Royal Society Publishing

A general HELP inequality connected with symmetric operators

Matthias Langer

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.

Keywords:

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):Embedded Imagefor fL2(0,∞) such that f and f′ are absolutely continuous and f″∈L2(0,∞). Everitt (1972) studied the more general inequalityEmbedded Image(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 inequalityEmbedded Image(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 inequalityEmbedded Imagefor Embedded Image, where Δxn=xn+1xn. 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 Embedded Image 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 Γ0, Γ1 from the domain of S* into another Hilbert space, Embedded Image, which satisfy the following abstract Lagrange or Green identityEmbedded Imagefor all f, g in the domain of S*, where (.,.) and 〈.,.〉 denote the inner products in Embedded Image and Embedded Image, respectively. For differential operators such boundary mappings are naturally given, e.g.Embedded Imagewith Embedded Image for the Sturm–Liouville operator on the half-lineEmbedded ImageThe right-hand side of (1.1) is Kf2S*f2, where ‖.‖ denotes the norm in the weighted space Embedded Image. Using the boundary mappings and integration by parts we can rewrite the left-hand side asEmbedded Image(1.3)which will serve as an abstract definition of a generalized Dirichlet form D[f]. For an arbitrary symmetric operator with boundary mappings Γ0, Γ1, the main theorem (theorem 3.1) establishes an equivalence of the existence of a constant K such thatEmbedded Imageholds 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 Γ0, Γ1. 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 Embedded Image be a Hilbert space with scalar product (.,.). A closed linear relation in Embedded Image is a closed subspace of Embedded Image, where we write {f; g} for elements in Embedded Image. 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 byEmbedded ImageThe inverse and the adjoint relations are given byEmbedded Imageand the sum of two linear relations T1 and T2 is given byEmbedded ImageA closed linear relation S is called symmetric if SS* and self-adjoint if S=S*.

Let S be a closed symmetric relation. Then, we define the deficiency spaces byEmbedded Imagewhere P1 is the projection onto the first component in Embedded Image. It is well known that Embedded Image is constant on the upper and the lower half-planes Embedded Image. The deficiency indices n+, n are defined by Embedded Image.

The triple Embedded Image is called a boundary triple for a closed symmetric relation S, if Embedded Image is a Hilbert space with scalar product 〈.,.〉 and Embedded Image are linear, continuous mappings such thatEmbedded Image(2.1)and the mapping Embedded Image from Embedded Image into Embedded Image is surjective. The Γi 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 Embedded Image are given by Embedded Image for some self-adjoint relation θ in Embedded Image. Here, we only prove that Embedded Image (which corresponds to Embedded Image) is self-adjoint.

Let S be a closed symmetric relation and Embedded Image a boundary triple. Then S=ker Γ0∩ker Γ1. Moreover, the relation A≔ker Γ0 is self-adjoint.

It follows from (2.1) that S=S**⊃ker Γ1∩ker Γ0. Let Embedded Image. Then, Embedded Image for every Embedded Image. Hence, Embedded Image. Since Embedded Image is surjective, we can choose Embedded Image such that Embedded Image and Embedded Image is arbitrary in Embedded Image, which implies Embedded Image. Similarly, one gets Embedded Image.

Now we show that A is self-adjoint. It follows from (2.1) that A is symmetric. As Embedded Image is surjective, the mapping Γ1A is also surjective. Let Embedded Image. Similar as above it follows thatEmbedded Imagefor all Embedded Image. Since, Γ1 is surjective on A, we obtain Embedded Image, i.e. Embedded Image, which shows A=A*. ▪

Since Embedded Image (Embedded Image denotes the direct sum of subspaces of Embedded Image) for Embedded Image, the mapping Embedded Image is bijective from Embedded Image onto Embedded Image. Hence, we can defineEmbedded Imagewhere again P1 denotes the projection onto the first component in Embedded Image. The Titchmarsh–Weyl function M is now defined byEmbedded Image(2.2)for Embedded Image, which is a bijective operator in Embedded Image. Note thatEmbedded Imagefor all Embedded Image. Moreover, the following symmetry relation holds:Embedded Image(2.3)which follows fromEmbedded Imagefor Embedded Image.

We see in §5a 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 (Aλ)−1 is an operator in Embedded Image for Embedded Image and thatEmbedded Image(2.4)for Embedded Image.

The following relations hold:Embedded Image(2.5)Embedded Image(2.6)

In order to show relation (2.5), let Embedded Image, Embedded Image and set fμγ(μ)x. Then Embedded Image. We have to show that Embedded Image. FromEmbedded Imageand relation (2.4) it follows that {fλ; λfλ}∈S* andEmbedded Imagewhich implies that fλ=γ(λ)x, i.e. (2.5) is true.

For the proof of (2.6), let Embedded Image be arbitrary and put Embedded Image and Embedded Image. Relation (2.1) impliesEmbedded Imagewhich is equivalent toEmbedded ImageSince x and y were arbitrary, relation (2.6) follows. ▪

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

If S is a densely defined operator, then S* is also an operator and we writeEmbedded Image

If the deficiency indices n+=nn are finite, then we can choose Embedded Image. Let ei, i=1, …, n, be the canonical basis in Embedded Image and set Embedded Image, which is the basis in Embedded Image for which Γ0{ψi(λ); λψi(λ)}=ei. The Titchmarsh–Weyl function is then an n×n matrix function Mij(λ) withEmbedded Image

3. The general inequality

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

If S is a densely defined operator, then we can setEmbedded Imagewhich yieldsEmbedded Image

Now we can formulate the main theorem.

Let S be a closed symmetric relation in Embedded Image and Embedded Image 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.

  1. There exists a positive constant C such thatEmbedded Image(3.2)for all Embedded Image.

  2. There exist Embedded Image such thatEmbedded Image(3.3)for all λ=ρ eiϕ with ρ>0 and θ+≤ϕ≤π−θ.

Let θ+, θ be minimal in (ii) and put θ0≔max+, θ}. If θ00, then the best possible constant in (3.2) is Embedded Image

Equality holds in (3.2) if and only if f=0 or Embedded Image orEmbedded Image

with arg  λ=θ+ (if θ0+) or arg λ=π−θ (if θ0) andEmbedded Image

Inequality (3.3) is always valid for λ=it, t>0, sinceEmbedded Image

It can be shown that θ0=0 if, and only if, M(λ)=(−1/λ)C, where C is a positive self-adjoint operator in Embedded Image. In this case, the Dirichlet form can vanish identically. Moreover, it can be shown that if θ0=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+iν with ν≠0 and define the following inner product on S*:Embedded Image

The corresponding norm is denoted by ‖.‖λ.

For Embedded Image, we have the following orthogonal decomposition:Embedded Imagewith respect to the inner product (.,.)λ. If Embedded Image, thenEmbedded Image

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

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

Let Embedded Image andEmbedded Imagewith Embedded Image, a,Embedded Image. ThenEmbedded Image(3.4)andEmbedded Image(3.5)Embedded Image(3.6)

Using lemma 3.4 and equations (2.6) and (2.3), we obtainEmbedded Imagewhich proves (3.4). Equation (3.5) is trivial and equation (3.6) follows fromEmbedded Image ▪

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

It follows from the definitions of (.,.)λ and D[.] thatEmbedded Image

Using lemma 3.5, we obtainEmbedded Imageand setting Embedded Imagewhich proves (3.7). The expression R(λ; a,b) is a quadratic form in μ, ν with corresponding matrixEmbedded Image(3.9)Its determinant is equal toEmbedded Imageaccording 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 Embedded Image, |α|=1. Moreover, the vector Embedded Image is in the kernel of the matrix in (3.9). An easy calculation shows that then Embedded Image. Hence, a=λx, Embedded Image for some Embedded Image. Conversely, it is clear that for such a,b, we have R(λ;a,b)=0. ▪

Proof of theorem 3.1. Set Embedded Image and let λ, Embedded Image, Embedded Image, a, and b be as in lemma 3.5. If we set λρ eiϕ with Embedded Image and ϕ∈(0, π), then equation (3.7) becomesEmbedded Image(3.10)Let θ+,θ∈[0,π/2] be minimal such that (3.3) holds and set θ0≔max{θ+, θ}. Condition (ii) in theorem 3.1 is then equivalent to θ0<π/2.

Assume that (ii) is satisfied, i.e. θ0<π/2. First, we consider the case that θ0>0. Then for ϕ=θ0 and πθ0 and all Embedded Image, we have F(λ)≥0 and henceEmbedded Imagefor every Embedded Image. Since the infimum of the left-hand side (if Embedded Image, then the minimum is attained for Embedded Image) is equal to Embedded Image, we obtainEmbedded Imagefor every Embedded Image, i.e. (i) with C=1/cosθ0 is true. In the case that θ0=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 Embedded Image, i.e. there exists an Embedded Image with 〈F(λ)x, x〉<0. If we setEmbedded Imagei.e. Embedded Image, a=λx, Embedded Image, then according to lemma 3.6Embedded ImageIf 0<ϕ<θ+, then there exists a ρ>0 such that Embedded Image; hence there exists an Embedded Image such thatEmbedded Image(3.11)If πθ<ϕ<π, then there exists an Embedded Image such thatEmbedded Image(3.12)In particular, if θ0=π/2, i.e. (ii) does not hold, then 1/cos ϕ or 1/cos(πϕ) can be arbitrarily large and (3.2) cannot be true for all Embedded Image. Hence, we have proved the implication (i)⇒(ii). If θ0>0, relations (3.11) and (3.12) also show that C=1/cosθ0 is the best constant.

We consider now the case of equality. Assume that equality holds for some Embedded Image, i.e. Embedded Image. If Embedded Image or f=0, we are done. Otherwise, set Embedded Image and ϕθ0, if Embedded Image or ϕπθ0, if Embedded Image. Let λρeiϕ and Embedded Image, with Embedded Image, Embedded Image. Then according to (3.10), we haveEmbedded ImageSince all three terms in the brackets on the right-hand side are non-negative, we obtainEmbedded Image(3.13)From the latter equality and lemma 3.6, it follows that a=λx, Embedded Image for some Embedded Image. 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 Embedded Image, for x∈kerF(λ).

Conversely, let Embedded Image with θ0=θ+ (if θ0=θ+) or ϕ0=πθ (if θ0=θ) be such that ker F(λ0)≠0. Set Embedded Image with x∈ker F(λ0) and Embedded Image. Moreover, let Embedded Image with arbitrary Embedded Image and let Embedded Image be as in lemma 3.6. Then the left-hand side of (3.10) is strictly convex in ρ and its minimum is Embedded Image. The right-hand side is Embedded Image and it is equal to Embedded Image, for ρ=ρ0. Hence, the minimum of the right-hand side is Embedded Image, which implies Embedded Image. ▪

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 Embedded Image and Titchmarsh–Weyl function M(λ).

Proposition 4.1—which can be proved easily—shows that if Embedded Image 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 Embedded Image is finite dimensional. If there exist r,R>0 and θ+,θ[0,π/2) such that Im(−λ2M(λ))≥0 for all λ=ρeiϕ with ϕ∈[θ+,πθ] 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 thatEmbedded Imagein a sector {λ: arg λ∈[θ+,π−θ]} with θ+,θ[0,π/2) for some Embedded Image and α∈[−1,1].

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

If α≠0 or Embedded Image, then there exist Embedded Image and R>0 such thatEmbedded Imagefor all λ=ρ eiϕ with ρ>R, Embedded Image.

Let α=0 and Embedded Image. ThenEmbedded Image

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

Now, let α≠0 or Embedded Image. Then arg(−(it)2c(it)α)=arg(c(it)α)∈(0,π) for Embedded Image since M(λ) is a Nevanlinna function. Hence, there exist Embedded Image and ϵ>0 such that arg(−λ2α)∈(ϵ,πϵ) for Embedded Image. So Embedded Image 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 thatEmbedded Image(4.1)

Let M be a scalar-valued Nevanlinna function.

  1. Assume that M is analytic at 0.

  2. If M(0)=0, then (4.1) holds; if M(0)0, then (4.1) does not hold.

  3. If M has a pole at 0, then condition (4.1) is satisfied.

  4. Assume that M(λ)→c non-tangentially for λ→0, where Embedded Image. 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 Γ0; M has a zero at 0 if and only if 0 is an isolated eigenvalue of ker Γ1.

The condition in (iii) can be expressed in terms of the spectral measure of M, which appears in the integral representation of MEmbedded Imagewhere Embedded Image, b≥0, and σ is a measure with Embedded Image. If the measure σ is absolutely continuous in a neighbourhood of 0 and its density h(t)≔dσ(t)/dt 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 byEmbedded Imagein the space Embedded Image with domainEmbedded Imagewhere w>0, 1/p, q, Embedded Image. For simplicity, we only consider limit point case, i.e. ker(S*λ) is one-dimensional for Embedded Image. 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 Embedded Image, for all f, Embedded Image. For conditions that guarantee strong limit point case see, e.g. Everitt et al. (1974).

It follows fromEmbedded Imagethat Embedded Image is a possible boundary triple, whereEmbedded Image

This choice of boundary mappings yields the right Dirichlet form: if the strong limit point case prevails, then the Dirichlet form is given byEmbedded Image(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 Γ0 (cf. lemma 2.1) is the Sturm–Liouville operator with Neumann boundary condition. The corresponding Titchmarsh–Weyl function is the Neumann Titchmarsh–Weyl coefficientEmbedded Image(5.2)where ψ(.;λ) is the Embedded Image solution of the equation (S*λ)y=0 withEmbedded Image(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, Embedded Image and assume that the strong limit point case prevails for the Sturm–Liouville operator 1/w((pf′)′+qf). Then, the following assertions are equivalent.

  1. There exists a constant K>0 such thatEmbedded Image(5.4)for all Embedded Image such that f and f′ are absolutely continuous and Embedded Image.

  2. There exist θ+[0,π/2) such thatEmbedded Image(5.5)for all λ=ρ eiϕ with ρ>0 and θ+≤ϕ≤π−θ.

Let θ+ be minimal in (5.5) and θ0max+,θ}, then the best constant in (5.4) is K=1/(cos θ0)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(λ2M(λ))=0 and arg λ=θ+ if θ0+ or arg λ=π−θ if θ0, and Embedded Image, and ψ(.;λ) is the Embedded Image 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 Embedded Image with Embedded Image, i.e. Embedded Image. ▪

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 Embedded Image, then the strong limit point assumption is not necessary. If condition (ii) is satisfied, one has a valid inequality with Embedded Image on the left-hand side. For functions satisfying Embedded Image, 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 Embedded Image, where the square root is chosen such that Embedded Image. It is easy to see that condition (ii) in corollary 5.1 is+ fulfilled with θ+=π/3 and θ=0; hence Embedded Image. This yields the classical Hardy–Littlewood inequality (see Hardy & Littlewood 1932; Hardy et al. 1934)Embedded Imagefor fL2(0,∞) with f, f′ absolutely continuous and f″∈L2(0,∞), where equality can only occur forEmbedded Imagewith Embedded Image and b>0. These equalizing functions correspond to λ=b2 eiπ/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)=xα with α>−1 there exists a HELP inequality with the constant Embedded Image

(b) A difference operator

In this section, we consider a HELP inequality which is related to difference equations. Let Embedded Image, Embedded Image and Embedded Image be real sequences with pn≠0 and wn>0. For notational convenience, we set q0≔0 and p−1≔0. In the space, Embedded Image with the scalar product Embedded Image, where Embedded Image and Embedded Image, we consider the self-adjoint operatorEmbedded Image(5.6)where Δxn=xn+1xn, with domain Embedded Image. For n=0 relation (5.6) means (Tx)0=−(1/w0)p0Δx0. The restriction S of T to the domain Embedded Image is a symmetric non-densely defined operator, whose adjoint is the relationEmbedded Imagecf. Brown & Evans (1992), lemma 2.2. Note that Embedded Image is arbitrary if Embedded Image, 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 Embedded Image. 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 Embedded Image for Embedded Image, Embedded Image. In order to get boundary mappings, we calculate for Embedded Image as above:Embedded ImageHenceEmbedded Imagewhich shows thatEmbedded Imageare possible boundary mappings. The Dirichlet form is then equal toEmbedded ImageNote that the self-adjoint extension A=ker Γ0 of S (cf. lemma 2.1) is an operator and equals T. In order to calculate the Titchmarsh–Weyl function M(λ) let Embedded Image, i.e. {ψ; λψ}∈S*. So ψ=(ψn) is a sequence in Embedded Image such thatEmbedded ImageWe normalize the sequence such that Embedded Image, i.e.Embedded Imagesince Embedded Image; the Titchmarsh–Weyl function is then given by Embedded Image. The inequality (3.2) is equivalent toEmbedded ImageSince Embedded Image is arbitrary, we get corollary 5.3 from theorem 3.1, which first appeared in Brown & Evans (1992).

Let Embedded Image, Embedded Image and Embedded Image be real sequences with pn0, wn>0 and q0=0 and assume that the strong limit point case prevails. Let M(λ) be defined as above. Then the following assertions are equivalent.

  1. There exists a constant K>0 such thatEmbedded Image(5.7)for all Embedded Image such that the right-hand side is finite.

  2. There exist θ+[0,π/2) such thatEmbedded Image(5.8)for all λ=ρ eiϕ with ρ>0 and θ+≤ϕ≤πθ.

Let θ+ be minimal in (5.8) and θ0max+, θ}, then the best constant in (5.7) is K=1/(cos θ0)2.

Equality holds if and only if x=0 or −Δ(pn−1Δxn−1)+qnxn=0 for n=1, 2, … or xn=a Im(λψn(λ)), where λ is such that Im(λ2M(λ))=0 and arg λ=θ+if θ0+ or arg λ=πθ if θ0, and Embedded Image, and (ψn) is defined above.

Similarly as detailed in remark 5.2 for the Sturm–Liouville case, one has a HELP inequality for all (xn) with Embedded Image if (ii) is satisfied and S is limit point but not strong limit point.

In the special case pn=1, qn=0, wn=1, we get Copson's inequality (Copson 1979)Embedded ImageIt is easy to calculate that in this case the Titchmarsh–Weyl function is equal toEmbedded Imagewhere the square root is such that Embedded Image, since a sequence ψ with Embedded Image is given by Embedded Image. 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 byEmbedded Imagewith Embedded Image in the space Embedded Image with domainEmbedded Imagewhere W2,2(0,∞) denotes the Sobolev space of second order on the half-line, and boundary mappingsEmbedded ImageThe operator S is the restriction of S* to those vectors Embedded Image that satisfy f1(0)=f1(0)=0. The Dirichlet form is given byEmbedded Imagesince Embedded Image for f1W2,2(0,∞). The eigenvalue equation for S* isEmbedded ImageFor the first component f1, we get a λ-rational Sturm–Liouville problem,Embedded Image(5.9)From this it is easy to calculate thatEmbedded Imagewhere Embedded Image, is the solution of (S*λ)ψ=0 with Embedded Image. Hence, the Titchmarsh–Weyl function is equal toEmbedded ImageNow we have to check in which sector Im F(λ)≥0, where F(λ)≔−λ2M(λ). For small λ, the function F(λ) behaves like −(1/|α|)λ5/2, which has non-negative imaginary part in the sector {λ=ρ eiϕ:ρ>0, 2π/5≤ϕ≤4π/5}. One can show that Im F(λ)>0 in this sector, which implies that θ+≤2π/5 and θπ/5. We only show that Im F(λ)>0 in the sector Λ≔{λ=ρ eiϕ:ρ>0,2π/5≤ϕ≤3π/5}, which also implies that θ0≤2π/5. Since Im F(it)>0, for Embedded Image, it suffices to show that Embedded Image for λΛ. We haveEmbedded Imageand with λ=μ+iν we obtainEmbedded ImageThe latter expression is positive for Embedded Image. The behaviour of F(λ) for small λ shows that θ0 cannot be smaller than 2π/5. Hence, θ0=2π/5 in theorem 3.1, and since Embedded Image, 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 Embedded Image, f1, f2∈L2(0,∞), f1, Embedded Image absolutely continuous, Embedded Image, we haveEmbedded ImageApart from the case f1=f2=0, there is no case of equality.

(d) A Hamiltonian system

Connected with Hamiltonian systems of the formEmbedded Imagethere 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 2n vector, A and B are 2n×2n matrices, where A is singular, andEmbedded ImageHere, we consider only one example of a HELP inequality that is connected with the Hamiltonian system with a regular matrix A and n=1, namelyEmbedded Imageon the interval [0,∞), whereEmbedded Imagei.e. A=I, B=0. The corresponding symmetric operator is given byEmbedded Image(5.10)in the space Embedded Image with domainEmbedded Image

The adjoint S* is given by the same expression in (5.10) but with no boundary conditions at 0. A simple computation shows thatEmbedded Imageare possible boundary mappings. The Dirichlet form is given byEmbedded Image(5.11)since limx→∞f(x)=0, for fW1,2(0,∞). The Titchmarsh–Weyl function is given by M(λ)=i, for Embedded Image and M(λ)=−i for Embedded Image. This follows from the fact that Embedded Image is a defect element for Embedded Image. It is easy to show that condition (ii) in theorem 3.1 is satisfied for θ+=θ=π/4, which yields Embedded Image. Therefore, the following HELP inequality is true.

For f1,f2∈W1,2(0,∞) we haveEmbedded ImageEquality holds exactly for the following pairs of functions:Embedded Imagewith Embedded Image, b>0.

The equalizing functions correspond to Embedded Image and to Embedded Image.

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 T1=λT2, where Ti 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.

References

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