Heisenberg uniqueness pairs and the Klein-Gordon equation

arXiv:0910.1246v1 [math.AP] 7 Oct 2009 Heisenberg uniqueness pairs and the Klein-Gordon equation Håkan Hedenmalm and Alfonso Montes-Rodríguez Abstract. A Heisenberg uniqueness pair (HUP) is a pair (Γ,Λ), where Γis a curve in the plane and Λis a set in the plane, with the following property: any bounded Borel measure µin the plane supported on Γ, which is absolutely continuous with respect to arc length, and whose Fourier transform b µvanishes on Λ, must automatically be the zero measure. We prove that when Γis the hyperbola x1x2=1, and Λis the lattice-cross Λ = (αZ×{0})∪({0}×βZ), where α, β are positive reals, then (Γ,Λ) is an HUP if and only if αβ ≤1; in this situation, the Fourier transform b µof the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that eπiαnt,eπiβn/t,n∈Z, span a weak-star dense subspace in L∞(R) if and only if αβ ≤1. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin. 1. Introduction Heisenberg uniqueness pairs. Let µbe a finite complex-valued Borel measure in the plane R2, and associate to it the Fourier transform b µ(ξ)=ZR2 eπihx,ξidµ(x), where x=(x1,x2) and ξ=(ξ1, ξ2), with inner product hx, ξi=x1ξ1+x2ξ2. The Heisenberg uncertainty principle states that both µand b µcannot both be too concentrated to a point (see [6] for the original paper of Heisenberg, and [5] for a more general treatment); in particular, they cannot both have compact support. Here, we shall study a variation on that theme. Let Γbe a smooth curve in R2, or, more generally, a finite disjoint union of smooth curves. 1991 Mathematics Subject Classification. Primary 42B10, 42A10, 58F11; Secondary 11K50, 31B35, 43A15, 81Q05. Key words and phrases. Trigonometric system, inversion, composition operator, Klein-Gordon equation, ergodic theory. Research partially supported by the Göran Gustafsson Foundation and by the Swedish Science Council (Vetenskapsrådet). Research partially supported by Plan Nacional I+D+I grant no. MTM2006-09060 and by Junta de Andalucía grants nos. FQM-260 and FQM06-02225. 2 Hedenmalm and Montes Suppose that supp µ⊂Γ, and that µis absolutely continuous with respect to arc length measure on Γ. Which sets Λ⊂R2have the property that b µ|Λ=0=⇒µ=0? If this is the case, we say that (Γ,Λ) is a Heisenberg uniqueness pair. A dual formulation is that (Γ,Λ) is a Heisenberg uniqueness pair if and only if the functions eξ(x)=eπihx,ξi, ξ ∈Λ, span a weak-star dense subspace in L∞(Γ). This concept of Heisenberg uniqueness pairs has many features in common with the notion of (weakly) mutually annihilating pairs of Borel measurable sets having positive area measure, which appears, for instance, in the book by Havin and Jöricke [5]. The properties of the Fourier transform with respect to translation and multiplication by complex exponentials show that for all points x∗, ξ∗∈R2, we have (inv-1) (Γ + {x∗},Λ + {ξ∗}) is an HUP ⇐⇒ (Γ,Λ) is an HUP, where HUP is short for “Heisenberg uniqueness pair”. Likewise, it is also straightforward to see that if T:R2→R2is an invertible linear transformation with adjoint T∗, then (inv-2) (T−1(Γ),T∗(Λ)) is an HUP ⇐⇒ (Γ,Λ) is an HUP. Algebraic curves and partial differential equations. Algebraic curves Γare of particular interest, because of their connection to partial differential equations. That connection follows from the observation that for polynomials pof two variables, p∂1 πi,∂2 πib µ(ξ)=ZR2 eπihx,ξip(x1,x2) dµ(x), so that if pis real-valued and Γis the locus of the equation p(x1,x2)=0, then p(x1,x2) dµ(x1,x2)=0 identically. Therefore b µsolves the PDE (1.1) p∂1 πi,∂2 πib µ(ξ)=0 in the plane. In fact, the equation (1.1) encodes the requirement that supp µ⊂Γ. Conic sections. We shall consider the case when Γis a conic section, that is, the locus of a quadratic equation ax2 1+bx2 2+cx1x2+dx1+ex2+f=0, where a,b,c,d,e,fare real constants. As we only consider the case when Γis a curve, this leaves us with the following cases: a straight line, two parallel straight lines, a cross, an ellipse, a parabola, or a hyperbola. The line. Let us look at the line first, as a model example. By the invariance properties (inv-1) and (inv-2), we may assume that Γ = R× {0}, the x1-axis. In this case, b µ(ξ) depends only on ξ1, and it is easy to see that (Γ,Λ) is a Heisenberg uniqueness pair if and only if π1(Λ), the orthogonal projection of Λto the ξ1-axis, is dense. Two parallel lines. If Γis the union of two parallel lines, we may without loss of generality assume that Γ = R×{0,1}. Heisenberg uniqueness pairs and the Klein-Gordon equation 3 In this case, we see from the example of the line that it is necessary for (Γ,Λ) to be a Heisenberg uniqueness pair that π1(Λ) be dense. But something more is needed. An absolutely continuous measure µon Γmay be written in the form dµ(x)=f(x1)dx1dδ0(x2)+g(x1)dx1dδ1(x2), where f,g∈L1(R) (δydenotes the unit point mass at the point y), so that b µ(ξ)=b f(ξ1)+eπiξ2b g(ξ1). Next, we split π1(Λ)=πa 1(Λ)∪πb 1(Λ), where the two sets are disjoint: t∈πa 1(Λ) if there are two lifted points ξ=(ξ1, ξ2) and η=(η1, η2) in Λ, with ξ1=η1=tand ξ2−η2<2Z, whereas t∈πb 1(Λ) if the latter does not happen. We quickly find that (1.2) b f(t)=b g(t)=0,t∈πa 1(Λ). On the other hand, for t∈πb 1(Λ), the expression eπiξ2is a well-defined function of ξ1=t, where ξ2 stands for any of the points with (ξ1, ξ2)∈Λ; we write χ(t) for this unimodular function. If Eis a closed subset of Rand t0∈E, we say that a function ϕ:E→Cis locally the Fourier transform of an L1(R)function around t0provided that there exists a small open interval Iaround t0and a function ψwhich is the Fourier transform of an L1(R) function, such that ψ=ϕon E∩I. Let πc 1(Λ) consist of those points t0∈πb 1(Λ) where χ:πb 1(Λ)→Cis locally the Fourier transform of an L1(R) function around t0. Theorem 1.1. (Γ,Λ)is a Heisenberg uniqueness pair if and only if πa 1(Λ)∪(πb 1(Λ)\πc 1(Λ)) is dense in R. Proof. We observe that (1.3) b f(t)=−χ(t)b g(t),t∈πb 1(Λ). If t∈πb 1(Λ)\πc 1(Λ), this is only possible if b g(t)=0, so that (1.4) b f(t)=b g(t)=0,t∈πb 1(Λ)\πc 1(Λ). A combination of (1.2) and (1.4) shows that f=g=0 (so that µ=0) if the set πa 1(Λ)∪(πb 1(Λ)\πc 1(Λ)) is dense in R. As for the other direction, suppose that π1(Λ) is dense in R, while πa 1(Λ)∪(πb 1(Λ)\πc 1(Λ)) fails to be dense in R. We then pick a point t0∈Rsuch that an open interval Jaround it has empty intersection with πa 1(Λ)∪(πb 1(Λ)\πc 1(Λ)). But then πc 1(Λ)∩Jis dense in J, and χis locally the Fourier transform of an L1(R) function around t0. We thus find a function χ1which coincides with χon some open interval I⊂Jwith t0∈I, while χ1is the Fourier transform of an L1(R) function. Next, we pick g∈L1(R) with b g(t0),0, such that suppb g⋐I, and define f∈L1(R) via b f=−χ1b g, so that (1.3) holds. This gives us a nontrivial measure µwith the required properties, and so (Γ,Λ) cannot be a Heisenberg uniqueness pair.  The cross. If Γis a cross, the PDE (1.1) expresses the wave equation. By the invariance properties (inv-1) and (inv-2), we may restrict our attention to the case when Γ = (R×{0})∪({0}×R) is the union of the two axes. Here, it appears that the characterization of uniqueness pairs (Γ,Λ) may get quite complicated. Obviously, it is a necessary condition that π1(Λ) and π2(Λ) be dense (π2(Λ) is the orthogonal projection to the ξ2-axis). This is far from sufficient, because if Λis contained in a smooth graph, we may run into trouble. For instance, if Λis contained in the diagonal ξ1=ξ2, then we may choose dµ(x1,x2)=f(x1) dx1dδ0(x2)−f(x2) dx2dδ0(x1), 4 Hedenmalm and Montes where f∈L1(R), which is supported on Γand nontrivial generically, whileb µ(ξ1, ξ2)=0 for ξ1=ξ2. The ellipse. If Γis an ellipse, the invariance of (inv-1) and (inv-2) allows us to focus on the circle Γ = {x=(x1,x2)∈R2:x2 1+x2 2=1}. The corresponding PDE (1.1) is the eigenvalue equation for the Laplacian. Here, the fact that Γis compact entails that b µ(ξ) extends to an entire function of exponential type in C2. It would seem that reasonable criteria on Λmay be found that are at least close to being necessary and sufficient for (Γ,Λ) to be a Heisenberg uniqueness pair. The parabola. If Γis a parabola, the invariance of (inv-1) and (inv-2) allows us to focus on the parabola Γ = {x=(x1,x2)∈R2:x2=x2 1}. The corresponding PDE (1.1) is the one-dimensional Schrödinger equation without potential. Here, the problem of characterizing the Heisenberg uniqueness pairs (Γ,Λ) appears quite challenging. The hyperbola. We shall focus most of our attention to the case when Γis a hyperbola. The corresponding PDE (1.1) is the one-dimensional Klein-Gordon equation. We will see that the situation with Heisenberg uniqueness pairs is dramatically different from that of the cross. By the invariance (inv-1) and (inv-2), we may assume that the hyperbola is given by x1x2=1. Theorem 1.2. Suppose Γis the hyperbola x1x2=1and that Λis the lattice-cross Λ = (αZ×{0})∪({0}×βZ), where α, β are positive reals. Then (Γ,Λ)is a Heisenberg uniqueness pair if and only if αβ ≤1. The remainder of this work is devoted to proving this assertion. But before we turn to the proof, let us consider a generalization which is more or less immediate. Corollary 1.3. Suppose Γεis the hyperbola x1x2=ε, where ε,0is real, and that Λis the lattice-cross Λ = (αZ×{0})∪({0}×βZ), where α, β are positive reals. Then (Γε,Λ)is a Heisenberg uniqueness pair if and only if αβ ≤1/|ε|. The eccentricity of the hyperbola Γεis √2 independently of ε. The condition of the corollary (αβ ≤1/|ε|) gets weaker as |ε|decreases. However, in the limit situation ε=0 – the cross – the situation changes dramatically: if Λis contained in the dual cross (R×{0})∪({0}×R),then Λmust actually be dense in the cross for (Γ0,Λ) to be a Heisenberg uniqueness pair. Remark 1.4.Consider for a moment the sets Λ′=([θ, +∞[×]−∞,0]) ∪(] −∞,0] ×[0,+∞[) and Λ′′ =n(ξ1, ξ2)∈R2:a1ξ1+a2ξ2=0o, where θ, a1,a2are all real parameters, subject to θ > 0 and a1a2>0. The set Λ′is arguably more massive than the lattice-cross Λof Corollary 1.3. Nevertheless, if Γεis as in Corollary 1.3, with εpositive, it can be shown that (Γε,Λ′) fails to be a Heisenberg uniqueness pair, no matter what positive values εand θassume. Analogously, (Γε,Λ′′) also fails to be a Heisenberg uniqueness pair, for all ε > 0 and a1a2>0 (but it can be shown that (Γε,Λ′∪Λ′′) is a Heisenberg uniqueness pair, however). This suggests that it is crucial that the points of the lattice-cross Λof Corollary 1.3 are located along the characteristic directions for the Klein-Gordon equation (the two axes). We need a result of algebraic nature. Heisenberg uniqueness pairs and the Klein-Gordon equation 5 Lemma 1.5. Let z1,z2∈Cbe two points such that z1−z2=am ∈aZ,1 z1−1 z2 =bn ∈bZ, for some positive reals a,b. Then, unless z1=z2, we have z1=am 21±r1−4 abmn,z2=z1−am. The proof is a simple exercise, and therefore omitted. Remark 1.6.Let us consider the singular measure µ=δu−δv, where u=(u1,1/u1)∈Γ,v=(v1,1/v1)∈Γ. Then b µ(ξ)=eπi(ξ1u1+ξ2/u1)−eπi(ξ1v1+ξ2/v1), so that b µ(ξ1,0) =eπiξ1u1−eπiξ1v1,b µ(0, ξ2)=eπiξ2/u1−eπiξ2/v1. Suppose we try to achieve that (1.5) b µ(αj,0) =b µ(0, βk)=0,j,k∈Z, for some positive reals α, β. We see that this amounts to eπiαu1=eπiαv1,eπiβ/u1=eπiβ/v1, which we rewrite in the form u1−v1∈2 αZ,1 u1−1 v1∈2 βZ. In view of Lemma 1.5, there are plenty of such points u1,v1∈Rwith u1,v1, for any given α, β. This shows that the requirement that the measure µbe absolutely continuous with respect to arc length measure on Γis essential; without it, Theorem 1.2 would simply not be true. 2. Dynamics of a Gauss-type map A Gauss-type map. In order to prove our main theorem (Theorem 1.2), we will need to study the invariant measures of a particular map. We shall consider a map on the interval ] −1,1], which we think of as R/2Z(topologically as well). The map in question is defined by U(0) =0 and U(x)=−1 x2 ,x,0, where for real t, the expression {t}2∈]−1,1] is the unique number such that t− {t}2∈2Z. The function Uis locally strictly increasing and continuous, except for being interrupted by jumps. The map U:] −1,1] →]−1,1] is associated with continued fractions with even partial quotients (see [10], [11], [7], [3]). We see that, for j=±1,±2,±3,..., U(x)=−1 x+2j,1 2j+1<x≤1 2j−1, and hence Umaps the interval ] 1 2j+1,1 2j−1] onto ] −1,1] in a one-to-one fashion. The derivative of Uis locally U′(x)=1 x2,x∈]−1,1] \1 2Z+1. The point 1 is a fixed point for U, and U′(1−)=U′(−1+)=1, which makes 1 is a weakly repelling fixed point. This means that when we iterate U, once we are close to 1 (which is the same point as −1 in R/2Z), the successive iterates will remain near 1 for a long time. If x∈]−1,1] is rational, 6 Hedenmalm and Montes then after a finite number of steps, the U-iterate of xis either 0 or 1 (see, for instance [7]). This illuminates why irrational numbers tend to spend a large portion of their U-orbits near 1. Invariant measures. If ϕis a continuous function on R/2Zand νis a bounded complex Borel measure on ] −1,1], then the integral (2.1) Z]−1,1] ϕ(x) dν(x) is well-defined. However, the integral (2.1) makes sense under weaker assumptions on ϕ. Suppose Eis an open subset of ] −1,1] such that the complement ] −1,1] \Eis countable, and that ϕis bounded on ] −1,1] and continuous on E. Then (2.1) makes sense for ϕ, and we call the function ϕpseudo-continuous. We recall the familiar notion that a bounded complex Borel measure νon ]−1,1] is U-invariant provided that (2.2) Z]−1,1] ϕ(U(x)) dν(x)=Z]−1,1] ϕ(x) dν(x) holds for all pseudo-continuous test functions ϕ; it is easy to see that ϕ◦Uis pseudo-continuous if ϕis pseudo-continuous, so that (2.2) makes sense. We shall reformulate this criterion in more concrete terms. First, we note that Z]−1,1]\{0} ϕ(U(x)) dν(x)=X j∈Z∗Z]1 2j+1,1 2j−1] ϕ(U(x)) dν(x)=X j∈Z∗Z]1 2j+1,1 2j−1] ϕ−1 x+2jdν(x), where Z∗=Z\{0}, and that Z]1 2j+1,1 2j−1] ϕ−1 x+2jdν(x)=Z]−1,1] ϕ(t) dνj(t), where (2.3) dνj(t)=dν1 2j−t,−1<t≤1, so that we have Z]−1,1]\{0} ϕ(U(x)) dν(x)=X j∈Z∗Z]−1,1] ϕ(t) dνj(t). It follows that νis U-invariant if and only if (2.4) ν=ν({0})δ0+X j∈Z∗ νj. More generally, given λ∈C, we want to talk about (U, λ)-invariant measures, defined by the requirement that (2.5) Z]−1,1] ϕ(U(x)) dν(x)=λZ]−1,1] ϕ(x) dν(x) hold for all test functions ϕ; specifically, this means that (2.6) λν =ν({0})δ0+X j∈Z∗ νj. It is easy to see that for |λ|>1, there are no (U, λ)-invariant measures except for the zero measure. Proposition 2.1. Suppose νis a bounded (U, λ)-invariant measure on ]−1,1], and write ν=νa+νs, where νais absolutely continuous, while νsis singular. Then νaand νsare also (U, λ)-invariant. Moreover, if |λ|=1, then |ν|,|νa|, and |νs|are all U-invariant measures. Heisenberg uniqueness pairs and the Klein-Gordon equation 7 Proof. The relation (2.6) splits: (2.7) νa=λX j∈Z∗ (νj)a, νs=λν({0})δ0+X j∈Z∗ (νj)s, where the subscripts aand sindicate the absolutely continuous and singular parts, respectively, of the measure in question. We easily realize that (νj)a=(νa)jand (νj)s=(νs)j, so that (2.7) expresses that νaand νsare both U-invariant. Next, we suppose |λ|=1, and turn to the assertion that |ν|is U-invariant. Taking absolute values, we have (2.8) |dν(t)| ≤ |ν({0})|dδ0(t)+X j∈Z∗|dνj(t)|, and so Z]−1,1] |dν(t)|≤ X j∈Z∗Z]−1,1] |dνj(t)|=|ν({0})|+X j∈Z∗Z]1 2j+1,1 2j−1]|dν(t)| =|ν({0})|+Z]−1,1]\{0}|dν(t)|=Z]−1,1] |dν(t)|. This is only possible if we have in fact equality in (2.8): |dν(t)|=|ν({0})|dδ0(t)+X j∈Z∗|dνj(t)|. This relation expresses that |ν|is U-invariant; that |νa|and |νs|are U-invariant is a simple consequence of this fact.  An unbounded smooth invariant measure. We now consider the positive unbounded smooth measure dω(x)=dx 1−x2. The criterion (2.6) makes sense although ωis unbounded. The following assertion was essentially found by Schweiger [10]. Proposition 2.2. The measure ωis U-invariant. We supply the simple proof. Proof. We check that dωj(t)=dω1 2j−t=dt (2j−t)2−1, and since X j∈Z∗ 1 (2j−t)2−1=1 2X j∈Z∗1 2j−t−1−1 2j−t+1=1 21 1+t+1 1−t=1 1−t2, we find from (2.6) that ωis U-invariant.  Schweiger [10] actually focused on the related map |U|: [0,1] →[0,1] given by |U|(x)=|U(x)|. He obtained the following basic result. Proposition 2.3. The measure ωis invariant also with respect to |U|. Moreover, |U|is ergodic, that is, if E⊂[0,1] is a |U|-invariant set, then either ω(E)=0or ω([0,1] \E)=0. 8 Hedenmalm and Montes Consequences of Ergodic Theory. The BirkhoffErgodic Theorem – in this setting of an unbounded invariant ergodic measure [1] – states that if ϕis Borel measurable and even with Z1 −1 |ϕ(t)| 1−t2dt<+∞, then 1 N N−1 X k=0 ϕ(Uhki(t)) →0 as N→+∞ almost everywhere on ] −1,1]. Here, Uhkistands for the k-th iterate of U. We observe that we do not need to know whether Uis ergodic, just that |U|is, if we use that |U(−x)|=|U(x)|. We pick ϕ(t)=1−t2, and get: (2.9) 1 N N−1 X k=01−|Uhki(t)|2→0 as N→+∞ almost everywhere on ]−1,1]. Suppose νis a positive, bounded, and absolutely continuous U-invariant measure on ] −1,1]. By the U-invariance, we have (2.10) Z]−1,1] 1−|Uhki(t)|2dν(t)=Z]−1,1] (1 −t2) dν(t), and so Z]−1,1] 1 N N−1 X k=01−|Uhki(t)|2dν(t)=Z]−1,1] (1 −t2) dν(t). By the Lebesgue dominated convergence theorem, it follows from (2.9) that Z]−1,1] 1 N N−1 X k=01−|Uhki(t)|2dν(t)→0,as N→+∞, which combined with the (2.10) leads to Z]−1,1] (1 −t2) dν(t)=0. This is only possible if ν=0. We formalize this in a proposition. Proposition 2.4. Suppose λ∈Chas |λ|=1, and that νis an absolutely continuous bounded complex (U, λ)-invariant Borel measure on ]−1,1]. Then ν=0. Proof. By Proposition 2.1, |ν|is a U-invariant measure. By the above argument, |ν|=0, and so ν=0.  3. Extension of the trigonometric system The trigonometric system. The trigonometric system {en(x)}n∈Z, with en(x)=eπinx, is very successful in describing 2-periodic functions on the line. Harald Bohr – the brother of Niels Bohr, the physicist – developed over a number of years in the 1920s and 1930s the theory of almost periodic functions based on more general real frequencies rather than the integer frequencies of the trigonometric system. An extension of the trigonometric system. Here, we consider another extension of the trigonometric system, connected with the theory of composition operators. Let βbe a positive real parameter. We introduce, for integers n, ehβi n(x)=enβ x=eπiβn/x, Heisenberg uniqueness pairs and the Klein-Gordon equation 9 and note that these functions are bounded on the real line. After a dilation of the line, Theorem 1.2 is equivalent to the following statement. Theorem 3.1. As n ranges over the integers, the functions en(x)and ehβi n(x)form a weak-star-spanning system in L∞(R)if and only if 0< β ≤1. If µis a positive bounded absolutely continuous Borel measure on R, then a bounded function in L∞(R) is automatically in Lp(R, µ) for 1 <p<+∞, and the weak-star closure of a subspace in L∞(R) is contained in the norm closure in Lp(R, µ). We then have the following consequence of Theorem 3.1. The necessity part just requires mimicking the corresponding argument involving harmonic extensions in Section 4 below. Corollary 3.2. Suppose 1<p<+∞, and that dµ(x)=M(x)dx, where M(x)≥0is Borel measurable, with 0<Z+∞ −∞ M(x) dx<+∞. Then, as n ranges over the integers, the functions en(x)and ehβi n(x)form a spanning system in Lp(R, µ) provided that 0< β ≤1. If, in addition, Z+∞ −∞ dx (1 +x2)p/(p−1)M(x)1/(p−1) <+∞, the condition 0< β ≤1is also necessary in order to have a spanning system. 4. Necessity of the condition 0< β ≤1 Harmonic extension. We extend the functions enharmonically and boundedly to the upper half plane C+={z∈C: Im z>0}: en(z)=eπinz,Im z≥0,n≥0, while en(z)=eπin¯ z,Im z≥0,n<0. Likewise, the harmonic extension of ehβi nis ehβi n(z)=eπiβn/¯ z,Im z≥0,n≥0, and ehβi n(z)=eπiβn/z,Im z≥0,n<0. Point separation. A general L∞(R) function is extended harmonically and boundedly to C+via the Poisson kernel; for each z0=x0+iy0∈C+, the point evaluation functional f7→ f(z0) is given by f(z0)=1 πZ+∞ −∞ P(t,z0)f(t) dt,P(t,z0)=y0 (x0−t)2+y2 0 , where t7→ P(t,z0) is in L1(R), and the functional is therefore weak-star continuous on L∞(R). As we harmonically extend all the functions in L∞(R), we get the space of all bounded harmonic functions in C+. The bounded harmonic functions in C+separate the points of C+, so if we can find two points z1,z2∈C+with z1,z2, such that (4.1) en(z1)=en(z2),ehβi n(z1)=ehβi n(z2), for all n∈Z, then the linear span of en,ehβi n, cannot be weak-star dense in L∞(R). The condition (4.1) boils down to z1−z2∈2Z,1 z1−1 z2∈2 βZ, 16 Hedenmalm and Montes Hedenmalm: Department of Mathematics, The Royal Institute of Technology, S – 100 44 Stockholm, SWEDEN E-mail address:[email protected] Montes-Rodríguez: Department of Mathematical Analysis, University of Sevilla, Sevilla, SPAIN E-mail address:[email protected]