A DERIVATIVE-HILBERT OPERATOR ACTING ON HARDY SPACES?

(葉善力) (馮光豪)

School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

E-mail: slye@zust.edu.cn; gh945917454@foxmail.com

Abstract Let μ be a positive Borel measure on the interval [0,1).The Hankel matrix Hμ=(μn,k)n,k≥0 with entries μn,k= μn+k,where μn=∫[0,1)tndμ(t),induces formally the operator aswhereis an analytic function in D.We characterize the positive Borel measures on [0,1) such thatfor all f in the Hardy spaces Hp(0< p < ∞),and among these we describe those for which DHμ is a bounded (resp.,compact) operator from Hp(0< p < ∞) into Hq(q > p and q ≥1).We also study the analogous problem in the Hardy spaces Hp(1 ≤p ≤2).

Key words Derivative-Hilbert operators;Hardy spaces;Carleson measures

1 Introduction

Letμbe a positive Borel measure on the interval [0,1).The Hankel matrix isHμ=(μn,k)n,k≥0,with entriesμn,k=μn+k,whereFor an analytic functionthe generalized Hilbert operatorHμis defined by

whenever the right hand side makes sense and defines an analytic function in D.

In recent decades,the generalized Hilbert operatorHμ,which is induced by the Hankel matrixHμ,has been studied extensively.For example,Galanopoulos and Peláez [12]characterized the Borel measuresμfor which the Hankel operatorHμis a bounded (resp.,compact)operator onH1.Then Chatzifountas,Girela and Peláez [2]extended this work to all Hardy spacesHpwith 0

In 2021,Ye and Zhou[18]first considered the Derivative-Hilbert operatorDHμ,defined by

Another generalized Hilbert-integral operator related toDHμ,denoted byIμα(α ∈N+),is defined by

whenever the right hand side makes sense and defines an analytic function in D.We can easily see that the caseα=1 is the integral representation of the generalized Hilbert operator.Ye and Zhou characterized the measuresμfor whichIμ2andDHμare bounded (resp.,compact)on the Bloch space [18]and on the Bergman spaces [19].

In this paper,we consider the operators

Our aim is to study the boundedness (resp.,compactness) ofIμ2andDHμ.

In this article we characterize the positive Borel measuresμfor which the operatorsIμ2andDHμare well defined in the Hardy spacesHp.Then we give the necessary and sufficient conditions such that the operatorDHμis bounded (resp.,compact) from the Hardy spaceHp(0

2 Notation and Preliminaries

Let D denote the open unit disk of the complex plane,and letH(D) denote the set of all analytic functions in D.

If 0

For 0

We refer to [9]for the notation and results regarding Hardy spaces.

For 0

The Banach spaceBqis the “containing Banach space” ofHq;that is,Hqis a dense subspace ofBq,and the two spaces have the same continuous linear functionals.(We mention [10]as a general reference for theBqspaces.)

The space BMOA consists of those functionsf ∈H1whose boundary values has bounded mean oscillation on?D,as defined by John and Nirenberg.There are many characterizations of BMOA functions.Let us mention the following: fora ∈D,let?abe the M?bius transformation defined byIffis an analytic function in D,thenf ∈BMOA if and only if

where

It is clear that the seminorm||||?is conformally invariant.If

then we say thatfbelongs to the space VMOA (analytic functions of vanishing mean oscillation).We refer to [13]for the theory of BMOA functions.

Finally,we recall that a functionf ∈H(D) is said to be a Bloch function if

The space of all Bloch functions is denoted byB.Classical references for the theory of Bloch functions are [1,15].The relation between the spaces we introduced above is well known:

Let us recall some things about the Carleson measure,which is a very useful tool in the study of the Banach spaces of analytic functions.For 0

for every setS(I) of the form

whereIis an integral of?D and|I| denotes the length ofI.Ifμsatisfieswe say thatμis a vanishings-Carleson measure.

Letμbe a positive Borel measure on D.For 0≤α<∞and 0

Ifμ(S(I))as|I|→0,we say thatμis a vanishingα-logarithmics-Carleson measure [6,17,20].

A positive Borel measure on [0,1) can also be seen as a Borel measure on D by identifying it with the measureμdefined by

for any Borel subsetEof D.Then a positive Borel measureμon [0,1) can be seen as ans-Carleson measure on D if

Also,we have similar statements for vanishings-Carleson measures,α-logarithmics-Carleson measures and vanishingα-logarithmics-Carleson measures.

Throughout this paper,Cdenotes a positive constant which depends only on the displayed parameters,but which is not necessarily the same from one occurrence to the next.For any givenp>1,p′will denote the conjugate index ofp;that is,1/p+1/p′=1.

3 Conditions such that DHμ is well Defined on Hardy Spaces

In this section,we find a sufficient condition such thatDHμare well defined inHp(0

Lemma 3.1([9,p98]) If

thenan=o(n1/p-1),and|an|≤Cn1/p-1||f||Hp.

Lemma 3.2([9,p95]) If

Theorem 3.3Suppose that 0

(i) the measureμis a 1/p-Carleson measure if 0

(ii) the measureμis a 1-Carleson measure if 1

Furthermore,in cases such as these we have that

ProofFirst,recall the following well known results of Carleson [3]and Duren [8](see also [9,Theorem 9.4]): for 0

Thus,if 0

Thenμis a vanishingq/p-Carleson measure if and only if

(i) Suppose that 0

This implies that the integraldμ(t) uniformly converges on any compact subset of D,the resulting function is analytic in D and,for everyz ∈D,

Then it follows that,for everyn,

and so by Lemma 3.2,we deduce that

This implies thatDHμis well defined for allz ∈D and that

This gives thatDHμ(f)=Iμ2(f).

(ii) When 1

Sinceμis 1-Carleson measure,by [2,Theorem 3],we have that

which implies thatDHμis well defined for allz ∈D,and thatDHμ(f)=Iμ2(f).

4 Bounededness of DHμ Acting on Hardy Spaces

In this section,we mainly characterize those measuresμfor whichDHμis a bounded(resp.,compact) operator fromHpintoHqfor somepandq.

Theorem 4.1Suppose that 0

(i)ifq>1,DHμis a bounded operator fromHpintoHqif and only ifμis a(1/p+1/q′+1)-Carleson measure;

(ii) ifq=1,DHμis a bounded operator fromHpintoH1if and only ifμis a (1/p+1)-Carleson measure;

(iii) if 0

ProofSuppose that 0

Hence,it follows that

Using Theorem 3.3,(4.1)and Fubini’s theorem,and Cauchy’s integral representation ofH1[9],we obtain that

(i)First we considerq>1.Using(4.2)and the duality theorem[9],forHqwhich says that(Hp)??Hp′and (Hp′)??Hp(p>1),under the Cauchy pairing we have that

We obtain thatDHμis a bounded operator fromHpintoHqif and only if there exists a positive constantCsuch that

Assume thatDHμis a bounded operator fromHpintoHq.Take the families of the text functions

A calculation shows that{fa}?Hp,{ga}?Hq′and

It follows that

This is equivalent to saying thatμis a (1/p+1/q′+1)-Carleson measure.

On the other hand,suppose thatμis a (1/p+1/q′+1)-Carleson measure.It is well known that any functiong ∈Hq′[9]has the property that

By the Cauchy formula,we can obtain that

Hence (4.4) holds,andDHμis a bounded operator fromHpintoHq.

(ii) We shall use Fefferman’s duality theorem,which says that (H1)??BMOA and(VMOA)??H1,under the Cauchy pairing

Using the duality theorem and (4.2),it follows thatDHμis a bounded operator fromHpintoH1if and only if there exists a positive constantCsuch that

Suppose thatDHμis a bounded operator fromHpintoH1.Take the families of text functions

A calculation shows that{fa}?Hp,{ga}?VMOA and that

We letr ∈[a,1),and obtain that

This is equivalent to saying thatμis a (1/p+1)-Carleson measure.

On the other hand,suppose thatμis a (1/p+1)-Carleson measure.It is well known that any functiong ∈B[1]has the properties that

Hence (4.10) holds,andDHμis a bounded operator fromHpintoH1.

(iii) Set that 0

This,together with (4.2) and (4.14),gives thatDHμis a bounded operator fromHpintoBqif and only if there exists a positive constantCsuch that

Suppose thatμis a (1/p+1)-Carleson measure.Thenis a 1/p-Carleson measure,and we have that

Hence (4.15) holds,andDHμis a bounded operator fromHpintoBq.

Next,we will consider 1

We first give Lemma 4.2,which is useful for the proof the Theorem 4.3.

Lemma 4.2Forγ>0 andα>0,letμbe a positive measure on [0,1).Ifμis a(α+γ)-Carleson measure,then

The result is obvious,so we omit the details.

Theorem 4.3Let 1

(i)ifμis a(1/p+1/q′+1+γ)-Carleson measure for anyγ>0,DHμis a bounded operator fromHpintoHq;

(ii)ifDHμis a bounded operator fromHpintoHq,μis a(1/p+1/q′+1)-Carleson measure.

ProofSuppose thatμis a (1/p+1/q′+1 +γ)-Carleson measure.Letting dν(t)=we have thatνis a (1/p+1/q′+γ)-Carleson measure.Settings=1+p/q′,the conjugate exponent ofsiss′=1+q′/pand 1/p+1/q′=s/p=s′/q′.Then,by [9,Theorem 9.4],Hpis continuously embedded inLs(dν),that is,

and,by Lemma 4.2,

Using H?lder’s inequality with the exponentssands′,and (4.17) and (4.18),we obtain that

Hence,(4.4) holds,and it follows thatDHμis a bounded operator fromHpintoHq.

Conversely,ifDHμis a bounded operator fromHpintoHq,thenμis a (1/p+1/q′+1)-Carleson measure.The proof is the same as that of Theorem 4.1(i),so we omit the details here.

We also findDHμinHp(1≤p ≤2) has a better conclusion.

Theorem 4.4Let 1≤p ≤2,and thatμbe a positive Borel measure on [0,1),which satisfies the condition in Theorem 3.3.ThenDHμis a bounded operator inHpif and only ifμis a 2-Carleson measure.

ProofFirst,ifp=1,by Theorem 4.1 we obtain thatDHμis a bounded operator inH1if and only ifμis a 2-Carleson measure.

Next,ifp=2,then,according to Theorem 4.3,we only need to prove that ifμis a 2-Carleson measure thenDHμis a bounded operator inH2.

By using the classical Hilbert inequality,(1.1),and (4.20),we obtain that

ThusDHμis a bounded operator inH2.

Finally,we shall use complex interpolation to prove our results.We know that

Using (4.22) and Theorem 2.4 of [22],it follows thatDHμis a bounded operator inHp(1≤p ≤2).

Conjecture 4.5We conjecture that ifμis a 2-Carleson measure,thenDHμis a bounded operator inHpfor all 2

5 Compactness of DHμ Acting on Hardy Spaces

In this section we characterize the compactness of the Derivative-HilbertDHμ.We begin with the following lemma,which is useful for dealing with the compactness:

Lemma 5.1For 0

The proof is similar to that proof of [4,Proposition 3.11],so we omit the details.

Theorem 5.2Suppose that 0

(i) ifq>1,DHμis a compact operator fromHpintoHqif and only ifμis a vanishing(1/p+1/q′+1)-Carleson measure;

(ii) ifq=1,DHμis a compact operator fromHpintoH1if and only ifμis a vanishing(1/p+1)-Carleson measure;

(iii) if 0

Proof(i) First,considerq>1.Suppose thatDHμis a compact operator fromHpintoHq.Let{an}?(0,1) be any sequence withan →1.We set that

Thenfan(z)∈Hp,andfan →0,uniformly on any compact subset of D.Using Lemma 5.1,and bearing in mind thatDHμis a compact operator fromHpintoHq,we obtain that{DHμ(fan)} converges to 0 inHq.This,together with(4.2),implies that

It is obvious to find thatg ∈Hq′.For everyn,fixr ∈(an,1).Thus,

By (5.1) and the fact that{an} ?(0,1) is a sequence withan →1,asn →∞,we obtain that

Thusμis a vanishing (1/p+1/q′+1)-Carleson measure.

On the other hand,suppose thatμis a vanishing (1/p+1/q′+1)-Carleson measure.Letbe a sequence ofHpfunctions with,and let{fn}→0 uniformly on any compact subset of D.Then,by Lemma 5.1,it is enough to prove that{DHμ(fn)}→0 inHq.

Takingg ∈Hq′andr ∈[0,1),we obtain that

By way of conclusion,in the proof of the boundedness in Theorem 4.1(i),let dν(t)=We know thatνis a vanishing 1/p-Carleson measure.Then it is implied that

This also tends to 0,by (3.3).Thus,

This means thatDHμ(fn)→0 inHq,and by Lemma 5.1,we obtain thatDHμis a compact operator fromHpintoHq.

(ii)Letq=1.Suppose thatDHμis a compact operator fromHpintoH1.Let{an}?(0,1)be any sequence withan →1,withfandefined as in (i).Lemma 5.1 implies that{DHμ(fan)}converges to 0 inH1.Then we have that

It is well known thatgan ∈VMOA.Forr ∈(an,1),we deduce that

Lettingan →1-asn →∞,we have that

This implies thatμis a vanishing (1/p+1)-Carleson measure.

On the other hand,suppose thatμis a vanishing (1/p+1)-Carleson measure.Letting dν(t)=(1-t)-1dμ(t),we know thatνis a vanishing 1/p-Carleson measure.Letbe a sequence ofHpfunctions withand let{fn} →0 uniformly on any compact subset of D.Then,by Lemma 5.1,it is enough to prove that{DHμ(fn)}→0 inH1.For everyg ∈VMOA,0

This also tends to 0 by (3.3).Thus

This means thatDHμ(fn)→0in H1.By Lemma 5.1,we obtain thatDHμis a compact operator fromHpintoH1.

(iii) The proof is the same as that of Theorems 4.1(iii) and 5.2(i),so we omit the details here.

Finally,we consider the situation ofp>1,characterize those measuresμfor whichDHμis a compact operator fromHpintoHq,and give sufficient and necessary conditions.

Theorem 5.3Let 1

(i)ifμis a vanishing(1/p+1/q′+1+γ)-Carleson measure for anyγ>0,DHμis a compact operator fromHpintoHq;

(ii)ifDHμis a compact operator fromHpintoHq,μis a vanishing(1/p+1/q′+1)-Carleson measure.

Proof(i) The proof is the same as that of Theorem 5.2(i),so we omit the details here.

(ii) The proof is similar to that of Theorems 4.3(ii) and 5.2(i),so we omit the details here.

Similarly,DHμinHp(1≤p ≤2) also has a better conclusion.

Theorem 5.4Let 1≤p ≤2,and letμbe a positive Borel measure on [0,1) which satisfies the condition in Theorem 3.3.ThenDHμis a compact operator inHpif and only ifμis a vanishing 2-Carleson measure.

ProofFirst,lettingp=1,we know thatDHμis a compact operator inH1if and only ifμis a vanishing 2-Carleson measure by Theorem 5.2.

Next,letp=2.According to Theorem 5.3,we only need to prove that ifμis a vanishing 2-Carleson measure,thenDHμis a compact operator inH2.

Assume thatμis a vanishing 2-Carleson measure and let{fj} be a sequence of functions inH2with||fj||H2≤1,for allj,and letfj →0 uniformly on compact subsets of D.Sinceμis a vanishing 2-Carleson measure,

then{εn}→0.If,for everyj,

then,using the classical Hilbert inequality,we have that

Takeε>0 and then take a natural numberNsuch that

We have that

Now,sincefj →0 uniformly on compact subsets of D,it follows that

Then it follows that that there existj0∈Nsuch thatThus,we have proven thatThe compactness ofDHμonH2follows.

We have proven that whenp=1,we have the compactness ofDHμonH1.To deal with the cases 1

We have also that,if 2

for a certainα ∈(0,1),namely,α=(1/2-1/s)/(1-1/s).SinceH2is reflexive,andDHμis compact fromH2into itself and fromH1into itself,Theorem 10 of [5]gives thatDHμis a compact operator inHp(1≤p ≤2).

Conflict of InterestThe authors declare no conflict of interest.