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A generalization of the First Beurling--Malliavin theorem

In this article we prove a generalization of the First Beurling--Malliavin Theorem. In more detail, we establish a new sufficient condition for a function to be a Beurling--Malliavin majorant. Our result is strictly more general than that of the First Beurling--Malliavin Theorem. We also show that our result is sharp in a number senses.


Introduction
Let Lip(R) denote the space of Lipschitz functions in R (i.e. functions f satisfying for all x, y ∈ R the following inequality: |f (x) − f (y)| ≤ C|x − y| with C > 0 independent of x, y). By Lip(R, ξ) we shall denote all Lipschitz functions in R with the Lipschitz constant C satisfying C ≤ ξ.
The following theorem was first proved by A. Beurling and P. Malliavin, see [6].
Remark. For a function f ∈ L 2 (R), by spec(f ) we mean the spectrum of f , i.e. the support of its Fourier transformation. Note that the spectrum is defined up to a set of the Lebesgue measure zero.
Remark. Here, the term "not identically zero" means "not zero almost everywhere". We shall further sometimes write just "nonzero" for brevity.
Remark. The expression log(1/ω) will be sometimes further referred to as the logarithmic integral of ω.
The Beurling-Malliavin theorems are considered by many experts to be among the most deep and important results of the 20th century Harmonic Analysis. Theorem A above is called the First Beurling-Malliavin Theorem (we shall further write First BM Theorem to save space). This result gives conditions for the majorant ω ensuring existence of nonzero function whose spectrum lies in an arbitrary small interval and whose modulus is majorized by ω. This theorem is a crucial tool in the proof of the Second BM Theorem about the radius of completeness of an exponential system. Moreover, Theorem A was recently used by J. Bourgain and S. Dyatlov in the theory of resonances for hyperbolic surfaces, see [9]. Deep connections of the First BM Theorem with nowadays popular gap and type problems are discussed in papers [8], [24] and [20].
Note that Theorem A is in a certain sense a contradiction to the following postulate, called the uncertainty principle: "It is impossible for a nonzero function and its Fourier transform to be simultaneously very small, unless the function is zero". Indeed, Theorem A shows that there exist nonzero functions that are "small" and whose Fourier transforms are also "small". Of course these smallnesses are different from each other and from the smallness in L 2 (R). So in fact Theorem A does not contradict the most well known variant of uncertainty principle, the Heisenberg inequality. For recent violations of the uncertainty principle of a completely different nature, see papers [15] and [22].
In addition to the original proof of A. Beurling and P. Malliavin, there are many approaches to the proof of BM theorems due to H. Redheffer (see [25]), L. De Branges (see [10]), P. Kargaev (see [18]), N. Makarov and A. Poltoratskii (see [20]), to name just some of them. In 2005 V. Havin, J. Mashreghi and F. Nazarov [21] suggested a new proof of the First BM Theorem. An essential novelty of their proof was that it was done by (almost) purely real methods and did not use complex analysis except at one place, see [21] and the Remark right after the formulation of Theorem B below.
Among the goals of the present paper is to give a proof of a new nontrivial generalization of Theorem A. Before stating our main results, we recall some classical definitions and fix some notations.
One of the principle objects of this paper is the class of BM majorants.

Definition 1.
Let ω be a bounded nonnegative function on R. This function is called a Beurling-Malliavin majorant (we shall further write "BM majorant" to save space), if for any σ there exists a nonzero function f ∈ L 2 (R) such that The set of all BM majorants will be further referred to as the BM class. If the conditions (a) and (b) just above are satisfied for a function ω with some fixed σ > 0, then we call such function ω a σ admissible majorant.
Remark. The Poisson measure dP on R is defined by the following formula The corresponding weighted Lebesgue space L 1 (dP ) is the space of all functions f satisfying R |f |dP < ∞.
Note that the condition log(1/ω) ∈ L 1 (dP ) is necessary for ω to be a BM majorant, but not sufficient, see [21]. What Theorem A suggests is that some additional regularity is required for admissibility.
We remind the reader of how one should modify the Cauchy kernel in order to extend the definition of the Hilbert transformation up to the space L 1 (dP ). Definition 2. The Hilbert transformation of a function f ∈ L 1 (dP ) is defined as the following principal value integral It is worth noting that the integral above converges for almost all x ∈ R.
To avoid ambiguity, we stress that this definition coincides with the classical one for functions in L 1 (R).
Let us now introduce function classes that will play an important role in what follows.
Definition 3. Let j ∈ N and let r be a positive number. Consider the following system of intervals: J 0 := [−2, 2), and for j ∈ N, J j := [2 j , 2 j+1 ) and J −j := [−2 j+1 , −2 j ). We say that an absolutely continuous function ϕ belongs to the class The expression in the left hand part of the last line will be further denoted ϕ Vr .
In case when r = ∞, we use the following convention V ∞ = Lip(R).
Note that these classes resemble homogeneous weighted Sobolev spaces. We are going to work with functions that belong to intersections L 1 (dP ) V r with r > 1. From the functional-analytic point of view, these intersections are Banach spaces with respect to the norms · L 1 (dP ) + · Vr .
In order to get some intuition of what an "average" function satisfying log(1/ω) ∈ L 1 (dP ) and log(1/ω) ∈ V r looks like, the reader is welcomed to think of a function, whose graph consists of an infinite number of "pits" and "hills", see the pictures of Section 1.5 in paper [21]. Of course, the same intuition applies to the functions with Lipschitz logarithm and finite logarithmic integral (i.e. those satisfying the conditions of the First BM Theorem). However, we shall shortly see that there are drastic differences between these classes of functions.
Let us compare our sufficient condition of Theorem 1 with these already known. First, it is obvious that our theorem is stronger than the First BM Theorem. Another available in literature sufficient condition for being a BM majorant belongs to A. Baranov and V. Havin, see article [1], Theorem 1.4. That result stemmed from an approach, earlier developed in papers [12] and [13]. In their paper, A. Baranov and V. Havin proved that a function is a BM majorant, if its Hilbert transform is mainly increasing (see [1] for the definition). We shall see in Proposition 1 that follows, that our result applies to some majorants, that are not covered by the aforementioned Baranov-Havin Theorem. Proposition 1. There exist functions in classes V r whose Hilbert transform is not mainly increasing.
Among other things, Theorem 1 infers that the logarithm of a function in the class BM can possibly have sublinear "hills" at the points 2 n , n ∈ N (see the propositions below). Note that there exists no function with Lipschitz logarithm and finite logarithmic integral having this property. For reader's convenience, this is explained in Proposition 2. On the other hand, there are functions whose logarithm is: Lipschitz, belongs to the space L 1 (dP ) and has a sublinear growth at infinity. However, in order to build upon an example of such a function one has to consider "hills" at points 2 n γ with γ > 1 and natural n. In other words, our Theorem 1 provides examples of functions in the BM class with more frequent "hills" than in the Lipschitz case.
Remark. One might think that Theorem 1 could be derived directly from Theorem A in the following way. Take a function ω as in Theorem 1. If there was another function ̟ such that: log(1/̟) is Lipschitz, log(1/̟) ∈ L 1 (dP ) and log(1/ω) ≥ log(1/̟), then, obviously, we would have ω ≥ ̟, and, thanks to Theorem A, ω would be σ admissible for any σ > 0. However, this argument does not apply (at least as is), in view of two propositions that follow.
By reasons, similar to these discussed in the remark right before Proposition 2, Theorem 1 can not be proved via majorisation by β Hölder functions with β ∈ (0, 1). We mean the method disclosed in Sections 1.16 and 1.17 of the paper by Havin, Mashreghi and Nazarov, see [21].
Let ω be a function satisfying conditions of our Theorem 1. The author has been asked independently by Anton Baranov and Konstantin Dyakonov, whether then the function (log ω(·))/(1 + (·) 2 ) 1/2 necessarily belongs to the homogeneous fractional Sobolev-Slobodecki spaceẆ 1/2,2 (R). If this condition is fulfilled for some function that has convergent logarithmic integral, then some regularization of this function is an admissible majorant, see [6]. This is hence a very natural question, since the regularity condition of our Theorem 1 also resembles one of the definitions of homogeneous Sobolev spaces. In fact, it turns out that there exist functions to which our Theorem 1 can be applied and that do not verify this condition.
Proposition 4. For each r > 1, there exists a function Υ ∈ L 1 (dP ) V r such that Υ(x) = 0 for x ∈ [−1, 1] and such that the quotient K defined for real x by Note that the following approximation property of the spaces V r with r > 1 is a direct consequence of our Theorem 1.
Corollary. Thanks to a theorem by A. Baranov and V. Havin (see paper [1], Section 6), we get that for any ω ∈ V r with r > 1, the space of all functions in L 2 (R) with the spectrum in R\[0, σ] is not dense in the weighted Lebesgue space L 1 (ω) for any σ > 0.
We hope that our main results will find other applications in harmonic and complex analysis, in particular in theories of uncertainty principle and of exponential systems.
The main step of the proof of Theorem 1 is the following lemma.

Lemma 1. (a new variant of the global Nazarov lemma)
Let Ω ∈ L 1 (R, dx/(1 + x 2 )) be a nonnegative absolutely continuous function such that Ω ∈ V r . Then for each ε > 0 there exists a function Ω 1 , satisfying where H is the Hilbert transform on the real line.
Indeed, Theorem 1 follows from Lemma 1, thanks to the following sufficient condition for a function to be a BM majorant, which is a consequence of a more general result, proved by Mashreghi and Havin.
Remark. The proof of Theorem B uses a one-dimensional construction coming from the classical (complex) theory of Hardy spaces on the unit circle. Namely, given a nonnegative function on the unit circle with convergent logarithmic integral there exists an analytic function whose modulus coincides with the former function. Such functions are called outer, see [23] for details.
For necessary conditions for σ admissible majorants, see papers [3] and [4] by Y. Belov and paper [2] by A. Baranov and V. Havin.
We briefly discuss main ideas lying behind our proof of Lemma 1. Our proof is inspired by that of the Nazarov Lemma from paper [21]. Indeed, we use the beautiful idea of a so-called regularized system of intervals, which was first introduced by F. Nazarov and fruitfully used in [21]. Another important feature of the proof of the Nazarov Lemma in [21] is a version of the Hadamard-Landau inequality. We have had to modify this result drastically in order for it to fit the conditions of our Lemma 1. This culminated in Lemma 4 of this paper. On top of that, most estimates from the proof in [21] become considerably harder under the assumption log(1/ω) ∈ V r in comparison to the Lipschitz condition from [21].
Note that Nazarov's Lemma is by itself a highly nontrivial and very interesting result in Harmonic Analysis. To illustrate this, we mention paper [26] by P. Zatitsky and D. Stolyarov, where the authors have utilized the main object of the Nazarov Lemma, the regularized system of intervals, in some special form. For the multidimensional version of the classical Nazarov Lemma, see paper [28].
Let us now discuss the second main result of this article. Our Theorem 2, gives an answer to the following question: "Can one put r = 1 in the formulation of Theorem 1 ?" The answer to this question is negative, which is shown in the following result.
In fact, in this case, both alternatives (ω in and not in BM class) may occur. This can be easily derived from our Theorems 1 and 2 and the observation that V r ⊂ V 1 once r > 1. However, in the space V 1 , there are BM majorants that do not belong to any class V r with r > 1.
The main reason why Theorem 2 holds is, vaguely speaking, that a function in the class V 1 can have linear growth at sufficiently long intervals, tending to infinity. Indeed, it is not hard to give an example of a system of intervals {J j } j∈Z and the corresponding space V r which is strictly wider than V r . Moreover, one can give an example of a Hadamard lacunary sequence of positive reals, that can not play the role of the lengths of intervals {J j } from the definition of spaces V r , see the two remarks right after the proof of Theorem 2. The same applies to the weight in front of the integrals in this definition. The reason why it is so is that under such general conditions the function Ω need not have linear growth at infinity, whereas this is indispensable within our framework, see estimate (3) in our Section 2 below. However, in most cases like that of which we can imagine, the corresponding functions growing faster than linear at infinity will actually not be BM majorants, see the second remark of our fourth section.
Remark. On the positive side, one can of course replace 2 n in the definition of the spaces V r by any sequence of reals {λ n } satisfying λ ≤ λ n+1 /λ n ≤ Λ with 1 < λ ≤ Λ < ∞.
Note that the discussion just above and our Theorem 2 and Proposition 5 show that the condition log(1/ω) ∈ V r with r > 1 in our Theorem 1 is sharp in a number of different senses.
The proof of Theorem 2 builds upon one construction from paper [2]. This construction says that smallness of a bandlimited function is "contagious": if such a function is small on an interval it is also small on a much larger concentric interval. This construction is due to A. Borichev and it works only for majorants that have a growth strictly greater than linear at a sequence tending to infinity. Majorants in class V 1 have at most linear growth at infinity. Nevertheless, for some functions in class V 1 , we were able to use a combination of Borichev's construction with an iteration method to prove Theorem 2.
The very same "stability of smallness" argument of Borichev yields the fact that "typical" functions in classes V r \Lip(R) are not strongly admissible majorants, see [5] for a definition of the strong admissibility. This is discussed in our third remark after the proof of Theorem 2.
The paper is organized as follows. Theorem 1 is proved in Sections 2 and 3. The fourth Section is devoted to the proofs of Theorem 2 and Proposition 5. Elementary proofs of the first three propositions are left to the reader and are not detailed here. The Appendix contains examples of functions verifying conditions of our Theorem 1 and whose weighted logarithms have unbounded fractional homogeneous Sobolev spaceẆ 1/2,2 (R) seminorms.
We finally mention four open questions concerning Theorems 1 and 2. The first question consists in determining whether the condition log(1/ω) ∈ V r in Theorem 1 can be weakened down to, roughly speaking, a condition of the kind "ω belongs to some Orlicz type space, defined in the spirit of V r spaces". The second question concerns the system of intervals that are used in the definition of the spaces V r . Indeed, still open is the question of finding a necessary and sufficient condition on the alluded system of intervals, under which we have the first theorem. Yet another question is to find a multidimensional version of Theorem 1 which seems unavailable at the present time, according to [27]. The fourth and the final question reads as follows. It would be also interesting to find counterparts of the main results of this paper in the context of the so-called model spaces, in spirit of Yu. S. Belov's early papers. The author plans to attack the first three of the aforementioned questions in the nearest future. D. Baranov, Evgueni S. Doubtsov and to Konstantin M. Dyakonov for a number of helpful discussions.

A new local Nazarov lemma
Remark. We accumulate here the list of the frequently used technical abbreviations and notations. For an interval a ⊂ R its length is denoted by l(a), c a will stand for the center of a and λa with λ positive will be the interval centered at c a and whose edge length equals λl(a).
The main step of the proof of our new global Nazarov lemma is its following local variant.
Remark. Until the end of this section, the signs and indicate that the lefthand (right-hand) part of an inequality is less than the right-hand (left-hand) part multiplied by a constant independent of δ, f, I and κ. The sign ≍ means that we have both bounds and simultaneously.
Remark. The rest of this section is entirely devoted to the proof of Lemma 2. . Note that then f L ∞ (I * ) ≤ l(I * ) ≡ 1 and f ′ L r (I * ) ≤ κ/δ. If we manage to prove that there exists a majorant F of the function f which equals 0 outside 1.5I * and satisfying properties 3 and 4 with δ = 1 and κ replaced by κ/δ, then the function F (·) := δl(I) F (((·) − c I )/l(I)) will be the needed majorant of the function f . Thus, it is sufficient to prove Lemma 2 for δ = 1 and I * . The rest of this section is devoted to the proof of Lemma 2 in this case.
The following definition is very important.
Definition 4. We say that a dyadic interval I ⊂ I * is essential, if f L ∞ (I) ≥ l(I)/2. Denote A the set of essential intervals.
It is straightforward to see that holds

Consider, A M the set of maximal by inclusion elements of A.
To each interval a ∈ A M we associate its tail t(a). For the definition of a tail, see paper [21], Section 2.6.5. In fact, after that we will have added these tails, we will get a regularized system of intervals, see the discussion in the introduction to this paper. Next, we define B := a∈A M t(a), and then pose τ := {c ∈ B M : c ⊆ I * }.
Here, B M stands for the set of maximal by inclusion elements of B. Define for an interval a ∈ τ its neighborhood N(a) by Note that #N(a) ≤ 9. We shall need the following property of the system τ. Proof: The proof of this lemma is not detailed here, since it can be found in paper [21], Section 2.6.6.
Denote 2τ := {2c : c ∈ τ }. As a direct consequence of the lemma, we deduce that the multiplicity #{b ∈ 2τ : x ∈ b} is uniformly bounded in

Simple calculation shows that Hφ
1. We finally define F by F := a∈τ φ a . Now we have to check the required properties of the majorant F. The first one follows readily from the definition of F. To prove the second one, note that for all a ∈ τ holds f L ∞ (a) ≤ l(a). Indeed, suppose the contrary, i.e. that f L ∞ (a 0 ) > l(a 0 ) for some a 0 ∈ τ . This means that which in turn signifies that 2a 0 is an essential interval and hence 2a 0 ∈ τ . This contradicts to the definition of τ. From here we deduce that if x ∈ a ∈ τ , then Next we estimate the integral of the function F . To this end, we prove a variant of the Hadamard-Landau inequality which is appropriate for our goals.
Let x 0 ∈ a be a point such that f L ∞ (a) = f (x 0 ). Suppose with no loss of generality that a + − x 0 ≥ l(a)/2, where a + is the right end of the interval a. Consider a point x ∈ (x 0 , a + ). We make use of the Newton-Leibniz formula and of the Hölder inequality to infer the following estimates Denote υ := (f (x 0 )/κ a ) r/(r−1) . Observe that the point x 0 + υ/2 belongs to the interval a. This follows from the inequality f L ∞ (a) ≤ l(a), which is true since a ∈ τ . We integrate the inequality f (x) ≥ f (x 0 ) − κ a |x − x 0 | 1−1/r using this observation and deduce that and Lemma 4 is proved.
We use this result to estimate the integral of the function F : The last and the last but one inequalities just above are in need of explanation. The last estimate uses the facts that κ c < κ and that 2 − α = 2r/(2r − 1), whereas the penultimate bound follows from the Hölder inequality with exponents 1/α > 1 and its conjugate 1/(1 − α).
It is now left to derive the inequality on the derivative of the Hilbert transformation of the function F . First, we shall obtain this estimate for x ∈ b∈τ 2b. Let a(= a(x)) denote the interval from τ such that x ∈ 2a. We isolate the neighborhood N(a) from its complement in τ and infer the following inequality We shall estimate the terms S 1 , S 2 and S 3 separately. We start with the sum S 1 , whose estimate turns out to be easy

1.
We further proceed to the second term. We use a simple estimate on the kernel of the Hilbert transformation, the discussed property of the covering τ and the fact that the system of intervals {2b} b∈τ (we mean Lemma 3) has finite multiplicity to write The third term can be estimated using the same property of τ : and the lemma for x ∈ b∈τ 2b follows. Hence, the needed variant of the local Nazarov lemma is proved.

Proof of a new global Nazarov lemma
In this section, we shall derive the global Nazarov lemma from the local one. Proof: Denote µ := Ω Vr . Note that we may assume in the global Nazarov lemma that Ω(x) = 0 for |x| ≤ R with R being an arbitrary large positive number. Indeed, if it is not the case, then consider the function Ω (·) = max(0, Ω − M)(·), where M := max x∈B(0,R) Ω(x). If Ω 1 is a majorant of Ω , then the function Ω 1 +M will be the desired majorant of Ω.
Next, we shall prove the following inequality Indeed, according to the previous paragraph this is obvious once |x| 1, and for 1 |x| we argue as in Lemma 4 to infer the following chain of inequalities Recall the above defined system of intervals: J 0 = [−2, 2), and for j ∈ N, and apply the local lemma to each interval J j and the corresponding restriction f j = Ω J j . Indeed, Lemma 2 can be applied since these functions satisfy f j ∞ ≤ ε2 j ≤ εl(J j ) by the line (2) and f ′ j L r (J j ) ≤ µl(J j ) 1/r , thanks to the fact that Ω is in V r . This yields functions F j for j ∈ Z. The needed majorant Ω 1 is defined by Now, we shall check the required properties of Ω 1 . The first property follows obviously from the local Lemma. We proceed to the second one: where in the fourth inequality above we have used the fact that α ∈ (0, 1). So, it is left to check that the third conclusion holds. First, fix a point x ∈ R. Second, denote by S(x) the interval from the system F = {J j } j∈Z such that x ∈ S(x). Next, denote by U(x) the subset of F consisting of the neighbor intervals of S(x) and by W (x) its complement: W (x) = F \U(x). Finally, write the function Ω 1 as a sum of two functions as follows: Since there is only a finite number of intervals in the family U(x), we see that where we have just used condition 3 of Lemma 2 in the last estimate. On the other hand, since supp(ω 1 ) ⊆ j∈W (x) 1.5J j we deduce that Therefore, we arrive at the following chain of inequalities Hence, the needed variant of the Nazarov lemma is proved.
Thus, Theorem 1 is also proved, recall Theorem B.

The class V 1
We begin with the proof of Theorem 2. Proof: Fix γ > 1. Denote I n := [2 n − 2 n /n γ , 2 n + 2 n /n γ ], for n ≥ 3 and let T In (x) stand for the distance from x to R\I n . Consider for x ∈ R the following function ω(x) := exp(−n γ T In (x)), if x ∈ I n with n ≥ 3, 1, otherwise.
We claim that log(1/ω) ∈ L 1 (dP ) and log(1/ω) ∈ V 1 , though ω is not a BM majorant. The first two claims are easy to verify, so we omit their proofs.
Let us prove that ω is not in the BM class. Suppose the contrary. Hence, for a fixed σ > 0 there exists a function, not identically zero, satisfying f ∈ L 2 (R), spec(f ) ⊂ [0, σ] and |f (x)| ≤ ω(x) for all real x. We shall now use Lemma 1 from paper [2]. Notice that the function f satisfies the conditions of this lemma with h := n γ and I := I n for n ≥ 3. We deduce from this lemma that there exists a universal constant C and a power α ∈ (0, 1/2), depending only on σ such that |f (x)| ≤ exp(−C2 n ) on the interval I n,1 := (n γα /2)I n .
Remark. From now on until the end of the present article, the sign X ≍ Y means that C 1 Y ≤ X ≤ C 2 Y for some constants C 1 and C 2 depending only on γ, α, σ and C. In this case, we shall say that X is of order Y .
Acting inductively, after m steps, we arrive at the estimate |f (x)| ≤ exp(−C m 2 n ), verified by f for x ∈ I n,m with |I n,m | ≍ n −γ(1−α) m 2 n .
Maybe, it is worth noting that I k,m ∩ I n,m = ∅ for any natural m, once n = k. This results from the fact that we assume, as we can, that C < 1.
We are now in position to prove that f = 0 identically, which will lead to a contradiction. To this end, we estimate the logarithmic integral of f . For each natural number m holds Choosing m sufficiently large and recalling that (1 − α) ∈ (0, 1), we arrive at the formula R log |f (x)|dP (x) = −∞. Since f ∈ L 2 (R) has the spectrum in the interval [0, σ], it hence belongs to the Hardy class H 2 (R). From the Jensen inequality, see [14] we deduce that f = 0 identically, which contradicts our assumption. Hence, the second theorem is proved.
Remark. Alas, our proof above does not work, if one considers functions with larger apertures of the "walls" of graphs.
The proof just above can be used to show that this function is not a BM majorant. This means that one can not replace 2 n in the definition of spaces V r by a general Hadamard lacunary sequence.
Remark. It can be seen exactly as above that the function ω * is not a strictly admissible majorant.
We next give a proof of Proposition 5. Proof: We shall construct desired functions with help of (a part of) the proof of the Second BM theorem.
Let Λ be a sequence of reals. Recall the definition of the Beurling-Malliavin density of Λ, denoted here BM(Λ) (see [20], Section 1.1). A part of the proof of the second BM theorem (see [21], Section 3.3) says exactly that if r > BM(Λ), then the function χ R\Λ is a 2πr admissible majorant.
Note that holds log(1/ω) ∈ L 1 (dP ), log(1/ω) ∈ V 1 and log(1/ω) / ∈ V r for all r > 1. Let us now show that ω ∈ BM. Since ω 0 ∈ BM, we know that for any σ > 0 there is f ∈ L 2 (R), not identically zero with spec(f ) ⊂ [0, σ] and |f (x)| ≤ ω 0 (x) for all real x. Fix σ > 0 and a function f as just above. Note that we can assume that f L 2 (R) ≤ 1. Indeed, if it is not the case, then it suffices to consider the function f 0 := f / f L 2 (R) .
We end up the proof of the proposition via minorizing the logarithm inside the maximum just above by a piecewise linear function. This gives log 1 |f (x)| ≥ max (2 n − exp(2 n ) · |x − 2 n |, 0) , and we are done.
Remark. The angles of the walls of the graphs of functions constructed in this proposition are even more acute than these of Theorem 2. It would be interesting to find an optimal borderline bound for the angle that still works.

Appendix
We finally prove Proposition 4. Proof: For a given r > 1, let us construct a function Υ that is zero in a neighborhood of the origin, that is integrable against the Poisson measure, that belongs to the class V r and such that E(K) = ∞. Recall that K stands for the following quotient K(x) = Υ(x)/x. Choose α ∈ (0, 1/2] and β > 0 satisfying α + β > 1. Define intervals I n := [2 n , 2 n + 2 n /n β ]. Let T In (x) as before stand for the distance from x to R\I n . Consider for x ∈ R the following function (1/n α ) 2 (2 n /n β ) 2 dxdy ≥ n≥3 1 n 2α .
The series from the right-hand side of the last line diverges, so E(K) = ∞ and the result follows.