A DERIVATIVE-HILBERT OPERATOR ACTING ON HARDY SPACES∗

(叶善力) (冯光豪)

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,letϕabe the Möbius 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 Hölder’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.