Spectral gaps for sets and measures

c 2012 by Institut Mittag-Leffler. All rights reserved

Spectral gaps for sets and measures

by

Alexei Poltoratski

Texas A&M University College Station, TX, U.S.A.

1. Introduction

Letµbe a non-zero finite complex measure on the real line. By ˆµwe denote its Fourier transform

ˆ µ(z) =

Z

R

e^−iztdµ(t).

Various properties of the Fourier transform of a measure have been studied by harmonic analysts for more than a century. One of the reasons for such a prolonged interest is the natural physical sense of the quantity ˆµ(t). For instance, in mathematical models of quantum mechanics, ifµis a spectral measure of a Hamiltonian, then|µ(t)|ˆ ²represents the so-called survival probability of the particle, i.e. the probability to find the particle in its initial state, at the momentt. The problems considered in the present paper belong to the area of the uncertainty principle in harmonic analysis, whose name itself suggests relations and similarities with physics.

The uncertainty principle in harmonic analysis, as formulated in [13], says that a measure (function, distribution) and its Fourier transform cannot be simultaneously small. This broad statement gives rise to a multitude of exciting mathematical problems, each corresponding to a particular sense of “smallness”.

One such problem is the well-known gap problem. Here the smallness ofµand ˆµis understood in the sense of porosity of their supports. The statement that one hopes to obtain is that if the support of ˆµ has a large gap, then the support ofµcannot be too

“rare”. As usual, the ultimate challenge is to obtain quantitative estimates relating the two supports, something that we will attempt to do in this paper.

Beurling’s gap theorem says that if the sequence of gaps in the support ofµis long, in the sense given by (4.3), then the support of ˆµcannot haveanygaps, unlessµis trivial,

The author was supported by N.S.F. Grant No. 0800300.

see [3] or [15, Volume 1, p. 237]. We discuss this classical result in §5. Beurling’s proof used some of the methods of an earlier gap theorem by Levinson [19]. In [7, Theorem 66, p. 271] de Branges proved that the existence of a measure with a given spectral gap is equivalent to the existence of a certain entire function of exponential type. In §8 we look at this result from the point of view of Toeplitz kernels and formulate its extension.

Further results and references concerning the gap problem can be found in [2], [3], [6], [10], [15], [19] and [32].

To state the gap problem more precisely let us give the following definition. IfX is a closed subset of the real line, denote byG_X itsgap characteristic, i.e. the supremum of the size of the gap in the support of ˆµ, taken over all non-trivial finite complex measures µsupported onX, see§2. The gap problem is the problem of findingG_X in terms ofX. To formulate a solution, we introduce a new metric characteristic of a closed set, C_X, see§4. Our main result is Theorem 4.7, which says thatG_X is equal to 2πC_X.

The definition ofCX contains two conditions that, for the purposes of this paper, we call the density condition and the energy condition. The density condition is similar to some of the definitions of densities used in the area of the uncertainty principle, see§4.

The physical flavor of the energy condition seems to suggest new connections for the gap problem that are yet to be fully understood.

As discussed in§2, the gap problem can be equivalently reformulated as follows. Let µbe a finite complex measure onR. Find the supremumGµof the size of the spectral gap of the measuref µ, taken over all non-trivialf∈L¹(|µ|). In this reformulation Gµ=GX

forX=suppµ, see Proposition 2.1.

A close relative of the gap problem is another classical question of harmonic analysis, the so-called type problem. It can be stated in several equivalent ways. For instance, if one replaces L¹ with L² in the above definition of Gµ one can introduce a similar quantity G²_µ, the supremum of the size of the gap in the support of f µ, taken over allc non-zerof∈L²(|µ|). Via duality,G²_µ can also be defined as the infimum ofa such that the family of exponential functions

Ea={e^iλx:λ∈[0, a]}

spansL²(µ). The type problem asks for findingG²_µ in terms ofµ.

The type problem dates back to the works of Wiener, Kolmogorov and Kre˘ın on sta- tionary Gaussian processes. In that context the property that the family of exponentials Ea is complete inL²(µ), whereµis the spectral measure of the process, is equivalent to the property that the process at any time can be predicted from the data for the time period from 0 toa. As any positive even measure is a spectral measure of a stationary Gaussian process and vice versa, this reformulation is practically equivalent, see [11], [16]

and [18].

Since for finite measures

G²_µ6Gµ=Gsuppµ,

Theorem 4.7 gives an upper estimate forG²_µ. The methods we develop in this paper can be applied to give further results for the type problem, see [28].

In addition to G¹_µ=Gµ and G²_µ, one can define and study similar quantities for otherp, see for instance the book by Koosis [15] for the case p=∞, or [28].

Such problems can also be restated in terms of the Bernstein uniform weighted approximation, see for example [15]. From that point of view, GX is the minimal size of the interval such that continuous functions on X admit weighted approximation by trigonometric polynomials with frequencies from that interval. We give a short example of a similar connection at the end of§3. Important relations with spectral theory of second order differential operators were studied by Kre˘ın [17], [18], Gelfand and Levitan [12], and Borichev and Sodin [6].

Our methods are based on the approach developed by N. Makarov and the author in [20] and [21]. We utilize close connections between most problems from this area of harmonic analysis and the problem of injectivity of Toeplitz operators. In the case of the gap problem, this connection is expressed by Theorem 3.2 below. The Toeplitz approach for similar problems was first suggested by Nikol⁰skii in [24], see also [25]. Our main proof utilizes several important ideas of the Beurling–Malliavin theory [4], [5], [21], including its famous multiplier theorem.

One of the advantages of the Toeplitz approach is that it reveals hidden connections between various problems of analysis and mathematical physics, see [20]. The relations between the gap problem and the Beurling–Malliavin theory on the completeness of exponentials inL²on an interval have been known to experts, at a rather intuitive level, for several decades. Now we can see this connection formulated in precise mathematical terms. Namely, the Beurling–Malliavin problem is equivalent to the problem of triviality of the kernel of a Toeplitz operator with the symbol

φ=e^−iaxθ,

for a suitable meromorphic inner functionθ, while the gap problem reduces to the trivi- ality of the kernel of the Toeplitz operator with the symbol

φ¯=e^iaxθ,¯ see [20,§4.6] and Theorem 3.2 below.

The paper is organized as follows:

• In §2 we discuss an alternative formulation of the gap problem and show that the supremum of the size of the spectral gap for a fixed measure, taken over all possible densities, is determined by the support of the measure.

• In §3 we restate the gap problem in terms of kernels of Toeplitz operators and introduce the approach that will be used in the main proof. We point out a connection with problems on uniform approximation.

• §4 contains the main definition and the main result of the paper. For a closed real set X we define a metric characteristicCX that determines the maximal size of the gap over all non-zero complex measures supported onX.

• In§5 we discuss examples pertinent to the main theorem as well as its relations with some of the known results.

• §6 contains the main proof.

• In§7 we prove several technical lemmas and corollaries used in§4 and§6.

• §8 can be viewed as an appendix. It contains a Toeplitz version of the statement and proof of de Branges’ theorem [7, Theorem 66].

Acknowledgments. I would like to thank Nikolai Makarov whose mathematical in- tuition led to the development of the approach used in this paper. I am also grateful to Misha Sodin for getting me interested in the gap and type problems and for numerous invaluable discussions.

2. Spectral gap as a property of the support

Let M be a set of all finite Borel complex measures on the real line. If X is a closed subset of the real line we set

GX= sup{a: there isµ∈M, withµ6≡0 and suppµ⊂X, such that ˆµ= 0 on [0, a]}.

Now letµ∈M. Denote

Gµ= sup{a: there isf∈L¹(|µ|) such thatf µc= 0 on [0, a]}.

Proposition 2.1. G_µ=G_suppµ.

Proof. Obviously, Gsuppµ>Gµ. To prove the opposite inequality, notice that by Lemma 8.2 below there exists a finite discrete measure

ν=X

n

α_nδ_xn, {xn}n⊂suppµ,

such that ˆµ has a gap of size greater than Gsuppµ−ε. Around each xn choose a small neighborhoodV_n=(a_n, b_n) such that for any sequence of points

Y ={yn}n, yn∈Vn,

there exists a non-trivial measure ηY=P

nβnδyn such that ˆη has a gap of size greater than Gsuppη−ε. The existence of such a collection of neighborhoods follows from the results of [2] (for some sequences) and [6], as well as from Theorem 4.7 below.

Now one can choose a family of finite measures η_τ, τ∈[0,1], with the following properties:

• For eachτ,

ητ=X

n

β_n^τδy_n^τ

wherey_n^τ∈Vn and ˆητ has a gap of size greater thanGsuppη−εcentered at 0;

• The measure

γ= Z 1

0

η_τdτ

is non-trivial and absolutely continuous with respect toµ.

It remains to observe that the support of ˆγhas a gap of size at leastG_suppη−ε.

3. Clark measures, Toeplitz kernels and uniform approximation By H² we denote the Hardy space in the upper half-plane C⁺. We say that an inner function θ(z) in C⁺ is meromorphic if it has a meromorphic extension to the whole complex plane. The meromorphic extension to the lower half-planeC⁻is given by

θ(z) = 1 θ(¯¯z).

Each inner functionθ(z) determines a model subspace K_θ=H² θH²

of the Hardy space H²(C⁺). These subspaces play an important role in complex and harmonic analysis, as well as in operator theory, see [25].

Each inner functionθ(z) determines a positive harmonic function Re1+θ(z)

1−θ(z)

and, by the Herglotz representation, a positive measureσsuch that Re1+θ(z)

1−θ(z)=py+1 π

Z

R

y dσ(t)

(x−t)²+y², z=x+iy, (3.1)

for some p>0. The numberpcan be viewed as a point mass at infinity. The measure σ is singular, supported on the set where non-tangential limits of θ are equal to 1 and satisfies

Z

R

dσ(t)

1+t²<∞. (3.2)

The measureσ+pδ_∞ onRb is called theClark measure forθ(z). (Following a standard notation, we will sometimes denote the Clark measure defined in (3.1) byσ1.)

Conversely, for every positive singular measureσsatisfying (3.2) and a numberp>0, there exists an inner functionθ(z) satisfying (3.1).

Every functionf∈Kθ has non-tangential boundary values σ-a.e. and can be recov- ered from these values via the formula

f(z) = p

2πi(1−θ(z)) Z

R

f(t)(1−θ(t))dt+1−θ(z) 2πi

Z

R

f(t)

t−zdσ(t), (3.3) see [26]. If the Clark measure does not have a point mass at infinity, the formula is simplified to

f(z) = 1

2πi(1−θ(z))Kf σ, whereKf σ stands for the Cauchy integral

Kf σ(z) = Z

R

f(t) t−zdσ(t).

This gives an isometry ofL²(σ) ontoK_θ. In the case of meromorphicθ(z), every function f∈Kθ also has a meromorphic extension inC, which is given by the formula (3.3). The corresponding Clark measure is discrete with masses at the points of {z:θ(z)=1} given by

σ({x}) = 2π

|θ⁰(x)|.

For more details on Clark measures the reader may consult [29] or the references therein.

Each meromorphic inner function θ(z) can be written as θ(t)=e^iφ(t) on R, where φ(t) is a real-analytic and strictly increasing function. The functionφ(t)=argθ(t) is the continuous argument ofθ(z).

Recall that the Toeplitz operatorTU with a symbolU∈L^∞(R) is the map TU:H²−!H²,

F7−!P+(U F),

whereP+is the orthogonal projection inL²(R) onto the Hardy spaceH²=H²(C+).

We will use the following notation for kernels of Toeplitz operators (or Toeplitz kernels in H²):

N[U] = kerT_U.

For example,N[¯θ]=Kθifθis an inner function. Along withH²-kernels, one may consider Toeplitz kernelsN^p[U] in other Hardy classesH^p, the kernelN^1,∞[U] in the “weak” space H^1,∞=H^p∩L^1,∞, 0<p<1, or the kernel in the Smirnov classN⁺(C+):

N⁺[U] ={f∈ N⁺∩L¹_loc(R) :Uf¯∈ N⁺}.

Ifθ is a meromorphic inner function,K_θ⁺=N⁺[¯θ] can also be considered. For more on such kernels see [20] and [21].

For any inner functionθ in the upper half-plane we denote by spec_θ the closure of the set{z:θ(z)=1}, the set of points on the line where the non-tangential limit of θ is equal to 1, plus the infinite point if the corresponding Clark measure has a point mass at infinity, i.e. ifpin (3.1) is positive. If spec_θ⊂R, as in the next definition, then pin (3.1) is 0. Throughout the paper,S stands for the exponential inner functionS(z)=e^iz. We call a sequence of real points discrete if it has no finite accumulation points.

Note that{z:θ(z)=1}is discrete if and only ifθ is meromorphic.

Definition 3.1. If X⊂Ris a closed set, we define

TX= sup{a:N[¯θS^a]6= 0 for some meromorphic innerθwith spec_θ⊂X}.

The following theorem (see [22,§2.1]) shows the connection between the gap prob- lem and the problem of triviality of Toeplitz kernels. Such connections will be used throughout the paper.

Theorem 3.2. ([22]) G_X=T_X.

The gap problem is closely related to problems of uniform approximation of con- tinuous functions by trigonometric polynomials, see [6] for a more detailed discussion and further references. To give a simple example of such a connection we consider the following version of the problem.

Let again X be a closed subset of the line. Denote by C0(X) the space of all continuous functions onX tending to 0 at infinity, with the usual sup-norm. It is not possible to discuss approximation by trigonometric polynomials in this particular space directly, since finite linear combinations of exponential functions do not belong toC0(X).

The standard solution is to consider “generalized” linear combinations of exponentials, i.e. the Paley–Wiener space

PWa={fˆ:f∈L²([−a, a])}.

Let us define

AX= inf{a >0 : PWa is dense inC0(X)}

or AX=∞ if the set is empty. The following statement is a product of the standard duality argument.

Proposition 3.3. GX=2AX.

Together with Theorem 4.7 this statement gives a formula forAX.

4. The main theorem

Before stating our main result we need to give several definitions.

Let Λ={λ1, ..., λ_n}be a finite set of points onR. Consider the quantity E(Λ) = X

λ_k,λ_l∈Λ λk6=λl

log|λk−λl|. (4.1)

Physical interpretation

According to the 2-dimensional Coulomb law, E(Λ) is the energy of a system of “flat”

electrons placed at the points of Λ. The 2-dimensional Coulomb-gas formalism corre- sponds to the planar potential theory with logarithmic potential and assumes the poten- tial energy at infinity to be equal to −∞, see for example [9], [23] and [30].

Physically, the 2-dimensional Coulomb law can be derived from the standard 3- dimensional law via a method of “reduction”. According to this method, one replaces each electron in the plane with a uniformly charged string orthogonal to the plane. After that one applies the 3-dimensional law and a renormalization procedure.

Key example

Let I⊂Rbe an interval,C >0 and let Λ be a set of kpoints uniformly distributed on I:

Λ =I∩CZ={(n+1)C,(n+2)C, ...,(n+k)C}.

Then

E(Λ) =

k

X

m=1

log[C^k−1(m−1)!(k−m)!] =k²log|I|+O(|I|²) (4.2) as follows from Stirling’s formula. Here |I| stands for the length of I and the notation O(|I|²) corresponds to the direction |I|!∞ (with C remaining fixed).

Remark 4.1. The uniform distribution of points on the interval does not maximize the energyE(Λ) but comes withinO(|I|²) from the maximum, which is negligible for our purposes, see the main definition and its discussion below. It is interesting to observe that the maximal energy forkpoints is achieved when the points are placed at the endpoints ofI and at the zeros of the Jacobi (1,1)-polynomial of degreek−2, see for example [14].

We call a sequence of disjoint intervals {In}n on the real line long (in the sense of Beurling and Malliavin) if

X

n

|I_n|²

1+dist²(0, In)=∞. (4.3)

If the sum is finite, we call{I_n}_n short.

Let

... < a−2< a−1< a0= 0< a1< a2< ...

be a two-sided sequence of real points. We say that the intervalsI_n=(a_n, a_n+1] form a short partition ofRif|I_n|!∞asn!±∞and the sequence{I_n}_n is short.

Main definition

Let Λ={λn}nbe a sequence of distinct real points. We writeCΛ>aif there exists a short partition{In}n such that

∆n>a|In| for alln (density condition) (4.4) and

X

n

∆²_nlog|In|−En

1+dist²(0, In) <∞ (energy condition), (4.5) where

∆_n= #(Λ∩In) and

E_n=E(Λ∩I_n) = X

λ_k,λ_l∈In

λk6=λl

log|λ_k−λ_l|.

IfX is a closed subset of R, we put

CX= sup{a: there is a sequence Λ⊂X such thatCΛ>a}.

Remark 4.2. Notice that the series in the energy condition is positive. Indeed, every term in the sum definingEn is at most log|In|and there are less than ∆²_n terms.

The example before the definition shows that the numerator in the energy condition is (up to lower order terms) the difference between the energy of the optimal configura- tion, when the points are spread uniformly onI_n, and the energy of Λ∩In.

Thus the energy condition is a requirement that the placement of the points of Λ is close to uniform, in the sense that the work needed to spread the points of Λ uniformly on each interval is summable with respect to the Poisson weight.

Remark 4.3. The inequality log|I_n|−log|λ_k−λ_l|>0, which holds for anyλ_k, λ_l∈I_n, also implies that if a sequence Λ satisfies the energy condition (4.5), then any subsequence of Λ also satisfies (4.5) on{I_n}_n. A deletion of points from Λ will eliminate some positive terms from the numerator in (4.5) which can only make the sum smaller.

Remark 4.4. We say that a partition{In}n ismonotoneif|In|6|In+1|forn>0 and

|In+1|6|In| forn<0. Corollary 7.7 shows that in the above definition the words “short partition” can be replaced by “short monotone partition”. Since monotone partitions are easier to work with, this modified definition will be used in the proof of Theorem 4.7.

Remark 4.5. The requirement that the partitionIn=(an, an+1] satisfied|In|!∞is not essential and can be omitted if one slightly changes the definitions of ∆n and En

in (4.5). One could, for instance, use

∆n= #(Λ∩(an−1, an+1+1]) and

E_n= X

λ_k,λ_l∈(an−1,an+1+1]

λ_k6=λl

log|λk−λl|.

Remark 4.6. The density condition says simply that the lower (interior) density of the sequence in the sense of Beurling and Malliavin, is at leasta. Such a density can be defined in several different ways:

• If Λ is a real sequence defined₁(Λ) to be the supremum of alla such that there exists a short monotone partition{I_n}_n satisfying (4.4).

• Denote byd₂(Λ) the supremum of allasuch that there exists a short (not neces- sarily monotone) partition {I_n}_n satisfying (4.4).

• Defined₃(Λ) to be the supremum of allasuch that there exists a subsequence of Λ whose counting functionn(x) satisfies

Z

R

|n(x)−ax|

1+x² dx <∞.

This definition was used in [7].

• Finally, define d4(Λ) to be the infimum of all a such that there exists a long sequence of disjoint intervalsIn satisfying

#(Λ∩In)< a|In|.

This definition was used in [22].

One can easily show that all these definitions are equivalent, i.e.

d1(Λ) =d2(Λ) =d3(Λ) =d4(Λ).

As was mentioned above, such a density d was introduced in [5], where it was called interior density. A closely related notion of exterior density appears in the Beurling–

Malliavin theorem on the completeness of exponential functions in L² on an interval, see [5] or [22]. We give a definition of exterior density in§7.

Now we are ready to return to the gap problem and use our newly defined metric characteristicCX of a closed setX⊂R. Our main result is the following theorem.

Theorem 4.7. GX=2πCX. It will be proved in§6.

5. Examples and applications

In this section we discuss examples related to Theorem 4.7, including some of its relations with existing results.

Example 5.1. As discussed above, if the points of the sequence are spread uniformly over the interval, then

En= X

λk,λl∈In

log|λk−λl|

is roughly (up toO(|In|²), which is small for short sequences{In}n) equal to ∆²_nlog|In| as follows from Stirling’s formula. This happens for instance when the sequence Λ is separated, i.e. satisfies |λ_n−λ_n+1|>δ >0 for all n. Thus for separated sequences Λ the energy condition disappears and

GΛ= 2πdj(Λ),

where dj, j=1,2,3,4, is any of the equivalent densities defined in the last remark, i.e.

the interior density of Λ. This is one of the results of [22].

For example, as follows from Proposition 2.1, if the support of a measureµcontains a separated sequence of interior densityD, then for anyε>0 there existsf∈L¹(|µ|) such thatf µ=0 on [0,c 2πD−ε].

Example 5.2. Let Λ be a real sequence such that the density condition (4.4) holds for somea>0 and some partition{In}n satisfying the stronger shortness condition

X

n

|In|²log|In| 1+dist²(0, I_n)<∞.

Then we will automatically have that X

n

∆²_nlog|In|−P

λ_k,λ_l∈Inlog₊|λk−λl| 1+dist²(0, In) <∞.

Accordingly, condition (4.5) will be significantly simplified and one will only need to check that

X

λ_k,λ_l∈Λ λk6=λl

log₋|λk−λl| 1+λ²_k <∞ to conclude thatGΛ=2πCΛ>2πa.

Consider, for instance, the log-short partition

I₀= (−1,1], I_n= (n^α,(n+1)^α], I_−n= (−(n+1)^α,−n^α], n= 1,2, ..., for someα>1. Let an increasing discrete sequence Λ={λn}n be such that

α|n|^α−16#(Λ∩In)6α|n|^α−1+1 (5.1) for allnand

λ_k+1−λk>conste^{−|k|/log2|k|} (5.2) for allk,|k|>1. Then by the previous discussionG_Λ=2π.

Similarly to the last example, ifµis a finite measure whose support contains Λ, then for anyε>0 there existsf∈L¹(|µ|) such thatf µ=0 on [0,c 2π−ε].

On the other hand, if (5.1) holds, but instead of (5.2) we have that on eachJn, λk+1−λk6conste^{−|k|/log|k|}

for any λk, λk+1∈In, |k|>1, then the interior density of Λ is still 1, but the energy condition is not satisfied by any subsequence of Λ of positive interior density on any short partition. ThusGΛ=0.

As was mentioned in the introduction, one of the classical results on the gap problem is the following theorem by Beurling.

Theorem 5.3. ([3], [15]) Let µ be a finite complex measure on R such that the complement of its support contains a long sequence of intervals (in the sense of (4.3)).

Suppose that µ=0ˆ on an interval of positive length. Then µ≡0.

To obtain this statement from Theorem 4.7, notice that ifµis a measure as in the statement, then for any short partition ofRone of the intervals of the partition will be contained in a gap ofX=suppµ. HenceX does not contain a sequence Λ that satisfies the density condition (4.4) on a short partition witha>0. ThereforeGX=0.

To see another application of Theorem 4.7 in the “positive” direction, let us formulate the following result proved by Benedicks in [2]. This statement provides one of the very few non-trivial examples of sets with positive gap characteristic that exist in the literature.

Theorem 5.4. ([2]) Let ...<a₋₁<a0<a1<a2<... be a discrete sequence of points and let In=(an, an+1] be the corresponding partition of R. Suppose that there exist positive constants C1,C2 and C3 such that

(1) if C₁⁻¹a2n+1<a2k+1<C1a2n+1, then

C₂⁻¹|I2n+1|<|I2k+1|< C₂|I2n+1|;

(2) for all n,

C₁⁻¹|a2n+1|<|a_2n−1|< C1|a2n+1|;

(3) for all n,

|I_2n+1|> C₃max{|I_2n|,1};

(4)

X

n

|I2n+1|² 1+a²_2n+1

log₊|I2n+1|

|I2n| +1

<∞.

Then for any real number A>0 and 16p<∞there exists a non-zero function f∈L¹(R)∩L^p(R)∩C^∞(R), suppf⊂[

n

I2n,

such that fˆ=0on [0, A].

One can show that the part pertaining to the gap problem, i.e. the existence of f∈L¹ S

nI2n

with a spectral gap of sizeA, follows from Theorem 4.7.

For simplicity, let us consider the case when|In|!∞. In this case conditions (1) and (2) of the theorem prove to be redundant. Indeed, put X=clos S

nI2n

and let µ be the restriction of the Poisson measuredx/(1+x²) toX. By Proposition 2.1, we need to show thatGµ>A.

Let

Jn= (a2n, a2n+2] =I2n∪I2n+1.

Then, by conditions (3) and (4) of the theorem,J_n is a short partition.

Consider the sequence Λ⊂S

nI2n such that

2A|Jn|6#(Λ∩I2n)62A|Jn|+1

for each n and the points of Λ are spread uniformly on each I2n. Then Λ satisfies the density condition (4.4) on{Jn}nwitha=2A. As for the energy condition, the numerator in (4.5) is at most

const(|Jn|²log₊|Jn|−|Jn|²log₊|I2n|)+O(|Jn|²)

by (4.2), and the energy condition follows immediately from conditions (3) and (4) of the theorem. HenceGµ=GX>2A.

Similar examples with 16p6∞are discussed in [28].

6. The proof of the main theorem

Before starting the proof, let us introduce the following notation. Iff is a function onR andI⊂Rwe denote byf|I the function that is equal tof onI and to 0 onR\I.

In our estimates we write a(n).b(n) ifa(n)<Cb(n) for some positive constant C, not depending onn, and large enough|n|. We writea(n)b(n) ifca(n)<b(n)<Ca(n) for someC >c>0. Some formulas will have other parameters in place ofnor no parameters at all. For instance,may be put between two improper integrals to indicate that they either both converge or both diverge.

By Π we denote the Poisson measure dx/(1+x²) on the real line. In particular, L^p_Π=L^p(R, dx/(1+x²)).

We will denote by D(R) the standard Dirichlet space on R (in C⁺). Recall that the Hilbert spaceD=D(R) consists of functionsh∈L¹_Πsuch that the harmonic extension u=u(z) ofhto C+ has a finite gradient norm,

khk²_D≡ kuk²_∇^def= Z

C+

|∇u|²dA <∞,

wheredAis the area measure. Ifh∈D(R) is a smooth function, then we also have khk²_D=

Z

R

¯hh˜⁰dx, where ˜hdenotes a harmonic conjugate function.

6.1. Proof of the main theorem, part I

First suppose thatCX>1/2π. We will show thatGX>1.

Chooseε>0. AsC_X>1/2π, there exists a sequence Λ={λ_n}_n⊂X,C_Λ>1/2π. Let I_n= (a_n, a_n+1]

be the corresponding short monotone partition, see Remark 4.4. Without loss of gener- ality, we may assume that

1

2π|In|<#(Λ∩In)6 1 2π|In|+1

(otherwise just delete some of the points from Λ, see Remark 4.3). We will assume that

|I_n|1/ε1 for alln.

By Lemma 7.1 and Corollary 7.5, we may assume that the lengths of the intervals (λ_n, λ_n+1) are bounded from above. It will be convenient for us to assume that the endpoints ofI_n belong to Λ, i.e. thatI_n=(λ_kn, λ_kn+1] for some λ_kn, λ_kn+1∈Λ. We will also include the endpoints of the intervals into the energy condition, by definingE_n as

En= X

λ_kn6λk,λl6λ_kn+1 λ_k6=λl

log|λk−λl|, (6.1)

and assuming that (4.5) is satisfied with these En. Such an assumption can be made because if the sum in (4.5) becomes infinite with E_n defined by (6.1), one can, for instance, delete the first pointλ_nk+1 from Λ on allI_n for largen. After the addition of λ_knand deletion ofλ_nk+1 in the sum definingE_n, each term in (4.5) will become smaller and the sum will remain finite. At the same time, since

|In| #(Λ∩In)!∞,

the subsequence will still have more than |In| points on each In and will satisfy the density condition.

Our goal is to show thatGΛ>1 by producing a measure on Λ with spectral gap of size arbitrarily close to 1. Due to connections discussed in §3, the existence of such a measure will follow from the non-triviality of a certain Toeplitz kernel.

Since the lengths of (λn, λn+1) are bounded from above, we can apply Lemma 7.8.

Denote byθthe corresponding meromorphic inner function with spec_θ=Λ.

Letu=arg(θS)=argθ−x. First, we choose a larger partition Jn=(bn, bn+1) and a small “correction” functionv such thatu−v becomes an atom on eachJn.

Claim 6.1. There exists a subsequence {bn}n of the sequence {an}n and smooth functions v1 and v2 such that

(1) |v₁⁰|<¹₂εand u−v1=0 at all a_n;

(2) J_n=(b_n, b_n+1)is a short monotone partition; (3) |v₂⁰|<¹₂εand u−v=u−(v1+v₂)=0at all b_n; (4) R

Jn(u−v)dx=0 for all n;

(5) ˜u−˜v∈L¹_Π.

Proof. First, choose a smooth function v1 satisfying (1). Such a function exists because

2π∆n−|In|

62π¹₂ε|In|.

Notice that because the sequenceIn is short and (u−v₁)⁰>−1−¹₂ε, condition (1) implies that

u−v1∈L¹_Π. (6.2)

Chooseb₀=a₀=0. Chooseb₁=a_n1>b₀to be the smallest element of{ak}k satisfying

Z a_n1 b₀

(u−v1)dx

<ε

8(an1−b0)².

Notice that, because of (6.2), such anan1 will always exist. After that proceed choosing b₂, b₃, ... in the following way: If b_j is chosen, choose b_j+1=a_nj+1 to be the smallest element of{ak}k satisfyinga_nj+1>b_j,

Z a_nj+1 b_j

(u−v1)dx

<ε

8(anj+1−bj)² (6.3)

and

an_j+1−bj>bj−b_j−1. Chooseb_k,k<0, in the same way.

We claim that the resulting sequence J_k=(b_k−1, b_k) forms a short monotone parti- tion.

Letkbe positive. By our construction,In_kis the last (rightmost) among the intervals Incontained inJk. Notice that, because of monotonicity,In_kis the largest interval among the intervalsIn contained in Jk. We will show that for eachk,

|J_k|<

10 ε

+1

|I_nk|, (6.4)

whereb · cstands for the integer part (floor function) of a real number.

This can be proved by induction. The basic step: By our constructionb1=an1 and

Z an1−1

b₀

(u−v1)dx >ε

8(a_n1−1−b0)².

Since (u−v1)⁰>−1−εand u−v1=0 at all an,|u−v1|6(1+ε)|In₁−1|on (b0, an₁−1).

Hence

(1+ε)|I_n1−1|(a_n1−1−b₀)>

Z an1−1

b₀

(u−v₁)dx >ε

8(a_n1−1−b₀)² and

an₁−1−b0681+ε ε |In₁−1|.

It follows that

|J₁|= (a_n1−1−b₀)+|I_n1|69

ε|I_n1−1|+|I_n1−1|610

ε |I_n1−1| (6.5) (ifεis small enough). For the inductional step, assume that (6.4) holds fork=l−1. For Jl=(b_l−1, bl),bl=an_l, there are two possibilities:

Z a_nl−1 bl−1

(u−v1)dx >ε

8(an_l−1−b_l−1)² or

a_nl−1−b_l−1< b_l−1−b_l−2.

In the first case we prove (6.5) in the same way as in the basic step. In the second case we note that, by the monotonicity of{In}n, the number of intervals In inside (bl−1, an_l−1) is at most (an_l−1−bl−1)/|Inl−1|, which is strictly less than|Jl−1|/|Inl−1|6b10/εc+1. Ac- cordingly the number of intervals in (bl−1, an_l−1) is at mostb10/εc. Therefore the number of intervals inJl=(bl−1, bl) is at mostb10/εc+1. Now, sinceIn_l is the largest interval in Jl, we again get (6.4), which implies the shortness of {Jn}n. The monotonicity follows from our construction.

Now define the functionv2on each Jk in the following way. First consider the tent functionT_k defined on Ras

T_k(x) =¹₄εdist(x,R\J_k).

Notice that, because of (6.3), for each k there exists a constant C_k, |C_k|61, such

that Z

J_k

[(u−v₁)−C_kT_k]dx= 0.

Now define v2 as a smoothed-out sumP

kCkTk that satisfies|v₂⁰|<¹₂εand still has the properties that v2(bk)=0 and

Z

Jk

[(u−v1)−v2]dx= 0

for eachk. Finally, letv=v1+v2. The last condition of the claim will be satisfied because the restrictions (u−v)|J_k form a collection of atoms with a finite sum ofL¹_Π-norms:

k(u−v)|J_kk_L1

Π. |Jk|² 1+dist²(0, Jk) (for more on atomic decompositions see [8]).

The functionvfrom the last claim is a smooth function satisfying|v⁰|6ε. Therefore it can be represented asv=v+−v−, wherev± are smooth growing functions, ε6v_±⁰ 62ε.

Hence one can choose two meromorphic inner functionsI± satisfying {x: argI±(x) =kπ}={x:v±(x) =kπ}

and

|I_±⁰|.ε.

The existence of suchI± follows from Lemma 7.8 below.

Note that then, automatically, |arg( ¯I+I−)−v|<2π. The function arg(θSI+I¯−), as well as its harmonic conjugate, still belongs toL¹_Π.

Without loss of generality, we may assume that arg(θSI+I¯−)=0 at 0.

Claim 6.2. The function

arg(θ(x)S(x)I+(x) ¯I−(x)) x

belongs to the Dirichlet class D(R).

Proof. We will actually prove that the function w/x, w=argθ−x−v, belongs to D(R) instead (again, without loss of generality, we may assume thatw(0)=0 with large multiplicity). The function

v−(argI+−argI−) x

is a bounded function with bounded derivative which obviously belongs toD(R).

Letq be the harmonic extension ofw/x in the upper half-plane. We need to show that the gradient norm ofq+iq˜inC+is finite, i.e. that

kq+i˜qk²_∇= lim

r!∞

Z

∂D(r)

q d˜q=− lim

r!∞

Z

∂D(r)

˜

q dq <∞,

whereD(r) is the semidisk{z∈C⁺:|z|<r}.

We first prove that the integrals over∂D(r)∩Rare uniformly bounded from above, i.e. that

− Z

R

˜

q dq <∞.

First, notice that the harmonic conjugate ofw/x=qisw/x= ˜e q(we may assume that w(0)=0) and (w/x)e ⁰=w⁰/x−w/x². Recall that, by our construction (see Claim 6.1), w⁰ is bounded from below and is zero at the endpoints of everyJn. Hence|w|.|Jn|on everyJn. Since the partition{Jn}nis short, it follows that|w(x)|=o(|x|) and thatw/x² is a bounded function. Therefore,

− Z

R

˜ q dq −

Z

R

w⁰wedx x² and we can estimate the last integral instead.

IfIis an interval, then 2I denotes the interval with the same center asIsatisfying

|2I|=2|I|.

Putw_n=w|_Jn. Then Z

R

w⁰wedx x²=X

n

X

k

Z

Jn

w⁰we_k dx

x². (6.6)

To estimate the last integral, let us first consider the case when the intervalsJn and Jk are far from each other:

max{|Jn|,|Jk|}6dist(Jn, Jk).

In this case Z

J_n

w⁰wek

dx x² .

Z

J_n

|w⁰| |Jk|³ dist²(Jk, x)

dx

x². |Jk|³ 1+dist²(Jn,0)

Z

J_n

dx

dist²(Jk, x). (6.7) Here we used the property that eachwk is an atom supported onJk whose L¹-norm is .|Jk|² and employed the standard estimates from the theory of atomic decompositions, see [8]. In the last inequality we used the property

Z

J_n

|w⁰(x)|dx.|Jn|. (6.8)

Now let us consider the “mid-range” case when

min{|Jn|,|Jk|}6dist(Jn, Jk)<max{|Jn|,|Jk|}.

Assume that 0<k<n(other cases are analogous). Then, by monotonicity|Jk|6|Jn|and

Z

J_n

w⁰we_kdx x²

. 1 1+dist²(Jk,0)

Z

J_n

|w⁰| |J_k|³ dist²(Jk, x)dx 6 |Jk|

1+dist²(Jk,0)

|Jk|² dist²(Jk, Jn)

Z

Jn

|w⁰|dx. |Jk| |Jn| 1+dist²(Jk,0).

(6.9)

Finally, the last case is

dist(Jn, Jk)<min{|Jn|,|Jk|}. (6.10) Again we assume thatn>k>0. Then, by monotonicity, eithern=kor|n−k|=1, i.e. the intervals are either the same or adjacent. The estimates in this case are more complicated and will be done differently. First, integrating by parts we get

− Z

J_n

w⁰we_kdx x² =

Z

J_n

w⁰ Z

J_k

w(t)dt t−x

dx x²=−

Z

J_n

w⁰ Z

J_k

log|t−x|w⁰(t)dt dx

x². By the first inequality of (7.5) in Lemma 7.10, applied to h=w⁰, f=(argθ)⁰ and g=(x+v)⁰, for 1k6nwe have

− Z

Jn

w⁰ Z

Jk

log|t−x|w⁰(t)dt dx

x²

.− 1

1+dist²(Jn,0) Z Z

Jn×Jk

log|t−x|w⁰(x)w⁰(t)dx dt+C|Jn|²

(6.11)

and we can work with the latter integral instead of the former. To verify the conditions of Lemma 7.10, note that if E=Jn and I=Jk, then, for large enough kand n, we will have E∪I⊂[d,2d] as a consequence of the shortness condition and (6.10). The relation (7.4) will be satisfied becausew=0 at the endpoints ofJk, see condition (3) of Claim 6.1.

The constantD1 satisfies D16

Z

J_n

(argθ)⁰dx+

Z

J_n

(x+v)⁰dx.|Jn|.

Finally,

D2

d² . Z

Jk

log|t−x|w⁰(t)dt _L1

Π

=kwekk_L1

Π.kwkk_L1

Π.|Jk|² d² , becausew_k is an atom.

To estimate the integral in the right-hand side of (6.11), let p= argθ−x−v1=w+v2,

where the functionsv1andv2 are from Claim 6.1. Also letpn=p|Jn andvⁿ₂=v2|Jn. The key properties ofv1 that we will use are that argθ−x−v1=0 at the endpoints of allIn, v₂=0 at the endpoints ofJ_n and|v⁰₁|,|v₂⁰|<ε. Then

− Z Z

Jn×Jk

log|t−x|w⁰(x)w⁰(t)dx dt=− Z Z

Jn×Jk

log|t−x|p⁰(x)p⁰(t)dx dt

− Z

J_k

(˜p_nv₂⁰+ ˜v₂ⁿp⁰+ ˜v₂ⁿv₂⁰)dx.

Notice that Z

J_k

˜ p_nv₂⁰dx

6εkp˜_nk₂p

|J_k|6εkp_nk₂p

|J_k|.|J_n|², because|pn|.|Jn|onJ_n andp_n=0 outside. Also,

Z

J_k

p⁰v˜₂ⁿdx

=|hpk, v₂ⁿiD|= Z

J_n

˜ pkv₂⁰dx

.|Jn|²,

by the same estimate. Similarly, since|vⁿ₂|.|Jn|onJn and equals zero outside,

Z

Jk

˜ v₂ⁿv⁰₂dx

.|Jn|². Hence

− Z Z

J_n×Jk

log|t−x|w⁰(x)w⁰(t)dx dt=− Z Z

J_n×Jk

log|t−x|p⁰(x)p⁰(t)dx dt+O(|Jn|²).

(6.12) For the last integral we have

− Z Z

J_n×J_k

log|t−x|p⁰(x)p⁰(t)dx dt=− X

I_j⊂Jk

X

I_j0⊂Jn

Z Z

I_j×I_j0

log|t−x|p⁰(x)p⁰(t)dx dt.

(6.13) To estimate

− Z Z

Ij×I_j0

log|t−x|p⁰(x)p⁰(t)dx dt= Z Z

Ij×I_j0

log₋|t−x|p⁰(x)p⁰(t)dx dt

− Z Z

I_j×I_j0

log₊|t−x|p⁰(x)p⁰(t)dx dt

(6.14)

we consider three cases. First, to estimate the integral in the case when j=j⁰, notice that, since 1+v₁⁰ is bounded,

Z

Ij

log₋|x−t|(1+v⁰₁(x))dx <const

for anyt∈Ij. Once again, the positive functions (argθ)⁰ andv⁰₁+1 satisfy Z

I_l

(argθ)⁰dx= Z

I_l

(v₁⁰+1)dx= 2π∆_l+O(1) =|Il|+O(1). (6.15) Hence

Z Z

I_j×Ij

log₋|t−x|p⁰(x)p⁰(t)dx dt

= Z Z

I_j×I_j

log₋|t−x|(argθ)⁰(x)(argθ)⁰(t)dx dt

−2 Z Z

I_j×I_j

log₋|t−x|(1+v₁⁰(x))(argθ)⁰(t)dx dt +

Z Z

Ij×Ij

log₋|t−x|(1+v₁⁰(x))(1+v⁰₁(t))dx dt

= Z Z

Ij×Ij

log₋|t−x|(argθ)⁰(x)(argθ)⁰(t)dx dt+O(|Ij|).

For the last integral we have Z Z

I_j×I_j

log₋|t−x|(argθ)⁰(x)(argθ)⁰(t)dx dt

= X

(λ_l,λ_l+1)⊂Ij

X

(λ_m,λ_m+1)⊂Ij

Z λ_l+1 λ_l

Z λ_m+1 λ_m

(argθ)⁰(x)(argθ)⁰(t)dx dt.

Using that

Z λ_s+1 λ_s

(argθ)⁰dx= 2π and that

(argθ)⁰. 1

min{|I_s−1|,|Is|,|Is+1|}+ 1

|Is|² on (λs, λs+1),

for all s by Lemma 7.8, we can apply Lemma 7.9 (1)–(3). Assuming that λl6λm, we conclude that

Z λ_l+1 λ_l

Z λm+1

λ_m

log₋|t−x|(argθ)⁰(x)(argθ)⁰(t)dx dt

.















log₋(λ_m−λ_l+1), ifλ_m> λ_l+1,

max{log−(λ_l−λl−1),log₋(λ_l+1−λl),log₋(λ_l+2−λl+1),log₋(λ_l+3−λl+2)}+1, ifλ_m=λ_l+1, max{log−(λl−λl−1),log−(λl+1−λl),log−(λl+2−λl+1)}+1, ifλm=λl,

which implies that Z Z

Ij×Ij

log₋|t−x|p⁰(x)p⁰(t)dx dt. X

aj6λ_k,λ_l6aj+1

λ_k6=λ_l

log₋|λk−λl|+|Ij|. (6.16)

To estimate the integral of log₊, first notice that, by Lemma 7.9 (5) and (6.15),

Z

I_j

log₊|x−t|(1+v₁⁰(x))dx=|Ij|log₊|Ij|+O(|Ij|)

for anyt∈Ij.

Together with Lemma 7.9 (4) and (6.15), we get Z Z

I_j×I_j

log₊|t−x|p⁰(x)p⁰(t)dx dt

= Z Z

I_j×Ij

log₊|t−x|(argθ)⁰(x)(argθ)⁰(t)dx dt

−2 Z Z

I_j×Ij

log₊|t−x|(v⁰₁(x)+1)(argθ)⁰(t)dx dt +

Z Z

I_j×I_j

log₊|t−x|(v⁰₁(x)+1)(v₁⁰(t)+1)dx dt

= X

aj6λ_k,λ_l<aj+1

Z λ_k+1 λk

Z λ_l+1 λl

log₊|t−x|(argθ)⁰(x)(argθ)⁰(t)dx dt

−|Ij|²log|Ij|+O(|Ij|²)

>4π² X

a_j6λ_k,λ_l<a_j+1

log₊|λk−λl|−|Ij|²log|Ij|+O(|Ij|²).

(6.17)

Next, let us consider the case whenj6=j⁰and the intervalsIjandIj⁰are not adjacent.

This estimate is similar to (6.9), but we will treat it using a different technique. Assume for example thatj⁰>j+1.

For log₋, recalling that|Ik|>1 for allk, we get

− Z Z

I_j×I_j0

log−|t−x|p⁰(x)p⁰(t)dx dt= 0. (6.18)