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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ Abstract . A pair (Γ, Λ), where Γ R 2 is a locally rectifiable curve and Λ R 2 is a Heisenberg uniqueness pair if an absolutely continuous finite complex-valued Borel measure supported on Γ whose Fourier transform vanishes on Λ necessarily is the zero measure. Here, absolute continuity is with respect to arc length measure. Recently, it was shown by Hedenmalm and Montes that is Γ is the hyperbola x 1 x 2 = (with , 0), and Λ is the lattice-cross (αZ ×{0}) ({0βZ), where α, β are positive reals, then (Γ, Λ) is a Heisenberg uniqueness pair if and only if αβ||≤ 1. The Fourier transform of a measure supported on a hyperbola solves the one-dimensional Klein- Gordon equation, so in a sense the theorem supplies a kind of really thin uniqueness sets for a class of solutions to this equation. The above-mentioned theorem is equivalent to the following assertion: the functions e iαmt , e -iβn/t , m, n Z, span a weak-star dense subspace of L (R) if and only if 0 < αβ π 2 . In the critical case αβ = π 2 , the proof involved ideas from Dynamical Systems. More precisely, the crucial fact was that the Gauss-type map t 7→-1/t modulo 2Z on [-1, 1] has an ergodic absolutely continuous invariant measure with infinite total mass. Here, we investigate two variations on the same basic theme. Both have interpretations in terms of Heisenberg uniqueness, but here, we prefer the concrete formulation. First, we show that the functions e iαmt , e -iβn/t , m, n Z + ∪{0}, span a weak-star dense subspace of H + (R) if and only if 0 < αβ π 2 . Here, H + (R) is the subspace of L (R) which consists of those functions whose Poisson extensions to the upper half-plane are holomorphic. This theorem is strictly stronger than the above-mentioned result (which implies density of the linear span in BMOA(R) only). In the critical case αβ = π 2 , the proof relies on the nonexistence of a certain invariant distribution for the above-mentioned Gauss-type map, a fact not known previously. Second, we show that the restriction to R + of the functions e iαmt , e -iβn/t , m, n Z, span a weak-star dense subspace of L (R + ) if and only if 0 < αβ 4π 2 . In the critical case αβ = 4π 2 , the weak-star span misses the mark by one dimension only; the proof is based on the dynamics of the standard Gauss map t 71/t mod Z on the interval [0, 1]. By a combination of two of the above results, we find that for π 2 < αβ < 4π 2 , any function f L 1 (R) with Z R e iαmt f (t)dt = Z R e -iβn/t f (t)dt = 0, m, n Z, cannot vanish a.e. on R - unless it vanishes a.e. on R. This is an instance of dynamical unique continuation. 2000 Mathematics Subject Classification. Primary . Key words and phrases. transfer operator, Hilbert transform. The research of the author was supported by the Göran Gustafsson Foundation (KVA) and by Vetenskapsrådet (VR). 1

Transcript of KTHhaakanh/publications/NEW-alfonso2-2.pdf · THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR...

Page 1: KTHhaakanh/publications/NEW-alfonso2-2.pdf · THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINESWITH THE HILBERT TRANSFORM

HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

Abstract. A pair (Γ,Λ), where Γ ⊂ R2 is a locally rectifiable curve and Λ ⊂ R2 is a Heisenberguniqueness pair if an absolutely continuous finite complex-valued Borel measure supported on Γ

whose Fourier transform vanishes on Λ necessarily is the zero measure. Here, absolute continuityis with respect to arc length measure. Recently, it was shown by Hedenmalm and Montes thatis Γ is the hyperbola x1x2 = ε (with ε , 0), and Λ is the lattice-cross (αZ × {0}) ∪ ({0} × βZ),where α, β are positive reals, then (Γ,Λ) is a Heisenberg uniqueness pair if and only if αβ|ε| ≤ 1.The Fourier transform of a measure supported on a hyperbola solves the one-dimensional Klein-Gordon equation, so in a sense the theorem supplies a kind of really thin uniqueness sets for aclass of solutions to this equation. The above-mentioned theorem is equivalent to the followingassertion: the functions

eiαmt, e−iβn/t, m,n ∈ Z,span a weak-star dense subspace of L∞(R) if and only if 0 < αβ ≤ π2. In the critical case αβ = π2,the proof involved ideas from Dynamical Systems. More precisely, the crucial fact was that theGauss-type map t 7→ −1/t modulo 2Z on [−1, 1] has an ergodic absolutely continuous invariantmeasure with infinite total mass. Here, we investigate two variations on the same basic theme.Both have interpretations in terms of Heisenberg uniqueness, but here, we prefer the concreteformulation. First, we show that the functions

eiαmt, e−iβn/t, m,n ∈ Z+ ∪ {0},

span a weak-star dense subspace of H∞+ (R) if and only if 0 < αβ ≤ π2. Here, H∞+ (R) is the subspaceof L∞(R) which consists of those functions whose Poisson extensions to the upper half-plane areholomorphic. This theorem is strictly stronger than the above-mentioned result (which impliesdensity of the linear span in BMOA(R) only). In the critical case αβ = π2, the proof relies on thenonexistence of a certain invariant distribution for the above-mentioned Gauss-type map, a factnot known previously. Second, we show that the restriction to R+ of the functions

eiαmt, e−iβn/t, m,n ∈ Z,

span a weak-star dense subspace of L∞(R+) if and only if 0 < αβ ≤ 4π2. In the critical case αβ = 4π2,the weak-star span misses the mark by one dimension only; the proof is based on the dynamicsof the standard Gauss map t 7→ 1/t mod Z on the interval [0, 1].

By a combination of two of the above results, we find that for π2 < αβ < 4π2, any functionf ∈ L1(R) with ∫

Reiαmt f (t)dt =

∫R

e−iβn/t f (t)dt = 0, m,n ∈ Z,

cannot vanish a.e. on R− unless it vanishes a.e. on R. This is an instance of dynamical uniquecontinuation.

2000 Mathematics Subject Classification. Primary .Key words and phrases. transfer operator, Hilbert transform.The research of the author was supported by the Göran Gustafsson Foundation (KVA) and by Vetenskapsrådet

(VR).1

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2 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

1. Introduction

1.1. Heisenberg uniqueness pairs. Let µ be a finite complex-valued Borel measure in theplane R2, and associate to it the Fourier transform

µ(ξ) :=∫R2

eiπ〈x,ξ〉dµ(x),

where x = (x1, x2) and ξ = (ξ1, ξ2), with inner product

〈x, ξ〉 = x1ξ1 + x2ξ2.

The Fourier transform µ is a continuous and bounded function on R2. In [15], the concept ofa Heisenberg uniqueness pair (HUP) was introduced. It is similar to the notion of (weakly)mutually annihilating pairs of Borel measurable sets having positive area measure, whichappears, e.g., in the book by Havin and Jöricke [14]. For Γ ⊂ R2 which is finite disjoint unionof smooth curves in R2, let M(Γ) denote the Banach space of Banach space of complex-valuedfinite Borel measures in R2, supported on Γ. Moreover, let AC(Γ) denote the closed subspaceof M(Γ) consisting of the measures that are absolutely continuous with respect to arc lengthmeasure on Γ.

Definition 1.1.1. Let Γ be a finite disjoint union of smooth curves in R2. For a set Λ ⊂ R2, wesay that (Γ,Λ) is a Heisenberg uniqueness pair provided that

∀µ ∈ AC(Γ) : µ|Λ = 0 =⇒ µ = 0.

1.2. The Zariski closure operation. We turn to the notion of the Zariski closure operation.First, we need to let AC(Γ; Λ) be the subspace of AC(Γ) consisting of those measures µ whoseFourier transform vanishes on Λ.

Definition 1.2.1. Let Γ be a finite disjoint union of smooth curves in R2, and let let Λ ⊂ R2 bearbitrary. With respect to AC(Γ), the Zariski closure of Λ is the set

zclosΓ(Λ) := {ξ ∈ R2 : [∀µ ∈ AC(Γ; Λ) : µ(ξ) = 0]}.

As the Fourier image of AC(Γ) does not form an algebra with pointwise multiplication offunctions, we cannot expect the Zariski closure operation to correspond to a topology. Thismeans that the intersection of two Zariski closures need not be a closure itself. It is easy tosee that the closure operation is idempotent: zclos2

Γ = zclosΓ. In terms of the Zariski closureoperation, we may express the uniqueness pair property conveniently: (Γ,Λ) is a Heisenberguniqueness pair if and only if

zclosΓ(Λ) = R2.

1.3. The Klein-Gordon equation. In natural units, the Klein-Gordon equation reads

(1.3.1) ∂2t u − ∆xu + M2u = 0,

where M > 0 is a constant (it is the mass of the particle), and

∆x = ∂2x1

+ . . . + ∂2xd

is the d-dimensional Laplacian. We shall here restrict to the case of d = 1, one spatial dimension.So, our equation reads

∂2t u − ∂2

xu + M2u = 0.

In terms of the (preferred) coordinates

ξ1 := t + x, ξ2 := t − x,

the Klein-Gordon equation becomes

(KG) ∂ξ1∂ξ2 u +M2

4u = 0.

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 3

Remark 1.3.1. Since t2− x2 = ξ1ξ2, the time-like vectors (those vectors (t, x) ∈ R2 with t2

− x2 > 0)correspond to the union of the first quadrant ξ1, ξ2 > 0 and the third quadrant ξ1, ξ2 < 0 in the(ξ, ξ2)-plane). Likewise, the space-like vectors correspond to the union of the second quadrantξ1 > 0, ξ2 < 0 and the fourth quadrant ξ1 < 0, ξ2 > 0.

1.4. Fourier analytic treatment of the Klein-Gordon equation. In the sequel, we will notneed to talk about the time and space coordinates (t, x) as such. So, e.g., we are free to use thenotation x = (x1, x2) for the Fourier dual coordinate to ξ = (ξ1, ξ2).

LetM(R2) denote the (Banach) space of all finite complex-valued Borel measures inR2. Wesuppose that u is the Fourier transform of a µ ∈ M(R2):

(1.4.1) u(ξ) = µ(ξ) :=∫R2

eiπ〈x,ξ〉dµ(x), ξ ∈ R2.

The assumption that u solves the Klein-Gordon equation (KG) would mean that(x1x2 −

M2

4π2

)dµ(x) = 0

as a measure on R2, which we see is the same as a requirement on the support set of themeasure µ:

(1.4.2) suppµ ⊂ ΓM :={x ∈ R2 : x1x2 =

M2

4π2

}.

The set ΓM is a hyperbola. We may use the x1-axis to supply a global coordinate for ΓM, anddefine a complex-valued finite Borel measure π1µ on R by putting

(1.4.3) π1µ(E) =

∫E

dπµ(x1) := µ(E ×R) =

∫E×R

dµ(x).

We shall at times refer to π1µ as the compression of µ to the x1-axis. It is easy to see that µ maybe recovered from π1µ; indeed,

(1.4.4) u(ξ) = µ(ξ) =

∫R×

eiπ[ξ1t+M2ξ2/(4π2t)]dπ1µ(t), ξ ∈ R2.

Here, we use the standard notational convention R× := R \ {0}. We note that µ is absolutelycontinuous with respect to arc length measure on ΓM if and only if π1µ is absolutely continuouswith respect to (Lebesgue) length measure on R×.

1.5. The lattice-cross as a uniqueness set for solutions to the Klein-Gordon equation. Forpositive reals α, β, let Λα,β denote the lattice-cross

(1.5.1) Λα,β := (αZ × {0}) ∪ ({0} × βZ),

so that the spacing along the ξ1-axis is α, and along the ξ2-axis it is β. In the recent paper [15],Hedenmalm and Montes-Rodríguez found the following.

Theorem 1.5.1. (Hedenmalm, Montes) For positive reals M, α, β, the tuple (ΓM,Λα,β) is a Heisenberguniqueness pair if and only if αβM2

≤ 4π2.

In terms of the Zariski closure operation, the theorem says that

zclosΓM (Λα,β) = R2

holds if and only if αβM2≤ 4π2. By taking the relation (1.4.4) into account, and by reducing the

redundancy of the constants (i.e., we may without loss of generality consider M = 2π and α = 1only), Theorem 1.5.1 is equivalent to the following statement: the linear span of the functions

eiπmt, e−iπβn/t, m,n ∈ Z,

is weak-star dense in L∞(R) if and only if β ≤ 1. Here, we consider two possible extensions orgeneralizations of Theorem 1.5.1, outlined below in Theorems 1.7.2 and 1.8.1.

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4 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

1.6. The Zariski closure of the axes and half-axes. We first consider the Zariski closure ofthe two axes R × {0} and {0} × R with respect to the space AC(ΓM) of absolutely continuousmeasures (with respect to arc length) on the hyperbola ΓM.

Proposition 1.6.1. Fix a positive real M. Then the Zariski closures of the axesR× {0} and {0} ×R are:

zclosΓM (R × {0}) = zclosΓM ({0} ×R) = R2.

Our next result will show the difference between time-like and space-like quarterplanes.First, we need some notation. Let R+ := {t ∈ R : t > 0} and R− := {t ∈ R : t < 0} be the positiveand negative half-lines, respectively. We write R+ := {t ∈ R : t ≥ 0} and R− := {t ∈ R : t ≤ 0}for the corresponding closed half-lines.

Proposition 1.6.2. Fix a positive real M. Then the Zariski closures of each of the four semi-axesR+ × {0}, R− × {0}, {0} ×R+, and {0} ×R−, are as follows:

zclosΓM (R+ × {0}) = zclosΓM ({0} ×R−) = R+ × R−

andzclosΓM (R− × {0}) = zclosΓM ({0} ×R+) = R− × R+.

We notice that in each case, the Zariski closure of a semi-axis equals the (topological) closureof of a quadrant of space-like vectors. The proofs of the above two propositions are supplied inSection 2.

1.7. The Zariski closure of the lattice-cross restricted to a quadrant. Let

Z+ := {1, 2, 3, . . .}, Z− := {−1,−2,−3, . . .}, Z+,0 := {0, 1, 2, . . .}, Z−,0 := {0,−1,−2, . . .}

denote the sets of positive, negative, nonnegative, and nonpositive integers, respectively. Weconsider the following four portions of the lattice-cross Λα,β given by (1.5.1):

Λ++α,β := (αZ+,0 × {0}) ∪ ({0} × βZ+), Λ+−

α,β := (αZ+,0 × {0}) ∪ ({0} × βZ−),

andΛ−+α,β := (αZ−,0 × {0}) ∪ ({0} × βZ+), Λ−−α,β := (αZ−,0 × {0}) ∪ ({0} × βZ−).

We first calculate the Zariski closure of the two of these (first and third “quadrants”).

Theorem 1.7.1. For positive reals M, α, β, the Zariski closures of Λ++α,β and Λ−−α,β are as follows:

zclosΓM (Λ++α,β) = Λ++

α,β, zclosΓM (Λ−−α,β) = Λ−−α,β.

The proof requires careful handling of the H1-BMO duality and the explicit calculation ofthe Fourier transform of the unimodular function t 7→ ei/t as a tempered distribution. We turnto the Zariski closures of the remaining two portions (second and fourth “quadrants”). Herethe situation involves a lot more mixing, and we can only say when the Zariski closure of aportion of the lattice-cross equals the corresponding quadrants. We consider this our mainresult.

Theorem 1.7.2. For positive reals M, α, β, the following three assertions are equivalent:(i) zclosΓM (Λ+−

α,β) = R+ × R−,(ii) zclosΓM (Λ−+

α,β) = R− × R+,(ii) αβM2

≤ 4π2.

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 5

Let us explain how Theorem 1.5.1 follows from Theorem 1.7.2. First, an elementary argument(see [15], [4]) shows that zclosΓM (Λα,β) , R2 for αβM2 > 4π2, so that we just need to obtain theimplication

αβM2≤ 4π2 =⇒ zclosΓM (Λα,β) = R2.

In view of Theorem 1.7.2,

αβM2≤ 4π2 =⇒ zclosΓM (Λα,β) = zclosΓM (Λ+−

α,β ∪Λ−+α,β) ⊃ (R+ × R−) ∪ (R− × R+) ⊃ R × {0},

and Theorem 1.5.1 becomes a consequence of Proposition 1.6.1 together with the idempotentproperty zclos2

Γ = zclosΓ.By taking the relation (1.4.4) into account, and by reducing the redundancy of the constants

(i.e., we may without loss of generality consider m = 2π and α = 1 only), it is easy to see thatTheorem 1.7.2 is equivalent to the following statement: the linear span of the functions

eiπmt, e−iπβn/t, m,n = 0, 1, 2, . . . ,

is weak-star dense in H∞+ (R) if and only if β ≤ 1. Here, H∞+ (R) is the subspace of L∞(R) whichconsists of those functions whose Poisson extensions to the upper half-plane are holomorphic.This assertion cannot be derived from Theorem 1.5.1 directly. It is a finer statement whichneeds a more delicate argument related to the theory of Dynamical Systems. The technicalresult which we need goes beyond the current state-of-the art of Gauss-type transformationson the interval [−1, 1]. For instance, some properties of the Hilbert transform and the transferoperator of the related Gauss-type map turn out to play a crucial role. It is also very importantthat the Hilbert transform can be understood both as an operator on L1 taking values that are (1)Schwartzian distributions, or (2) elements of the weak-L1 space. Note that although weak-L1

functions have no immediate interpretation as distributions, (1) and (2) are in a one-to-onecorrespondence.

1.8. Heisenberg uniqueness pairs for a branch of the hyperbola. A variant of Theorem 1.5.1is obtained if we replace the hyperbola ΓM by one of its two branches, say

Γ+M := ΓM ∩ (R+ ×R+) =

{x ∈ R2 : x1x2 =

M2

4π2 and x1 > 0}.

We can now formulate our second main result.

Theorem 1.8.1. For positive reals M, α, β, the tuple (Γ+M,Λα,β) is a Heisenberg uniqueness pair if

and only if αβM2 < 16π2. Moreover, in the critical case αβM2 = 16π2, the space AC(Γ+M,Λα,β) is

one-dimensional.

Again, by taking the relation (1.4.4) into account, and by reducing the redundancy of theconstants (i.e., we may without loss of generality consider m = 2π and α = 1 only), it is easy tosee that Theorem 1.8.1 is equivalent to the following statement: the restriction to R+ of the linearspan of the functions

eiπmt, e−iπβn/t, m,n ∈ Z,

is weak-star dense in L∞(R+) if and only if β < 4. Moreover, if β = 4 the weak-star closure of this linearspan has codimension one in L∞(R+).

2. Calculations of the Zariski closures of the axes or semi-axes

2.1. Calculation of the Zariski closures of the axes. We now supply the proofs of Propositions1.6.1 and 1.6.2.

Proof of Proposition 1.6.1. By symmetry, it is enough to show that zclosΓM (R × {0}) = R2. Moreconcretely, we need to show that if µ ∈ AC(ΓM) and

µ(ξ1, 0) = 0, ξ1 ∈ R,

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6 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

then µ = 0 as a measure. In view of (1.4.4),

µ(ξ1, 0) =

∫R×

eiπξ1tdπ1µ(t),

where π1µ is the compression of µ to the real line. The uniqueness theorem for the Fouriertransform gives that π1µ = 0, and hence that µ = 0, since µ and its compression π1µ are in aone-to-one correspondence. �

Proof of Proposition 1.6.2. By symmetry, it is enough to show that

zclosΓM (R+ × {0}) = R+ × R−

To this end, we consider a measure µ ∈ AC(ΓM) with (use (1.4.4))

µ(ξ1, 0) =

∫R

eiπξ1tdπ1µ(t) = 0, ξ1 ∈ R+.

This condition is eqivalent to asking that dπ1µ(t) = f (t)dt, where f ∈ H1+(R). Here, H1

+(R) isthe standard Hardy space of all functions in L1(R) whose Poisson extensions to the upper halfplane are holomorphic. If follows from standard arguments that∫

R

g(t)dπ1µ(t) =

∫R

f (t)g(t)dt = 0

for all g ∈ H∞+ (R). Here, H∞+ (R) is the standard Hardy space of all functions in L∞(R) whosePoisson extensions to the upper half plane are holomorphic. We observe that for ξ1 ≥ 0 andξ2 ≤ 0, the function

g(t) := eiπ[ξ1t+M2ξ2/(4π2t)]

is in H∞+ (R), and so

µ(ξ1, ξ2) =

∫R×

eiπ[ξ1t+M2ξ2/(4π2t)]dπ1µ(t) = 0, (ξ1, ξ2) ∈ R+ × R−.

In conclusion, this argument proves the inclusion

zclosΓM (R+ × {0}) ⊃ R+ × R−.

To obtain the equality of the two sides, we need to show that if (ξ1, ξ2) ∈ R2\ (R+ × R−), then

there exists a µ ∈ AC(ΓM) with dπ1µ(t) = f (t)dt, where f ∈ H1+(R), such that µ(ξ1, ξ2) , 0. But

then the bounded function

g(t) = eiπ[ξ1t+M2ξ2/(4π2t)], t ∈ R,

is not an element of H∞+ (R), and by the standard Hardy space duality theory,

sup{∣∣∣∣∣ ∫

R

f (t)g(t)dt∣∣∣∣∣ : f ∈ ball(H1

+(R))}

= inf{‖g − h‖L∞(R) : h ∈ H∞+ (R)

}> 0.

In particular, there must exist an f ∈ H1+(R) with

µ(ξ1, ξ2) =

∫R

f (t)g(t)dt , 0.

This completes the proof. �

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3. Calculations of the Zariski closures of two portions of the lattice-cross

3.1. The Hardy and BMO spaces. For a reference on the basic facts of Hardy spaces and BMO(bounded mean oscillation), we refer to, e.g., the monographs of Duren and Garnett [8], [12].

We previously introduced the space H1+(R) as consisting of those functions in L1(R) whose

Poisson extensions to the upper half plane

C+ := {z ∈ C : Im z > 0}

are holomorphic. It is well-known that any function f ∈ H1+(R) has vanishing integral,

(3.1.1)∫R

f (t)dt = 0, f ∈ H1+(R).

In other words, H1+(C) ⊂ L1

0(R), where L10(R) denotes the subspace of L1(R) consisting of all

f ∈ L1(R) with (3.1.1). In fact, a Fourier analysis characterization of H1+(R) is that for f ∈ L1(R),

(3.1.2) f ∈ H1+(R) ⇐⇒ ∀y ≥ 0 :

∫R

eiyt f (t)dt = 0.

In a similar manner, the space H1−

(R) – which consists of the complex conjugates of the functionsin H1

+(R) – is characterized by

(3.1.3) f ∈ H1−(R) ⇐⇒ ∀y ≤ 0 :

∫R

eiyt f (t)dt = 0.

We also need the space H1~(R) := H1

+(R) ⊕H1−

(R); then f ∈ H1~(R) means that

(3.1.4) f = f1 + f2, where f1 ∈ H1+(R), f2 ∈ H1

−(R).

Clearly, we have the inclusion H1~(R) ⊂ L1

0(R). We observe that the above decomposition(3.1.4) is unique. Indeed, if f1 + f2 = g1 + g2, where f1, g1 ∈ H1

+(R) and f2, g2 ∈ H1−

(R), thenf1 − f2 = g2 − f2 ∈ H1

+(R) ∩ H1−

(R) = {0}. This is why we use an “⊕” in place of “+” in thedefinition of H1

~(R). We let P+ and P− denote the projections P+ f := f1 and P− f := f2 in thedecomposition (3.1.4). With respect to the dual action

〈 f , g〉R :=∫R

f (t)g(t)dt,

the dual space of H1~(R) may be identified with BMO(R), the space of functions (modulo the

constants) of bounded mean oscillation; this is the celebrated Fefferman duality theorem [9], [10].It is well-known that g ∈ BMO(R) if and only if g may be written in the form g = g1 + Hg2,where g1, g2 ∈ L∞(R). Here, H denotes the (modified) Hilbert transform, defined for f ∈ L∞(R)by the formula

(3.1.5) H f (x) := pv1π

∫R

f (t){ 1

x − t+

t1 + t2

}dt = lim

ε→0+

∫R\[x−ε,x+ε]

f (t){ 1

x − t+

t1 + t2

}dt.

This is a modified version of the standard Hilbert transform of a function in Lp(R), for p with1 ≤ p < +∞. We recall that H f for f ∈ Lp(R) (1 ≤ p < +∞) is given by

(3.1.6) H f (x) := pv1π

∫R

f (t)dt

x − t= limε→0+

∫R\[x−ε,x+ε]

f (t)dt

x − t.

So, for f ∈ Lp(R) (1 ≤ p < +∞), H f − H f equals a constant, which reflects the fact that weshould think of BMO(R) as a space modulo the constants. The Hilbert transforms H f , H f havenatural harmonic extensions to the upper half-plane C+:

(3.1.7) H f (z) :=1π

∫R

Re z − t|z − t|2

f (t)dt, H f (z) :=1π

∫R

{Re z − t|z − t|2

+t

t2 + 1

}f (t)dt,

so that H f (i) = 0 always. Returning to the real line, we note that on the dual side, it iswell-known that for f ∈ L1(R), we have the equivalence

f ∈ H1~(R) ⇐⇒ f ∈ L1

0(R) and H f ∈ L1(R).

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8 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

The projections P+,P− may be expressed in terms of the Hilbert transform:

(3.1.8) P+ f =12

( f + iH f ), P− f =12

( f − iH f ).

For f ∈ L∞(R), the modified Hilbert transform H permits us to define the correspondingprojections

(3.1.9) P+ f =12

( f + iH f ), P− f =12

( f − iH f ),

so that f = P+ f + P− f . For f ∈ H1~(R) and g ∈ L∞(R), the dual action 〈·, ·〉R naturally splits in

holomorphic and conjugate-holomorphic parts:

(3.1.10) 〈 f , g〉R = 〈P+ f , P−g〉R + 〈P− f , P+g〉R.

The space BMO(R) naturally splits in holomorphic and conjugate-holomorphic components:

(3.1.11) BMO(R) = BMOA+(R) ⊕ BMOA−(R).

To be precise, BMOA+(R) and BMOA−(R) stand for the subspaces of BMO(R) that consist offunctions whose Poisson extensions to the upper half-planeC+ are holomorphic and conjugate-holomorphic, respectively. Another useful decomposition of BMO(R) is that of Fefferman andStein [10]:

(3.1.12) BMO(R) = L∞(R) + HL∞(R).

This decomposition is of course not unique. The operator H makes sense also on functionsfrom BMO(R). It is then natural to ask what is H2:

Lemma 3.1.1. For f ∈ Lp(R), 1 < p < +∞, we have that H2 f = − f . Moreover, for f ∈ L∞(R), wehave that H2 f = − f + c( f ), where c( f ) is the constant

c( f ) :=1π

∫R

f (t)t2 + 1

dt.

Proof. The assertion for 1 < p < +∞ is completely standard (see any textbook in HarmonicAnalysis). We turn to the assertion for p = +∞. First, we observe that without loss of generality,we may assume f is real-valued. Then the function 2P+ f is the holomorphic function in theupper half-plane whose real part is the Poisson extension of f , and the choice of the imaginarypart is fixed by the requirement 2 Im P+ f (i) = H f (i) = 0. The function

−2iP+ f = H f − i f

extends to a holomorphic function in the upper half-plane C+, with real part H f . So we mayidentify − f with H2 f up to an additive constant. The additive constant is determined by therequirement that H2 f (i) = 0, and so H2 f (i) = − f + f (i) = − f + c( f ). Here, f (i) is understood interms of Poisson extension. �

For a positive real parameter β, let Jβ be the composition operator defined by

(3.1.13) Jβ[ f ](x) := f (−β/x), x ∈ R×.

Lemma 3.1.2. For f ∈ BMO(R) and a positive real β, we have that

(JβH f )(x) = (HJβ f )(x) + cβ( f ),

where cβ( f ) is the constant

cβ( f ) := H f (iβ) = (β2− 1)

∫R

t f (t) dt(1 + t2)(β2 + t2)

.

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 9

Proof. Without loss of generality, we may assume that f is real-valued. The mapping x 7→ −β/xextends to a conformal automorphism of the upper half-plane given by z 7→ −β/z, and thefunction 2P+ f is a holomorphic function in the upper half plane with real part equal to thePoisson extension of f . We realize that the functions JβP+ f and JβP+ f differ by an imaginaryconstant. Taking imaginary parts, the result follows by plugging in the point z = i. �

In addition to the “hat” notation, let us use F to denote the Fourier transform. In onevariable, the Fourier transform is given by

F f (x) :=∫R

eiπxt f (t)dt,

for f ∈ L1(R), and it is well-known how to extend the Fourier transform to the setting oftempered distributions [16]. It is well understood how to analyze [in terms of the Foriertransform] the spaces BMOA+(R),BMOA−(R) as subspaces of BMO(R). We state the assertionas a lemma.

Lemma 3.1.3. Suppose f ∈ BMO(R). Then f ∈ BMOA+(R) if and only if F f is supported on theinterval R− =] − ∞, 0]. Likewise, f ∈ BMOA−(R) if and only if F f is supported on the intervalR− = [0,+∞[.

Proof. This result is well-known; the necessary details are left to the interested reader. �

Let us write BMO(R/2Z) for the subspace of 2-periodic functions in BMO(R). Effectively,BMO(R/2Z) is the BMO space of functions on the “circle” R/2Z. As such, the functions inBMO(R/2Z) have Fourier series expansion, which means that the Fourier transform of sucha function is [as a distribution] a sum of Dirac point masses along Z. We formalize thisobservation as a lemma.

Lemma 3.1.4. Suppose f ∈ BMO(R). Then f ∈ BMO(R/2Z) if and only if the distribution F f issupported on the integers Z, and at each point of Z, it is a Dirac point mass.

3.2. The calculation of the Zariski closure of the portions of the lattice-cross on the space-like cone boundary. Let 1E denote the characteristic function of the set E, which equals 1 onE and 0 off of E. The following calculation of the Fourier transform of the function ei/t in thesense of Schwartzian distributions may be known, but we have no specific reference.

Proposition 3.2.1. We have that in the sense of distribution theory on the real line R,

limε→0+

∫R

ei/t+itx−ε|t| dt2π

= δ0(x) − 1[0,+∞[(x)x−1/2 J1(2x1/2) = δ0(x) − 1[0,+∞[(x)+∞∑j=0

(−1) j

j!( j + 1)!x j,

where J1 denotes the standard Bessel function and δ0 is the unit Dirac point mass at 0.

Proof. The calculation follows from a sequence of rather tedious calculations based on formula3.324 in [13]. The details are left to the interested reader. �

Remark 3.2.2. We will not need the precise expression of the Fourier transform of t 7→ ei/t, onlythe fact that apart from the point mass at 0, it is given by a smooth density whose support isthe whole positive half-line R+ = [0,+∞[.

Proof of Theorem 1.7.1. We obviously have the inclusions

Λ++α,β ⊂ zclosΓM (Λ++

α,β), Λ−−α,β ⊂ zclosΓM (Λ−−α,β),

and it remains to show that the Zariski closure contains no extraneous points. We will focusour attention to the set Λ++

α,β; the treatment of the set Λ−−α,β is analogous. In view of (1.4.4) –which relates µ(ξ) to the compressed measure π1µ – we need to do the following. Given a

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10 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

point ξ0 = (ξ01, ξ

02) ∈ R2

\ Λ++α,β, we need to find a finite complex-valued absolutely continuous

Borel measure ν on R×, such that∫R×

eiπ[ξ01t+M2ξ0

2/(4π2t)]dν(t) , 0,

while ∫R×

eiπαmtdν(t) =

∫R×

eiM2βn/(4πt)dν(t) = 0, m,n ∈ Z+,0.

By normalization, we may without loss of generality restrict to the case α := 1 and M := 2π.As ν is absolutely continuous, we may write dν(t) := g(t)dt, where g ∈ L1(R). Given the abovenormalization, we need g to satisfy

(3.2.1)∫R×

eiπ[ξ01t+ξ0

2/t]g(t)dt , 0,

where (ξ01, ξ

02) ∈ R2

\ [(Z+,0 × {0}) ∪ ({0} × βZ+)], while

(3.2.2)∫R×

eiπmtg(t)dt =

∫R×

eiπβn/tg(t)dt = 0, m,n ∈ Z+,0.

We try to find g in the smaller space H1~(R). The weak-star closure of the linear subspace of

BMO(R) spanned by the functions t 7→ eiπmt, with m ∈ Z+,0, is identified with the subspaceBMOA+(R/2Z) [of 2-periodic functions]. The weak-star closure of the linear subspace ofBMO(R) spanned by the functions t 7→ eiπβn/t, with n ∈ Z+,0, is denoted by BMOA−

〈β〉(R). Ifwe let Jβ denote the composition operator Jβ[ f ](t) = f (−β/t), then f ∈ BMOA−

〈β〉(R) meansthat Jβ[ f ] ∈ BMOA−(R/2Z); here BMOA−(R/2Z) denotes the space of complex conjugates offunctions in BMOA+(R/2Z). For g ∈ H1

~(R), (3.2.2) expresses that g annihilates the sum spaceBMOA+(R/2Z) + BMOA−

〈β〉(R). To simplify the notation, let F0(t) := eiπ[ξ01t+ξ0

2/t]; then in view of(3.1.10),

〈g,F0〉R = 〈P+g, P−F0〉R + 〈P−g, P+F0〉R.

We realize that if we can show that

(3.2.3) P+F0 < BMOA+(R/2Z) or P−F0 < BMOA−〈β〉(R),

then we are done. Indeed, if P+F0 < BMOA+(R/2Z), then we just pick a g ∈ H1−

(R) which doesnot annihilate BMOA+(R/2Z), and if P−F0 < BMOA−

〈β〉(R), then we just pick a g ∈ H1+(R) which

does not annihilate BMOA−〈β〉(R). In each case, we achieve (3.2.1). Using some properties of Jβ

(see, e.g., Lemma 3.1.2), we see that (3.2.3) is equivalent to having

(3.2.4) P+F0 < BMOA+(R/2Z) or P−JβF0 < BMOA−(R/2Z).

Moreover, the function F1 := JβF0 is of the same general type as F0: F1(t) = e−iπ[ξ11t+ξ1

2/t], whereξ1

1 := ξ02/β and ξ1

2 := βξ01. We can bring this one step further, and consider F2(t) := eiπ[ξ1

1t+ξ12/t]

[the complex conjugate of F1(t)], and express (3.2.4) in the form

(3.2.5) P+F0 < BMOA+(R/2Z) or P+F2 < BMOA+(R/2Z).

By combining Lemmas 3.1.3 and 3.1.4 with Proposition 3.2.1 in an appropriate manner, wederive that

P+F0 ∈ BMOA+(R/2Z) ⇐⇒ (ξ01, ξ

02) ∈ (R− × R+) ∪ (Z+ × {0}).

The analogous case of F2 in place of F0 gives that

P+F2 ∈ BMOA+(R/2Z) ⇐⇒ (ξ01, ξ

02) ∈ (R+ × R−) ∪ ({0} × βZ+).

If we put these assertion together, we see that

P+F0, P+F2 ∈ BMOA+(R/2Z) ⇐⇒ (ξ01, ξ

02) ∈ (Z+,0 × {0}) ∪ ({0} × βZ+).

This is precisely the forbidden set of (ξ01, ξ

02), and we conclude that (3.2.5) holds. This completes

the proof of the theorem. �

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 11

4. The Hilbert transform on L1 and two other operators

4.1. The Hilbert transform on L1. Let L1,∞(R) denote the weak-L1 space, i.e., the space ofLebesgue measurable functions f : R→ C such that the set

E f (λ) := {x ∈ R : | f (x)| > λ}, λ ∈ R+,

enjoys the estimate (the absolute value of a measurable subset of R stands for its [Lebesgue]length)

|E f (λ)| ≤C f

λ, λ ∈ R+;

the optimal constant C f is the L1,∞(R)-norm of f , which, strictly speaking, is a quasi-norm. Byidentifying functions that coincide almost everywhere (a.e. for short), the space L1,∞(R) is aquasi-Banach space. It is well-known that the Hilbert transform maps H : L1(R) → L1,∞(R).Note, however, that functions in L1,∞(R) are rather wild and, e.g., it is not immediately clearhow to associate such a function with a distribution. After, there is another interpretation of theHilbert transform as a mapping from L1(R) into a space of distributions onR, and it is good toknow that these interpretations of H f for a given f ∈ L1(R) are in a one-to-one correspondence.

The following result characterizes the space H1~(R).

Proposition 4.1.1. Suppose f ∈ L1(R). Then the following are equivalent:(i) f ∈ H1

~(R).(ii) H f ∈ L1(R), where H f is understood as a distribution on the line R.(iii) H f ∈ L1(R), where H f is understood as an almost everywhere defined function in L1,∞(R).

Proof. The implications (i)⇔(ii)⇒(iii) are trivial, so we turn to the remaining implication(iii)⇒(i). This result, however, is the real line analogue of the result for the circle in [19],p. 87. The transfer to the unit disk is handled by an appropriate Moebius map from D toC+. �

As a consequence, we obtain the following result.

Corollary 4.1.2. Suppose f ∈ L1(R), and that H f = 0 pointwise almost everywhere onR. Then f = 0almost everywhere.

Proof. Without loss of generality, f is real-valued. In view of Proposition 4.1.1, f ∈ H1~(R), and

as a consequence, the function F := f + iH f is in H1+(R). But on the real line, F is real-valued,

so that the Poisson extension of F to C+ is real-valued as well. But this Poisson extension isholomorphic in C+, so F must be constant, and the constant is seen to be 0. �

Remark 4.1.3. We note that there are the closely related theories of reflectionless measures (see,e.g., [24]) and of real outer functions [11].

4.2. The Hilbert transform on periodic L1. We let L1(R/2Z) denote the usual Lebesgue spaceof integrable 2-periodic functions, supplied with the norm

‖ f ‖L1(R/2Z) :=∫ 1

−1| f (t)|dt.

At times, we need also the codimension one subspace L10(R/2Z) consisting of all f ∈ L1(R/2Z)

with

〈 f , 1〉[−1,1] :=∫ 1

−1f (t)dt = 0.

The Hilbert transform H is well-defined on L1(R/2Z) as the two-sided principal value integral

(4.2.1) H f (x) := limR→+∞

limε→0+

∫[−R,R]\[x−ε,x+ε]

f (t)dt

x − t.

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12 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

We should state that it is also possible to express the Hilbert transform on L1(R/2Z) in termsof the cotangent kernel:

(4.2.2) H f (x) :=12

pv∫

[−1,1]f (t) cot

π(x − t)2

dt,

where “pv” means that we take the limit as ε→ 0+ of the integral where the the set {x} + 2Z +[−ε, ε] is removed from the interval [−1, 1]. The modified Hilbert transform H is of coursealso well-defined on L1(R/2Z); the difference is a matter of normalization: H f (i) = 0 whileH f (i∞) = 0. Here, i∞ is the point at infinity in the imaginary direction in the extended upperhalf plane. It is well-known that the Hilbert transform H maps L1(R/2Z) into the weak-L1

space of 2-periodic functions, which we denote by L1,∞(R/2Z) (see, e.g., [12]). Note that theaction of the Hilbert transform on the constant function 1 ∈ L1(R/2Z) is H1 = 0.

We define a (closed) subspace of L1(R/2Z) – denoted H1+(R/2Z) – as follows:

(4.2.3) g ∈ H1+(R/2Z) ⇐⇒ g ∈ L1(R/2Z) and ∀n ∈ Z+,0 :

∫ 1

−1eiπntg(t)dt = 0.

The space H1+(R/2Z) is the periodic analogue of the Hardy space H1

+(R). It can be understoodin terms of the Hardy H1-space of the disk. Then g ∈ H1

+(R/2Z) means that g(x) = f (eiπx) forsome f ∈ H1 with f (0) = 0. Moreover, g ∈ H1

−(R/2Z) means that g ∈ H1

+(R/2Z). We put

H1~,0(R/2Z) := H1

+(R/2Z) ⊕H1−(R/2Z), H1

~(R/2Z) := H1~,0(R/2Z) ⊕ C,

where the “⊕” is the the Banach space sense, and the “⊕C” means that we add the constantfunctions to the space. We have an analogue of Proposition 4.1.1 in this case as well.

Proposition 4.2.1. Suppose f ∈ L1(R/2Z). Then the following are equivalent:(i) f ∈ H1

~(R/2Z).(ii) H f ∈ L1(R/2Z), where H f is understood as a distribution on the line R.(iii) H f ∈ L1(R/2Z), where H f is understood as an almost everywhere defined function in L1,∞(R/2Z).

Proof. This is immediate from [19], p. 87. �

4.3. A weighted composition operator. For a positive real β, let J∗β denote the weighted com-position operator

(4.3.1) J∗β[ f ](x) :=β

x2 f (−β/x), x ∈ R×.

Then, by the change-of-variables formula, J∗β : L1(R)→ L1(R) acts isometrically.

Proposition 4.3.1. (0 < β < +∞) J∗β is an isometric isomorphism L1(R) → L1(R). In addition, J∗βmaps H1

+(R)→ H1+(R) and H1

−(R)→ H1

−(R).

Proof. The mapping z 7→ −β/z preserves the upper half-plane C+, and so that functions holo-morphic in C+ are sent to functions holomorphic in C+ under composition by z 7→ −β/z. Theisometric part is already settled, so it remains to check that the space H1

+(R) is preserved underJ∗β [the case of H1

−(R) is identical]. This can be done by checking the property on a dense

subspace (e.g. of rational functions). The details are left to the reader. �

4.4. The periodization operator. Let Π2 : L1(R) → L1(R/2Z) be the periodization operator, asdefined by

Π2 f (x) :=∑j∈Z

f (x + 2 j).

It has the following useful property, for f ∈ L1(R):

(4.4.1)∫ 1

−1eiπntΠ2 f (t)dt =

∑j∈Z

∫ 1

−1eiπnt f (t + 2 j)dt =

∫R

eiπnt f (t)dt, n ∈ Z.

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 13

Proposition 4.4.1. The operator Π2 acts contractively L1(R) → L1(R/2Z). Moreover, Π2 mapsH1

+(R)→ H1+(R/2Z) and H1

−(R)→ H1

−(R/2Z).

Proof. By the triangle inequality and Fubini’s theorem, Π2 is a contraction:∫ 1

−1|Π2 f (x)|dx ≤

∑j∈Z

∫ 1

−1| f (x + 2 j)|dx =

∑j∈Z

∫ 2 j+1

2 j−1| f (x)|dx =

∫R

| f (x)|dx,

It remains to check the mapping properties, which are immediate from (3.1.2) and (3.1.3)together with (4.4.1). �

4.5. Dual formulation of the spanning problem associated with Theorem 1.8.1. We considerthe following problem.

Problem 4.5.1. For which values of the positive real parameter β is the linear span of thefunctions

en(t) := eiπnt, e〈β〉m (t) := e−iπβm/t, m,n ∈ Z+,0,

weak-star dense in H∞+ (R)?

In dual terms, we are asking: When, given that f ∈ L1(R), we do we have the implication

(4.5.1) ∀m,n ∈ Z+,0 : 〈en, f 〉R = 〈e〈β〉m , f 〉R = 0 =⇒ f ∈ H1+(R)?

We first remark that the functions eiπnt and e−iπβm/t belong to H∞+ (R) (they have boundedholomorphic extensions to C+) for m,n ∈ Z+,0. Moreover, the argument involving pointseparation in C+ from [15] applies here as well, and it is a necessary condition that β ≤ 1 forthe implication (4.5.1) to hold. We make note of the observation that we may as well assumethat f ∈ L1

0(R) in (4.5.1), by testing n = 0. Here, we recall that L10(R) stands for the subspace of

L1(R) consisting of the functions f with total integral 0: 〈 f , 1〉R = 0. By (4.4.1),

(4.5.2) 〈en, f 〉R =

∫ 1

−1eiπntΠ2 f (t)dt =: 〈en,Π2 f 〉[−1,1],

so that for f ∈ L1(R), we have the equivalence

Π2 f ∈ H1+(R/2Z) ⇐⇒ ∀n ∈ Z+,0 : 〈en, f 〉R = 0.

We realize that Jβem = e〈β〉m , and, consequently,

〈e〈β〉m , f 〉R = 〈Jβem, f 〉R = 〈em, J∗β f 〉R,

which leads to the equivalence (for f ∈ L1(R))

Π2J∗β f ∈ H1+(R/2Z) ⇐⇒ ∀m ∈ Z+,0 : 〈e〈β〉m , f 〉R = 0.

We can now rephrase the question (4.5.1) and hence Problem 4.5.1.

Problem 4.5.2. (0 < β ≤ 1) Is it true that for f ∈ L1(R),

Π2 f , Π2J∗β f ∈ H1+(R/2Z) =⇒ f ∈ H1

+(R)?

The reverse implication holds (by Propositions 4.3.1 and 4.4.1). We may think of Π2 f andΠ2J∗β f as 2-periodic “shadows” of f , and the issue at hand is whether knowing that the twoshadows are in the right space we may conclude the function comes from the space H1

+(R).Note here that the main result of [15] asserts that f is determined uniquely by its “shadows”Π2 f and Π2J∗β f for 0 < β ≤ 1.

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14 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

4.6. The operatorsΠ2 and J∗β on a space of distributions. We shall be interested in the follow-ing space of distributions on the line R:

L(R) := L1(R) + HL1(R),

which we supply with the appropriate norm

‖u‖L(R) := inf{‖u1‖L1(R) + ‖u2‖L1(R) : u = u1 + Hu2, u1,u2 ∈ L1(R)

}.

We are interested in this space because it is the predual of H∞~ (R) := H∞+ (R) + H∞−

(R) [“realH∞”]. Each such distribution u ∈ L(R) has an associated “principal value” at almost everypoint of the line given by

(4.6.1) pv[u](x) := limε→0+〈Px+iε,u〉R, x ∈ R,

wherePx+iε(t) := π−1 ε

ε2 + (x − t)2

is the Poisson kernel. Here, we use that the adjoint of H on a suitable class of test functions(which ought to include the Poisson kernel) is H∗ = −H. If u = u1 + Hu2, where u j ∈ L1(R) forj = 1, 2, then pv[u] coincides a.e. with the understanding of u1 + Hu2 as a function in L1,∞(R).The following result allows us to identify u with pv[u].

Proposition 4.6.1. If u ∈ L(R) and pv[u] = 0 almost everywhere on R, then u = 0 as a distribution.

Proof. We write u = u1 + Hu2, where u j ∈ L1(R) for j = 1, 2. Since u2 ∈ L1(R) and pv[Hu2] =

−u1 ∈ L1(R), it follows from Proposition 4.1.1 that u2 ∈ H1~(R) and that Hu2 ∈ L1(R) as a

distribution. Since the Hilbert transform H leaves the space H1~(R) invariant, we also obtain

that u1 ∈ H1~(R), and then it follows that u = 0 as a distribution. �

We shall also need the analogous space of 2-periodic distributions:

L(R/2Z) := L1(R/2Z) + HL1(R/2Z),

supplied with the appropriate norm

‖u‖L(R/2Z) := inf{‖u1‖L1(R/2Z) + ‖u2‖L1(R/2Z) : u = u1 + Hu2 for u1,u2 ∈ L1(R/2Z)

}.

To each such distribution we associate a “principal value” according to formula (4.6.1), wherea little more care is needed because the tails do not decay near infinity due to periodicity.Instead, it is probably better to work with the Poisson kernel coming from the unit disk via theexponential map z 7→ eiπz and use the dual action

〈 f , g〉[−1,1] :=∫ 1

−1f (t)g(t)dt.

There is then an analogue of Proposition 4.6.1 in the 2-periodic setting.

Proposition 4.6.2. If u ∈ L(R/2Z) and pv[u] = 0 almost everywhere on R, then u = 0 as adistribution.

Proof. The proof is analogous to that of Proposition 4.6.1 but with Proposition 4.1.1 replacedby Proposition 4.2.1. �

The proof in [19], p. 87, may be localized to arc of the unit circle T. Using that, we may auniqueness result which combines Proposition 4.6.1 and 4.6.2. If u = v + w, where v ∈ L(R)and w ∈ L(R/2Z), then v and w are uniquely determined by the sum u (as a distribution onR).This is analogous to the classical theory of almost periodic functions. We write

L(R) ⊕ L(R/2Z)

for the space of such sums. Moreover, the “principal value” associated with each of thefunctions v and w gives a principal value for u: pv[u] = pv[v] + pv[w].

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 15

Proposition 4.6.3. If u ∈ L(R) ⊕ L(R/2Z) and pv[u] = 0 almost everywhere on R, then u = 0 as adistribution.

Proof. This is done using the local version of [19], p. 87; compare with the proof of Proposition4.6.1. �

We shall mainly work with the slighly smaller spaces

L0(R) := L10(R) + HL1

0(R), L0(R/2Z) := L10(R/2Z) + HL1

0(R/2Z),

where we recall that L10(R) is the codimension one subspace of functions with vanishing

total integral on R. The space L10(R/2Z) is defined analogously: f ∈ L1

0(R/2Z) means thatf ∈ L1(R/2Z) and that

〈 f , 1〉[−1,1] =

∫ 1

−1f (t)dt = 0.

Proposition 4.6.4. The Hilbert transform H maps L0(R)→ L0(R) and L0(R/2Z)→ L0(R/2Z). Assuch, its square is minus the identity: H2 = − I.

Proof. We show how to obtain the first assertion; the second one is analogous. It suffices toshow that H2 f = − f for f ∈ L1

0(R) in the sense of distributions. For a test function ϕ,

〈ϕ,H2 f 〉R = 〈(H∗)2ϕ, f 〉R〈H2ϕ, f 〉R = −〈ϕ, f 〉R,

since H∗ = −H and H2 = − I holds on test functions. The assertion follows. �

Proposition 4.6.5. The periodization operatorΠ2 maps L10(R)→ L1

0(R/2Z) andL0(R)→ L0(R/2Z).Moreover, HΠ2 f = Π2H f holds for f ∈ L1

0(R).

Proof. For f ∈ L10(R), we have that∫ 1

−1Π2 f (t)dt =

∫ 1

−1

∑j∈Z

f (t + 2 j)dt =

∫R

f (t)dt = 0,

so that Π2 maps L10(R) → L1

0(R/2Z). The assertion that HΠ2 f = Π2H f for f ∈ L10(R) can be

obtained by an argument based on the fact that H commutes with translations. To be morespecific, if f is smooth with compact support and 〈 f , 1〉R = 0, then the sum defining Π2 finvolves finitely many nonzero terms and we find that HΠ2 f = Π2H f . The general resultfollows by approximation. The remaining assertion concerning the mapping properties nowfollows as well. �

Proposition 4.6.6. (0 < β < +∞) The operator J∗β maps L10(R) → L1

0(R). Moreover, it commuteswith the Hilbert transform on L1

0(R): J∗βH f = HJ∗β f for f ∈ L10(R). As a consequence, J∗β maps

L0(R)→ L0(R).

Proof. By the change-of-variables formula,∫R

f (x)dx = β

∫R

f (−β/x)dxx2 =

∫R

J∗β f (x)dx,

so that f ∈ L10(R) if and only if J∗β f ∈ L1

0(R). Next, we see by a change of variables that

HJ∗β f (x) = pv∫R

β

t2(x − t)f (−β/t)dt = pv

∫R

txt + β

f (t)dt,

which we may compare with

J∗βH f (x) = −pv∫R

β

x(β + tx)f (t)dt.

We check thatt

xt + β+

β

x(β + tx)=

1x

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16 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

which is a constant with respect to the integration variable. We conclude that since f ∈ L10(R), we

have J∗βH f = HJ∗β f , as claimed. It should be mentioned that the principal value integrals involveremoval of infinitesimally short intervals which do not cause any additional difficulty. �

4.7. Second dual formulation of the spanning problem associated with Theorem 1.8.1. Weconsider the following problem.

Problem 4.7.1. (0 < β ≤ 1) Is it true that for f ∈ L0(R),

Π2 f = Π2J∗β f = 0 =⇒ f = 0?

Proposition 4.7.2. If the answer to Problem 4.7.1 is affirmative, then the answer to Problem 4.5.2 isaffirmative as well.

Proof. Let f ∈ L1(R) be such that Π2 f ∈ H1+(R/2Z) and Π2J∗β f ∈ H1

+(R/2Z). Then, as a first

step, f ∈ L10(R) by the identity (4.4.1) with n = 0. We recall the notation P− := 1

2 (I−iH) for theprojection to the conjugate-holomorphic functions in C+.

Next, we consider the distribution g := 2P− f = f − iH f which then belongs to L0(R). First,we calculate that

Π2g = Π2 f − iΠ2H f = Π2 f − iHΠ2 f = (I−iH)Π2 f = 2P−Π2 f = 0,

where we use that H andΠ2 commute (Proposition 4.6.5) and that P− projects the whole spaceH1

+(R/2Z) to 0. In a similar fashion, we obtain that

Π2J∗βg = Π2J∗β f − iΠ2J∗βH f = Π2J∗β f − iΠ2HJ∗β f

= Π2J∗β f − iHΠ2J∗β f = (I−iH)Π2J∗β f = 2P−Π2J∗β f = 0,

where we used the commutativity of H,Π2, J∗β as expressed in Propositions 4.6.5 and 4.6.6,and again the fact that P− projects the whole space H1

+(R/2Z) to 0. So, we have arrived atthe conclusion Π2g = Π2J∗βg = 0, and given that Problem 4.7.1 has an affirmative answer, weobtain that g = 0 as an element of L0(R). But g was the function g = P− f , so that P− f = 0. Fromf ∈ L1

0(R) and P− f = 0 it is easy to see that f ∈ H1+(R), which finishes the proof. �

5. A transfer operator on a space of distributions

5.1. Splitting of the periodization operator. We split the periodization operator Π2 in twoparts: Π2 = I +S2, where S2 is the operator defined by

S2 f (x) :=∑j∈Z×

f (x + 2 j),

whenever the right-hand side is meaningful in the sense of distributions. Here, we use thenotation Z× := Z \ {0}. In view of Proposition 4.6.5, S2 has the following mapping properties.We consider the spaces

L10(R) ⊕ L1

0(R/2Z) and L0(R) ⊕ L0(R/2Z)

as subspaces of L(R) ⊕ L(R/2Z), which consists of distributions on the line R.

Proposition 5.1.1. The operator S2 maps L10(R) → L1

0(R) ⊕ L10(R/2Z) and L0(R) → L0(R) ⊕

L0(R/2Z).

Let us consider the operator

Tβ :=: S2J∗β : L0(R)→ L0(R) ⊕ L0(R/2Z),

which also maps L10(R)→ L1

0(R) ⊕ L10(R/2Z).

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 17

5.2. A balancing function. Let us consider the function

h(x) := 1[−a,a](x) − abx−21R\[−b,b](x), x ∈ R,

for positive reals a, b. Then ∫R

h(x)dx = 0,

so that h ∈ L10(R). We quickly calculate the Hilbert transform of h:

Hh(x) =1π

log∣∣∣∣∣x + ax − a

∣∣∣∣∣ +abπx2 log

∣∣∣∣∣x + bx − b

∣∣∣∣∣ − 2aπx, x ∈ R.

Then Hh ∈ L1(R) by inspection, so that h ∈ H1~(R), and consequently, Hh ∈ H1

~(R); in particular,Hh ∈ L1

0(R) as well.

5.3. A weighted transfer operator and its extension.

6. The Hilbert transform on the hyperbola

Hilbert transforms. The Hilbert transform H of a function f ∈ L1(R) is

H[ f ](x) := pv1π

∫R

f (t)x − t

dt, x ∈ R,

wherever the integral makes sense. This may be thought of both as function in weak L1, andas a distribution. We shall need to think of it as a distribution. We easily extend the notion tomeasures: for a finite complex-valued Borel measure ν on R, we put

dH[ν](x) :=[

pv1π

∫R

f (t)x − t

dt]dx,

where the notation suggests that we get a measure; this is just a formality, as we generallyexpect only a distribution. As for the interpretation as a weak L1 function, we refer to the recentcontribution [25] by Poltoratski, Simon, and Zinchenko.

To simplify the notation, we restrict our attention to the hyperbola Γm with m = 2π, describedby the equation x1x2 = −1, and denote it by Γ (dropping the subscript). Let us consider thefollowing measure on Γ:

dλ(x1, x2) := |x1|−1dδ−1/x1 (x2)dx1,

which has the symmetry property

dλ(x1, x2) = dλ(x2, x1).

For a finite Borel measure ν supported on Γ, we put

dHΓ[ν](x1, x2) :=[

pv1π

∫Γ

dν(y1, y2)x1 − y1

]|x1|dλ(x1, x2),

which in general need not be a Borel measure, but rather can be interpreted as a distributionsupported on Γ. The way things are set up, π1 intertwines between HΓ and H:

(6.0.1) dπ1HΓ[ν] = dH[π1ν].

After a moment’s reflection we see that

dHΓ[ν](x1, x2) =1π

sgn(x1)ν(R2)dλ(x1, x2) +[

pv1π

∫Γ

dν(y1, y2)x2 − y2

]|x2|dλ(x1, x2).

Here, sgn(t) is the sign of t ∈ R (sgn(0) = 0, sgn(t) = 1 for t > 0, and sgn(t) = −1 for t < 0). Ifwe let AC0(Γ) denote the codimension one subspace of AC(Γ) consisting of measures ν withν(R2) = 0, we see that

dHΓ[ν](x1, x2) =[

pv1π

∫Γ

dν(y1, y2)x1 − y1

]|x1|dλ(x1, x2) =

[pv

∫Γ

dν(y1, y2)x2 − y2

]|x2|dλ(x1, x2),

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18 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

which means that if π2 is the compression to the x2-axis,

π2ν(E) =

∫E

dπ2ν(x2) :=∫R×E

dν(x),

then π2 intertwines H and HΓ as well:

(6.0.2) dπ2HΓ[ν] = dH[π2ν], ν ∈ AC0(Γ).

Since HΓ relates to the Hilbert transform of the compression to each of the two axes, it appears tobe a rather natural operator. We call it the Hilbert transform on Γ, and introduce the real H1 spaceon Γ, denoted ACH(Γ), which by definition consists of those ν ∈ AC0(Γ) with HΓ[ν] ∈ AC0(Γ).Supplied with the norm

‖ν‖ACH(Γ) := ‖ν‖M(R2) + ‖HΓ[ν]‖M(R2), ν ∈ ACH(Γ),

it is a Banach space, and the injection ACH(Γ) ↪→ M(Γ) is continuous, which makes ACH(Γ) aBanach subspace of M(Γ) which is contained in AC0(Γ).

Remark 6.0.1. In terms of the Fourier transform, we get from (6.0.1) and (6.0.2) that

(6.0.3) ∀ν ∈ AC0(Γ) : HΓ[ν](ξ1, 0) = i sgn(ξ1 )ν(ξ1, 0), HΓ[ν](0, ξ2) = i sgn(ξ2 )ν(0, ξ2),

for all ξ1, ξ2 ∈ R. For ν ∈ AC0(Γ), the function v := ν is continuous on R2, tends to 0 atinfinity (this is a consequence of the curvature of Γ), and has v(0, 0) = ν(0, 0) = ν(R2) = 0.So it is immediate from (6.0.3) that like v = ν, the Fourier transform v∗ := HΓ[ν] solves theKlein-Gordon equation (KG) and has v∗(ξ1, ξ2) = i sgn(ξ1 + ξ2)v(ξ1, ξ2) if ξ1ξ2 = 0, so v∗(ξ1, ξ2)makes sense as a continuous function on ξ1ξ2 = 0. Whether in general v∗ is automaticallycontinuous throughout R2 is not so clear. But if ν ∈ ACH(Γ), there is of course no problem.

7. Strong and weak Heisenberg uniqueness for the hyperbola

Strong Heisenberg uniqueness for the hyperbola. We recall the definition of strong Heisen-berg uniqueness pairs (Definition ??). First, we need some (standard) notation. Let R+, R−denote the sets of positive and negative reals, respectively, and put R+ := R+∪{0}, R− := R−∪{0}.We need the (standard) notion of a Riesz set E1 ⊂ R: E1 is a Riesz set if µ ∈ M(R) and µ = 0 onE1 implies that µ is absolutely continuous. Here, we write

µ(ξ) :=∫R

eiπξxdµ(x), ξ ∈ R.

By the well-known F. and M. Riesz theorem, any unbounded interval is a Riesz set (seeProposition 9.0.19 below). This suggests the following definition.

Definition 7.0.2. Fix m > 0. A set E ⊂ R2 is a Riesz set for the hyperbola Γm if every measureµ ∈M(Γm) with µ = 0 on E is absolutely continuous with respect to arc length measure.

Remark 7.0.3. We observe that given a Riesz set E1 ⊂ R, the lifted sets E1 × {0} and {0} × E1 areboth Riesz sets for the hyperbola Γm.

We consider the open quadrants

(7.0.4) R2++ := R+ ×R+, R

2−− := R− ×R−, R2

+− := R+ ×R−, R2−+ := R− ×R+,

and we write R2++, R

2−−, R2

+−, R2−+ for the corresponding closed quadrants. The quadrants R2

++

and R2−−

are space-like, while R2+− and R2

−+ are time-like.

Theorem 7.0.4. Fix positive reals α, β,m, and a Riesz set E ⊂ R2−−

for the hyperbola. Then ΛEα,β :=

Λα,β ∪ E is a Fourier uniqueness set for M(Γm) if and only if αβm2≤ 4π2.

In other words, for Riesz sets E ⊂ R−, (Γm,ΛEα,β) is a strong Heisenberg uniqueness pair

provided that αβm2≤ 4π2.

Remark 7.0.5. The assertion remains the same if we replace the assumption E ⊂ R2−−

by E ⊂ R2++.

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 19

Weak Heisenberg uniqueness for the hyperbola. For general m > 0, let AC0(Γm) denote thesubspace of AC(Γm) consisting of measures ν with ν(R2) = 0. A scaling argument allows us todefine the Hilbert transform HΓm on Γm for general m > 0, so that the analogue of (6.0.3) holds:

(7.0.5) ∀ν ∈ AC0(Γm) : HΓm [ν](ξ1, 0) = i sgn(ξ1 )ν(ξ1, 0), HΓm [ν](0, ξ2) = i sgn(ξ2 )ν(0, ξ2),

for all ξ1, ξ2 ∈ R. We define ACH(Γm) to be the (dense) subspace of AC0(Γm) of measures νwith HΓm [ν] ∈ AC0(Γm). This class of measures is better-behaved, and it is quite natural to useit to define a slightly bigger class of uniqueness sets.

Definition 7.0.6. The pair (Γm,Λ) is a weak Heisenberg uniqueness pair if Λ ⊂ R2 is a Fourieruniqueness set for ACH(Γm). Moreover, (Γm,Λ) is a K-local weak Heisenberg uniqueness pair if Λ(with Λ ⊂ K ⊂ R2) is a K-local Fourier uniqueness set for ACH(Γm).

Remark 7.0.7. (a) As the terminology suggests, it is easier for (Γm,Λ) to be a weak Heisenberguniqueness pair than to be a Heisenberg uniqueness pair. The same is true for the K-localvariant.

(b) As ACH(Γm) ⊂ AC0(Γm) automatically, we realize that for a set Λ ⊂ R2, we have theequivalence (WHUP = weak Heisenberg uniqueness pair)

(Γm,Λ) is a WHUP ⇐⇒ (Γm,Λ ∪ {0}) is a WHUP,

and, more generally, if 0 ∈ K, we have (WHUP(K) = K-local weak Heisenberg uniqueness pair)

(Γm,Λ) is a WHUP(K) ⇐⇒ (Γm,Λ ∪ {0}) is a WHUP(K).

Let Q be an open quadrant, i.e.,

Q ∈ {R2++,R

2−−,R

2+−,R

2−+};

we write Q for the closure of Q. The difference between space-like and time-like quarter-planesis made obvious by the the following.

Proposition 7.0.8. Fix a positive real m. Then:(a) For a time-like quarter-plane Q, the boundary ∂Q is a Fourier uniqueness set for M(Γm).(b) For a space-like quarter-plane Q, the set Q is not a Fourier uniqueness set for ACH(Γm).

We return to our lattice-cross Λα,β (see (1.5.1)), and keep Q as an open quadrant. Could itbe that Λα,β ∩ Q is a Q-local Fourier uniqueness set for AC(Γm), or at least for ACH(Γm)? Theanswer to this question, as it turns out, depends on whether the quadrant Q is space-like ortime-like. For the time-like quarter-planes, there is no analogue of Theorem 1.5.1, as can beseen from the following.

Theorem 7.0.9. (Time-like quarter-planes) Fix m, α, β > 0. Then, for Q ∈ {R2+−,R

2−+}, the pair

Λα,β ∩ Q is not a Q-local Fourier uniqueness set for ACH(Γm). In particular, Λα,β ∩ Q is not a Fourieruniqueness pair for ACH(Γm).

As regards the space-like quarter-planes, there is indeed an analogue.

Theorem 7.0.10. (Space-like quarter-planes) Fix positive reals m, α, β. Then, for Q ∈ {R2++,R

2−−},

Λα,β ∩ Q is a Q-local Fourier uniqueness set for ACH(Γm) if and only if αβm2≤ 4π2.

Remark 7.0.11. It is not known whether for αβm2≤ 4π2 the set Λα,β ∩ Q is a Q-local Fourier

uniqueness set for AC(Γm). This problem appears rather challenging, and it has an attractivereformulation (see Problems 10.0.23 and 10.0.24).

A family of weak Heisenberg uniqueness pairs for the hyperbola. For a point ξ0 = (ξ01, ξ

02) ∈

R2, we consider the distorted lattice-cross

Λ〈ξ0〉

α,β := (Λα,β ∩ R2−−) ∪ ((Λα,β ∩ R

2++) + {ξ0

}),

which has the general appearance of a “slanted lattice-waist” if ξ01 > 0 or ξ0

2 > 0.

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20 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

Theorem 7.0.12. Suppose α, β,m are positive reals, with αβm2≤ 4π2. Then (Γm,Λ

〈ξ0〉

α,β ) is a weakHeisenberg uniqueness pair if and only if min{ξ0

1, ξ02} < 0 or ξ0 = (0, 0).

Remark 7.0.13. For αβm2 > 4π2, we do not know what happens, but we suspect that Λ〈ξ0〉

α,β is nota weak Heisenberg uniqueness pair, independently of the location of ξ0.

8. Heisenberg uniqueness for one branch of the hyperbola

The branches of the hyperbola. The hyperbola Γm naturally splits into two connectivitycomponents:

(8.0.6) Γ+m :=

{x ∈ R2 : x1x2 = −

M2

4π2 , x1 > 0}, Γ−m :=

{x ∈ R2 : x1x2 = −

M2

4π2 , x1 < 0}.

In light of Theorem 1.5.1, it is natural to ask what happens if we replace Γm by one of Γ+m,Γ

−m.

In view of the invariance property (inv-2), it suffices to treat Γ+m.

Theorem 8.0.14. For positive reals m, α, β, (Γ+m,Λα,β) is a Heisenberg uniqueness pair if and only if

αβm2 < 16π2. Moreover, for αβm2 = 16π2, (Γ+m,Λα,β) is a Heisenberg uniqueness pair with defect 1.

Remark 8.0.15. For αβm2 > 16π2, (Γ+m,Λα,β) is not a Heisenberg uniqueness pair with a finite

defect d (i.e., the defect is infinite). This follows from the results of [4]; cf. Remark 11.0.25.

In the critical case αβm2 = 16π2, we can get rid of the defect by adding a point on the crosswhich does not lie on the lattice-cross.

Corollary 8.0.16. Suppose m, α, β are positive reals with αβm2 = 16π2. Pick a point ξ0∈ (R × {0}) ×

({0} ×R) on the cross, and put Λ0α,β := Λα,β ∪ {ξ0

}. Then (Γ+m,Λ

0α,β) is a Heisenberg uniqueness pair if

and only if ξ0 < Λα,β.

9. Elements of Hardy space theory

Hardy spaces. We shall need certain subspaces of L1(R) and L∞(R). If f is in L1(R) or in L∞(R),we define its Poisson extension to the upper half-plane

C+ := {z ∈ C : Im z > 0}

by the formula

UC+[ f ](z) :=

Im zπ

∫R

f (t)|z − t|2

dt, z ∈ C+.

The function UC+[ f ] is harmonic in C+, and its boundary values are those of f in the natural

sense. It is standard to identify the function f with its Poisson extension. We say that f ∈ H1+(R)

if f ∈ L1(R) and UC+[ f ] is holomorphic. Likewise, we say that f ∈ H∞+ (R) if f ∈ L∞(R) and

UC+[ f ] is holomorphic. Analogously, if f ∈ L1(R) and UC+

[ f ] is conjugate holomorphic (thismeans that the complex conjugate is holomorphic), we say that f ∈ H∞

−(R), while if f ∈ L∞(R)

and UC+[ f ] is conjugate holomorphic, we write f ∈ H∞

−(R). Clearly, f ∈ H1

−(R) if and only if its

complex conjugate is in H1+(R), and the same goes for H∞

−(R) and H∞+ (R).

We shall use the following bilinear form on R:

〈 f ,F〉R :=∫R

f (t)F(t)dt,

whenever it is well-defined. We shall frequently need the following well-known characteriza-tion of H1

+(R).

Proposition 9.0.17. Let us agree to write eτ(t) := eiπτt. Then the following are equivalent for a functionf ∈ L1(R): (a) f ∈ H1

+(R), and (b) 〈 f , eτ〉R = 0 for all τ > 0.

The following result is also standard.

Proposition 9.0.18. (a) If f ∈ H1+(R) and F ∈ H∞+ (R), then F f ∈ H1

+(R), and 〈 f ,F〉R = 0.(b) If f ∈ L1(R), then f ∈ H1

+(R) if and only if 〈 f ,F〉R = 0 for all F ∈ H∞+ (R).

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 21

We need also the next result, attributed to F. and M. Riesz.

Proposition 9.0.19. Suppose µ is a complex-valued finite Borel measure on R. If

∀τ > 0 :∫R

eiπτtdµ(t) = 0,

then µ is absolutely continuous, and dµ(t) = f (t)dt, where f ∈ H1+(R).

Applications of Hardy space methods. We now show (see Proposition 9.0.20 below) that thelinear span of the functions

t 7→ eπiξ1t, t 7→ eπiεξ2/t, ξ = (ξ1, ξ2) ∈ R2++,

is weak-star dense in L∞(R).

Proposition 9.0.20. Let ν be a complex-valued finite Borel measure on R. If∫R

eπiξ1tdν(t) =

∫R

eπiξ2/tdν(t) = 0, for all ξ1, ξ2 > 0,

then ν = 0.

Proof. By Proposition 9.0.19, the assumptions entail that dν(t) = f1(t)dt and dν(−1/t) = f2(−t)dt,where f j ∈ H1

+(R), j = 1, 2. By equating the two ways to represent dν, we see that

f1(t) = t−2 f2(1/t), t ∈ R,

so that f1 has an analytic pseudocontinuation to the lower half-plane (for Im t > 0, Im t−1 < 0).The pseudocontinuation is of course a genuine holomorphic continuation to C× := C \ {0} (wecan use, e.g., Morera’s theorem). In terms of g j(t) := t f j(t), for j = 1, 2, the above relation readsg1(t) = g2(1/t). The functions g j, j = 1, 2, extend holomorphically to C×, and have the estimate

|g j(t)| ≤ ‖ν‖|t|| Im t|

, t ∈ C \R, j = 1, 2,

Using the theory around the log log theorem (attributed to Levinson, Sjöberg, Carleman, Beurl-ing; see e.g. [21], pp. 374–383, also [3]) it is not difficult to show that such functions g j, j = 1, 2,must be constant. But then the constant must be 0, for otherwise, f j, j = 1, 2, would not be inH1

+(R). �

Proof of Proposition 7.0.8. We first consider time-like quarter-planes Q ∈ {R2+−,R

2−+}. In both

cases, the problem boils down to Proposition 9.0.20, which settles the issue.We turn to space-like quarter-planes Q ∈ {R2

++,R2−−}. Here, the matter is settled by Propo-

sition 9.0.18. The non-trivial measures µ ∈ ACH(Γm) whose Fourier transform vanishes on Qhave compressions to the x1-axis of the form f (t)dt, where f ∈ H1

+(R) or f ∈ H1−

(R) (which ofthe two it is depends on whether Q is R2

++ or R2−−

). �

We next show (see Proposition 9.0.21 below) that the linear span of the functions

t 7→ eπiξ1t, t 7→ eπiξ2/t, ξ = (ξ1, ξ2) ∈ R2+−,

is weak-star dense in H∞+ (R).

Proposition 9.0.21. Let ν be a complex finite absolutely continuous measure on R. Then∫R

eπiξ1tdν(t) =

∫R

e−πiξ2/tdν(t) = 0, for all ξ1, ξ2 > 0,

if and only if dν(t) = f (t)dt where f ∈ H1+(R).

Proof. The assertion is immediate from Propositions 9.0.17 and 9.0.18, once it is observed thatthe function F(t) = eπiξ2/t is in H∞(R) for ξ2 < 0. �

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22 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

The predual of H∞ on the line. It is well-known and can be seen from Proposition 9.0.18 thatthe predual of H∞+ (R) is the quotient space L1(R)/H1

+(R) with respect to the standard bilinearform 〈·, ·〉R. For f ∈ H1

+(R) and F ∈ H∞+ (R), we have – by Proposition 9.0.18 – 〈 f ,F〉R = 0, whichis why we need to mod out with respect to H1

+(R) in the predual.

The dual of H1 on the line. If we put

H1real(R) := H1

+(R) ⊕H1−(R),

and supply this space with the natural norm; as H1+(R)∩H1

−(R) = {0}, this is just the sum of the

two norms:

‖ f1 + f2‖H1real(R) := ‖ f1‖H1

+(R) + ‖ f2‖H1−

(R), f1 ∈ H1+(R), f2 ∈ H1

−(R).

The Cauchy projection

P+ : H1real(R)→ H1

+(R), P+[ f1 + f2] := f1 for f1 ∈ H1+(R), f2 ∈ H1

−(R),

is a thus norm contraction. It is related to the Hilbert transform H:

P+ f = 12 ( f + iH[ f ]), f ∈ H1

real(R).

The space H1real(R) is a Banach space, and a dense (Banach) subspace of

L10(R) := { f ∈ L1(R) : 〈 f , 1〉R = 0}.

Actually, the space H1real(R) has an alternative characterization in terms of the Hilbert transform:

H1real(R) =

{f ∈ L1

0(R) : H[ f ] ∈ L10(R)

}.

The dual space of L10(R) is L∞(R)/{constants}. The dual space of H1

real(R) is BMO(R), which isunderstood as the space of functions with bounded mean oscillation, modulo the constants. TheCauchy projection also acts on the dual side:

P+ : BMO(R)→ BMOA+(R),

where BMOA+(R) is the subspace of BMO(R) which is dual to H1−

(R) with respect to 〈·, ·〉R.

The predual of H∞ on the unit circle. We also need Hardy spaces in the context of the unitcircle (or the unit disk, if we talk about the harmonic extension). A function in L1(T) (T is theunit circle) has norm

‖ f ‖L1(T) :=∫ π

−π| f (eit)|

dt2π,

and we use the standard bilinear form

〈 f , g〉T :=∫ π

−πf (eit)g(eit)

dt2π, f ∈ L1(T), g ∈ L∞(T).

The Poisson extension to the unit diskD of f ∈ L1(T) is given by the formula

UD f (z) :=∫ π

−π

1 − |z|2

|1 − ze−it|2f (eit)

dt2π, z ∈ D.

If f ∈ L1(T) and UD f is holomorphic inD, we write f ∈ H1+(T). If, in addition, UD f (0) = 0, we

write f ∈ H1+,0(T). We frequently identify functions on the unit circle T with their harmonic

extensions to D. If H1(D),H10(D) are defined as the spaces of such extensions of boundary

functions, we thus identify H1+(T) � H1(D), H1

+,0(T) � H10(D). In a similar fashion, H∞+ (T) �

H∞(D). It is well-known that with respect to the standard bilinear form, the predual of H∞+ (T)may be identified with L1(T)/H1

+,0(T).

Periodic Hardy spaces and the exponential mapping. Let L∞(R/2Z) consist of those f ∈ L∞(R)which are 2-periodic: f (x+2) = f (x). Similarly, we let H∞+ (R/2Z) consist the 2-periodic functionsin H∞+ (R). The exponential mapping x 7→ eiπx provides an identification R/2Z � T, and theupper half space C+ modulo 2Z corresponds to the punctured diskD \ {0}. The results for theunit circle therefore carry over in a natural fashion to the 2-periodic setting. We let L1(R/2Z)

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 23

denote the space of locally integrable 2-periodic functions on R, supplied with the Banachspace norm

‖ f ‖L1(R/2Z) :=∫

[−1,1]| f (x)|dx.

We let H1+(R/2Z) denote the subspace of L1

(R/2Z) consisting of functions whose Poissonextension to the upper half plane C+ are holomorphic. The holomorphic extension is thenautomatically 2-periodic, and if, for f ∈ H1

+(R/2Z), the holomorphic extension (also denotedby f ) has f (z) → 0 as Im z → +∞, we write f ∈ H1

+,0(R/2Z). Via the exponential mapping,H1

+(R/2Z) corresponds to H1(T), and H1+,0(R/2Z) to H1

+,0(T). By carrying over the resultsavailable in the setting of the circle T, we see that with respect to the bilinear form

〈 f , g〉[−1,1] :=∫

[−1,1]f (x)g(x)dx, f ∈ L1

(2)(R), g ∈ L∞(2)(R),

H1+,0(R/2Z) is the pre-annilator of H∞+ (R/2Z), and we may identify[

L1(R/2Z)/H1+,0(R/2Z)

]∗= H∞+ (R/2Z).

10. Some reformulations and proofs

Strong Heisenberg uniqueness for the hyperbola. We may now supply the proof of Theorem7.0.4.

Proof of Theorem 7.0.4. We consider µ ∈ M(Γm) with µ = 0 on ΛEα,β. The assumption that E is a

Riesz set for Γm entails that the µ is absolutely continuous with respect to arc length measureon Γm. The main theorem of [15] – based on the dynamics of the Gauss-type map t 7→ −β/tmodulo 2 on the interval ] − 1, 1] – shows that for αβm2

≤ 4π2, the assumption that µ = 0 onΛα,β implies that µ = 0 identically. If we use that E ⊂ R−, we can adapt the counterexamplefrom [15] – involving harmonic extensions – to construct non-trivial measures µ ∈ ACH(Γm)with µ = 0 on ΛE

α,β in case αβm2 > 4π2. The proof is complete. �

The dual formulation for time-like quarter-planes. Theorem 7.0.9 deals with the time-likequarter-planes Q ∈ {R2

+−,R2−+}. If we take the invariance (inv-4) into account, with T as the

reflection in the origin (x1, x2) 7→ (−x1,−x2), we realize that it suffices to consider Q = R2++. The

dual formulation of the theorem runs as follows. For all triples α, β,m > 0, the linear span of thefunctions

eπiα jt, eiβm2k/(4πt), j, k = 0, 1, 2, . . . ,

fails to be weak-star dense in L∞(R). By a scaling argument, we may assume that

α = 1, m = 2π,

so that we are dealing with the linear span of

eπi jt, eπβik/t, j, k = 0, 1, 2, . . . .

This dual formulation of course requires that have Proposition 7.0.8 at our disposal. Actually,Proposition 7.0.8 may be deduced in a straightforward fashion from Propositions 9.0.20 and9.0.21. We leave the necessary details to the reader. This allows us to proceed with the proofof Theorem 7.0.9.

Proof of Theorem 7.0.9. First, we note that the functions eπi jt belong to H∞+ (R) for j = 0, 1, 2, . . .,while the functions eπβik/t instead belong to H∞

−(R) for k = 0, 1, 2, . . .. This means that the

spanning vectors live in rather different subspaces and have no chance to span BMO(R) evenafter weak-star closure. To make this more concrete, we pick a point z0 ∈ C+ in the upperhalf-plane and consider the function

fz0 (t) :=1

t − z0−

1t − 2 − z0

, t ∈ R.

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24 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

Clearly, fz0 (t) = O(t−2) as |t| → +∞, and so fz0 ∈ L1(R). Actually, we have fz0 ∈ H1−

(R) ⊂ H1real(R).

We may use the calculus of residue to obtain that∫R

fz0 (t) eπi jtdt = 2πi(eπi jz0 − eπi j(z0+2)) = 0, j = 0, 1, 2, . . . .

Next, we may show that ∫R

fz0 (t) eπβik/tdt = 0, k = 0, 1, 2, . . . ,

by appealing to Proposition 9.0.18 (we will need to take complex conjugates if we work inthe setting of the upper half-plane). So, for each z0 ∈ C+, fz0 annihilates the subspace, whichconsequently cannot be weak-star dense. �

Remark 10.0.22. The argument of the proof of theorem 7.0.9 actually shows that the weak-starclosure of the subspace spanned by eπi jt, eπβik/t, for j, k = 0, 1, 2, . . ., has infinite codimension inBMO(R).

The dual formulation for space-like quarter-planes. Theorem 7.0.10 and the open problemmentioned in Remark 7.0.11 deal with the space-like quarter-planes Q ∈ {R2

++,R2−−}. If we take

the invariance (inv-4) into account, with T as the inversion (x1, x2) 7→ (−x1, x2), we realize thatit suffices to consider Q = R2

+−. The dual formulation of the theorem runs as follows. For alltriples of positive numbers α, β,m, the linear span of the functions

eπiα jt, e−iβm2k/(4πt), j, k = 0, 1, 2, . . . ,

(taken modulo the constants) is weak-star weak-star dense in BMOA+(R) if and only if αβm2≤ 4π2.

Alternatively, given µ ∈ ACH(R), we consider its compression to the x1-axis π1µ, which hasdπ1µ(t) = f (t)dt, where f ∈ H1

real(R). We need to show that⟨t 7→ eπiα jt, f

⟩R

=⟨t 7→ e−iβm2k/(4πt), f 〉R = 0, j, k = 0, 1, 2, . . . ,

entails that f ∈ H1+(R) if and only if αβm2

≤ 4π2.

Proof of Theorem 7.0.10. The necessity of the condition αβm2≤ 4π2 is just as in [15], so we focus

on the sufficiency. We split f = f1 + f2, where f1 ∈ H1+(R) and f2 ∈ H1

−(R). Now, if we apply

Proposition 9.0.21 to f1, we conclude that

(10.0.7)⟨t 7→ eπiα jt, f2

⟩R

=⟨t 7→ e−iβm2k/(4πt), f2〉R = 0, j, k = 0, 1, 2, . . . .

Next, as f2 ∈ H1−

(R), (10.0.7) actually holds for all j, k ∈ Z. This puts us in the setting of [15],and we find that f2 = 0. The claim f ∈ H1

+(R) follows. �

An open problem for space-like quarter-planes. We now turn to the open problem mentionedin Remark 7.0.11. By a scaling argument, we may assume that

α = 1, m = 2π,

so that we are dealing with the linear span of

eiπ jt, e−iπβk/t, j, k = 0, 1, 2, . . . .

The issue at hand is whether this linear span is weak-star dense in H∞(R) for β ≤ 1. So, iff ∈ L1(R) has ⟨

t 7→ eiπ jt, f⟩R

=⟨t 7→ e−iπβk/t, f

⟩R

= 0, j, k = 0, 1, 2, . . .

may we then conclude (for β ≤ 1) that f ∈ H1+(R)? For j = 0, 1, 2, . . ., the functions t 7→ eiπ jt

belong to H∞+ (R), and they are 2-periodic: eiπ j(t+2) = eiπ jt. From the well-known theory ofFourier series we obtain that the linear span of these functions t 7→ eiπ jt, where j = 0, 1, 2, . . .,is weak-star dense in H∞+ (R/2Z), the subspace of 2-periodic H∞(R) functions. As for the

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 25

remaining spanning vectors e−iπβk/t a similar argument shows that their linear span is weak-star dense in H∞+ (R/〈β〉). Here, g ∈ H∞+ (R/〈β〉) if and only if g ∈ H∞(R) has {t 7→ g(−β/t)} ∈H∞(R/2Z). In other words, g ∈ H∞+ (R/〈β〉) means that g ∈ H∞+ (R) has the “Möbius periodicity”

g( βtβ − 2t

)= g(t).

We reformulate the problem in terms of these subspaces of H∞+ (R).

Problem 10.0.23. Is the sum H∞+ (R/2Z) + H∞+ (R/〈β〉) weak-star dense in H∞(R) for β ≤ 1?

We reformulate this problem in terms of the periodization operator Q2 : L1(R)→ L1(R/2Z):

(10.0.8) Q2 f (x) :=∑j∈Z

f (x + 2 j).

We first look at what it means for a function f ∈ L1(R) that

(10.0.9) 〈 f , g〉R = 0 for all g ∈ H∞+ (R/2Z).

For f ∈ L1(R) and g ∈ L∞(R/2Z), we see that

〈 f , g〉R =

∫R

f (x)g(x)dx =∑j∈Z

∫[2 j−1,2 j+1]

f (x)g(x)dx =

∫[−1,1]

Q2 f (x)g(x)dx = 〈Q2 f , g〉[−1,1],

Via the exponential map z 7→ eiπz the space H∞+ (R/2Z) can be identified with H∞+ (T), and inview of the identification of the pre-annihilator of H∞+ (R/2Z), we find that (10.0.9) is equivalentto having

(10.0.10) Q2 f ∈ H1+,0(R/2Z).

We turn to the interpretation of

(10.0.11) 〈 f , g〉R = 0 for all g ∈ H∞+ (R/〈β〉).

We recall that g ∈ H∞+ (R/〈β〉) means that g(x) = h(−β/x), for some function h ∈ H∞+ (R/2Z). Bythe change-of-variables formula, we have

(10.0.12) 〈 f , g〉R =

∫R

f (x)g(x)dx =

∫R

f (x)h(−β

x

)dx = β

∫R

f(−β

x

)h(x)

dxx2 = 〈Jβ f , h〉R,

where Jβ : L1(R)→ L1(R) denotes the isometric transformation

Jβ f (x) :=β

x2 f(−β

x

).

From (10.0.12) we see that (10.0.11) is equivalent to having

Q2Jβ f ∈ H1+,0(R/2Z).

It is easy to check that

f ∈ H1(R) =⇒ Q2 f , Q2Jβ f ∈ H1+,0(R/2Z).

Problem 10.0.23 asks whether, for 0 < β ≤ 1, the reverse implication holds: Is it true that

(10.0.13) f ∈ L1(R) and Q2 f , Q2Jβ f ∈ H1+,0(R/2Z) =⇒ f ∈ H1

+(R)?

We note that if we ask that, in addition, f ∈ H1real(R), the implication holds, by the preceding

argument.

A related open problem. It may shed light on (10.0.13) to formulate the analogous statementin the setting of Lp(R), for 0 < p < 1. From the well-known (quasi-triangle) inequality

|z1 + · · · + zn|p≤ |z1|

p + · · · + |zn|p, 0 < p ≤ 1,

we quickly see that Q2 : Lp(R)→ Lp(R/2Z) is bounded. It remains to define Jβ,p. We put

Jβ,p[ f ](x) := β1/p|x|−2/pθp(x) f

(−β

x

),

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26 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

where the phase factor θp(x) is defined as follows: θp(x) := 1 for x > 0, and θp(x) := e−i2π/p forx < 0. It is well understood how one defines the Hardy spaces Hp

+(R) and Hp+,0(R/2Z) as closed

subspaces of Lp(R) and Lp(R/2Z), respectively, also for 0 < p < 1. We are ready to formulatethe general problem.

Problem 10.0.24. (0 < p ≤ 1) For which positive β is it true that

f ∈ Lp(R) and Q2 f , Q2Jβ,p f ∈ Hp+,0(R/2Z) =⇒ f ∈ Hp

+(R)?

Distorted lattice-crosses. We consider the set Λ〈ξ0〉

α of Theorem 7.0.12. and assume α, β,m areall positive with αβm2

≤ 4π2.

Proof of Theorem 7.0.12. Let µ ∈ ACH(Γm) have µ = 0 on Λ〈ξ0〉

α . By Theorem 7.0.10, we have thatµ vanishes on the set

R2−− ∪ (R2

++ + {ξ0}).

In terms of the compressed measure π1µ ∈M(R×), this is equivalent to having dπ1µ(t) = f (t)dt,where f ∈ H1

−(R) and f/Uξ0 ∈ H1

+(R), where Uξ0 is the unimodular function

Uξ0 (t) := e−iπ[ξ01t−m2ξ0

2/(4π2t)], t ∈ R.

The given information allows us to conclude (e.g., we can use Morera’s theorem) that f has aholomorphic extension to C× = C \ {0}.

Now, if ξ01 < 0, then the extension must decay too quickly as we approach infinity in the

upper half plane, so f = 0 is the only possibility. If ξ01 = 0, then still the point at infinity must be

a removable singularity. If we look at the origin instead of infinity, we find that if ξ02 < 0, then

the decay prescribed is too strong unless f = 0. Moreover, if ξ02 = 0, we get at least a removable

singularity. So, if ξ0 = (0, 0), we get a removable singularity at the origin and at infinity, so byLiouville’s theorem, f must be constant, and the constant is 0, as f ∈ H1

−(R). Nest, if ξ0

1 > 0and ξ0

2 ≥ 0, we may pick a non-trivial f from a Paley-Wiener space of entire functions (this isa closed subspace of L1(R) of entire functions with the following properties: the functions arebounded in the lower half-plane, and have at most a given exponential growth in the upperhalf-plane). By applying the inversion x 7→ −1/x, we can find analogously non-trivial f ifξ0

2 > 0 and ξ01 ≥ 0. The proof is complete. �

11. Fourier uniqueness for a single branch of the hyperbola

Dual formulation of the theorem. We now turn to Theorem 8.0.14, and observe that a scalingargument allows us to suppose that

α = 2, m = 2π.

The dual formulation of Theorem 8.0.14 now reads as follows. The restriction to R+ of thefunctions

ei2π jt, eiπβk/t, j, k ∈ Z,span a weak-star dense dense subspace of L∞(R+) if and only if β < 2. Moreover, for β = 2, the weak-starclosure of the linear span has codimension 1 in L∞(R+).

Proof of Theorem 8.0.14. Let ν ∈ AC(R+); then ν may be written as dν(t) = f (t)dt, where f ∈L1(R+). When needed, we think of ν and f as defined to vanish on R−. We suppose that

(11.0.14)∫ +∞

0ei2π jtdν(t) =

∫ +∞

0ei2πγk/tdν(t) = 0, j, k ∈ Z,

where γ := β/2. We shall analyze the dimension of the space of solutions ν, depending on thepositive real parameter γ. We rewrite (11.0.14) in the form

(11.0.15)∫ +∞

0ei2π jtdν(t) =

∫ +∞

0ei2πktdν(γ/t) = 0, j, k ∈ Z,

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 27

which we easily see is equivalent to having (cf. [15])∑j∈Z

dν(t + j) =∑j∈Z

dν( γ

t + j

)= 0, t ∈ R.

Both expressions are 1-periodic, so it is enough to require equality on [0, 1[ (we remove termsthat are 0):

(11.0.16)+∞∑j=0

dν(t + j) =

+∞∑j=0

dν( γ

t + j

)= 0, t ∈ [0, 1[.

We single out the term with j = 0, and obtain that

(11.0.17) dν(t) = −

+∞∑j=1

dν(t + j), t ∈ [0, 1[,

and

(11.0.18) dν(t) = −

+∞∑j=1

dν( γtγ + jt

), t ∈]γ,+∞[.

If we take absolute values, apply the triangle inequality, and integrate, we get rather triviallyfrom (11.0.17) that

(11.0.19)∫

[0,1[d|ν|(t) ≤

+∞∑j=1

∫[0,1[

d|ν|(t + j) =

∫[1,+∞[

d|ν|(t),

and from (11.0.18) that

(11.0.20)∫

[γ,+∞[d|ν|(t) ≤

+∞∑j=1

∫[γ,+∞[

d|ν|( γtγ + jt

)=

+∞∑j=1

∫[ γ

j+1 ,γj [

d|ν|(t) =

∫]0,γ[

d|ν|(t).

For 0 < γ ≤ 1, we may combine (11.0.19) and (11.0.20), to arrive at

(11.0.21)∫

[0,1[d|ν|(t) ≤

∫[1,+∞[

d|ν|(t) ≤∫

[γ,+∞[d|ν|(t) ≤

∫]0,γ]

d|ν|(t),

which is only possible if we have equality everywhere in (11.0.21). But then |ν| takes no masson the interval [γ, 1], and we must also have (0 < γ ≤ 1)

(11.0.22) d|ν|(t) =

+∞∑j=1

d|ν|(t + j), t ∈ [0, 1[,

and

(11.0.23) d|ν|(t) =

+∞∑j=1

d|ν|( γtγ + jt

), t ∈]γ,+∞[.

Moreover, for some constant ζ ∈ Cwith |ζ| = 1, we must also have (0 < γ ≤ 1)

(11.0.24) dν(t) = ζd|ν|(t), t ∈ [0, 1[,

and

(11.0.25) dν(t) = −ζd|ν|(t), t ∈ [1,+∞[.

For 0 < γ ≤ 1, this allows us to focus on the positive measure d|ν|. Again for 0 < γ ≤ 1, we maycombine (11.0.22) and (11.0.23) and obtain as a result that

(11.0.26) d|ν|(t) =

+∞∑j,k=1

d|ν|( γ(t + j)γ + k(t + j)

), t ∈]0, 1[.

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28 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

For x ∈ R, let {x}1 be the fractional part of x; more precisely, {x}1 is the number in [0, 1[ suchthat x − {x}1 ∈ Z. We define Uγ : [0, 1[→ [0, 1[ as follows: Uγ(0) := 0, and

(11.0.27) Uγ(x) := {γ/x}1, x ∈]0, 1[.

As we already observed, |ν| takes no mass on [γ, 1]. If we integrate the left hand side of (11.0.26)on [γ, 1] to get 0, we should obtain 0 from the right hand side as well. But integration of theright hand side on [γ, 1] computes the |ν|-mass of the set

Eγ(2) := {t ∈ [0, 1[: U2γ(t) ∈ [γ, 1]},

where U2γ = Uγ ◦ Uγ, the composition square. So |ν| takes no mass on Eγ(2). By iterating this

argument, we see that µ assumes no mass on all sets of the form

Eγ(2n) := {t ∈ [0, 1[: U2nγ (t) ∈ [γ, 1]}, n = 1, 2, 3, . . . .

If 0 < γ < 1, the union of all the sets Eγ(n), n = 1, 2, 3, . . ., has full Lebesgue mass, which noplace for the mass of |ν|, and we get that |ν|([0, 1[) = 0. By (11.0.21), we get that |ν|(R) = 0, thatis, ν = 0 identically.

The case γ = 1 is a little different. Then (11.0.26) asserts that |ν| is an invariant measure forU2

1, the square of the standard Gauss map [7]. As U1 is ergodic with respect to the absolutelycontinuous probability measure

d$(t) :=dt

(1 + t) log 2,

we conclude that |ν|must be of the form

d|ν|(t) = C1d$(t), t ∈ [0, 1[,

for some real constant C1 ≥ 0. The analogous argument based on the interval [1,+∞[ in placeof [0, 1[ gives that

d|ν|(t) = C2d$(1/t) =C2dt

t(1 + t) log 2, t ∈ [1,+∞[.

We obtain that dν must be a complex constant multiple of the measure

1[0,1[(t)dt

1 + t− 1[1,+∞[(t)

dtt(1 + t)

.

This measure meets (11.0.17) and (11.0.18), so we really have a one-dimensional annihilator forγ = 1.

Finally, we need to consider γ > 1, and supply a non-trivial ν ∈ AC(R+) with (11.0.16). Inthis case, the Gauss-type map Uγ given in (11.0.27) is uniformly expanding, and therefore, ithas a non-trivial absolutely continuous invariant probability measure on [0, 1], which we call$γ (cf. [7], p. 169, and [6], [5]). We extend $γ to R+ trivially by putting it equal to the zeromeasure on R+ \ [0, 1] =]1,+∞[. Being invariant, $γ has the property

d$γ(t) =

+∞∑j=1

d$γ( γ

t + j

), t ∈ [0, 1].

We put

dν(t) := d$γ(t) − d$γ(γ/t), t ∈ R+,

so that ν gets to have the symmetry property

dν(t) = −dν(γ/t), t ∈ R+.

It is now a simple exercise to verify that ν meets (11.0.16), which completes the proof. �

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 29

Remark 11.0.25. In case γ > 1, it is of interest to know how to construct more general measuresν ∈ AC(R+) with (11.0.16). We could try with ν of the form

dν(t) = dω1(t) − dω1(γ/t) − dω2(t),

where ω1 is supported on [0, 1], while ω2 is supported on [1, γ]. We require that, in addition,dω2(γ/t) = −dω2(t). Then ν has the symmetry property dν(γ/t) = −dν(t), and we just need tocheck whether

+∞∑j=0

dν(t + j) = 0, t ∈]0, 1[.

We obtain the equation

dω1(t) =

+∞∑j=0

dω1

( γ

t + j

)+

]γ[∑j=1

dω2(t + j), t ∈]0, 1[

where ]γ[ denotes the largest integer < γ. In particular, if 1 < γ ≤ 2, this equation reads

dω1(t) =

+∞∑j=0

dω1

( γ

t + j

)+ dω2(t + 1). t ∈]0, 1[

This equation is a perturbation of the invariant measure equation (which is obtained forω2 = 0),and one would expect that there should exist many solutions ω1, ω2. In [4], we show that thereis an infinite-dimensional subspace of the absolutely continuous measures with (11.0.16), butwe have not checked whether they can be assumed to be of the above form.

Analysis of the critical case αβm2 = 16π2. Without loss of generality, we may take

α = β = 2, m = 2π,

which corresponds to γ = 1 in the above proof of Theorem 8.0.14. We now look at the cause ofthe defect 1, the one-dimensional subspace spanned by the measure

dν(t) = 1[0,1[(t)dt

1 + t− 1[1,+∞[(t)

dtt(1 + t)

,

as we see from the proof of Theorem 8.0.14. This measure has the symmetry property dν(1/t) =−dν(t), which means that∫

R+

ei2πx/tdν(t) = −

∫R+

ei2πxtdν(t), x ∈ R.

We will need to compute the one-dimensional Fourier transform

ν(x) :=∫R+

ei2πxtdν(t), x ∈ R.

We quickly find that

ν(x) := (1 − ei2πx)∫ +∞

0ei2πxt dt

t + 1, x ∈ R,

where the integral on the right hand side is understood in the generalized Riemann sense.

Proof of Corollary 8.0.16. By symmetry, we may take ξ0∈ R × {0}. It will be enough to establish

thatν(x) , 0, x ∈ R \Z.

It will be sufficient to obtain that∫ +∞

1ei2πxt dt

t=

∫ +∞

1cos(2πxt)

dtt

+ i∫ +∞

1sin(2πxt)

dtt, 0, x ∈ R×.

The real part of this expression equals∫ +∞

1cos(2πxt)

dtt

=

∫ +∞

2π|x|

cos yy

dy = − ci(2π|x|),

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30 HAAKAN HEDENMALM AND ALFONSO MONTES-RODRÍGUEZ

whereas the imaginary part equals∫ +∞

1sin(2πxt)

dtt

= sgn(x)∫ +∞

2π|x|

sin yy

dy = − sgn(x) si(2π|x|);

the sgn function was defined in Section 6, and the integral expression can be thought of asdefining the rather standard functions “si” and “ci”. It is well-known that the parametrization

ci(2πx) + i si(2πx), 0 < x < +∞,

forms the Nielsen (or sici) spiral which converges to the origin as x→ +∞, and whose curvatureis proportional to x (see, e.g. [1]). In particular, the spiral never intersects the origin, whichdoes it. �

12. Open problems in higher dimensions

The Klein-Gordon equation in dimension d. In space dimension d > 1, we consider a solutionu to (1.3.1) of the form

u(t, x) = µ(t, x) :=∫Rd+1

eπi(τt+〈x,ξ〉)dµ(τ, ξ),

where µ is a complex-valued finite Borel measure, and t, τ ∈ R, x, ξ ∈ Rd, and

〈x, ξ〉 = x1ξ1 + · · · + xdξd.

The assumption that u solves the Klein-Gordon equation means that(τ2− |ξ|2 −

M2

π2

)dµ(τ, ξ) = 0

as a measure on Rd+1, which we see is the same as having

suppµ ⊂ Γm(d) :={(τ, ξ) ∈ R ×Rd : τ2

− |ξ|2 =M2

π2

}.

The set Γm(d) is a two-sheeted d-dimensional hyperboloid. Let

Γ+m(d) := {(τ, ξ) ∈ Γm : τ > 0}, Γ−m(d) := {(τ, ξ) ∈ Γm : τ < 0},

be the two connectivity sheets of the hyperboloid Γm(d). We equip Γm with d-dimensionalsurface measure, and require of µ that it be absolutely continuous with respect to this surfacemeasure.

Light cones. We consider the light cone emanating from the origin:

Y0 :={(t, x) ∈ R ×Rd : |x| = |t|

}.

The light cone is a characteristic surface for the Klein-Gordon equation. For any ε ≥ 0, thesurface

Y0(ε) :={(t, x) ∈ R ×Rd : |x| = |t| + ε

}is characteristic as well. In connection with their study of the event horizon of Kerr blackholes, Ionescu and Klainerman [17] showed (for ε > 0) that if the function u – which solves theKlein-Gordon equation – vanishes on Y0(ε), then u = 0 for all (t, x) with |x| ≥ |t| + ε (so we getsuppression in the space-like direction); compare also with [20] and [23]. Klainerman (privatecommunication) has indicated that this should be true for ε = 0 as well. But then we shouldexpect Y0 to be a uniqueness set for u, as there is no width to the waist of Y0 which could bethe source for a wave. So, we suppose for the moment that it has been established that Y0 isa uniqueness set for u. Then it makes sense to ask for (small) subsets of Y0 that are sets ofuniqueness, too. This is what Theorem 1.5.1 supplies in d = 1. In analogy with Theorem 8.0.14,we would ask for even smaller subsets of Y0 that are sets of uniqueness for u, provided thatthe Borel measure µ (which u is the Fourier transform of) is supported on the branch Γ+

m(d).

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THE KLEIN-GORDON EQUATION: THE TRANSFER OPERATOR INTERTWINES WITH THE HILBERT TRANSFORM 31

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Hedenmalm: Department ofMathematics, KTH Royal Institute of Technology, S–10044 Stockholm, SwedenE-mail address: [email protected]

Montes-Rodriguez: Department ofMathematical Analysis, University of Sevilla, Sevilla, SpainE-mail address: [email protected]