## Abstract

In this paper, a general HELP (Hardy–Everitt–Littlewood–Pólya) inequality is considered which is connected with a symmetric operator in a Hilbert space and abstract boundary mappings. A criterion for the validity of such an inequality in terms of the abstract Titchmarsh–Weyl function is proved and applied to Sturm–Liouville operators, difference operators, a Hamiltonian system and a block operator matrix.

## 1. Introduction

In 1932, Hardy and Littlewood proved the following famous inequality (cf. Hardy & Littlewood (1932) and also the book Hardy *et al*. (1934), which contains three different proofs):for *f*∈*L*^{2}(0,∞) such that *f* and *f*′ are absolutely continuous and *f*″∈*L*^{2}(0,∞). Everitt (1972) studied the more general inequality(1.1)which is connected with the Sturm–Liouville equation −(*pf*′)′+*qf*=*λwf*. There, it was shown that the existence of a constant *K* such that (1.1) holds is equivalent to the validity of the inequality(1.2)in a certain sector in the complex plane, where *M*(*λ*) is the Titchmarsh–Weyl coefficient of the Sturm–Liouville equation with Neumann boundary condition at 0. Later analogous inequalities connected with Hamiltonian systems and higher-order differential equations were investigated (cf. Dias 1994; Brown & Dias 1997; Brown *et al*. 1999; Brown & Marletta 2000).

Copson (1979) proved a discrete analogue of the Hardy–Littlewood inequalityfor , where Δ*x*_{n}=*x*_{n+1}−*x*_{n}. Brown, Evans and Littlejohn generalized this to inequalities that are connected with second-order difference equations (Brown & Evans 1992; Brown *et al*. 1992, 1993).

In the present paper, we prove a criterion for the validity of a general inequality in an abstract operator theory setting using an abstract Titchmarsh–Weyl function. This generalizes the results mentioned above and can be used to derive new inequalities. Let *S* be a symmetric operator in a Hilbert space with equal deficiency indices. In general, *S* need not be densely defined, in which case *S*^{*} is not an operator but a relation, i.e. multivalued. However, in the introduction we restrict ourselves to the case of a densely defined operator. Let a pair of boundary mappings be given, that is, mappings *Γ*_{0}, *Γ*_{1} from the domain of *S*^{*} into another Hilbert space, , which satisfy the following abstract Lagrange or Green identityfor all *f*, *g* in the domain of *S*^{*}, where (.,.) and 〈.,.〉 denote the inner products in and , respectively. For differential operators such boundary mappings are naturally given, e.g.with for the Sturm–Liouville operator on the half-lineThe right-hand side of (1.1) is *K*‖*f*‖^{2}‖*S*^{*}*f*‖^{2}, where ‖.‖ denotes the norm in the weighted space . Using the boundary mappings and integration by parts we can rewrite the left-hand side as(1.3)which will serve as an abstract definition of a generalized Dirichlet form *D*[*f*]. For an arbitrary symmetric operator with boundary mappings *Γ*_{0}, *Γ*_{1}, the main theorem (theorem 3.1) establishes an equivalence of the existence of a constant *K* such thatholds for all *f* in the domain of *S*^{*} and the validity of inequality (1.2) in a certain sector, where *M* is replaced by the abstract Titchmarsh–Weyl function connected with *S* and the boundary mappings *Γ*_{0}, *Γ*_{1}. The size of the sector determines the best constant *K*.

In §2, we recall some facts about linear relations and abstract boundary mappings. Moreover, an abstract Titchmarsh–Weyl function that is connected with boundary mappings is introduced. The main theorem (theorem 3.1) and its proof is the contents of §3. In §4, some sufficient conditions are derived for the validity of the HELP inequality. Finally, in §5 several examples are considered, in particular the Sturm–Liouville operator, a difference operator, a block operator matrix (see corollary 5.4), and a Hamiltonian system (see corollary 5.5); the latter two yield new inequalities involving two functions on the half-line.

## 2. Boundary mappings for linear relations and the Titchmarsh–Weyl function

In this section, we recall the notion of boundary mappings for linear relations as it is contained, e.g. in Derkach & Malamud (1995). Most of the propositions below are in that paper, but for the convenience of the reader we present proofs. First, we recall some facts about closed, symmetric linear relations (multivalued operators) in a Hilbert space, cf., e.g. Coddington (1973).

Let be a Hilbert space with scalar product (.,.). A closed linear relation in is a closed subspace of , where we write {*f*; *g*} for elements in . A closed operator *T* can be identified with its graph and is, therefore, a closed linear relation with the property that {0; *f*}∈*T* implies *f*=0. The domain, the range and the kernel of a relation *T* are, respectively, defined byThe inverse and the adjoint relations are given byand the sum of two linear relations *T*_{1} and *T*_{2} is given byA closed linear relation *S* is called symmetric if *S*⊂*S*^{*} and self-adjoint if *S*=*S*^{*}.

Let *S* be a closed symmetric relation. Then, we define the deficiency spaces bywhere *P*_{1} is the projection onto the first component in . It is well known that is constant on the upper and the lower half-planes . The deficiency indices *n*_{+}, *n*_{−} are defined by .

The triple is called a *boundary triple* for a closed symmetric relation *S*, if is a Hilbert space with scalar product 〈.,.〉 and are linear, continuous mappings such that(2.1)and the mapping from into is surjective. The *Γ*_{i} are called *boundary mappings*.

There exists a boundary triple for a symmetric relation *S* if, and only if, the deficiency indices of *S* are equal (cf. Derkach & Malamud 1995, section 1). It can be shown that all self-adjoint extensions of *S* in are given by for some self-adjoint relation *θ* in . Here, we only prove that (which corresponds to ) is self-adjoint.

*Let S be a closed symmetric relation and* *a boundary triple*. *Then S=*ker *Γ*_{0}∩ker *Γ*_{1}. *Moreover, the relation A≔*ker *Γ*_{0} *is self-adjoint*.

It follows from (2.1) that *S*=*S*^{**}⊃ker *Γ*_{1}∩ker *Γ*_{0}. Let . Then, for every . Hence, . Since is surjective, we can choose such that and is arbitrary in , which implies . Similarly, one gets .

Now we show that *A* is self-adjoint. It follows from (2.1) that *A* is symmetric. As is surjective, the mapping *Γ*_{1}↾*A* is also surjective. Let . Similar as above it follows thatfor all . Since, *Γ*_{1} is surjective on *A*, we obtain , i.e. , which shows *A*=*A*^{*}. ▪

Since ( denotes the direct sum of subspaces of ) for , the mapping is bijective from onto . Hence, we can definewhere again *P*_{1} denotes the projection onto the first component in . The *Titchmarsh–Weyl function M* is now defined by(2.2)for , which is a bijective operator in . Note thatfor all . Moreover, the following symmetry relation holds:(2.3)which follows fromfor .

We see in §5*a* that for Sturm–Liouville operators this abstract Titchmarsh–Weyl function coincides with the Titchmarsh–Weyl coefficient.

Proposition 2.2 shows that the operator-valued function *M* is a *Q*-function for the pair *S*, *A* (for the definition of *Q*-functions, cf., e.g. Kreı̆n & Langer 1973). Note that (*A*−*λ*)^{−1} is an operator in for and that(2.4)for .

*The following relations hold:*(2.5)(2.6)

In order to show relation (2.5), let , and set *f*_{μ}≔*γ*(*μ*)*x*. Then . We have to show that . Fromand relation (2.4) it follows that {*f*_{λ}; *λf*_{λ}}∈*S*^{*} andwhich implies that *f*_{λ}=*γ*(*λ*)*x*, i.e. (2.5) is true.

For the proof of (2.6), let be arbitrary and put and . Relation (2.1) implieswhich is equivalent toSince *x* and *y* were arbitrary, relation (2.6) follows. ▪

It follows from (2.5) and (2.6) that *M*(.) is an analytic function on and . It may be analytically continued to parts of the real axis and we denote this continuation also by *M*. Relation (2.6) implies thatwhere Im *T*=(1/2i)(*T*−*T*^{*}); so *M*(.) is an operator-valued Nevanlinna function.

If *S* is a densely defined operator, then *S*^{*} is also an operator and we write

If the deficiency indices *n*_{+}=*n*_{−}≕*n* are finite, then we can choose . Let *e*_{i}, *i*=1, …, *n*, be the canonical basis in and set , which is the basis in for which *Γ*_{0}{*ψ _{i}*(

*λ*);

*λψ*

_{i}(

*λ*)}=

*e*

_{i}. The Titchmarsh–Weyl function is then an

*n*×

*n*matrix function

*M*

_{ij}(

*λ*) with

## 3. The general inequality

Let *S* be a closed symmetric relation and a boundary triple. Analogously to (1.3), we define a *Dirichlet form* by(3.1)It follows from (2.1) that *D* is a symmetric form on *S*^{*}. Moreover, set for .

If *S* is a densely defined operator, then we can setwhich yields

Now we can formulate the main theorem.

*Let S be a closed symmetric relation in* *and* *a boundary triple. Moreover, let D be the Dirichlet form defined in* *(3.1)* *and M be the Titchmarsh–Weyl function defined in* *(2.2)*. *Then the following assertions are equivalent.*

*There exists a positive constant C such that*(3.2)*for all*.*There exist**such that*(3.3)*for all λ=ρ*e^{iϕ}*with ρ>*0*and θ*_{+}*≤ϕ≤π−θ*_{−}.

*Let θ*_{+}*, θ*_{−} *be minimal in* (ii) *and put θ*_{0}≔max*{θ*_{+}, *θ*_{−}*}. If θ*_{0}*≠*0, *then the best possible constant in* *(3.2)* *is*

*Equality holds in* *(3.2)* *if and only if f=*0 *or* *or*

*with* arg *λ=θ*_{+} *(if θ*_{0}*=θ*_{+}*) or* arg *λ=π−θ _{−} (if θ*

_{0}

*=θ*

_{−}

*) and*

Inequality (3.3) is always valid for *λ*=i*t*, *t*>0, since

It can be shown that *θ*_{0}=0 if, and only if, *M*(*λ*)=(−1/*λ*)*C*, where *C* is a positive self-adjoint operator in . In this case, the Dirichlet form can vanish identically. Moreover, it can be shown that if *θ*_{0}=0 and the Dirichlet form does not vanish identically, then the best constant in (3.2) is 1.

The proof uses similar ideas as the proof in Evans & Everitt (1982). First, we need some lemmas. Let *λ*=*μ*+i*ν* with *ν*≠0 and define the following inner product on *S*^{*}:

The corresponding norm is denoted by ‖.‖_{λ}.

*For* *, we have the following orthogonal decomposition:**with respect to the inner product* (.,.)_{λ}*. If* *,* *then*

The decomposition itself follows from von Neumann's formula. We only have to show the orthogonality. Let and , then

The proofs of , and the last assertion are left to the reader. ▪

*Let* *and**with* *, a,*. *Then*(3.4)*and*(3.5)(3.6)

Using lemma 3.4 and equations (2.6) and (2.3), we obtainwhich proves (3.4). Equation (3.5) is trivial and equation (3.6) follows from ▪

*Let* *and* *,* *,* *a,* *b be as in* *lemma 3.5*. *Then*(3.7)*with*(3.8)*where* [*x*, *y*]≔1*/ν〈*Im *M*(*λ*)*x*, *y〉 is a positive definite inner product on* . *Moreover, R*(*λ*; *a*,*b*)*=*0 *if and only if a=λx,* *for some* .

It follows from the definitions of (.,.)_{λ} and *D*[.] that

Using lemma 3.5, we obtainand setting which proves (3.7). The expression *R*(*λ*; *a*,*b*) is a quadratic form in *μ*, *ν* with corresponding matrix(3.9)Its determinant is equal toaccording to Cauchy–Scharz's inequality; hence *R*(*λ*;*a*,*b*)≥0.

If *R*(*λ*; *a*,*b*)=0, then the determinant of (3.9) must vanish, i.e. [*a*,*a*]=[*b*,*b*] and [*a*,*a*][*b*,*b*]−[*a*,*b*][*b*,*a*]=0, which implies that *a* and *b* are linearly dependent. Hence, *b*=*αa* with , |*α*|=1. Moreover, the vector is in the kernel of the matrix in (3.9). An easy calculation shows that then . Hence, *a*=*λx*, for some . Conversely, it is clear that for such *a*,*b*, we have *R*(*λ*;*a*,*b*)=0. ▪

*Proof of* *theorem 3.1*. Set and let *λ*, , , *a*, and *b* be as in lemma 3.5. If we set *λ*≔*ρ* e^{iϕ} with and *ϕ*∈(0, *π*), then equation (3.7) becomes(3.10)Let *θ*_{+},*θ*_{−}∈[0,*π*/2] be minimal such that (3.3) holds and set *θ*_{0}≔max{*θ*_{+}, *θ*_{−}}. Condition (ii) in theorem 3.1 is then equivalent to *θ*_{0}<*π*/2.

Assume that (ii) is satisfied, i.e. *θ*_{0}<*π*/2. First, we consider the case that *θ*_{0}>0. Then for *ϕ*=*θ*_{0} and *π*−*θ*_{0} and all , we have *F*(*λ*)≥0 and hencefor every . Since the infimum of the left-hand side (if , then the minimum is attained for ) is equal to , we obtainfor every , i.e. (i) with *C*=1/cos*θ*_{0} is true. In the case that *θ*_{0}=0, we obtain in the same way the validity of (3.2) for every *C*>1; hence (3.2) holds also for *C*=1.

Now, let *λ* be such that , i.e. there exists an with 〈*F*(*λ*)*x*, *x*〉<0. If we seti.e. , *a*=*λx*, , then according to lemma 3.6If 0<*ϕ*<*θ*_{+}, then there exists a *ρ*>0 such that ; hence there exists an such that(3.11)If *π*−*θ*_{−}<*ϕ*<*π*, then there exists an such that(3.12)In particular, if *θ*_{0}=*π*/2, i.e. (ii) does not hold, then 1/cos *ϕ* or 1/cos(*π*−*ϕ*) can be arbitrarily large and (3.2) cannot be true for all . Hence, we have proved the implication (i)⇒(ii). If *θ*_{0}>0, relations (3.11) and (3.12) also show that *C*=1/cos*θ*_{0} is the best constant.

We consider now the case of equality. Assume that equality holds for some , i.e. . If or *f*=0, we are done. Otherwise, set and *ϕ*≔*θ*_{0}, if or *ϕ*≔*π*−*θ*_{0}, if . Let *λ*≔*ρ*e^{iϕ} and , with , . Then according to (3.10), we haveSince all three terms in the brackets on the right-hand side are non-negative, we obtain(3.13)From the latter equality and lemma 3.6, it follows that *a*=*λx*, for some . Because of the second equality in (3.13), we have 〈*F*(*λ*)*x*,*x*〉=0 and since *F*(*λ*) is non-negative, *F*(*λ*)*x*=0. Hence, *λ* must be such that ker *F*(*λ*)≠0 and then , for *x*∈ker*F*(*λ*).

Conversely, let with *θ*_{0}=*θ*_{+} (if *θ*_{0}=*θ*_{+}) or *ϕ*_{0}=*π*−*θ*_{−} (if *θ*_{0}=*θ*_{−}) be such that ker *F*(*λ*_{0})≠0. Set with *x*∈ker *F*(*λ*_{0}) and . Moreover, let with arbitrary and let be as in lemma 3.6. Then the left-hand side of (3.10) is strictly convex in *ρ* and its minimum is . The right-hand side is and it is equal to , for *ρ*=*ρ*_{0}. Hence, the minimum of the right-hand side is , which implies . ▪

## 4. Sufficient conditions

In this section, we collect some sufficient conditions for the existence of a HELP inequality, i.e. for condition (ii) in theorem 3.1. Most of them are known but have been formulated differently, see, e.g. Evans & Everitt (1982). Throughout this section, we assume that *S* is a symmetric operator with a boundary triple and Titchmarsh–Weyl function *M*(*λ*).

Proposition 4.1—which can be proved easily—shows that if finite dimensional, only the behaviour at 0 and at infinity of *M*(*λ*) is important for the existence of a constant *C* in theorem 3.1. However, this is not sufficient to determine the actual value of *C*.

*Assume that* *is finite dimensional. If there exist r,R>*0 *and θ*_{+,}*θ*_{−}*∈*[0,*π*/2) *such that* Im(*−λ*^{2}*M*(* λ))≥0 for all λ=ρ *e

^{iϕ}

*with ϕ∈*[

*θ*

_{+},

*π*−

*θ*

_{−}]

*and ρ<r or ρ>R, then condition (ii) in*

*theorem 3.1*

*is satisfied.*

First, we study the behaviour at infinity.

*Let M be scalar-valued and assume that**in a sector* {*λ*: arg *λ∈*[*θ _{+}*,

*π−θ*]}

_{−}*with θ*[0,

_{+,}θ_{−}∈*π*/2)

*for some*

*and α∈*[−1,1]

*.*

*If α=*0 *and* *, then condition (ii) in* *theorem 3.1* *is not satisfied.*

*If α≠*0 *or* *, then there exist* *and R>*0 *such that**for all λ=ρ* e^{iϕ} *with ρ>R,* .

Let *α*=0 and . Then

If |*λ*| is large enough and sgnRe *λ*=sgn *c*, then the latter expression is negative.

Now, let *α*≠0 or . Then arg(−(i*t*)^{2}*c*(i*t*)^{α})=arg(*c*(i*t*)^{α})∈(0,*π*) for since *M*(*λ*) is a Nevanlinna function. Hence, there exist and *ϵ*>0 such that arg(−*λ*^{2}*cλ*^{α})∈(*ϵ*,*π*−*ϵ*) for . So for *λ* in this sector and large enough. ▪

Next, we study the behaviour at 0, i.e. we look for conditions that imply the following assertion: there exist *θ*_{+,} *θ*_{−}∈[0,*π*/2) and *r*>0 such that(4.1)

*Let M be a scalar-valued Nevanlinna function.*

*Assume that M is analytic at*0.*If M*(0)=0*, then**(4.1)**holds; if M*(0)*≠*0*, then (4.1) does not hold.**If M has a pole at*0*, then condition**(4.1)**is satisfied.**Assume that M(**λ*)→*c non-tangentially for λ*→0,*where**. Then condition**(4.1)**is satisfied.*

The proof is straightforward and left to the reader. Under the condition that *S* is completely non-self-adjoint (i.e. there is no reducing subspace in which *S* is self-adjoint) it can be shown that *M* has a pole at 0 if and only if 0 is an isolated eigenvalue of *A*=ker *Γ*_{0}; *M* has a zero at 0 if and only if 0 is an isolated eigenvalue of ker *Γ*_{1}.

The condition in (iii) can be expressed in terms of the spectral measure of *M*, which appears in the integral representation of *M*where , *b*≥0, and *σ* is a measure with . If the measure *σ* is absolutely continuous in a neighbourhood of 0 and its density *h*(*t*)≔d*σ*(*t*)/d*t* is Hölder continuous there and *h*(0)≠0, then *M*(*λ*)→*c*, for *λ*→0 non-tangentially with Im *c*=*h*(0) (cf., e.g. Privalov 1950, section III.4.4). So by proposition 4.3 (iii), condition (4.1) is satisfied in this case.

## 5. Examples

In this section, we consider several examples. When applying theorem 3.1, one first has to choose the symmetric operator *S*. Then boundary mappings have to be selected to get the right Dirichlet form.

### (a) The Sturm–Liouville operator on the half-line

We consider the Sturm–Liouville operator *S* whose adjoint *S*^{*} is given byin the space with domainwhere *w*>0, 1/*p*, *q*, . For simplicity, we only consider limit point case, i.e. ker(*S*^{*}−*λ*) is one-dimensional for . For limit circle case, one needs an extra self-adjoint boundary condition at infinity for the domain of *S*^{*}; the HELP inequality is then only valid for functions satisfying this extra boundary condition, cf. Evans & Everitt (1991). In limit point case, we even need the following stronger assumption (as it was shown in Bennewitz (1984), but see remark 5.2 below), namely that *S*^{*} is strong limit point. This means that , for all *f*, . For conditions that guarantee strong limit point case see, e.g. Everitt *et al*. (1974).

It follows fromthat is a possible boundary triple, where

This choice of boundary mappings yields the right Dirichlet form: if the strong limit point case prevails, then the Dirichlet form is given by(5.1)where the integrals are understood to be limits for *X*→∞ of integrals over the intervals [0,*X*].

The self-adjoint extension of *S* with domain ker *Γ*_{0} (cf. lemma 2.1) is the Sturm–Liouville operator with Neumann boundary condition. The corresponding Titchmarsh–Weyl function is the Neumann Titchmarsh–Weyl coefficient(5.2)where *ψ*(.;*λ*) is the solution of the equation (*S*^{*}−*λ*)*y*=0 with(5.3)

If we apply theorem 3.1 to the operator *S*, we get the following criterion of Everitt for the validity of a Sturm–Liouville HELP inequality. Again the integral on the left-hand side of (5.4) has to be understood as the limit of integrals over [0,*X*].

*Let p, q and w be real-valued functions on* [0,*∞*) *with w>*0*,* 1*/p, q,* *and assume that the strong limit point case prevails for the Sturm–Liouville operator* 1*/w*(*−*(*pf′*)*′+qf*)*. Then, the following assertions are equivalent.*

*There exists a constant K>*0*such that*(5.4)*for all**such that f and f′ are absolutely continuous and*.*There exist θ*_{+}*,θ*_{−}*∈*[0,*π/*2)*such that*(5.5)*for all λ=ρ*e^{iϕ}*with ρ>*0*and θ*_{+}*≤ϕ≤π−θ*_{−}.

*Let θ*_{+}*,θ*_{−} *be minimal in* *(5.5)* *and θ*_{0}*≔*max*{θ*_{+},*θ*_{−}*}, then the best constant in* *(5.4)* *is K=*1/(cos *θ*_{0})^{2}.

*Equality holds if, and only if, f=*0 *or S*^{*}*f=*0 *or f*(*x*)*=a* Im(*λψ*(*x*;*λ*))*, where λ is such that* Im(*λ*^{2}*M*(* λ*))

*=*0

*and*arg

*λ=*

*θ*

_{+}

*if θ*

_{0}

*=θ*

_{+}

*or*arg

*λ=π−θ*

_{−}

*if θ*

_{0}

*=θ*

_{−}

*, and*

*, and ψ*(

*.*;

*λ*)

*is the*

*solution of the equation*(

*S*

^{*}

*−λ*)

*ψ=*0

*that satisfies*

*(5.3)*.

The equivalence of (i) and (ii) follows directly from theorem 3.1 since inequality (5.4) is just the square of (3.2). Equality holds if *f*=0, *S*^{*}*f*=0 or with , i.e. . ▪

In Bennewitz (1984), it was shown that if *S* is limit point, then it is necessary that *S* is strong limit point in order to have a valid HELP inequality (5.4) for all *f* in the domain of *S*^{*}. However, if one restricts the inequality (5.4) to functions *f* for which , then the strong limit point assumption is not necessary. If condition (ii) is satisfied, one has a valid inequality with on the left-hand side. For functions satisfying , one can rewrite this expression then as in (5.1).

As a special case, we consider *q*=0, *p*=1 and *w*=1, i.e. the operator *S*^{*}*f*=−*f*″. The Titchmarsh–Weyl coefficient is equal to , where the square root is chosen such that . It is easy to see that condition (ii) in corollary 5.1 is+ fulfilled with *θ*_{+}=*π*/3 and *θ*_{−}=0; hence . This yields the classical Hardy–Littlewood inequality (see Hardy & Littlewood 1932; Hardy *et al*. 1934)for *f*∈*L*^{2}(0,∞) with *f*, *f*′ absolutely continuous and *f*″∈*L*^{2}(0,∞), where equality can only occur forwith and *b*>0. These equalizing functions correspond to *λ*=*b*^{2} e^{iπ/3}.

There are many further examples in the literature, see, e.g. Benammar *et al*. (1998). We quote only one example: for *p*(*x*)=1, *q*(*x*)=0, *w*(*x*)=*x*^{α} with *α*>−1 there exists a HELP inequality with the constant

### (b) A difference operator

In this section, we consider a HELP inequality which is related to difference equations. Let , and be real sequences with *p*_{n}≠0 and *w*_{n}>0. For notational convenience, we set *q*_{0}≔0 and *p*_{−1}≔0. In the space, with the scalar product , where and , we consider the self-adjoint operator(5.6)where Δ*x*_{n}=*x*_{n+1}−*x*_{n}, with domain . For *n*=0 relation (5.6) means (*Tx*)_{0}=−(1/*w*_{0})*p*_{0}Δ*x*_{0}. The restriction *S* of *T* to the domain is a symmetric non-densely defined operator, whose adjoint is the relationcf. Brown & Evans (1992), lemma 2.2. Note that is arbitrary if , so *S*^{*} is not an operator but a proper relation. For simplicity, we assume that limit point case prevails, i.e. dim ker(*S*^{*}−*λ*)=1 for all . In limit circle case, one has to impose an extra boundary condition at infinity and all assertions hold for sequences that satisfy this boundary condition. To get a well-defined Dirichlet form we have to assume that the strong limit-point case prevails, i.e. that for , . In order to get boundary mappings, we calculate for as above:Hencewhich shows thatare possible boundary mappings. The Dirichlet form is then equal toNote that the self-adjoint extension *A*=ker *Γ*_{0} of *S* (cf. lemma 2.1) is an operator and equals *T*. In order to calculate the Titchmarsh–Weyl function *M*(*λ*) let , i.e. {*ψ*; *λψ*}∈*S*^{*}. So *ψ*=(*ψ*_{n}) is a sequence in such thatWe normalize the sequence such that , i.e.since ; the Titchmarsh–Weyl function is then given by . The inequality (3.2) is equivalent toSince is arbitrary, we get corollary 5.3 from theorem 3.1, which first appeared in Brown & Evans (1992).

*Let* *,* *and* *be real sequences with p*_{n}*≠*0*, w*_{n}*>*0 *and q*_{0}*=*0 *and assume that the strong limit point case prevails. Let M*(*λ*) *be defined as above. Then the following assertions are equivalent.*

*There exists a constant K>*0*such that*(5.7)*for all**such that the right-hand side is finite.**There exist θ*_{+}*,θ*_{−}*∈*[0,*π*/2)*such that*(5.8)*for all λ=ρ*e^{iϕ}*with ρ>*0*and θ*_{+}*≤ϕ≤π*−*θ*_{−}.

*Let θ*_{+}*,θ*_{−} *be minimal in* *(5.8)* *and θ*_{0}*≔*max*{θ*_{+}, *θ*_{−}*}, then the best constant in* *(5.7)* *is K=*1/(cos *θ*_{0})^{2}*.*

*Equality holds if and only if x=*0 *or* −Δ(*p*_{n−1}Δ*x*_{n−1})+*q*_{n}*x*_{n}*=*0 *for n=*1*, *2*, … or x*_{n}*=a* Im(*λψ*_{n}(*λ*)), *where λ is such that* Im(*λ*^{2}*M*(*λ*))=0 *and* arg *λ=θ*_{+}*if θ*_{0}*=θ*_{+} *or* arg *λ=π*−*θ*_{−} *if θ*_{0}*=θ*_{−}*, and* *, and* (*ψ*_{n}) *is defined above.*

Similarly as detailed in remark 5.2 for the Sturm–Liouville case, one has a HELP inequality for all (*x*_{n}) with if (ii) is satisfied and *S* is limit point but not strong limit point.

In the special case *p*_{n}=1, *q*_{n}=0, *w*_{n}=1, we get Copson's inequality (Copson 1979)It is easy to calculate that in this case the Titchmarsh–Weyl function is equal towhere the square root is such that , since a sequence *ψ* with is given by . In Brown & Evans (1992), it was proved that condition (ii) in corollary 5.3 is satisfied for *θ*_{+}=*π*/3 and *θ*_{−}=0. For further examples and connections with orthogonal polynomials see Brown *et al*. (1992, 1993).

### (c) A block operator matrix

In this section, we consider the HELP inequality for a block operator matrix *S* whose adjoint *S*^{*} is given bywith in the space with domainwhere *W*^{2,2}(0,∞) denotes the Sobolev space of second order on the half-line, and boundary mappingsThe operator *S* is the restriction of *S*^{*} to those vectors that satisfy *f*_{1}(0)=*f*_{1}^{′}(0)=0. The Dirichlet form is given bysince for *f*_{1}∈*W*^{2,2}(0,∞). The eigenvalue equation for *S*^{*} isFor the first component *f*_{1}, we get a *λ*-rational Sturm–Liouville problem,(5.9)From this it is easy to calculate thatwhere , is the solution of (*S*^{*}−*λ*)*ψ*=0 with . Hence, the Titchmarsh–Weyl function is equal toNow we have to check in which sector Im *F*(*λ*)≥0, where *F*(*λ*)≔−*λ*^{2}*M*(*λ*). For small *λ*, the function *F*(*λ*) behaves like −(1/|*α*|)*λ*^{5/2}, which has non-negative imaginary part in the sector {*λ*=*ρ* e^{iϕ}:*ρ*>0, 2*π*/5≤*ϕ*≤4*π*/5}. One can show that Im *F*(*λ*)>0 in this sector, which implies that *θ*_{+}≤2*π*/5 and *θ*_{−}≤*π*/5. We only show that Im *F*(*λ*)>0 in the sector *Λ*≔{*λ*=*ρ* e^{iϕ}:*ρ*>0,2*π*/5≤*ϕ*≤3*π*/5}, which also implies that *θ*_{0}≤2*π*/5. Since Im *F*(i*t*)>0, for , it suffices to show that for *λ*∈*Λ*. We haveand with *λ*=*μ*+i*ν* we obtainThe latter expression is positive for . The behaviour of *F*(*λ*) for small *λ* shows that *θ*_{0} cannot be smaller than 2*π*/5. Hence, *θ*_{0}=2*π*/5 in theorem 3.1, and since , we have proved the following inequality. That there is no non-trivial case of equality follows from the fact that Im *F*(*λ*) is strictly positive on *Λ*.

*For* *, f*_{1}*, f*_{2}*∈L*^{2}(0,∞)*,* *f*_{1}*,* *absolutely continuous,* *, we have**Apart from the case f*_{1}*=f*_{2}*=*0*, there is no case of equality.*

### (d) A Hamiltonian system

Connected with Hamiltonian systems of the formthere also exist HELP inequalities (cf. Brown *et al*. 1999; Brown & Marletta 2000), which can be proved using theorem 3.1. In these papers, *f* is a 2*n* vector, *A* and *B* are 2*n*×2*n* matrices, where *A* is singular, andHere, we consider only one example of a HELP inequality that is connected with the Hamiltonian system with a regular matrix *A* and *n*=1, namelyon the interval [0,∞), wherei.e. *A*=*I*, *B*=0. The corresponding symmetric operator is given by(5.10)in the space with domain

The adjoint *S*^{*} is given by the same expression in (5.10) but with no boundary conditions at 0. A simple computation shows thatare possible boundary mappings. The Dirichlet form is given by(5.11)since lim_{x→∞}*f*(*x*)=0, for *f*∈*W*^{1,2}(0,∞). The Titchmarsh–Weyl function is given by *M*(*λ*)=i, for and *M*(*λ*)=−i for . This follows from the fact that is a defect element for . It is easy to show that condition (ii) in theorem 3.1 is satisfied for *θ*_{+}=*θ*_{−}=*π*/4, which yields . Therefore, the following HELP inequality is true.

*For f*_{1}*,f*_{2}*∈W*^{1,2}(0*,∞*) *we have**Equality holds exactly for the following pairs of functions:**with* , *b>*0.

The equalizing functions correspond to and to .

Note that derivatives appear in the Dirichlet form (5.11) in contrast to the Dirichlet form defined in Brown *et al*. (1999) and Brown & Marletta (2000). This is due to the fact that the weight matrix *A* is regular.

## 6. Concluding remarks

The results of the paper can be applied to many other situations, e.g. to higher-order differential equations (cf. Dias 1994) or to problems connected with eigenvalue problems of the form *T*_{1}=*λT*_{2}, where *T*_{i} are differential operators of different orders (cf. Bennewitz 1984). When one wants to apply theorem 3.1, one has first to choose the correct space and a symmetric operator according to the right-hand side of a possible HELP inequality, then choose boundary mappings that yield the right Dirichlet form, and finally compute the Titchmarsh–Weyl function and check whether condition (ii) in the theorem is fulfilled.

## Acknowledgements

The author gratefully acknowledges the support of the ‘Fonds zur Förderung der wissenschaftlichen Forschung’ of Austria, FWF, grant no. P 15540-N 05.

## Footnotes

- Received June 14, 2005.
- Accepted September 12, 2005.

- © 2005 The Royal Society