‘›=āˆ‘(1āˆ’š‘Ÿ)āˆžš‘›=0ī(š‘›+1)š‘“(š‘›+1)š‘’intš‘Ÿš‘›=ī‚š·1āˆ’š‘Ÿ{(š‘›+1)īš‘“(š‘›+1)š‘’int}ī‚€āˆšš‘Ÿī‚,1āˆ’š‘Ÿ(2.7) thus š‘“ī…žī‚š·(š‘§)={(š‘›+1)īš‘“(š‘›+1)š‘’int}ī‚€āˆšš‘Ÿī‚1āˆ’š‘Ÿ.(2.8) Therefore, by using formula (2.8), we have that ī€œš”»ī€·1āˆ’|š‘§|2ī€øš‘(š‘”)||š‘“ī…ž(||š‘§)š‘(š‘”)š‘‘š“(š‘§)ī€·1āˆ’|š‘§|2ī€ø2<+āˆž(2.9) if and only if ī€œ02šœ‹ī€œ10|||ī‚š·{(š‘›+1)īš‘“(š‘›+1)š‘’int}ī‚€āˆšš‘Ÿī‚|||š‘(š‘”)ī€·1āˆ’š‘Ÿ2ī€øš‘(š‘”)(1āˆ’š‘Ÿ)š‘(š‘”)š‘Ÿī€·1āˆ’š‘Ÿ2ī€ø2š‘‘š‘Ÿš‘‘š‘”<+āˆž,(2.10) that is, ī€œ02šœ‹ī€œ10|||ī‚š·{(š‘›+1)īš‘“(š‘›+1)š‘’int}ī‚€āˆšš‘Ÿī‚|||š‘(š‘”)(1+š‘Ÿ)š‘(š‘”)āˆ’2(1āˆ’š‘Ÿ)2š‘Ÿš‘‘š‘Ÿš‘‘š‘”<+āˆž,(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 š»š‘ function can be recognized by the behavior of its Taylor coefficients?