BEURLING-MALLIAVIN THEORY FOR TOEPLITZ KERNELS

N. MAKAROV AND A. POLTORATSKI

Abstract. We consider the family of Toeplitz operators T_{JS̄^{a}} acting in the Hardy space H^{2} in the upper halfplane; J and S are given meromorphic inner functions, and a is a real parameter. In the case where the argument of S has a power law type behavior on the real line, we compute the critical value c(J, S) = inf {a : ker T_{JS̄^{a}} ≠ 0} . The formula for c(J, S) generalizes the Beurling-Malliavin theorem on the radius of completeness for a system of exponentials.

1. Introduction and results

1.1. Completeness of complex exponentials. For Λ ⊂ C denote

E_{Λ} = {e^{iλx} : λ ∈ Λ} .

By definition, the radius of completeness for the family E_{Λ} is the number

R(Λ) = sup{a : E_{Λ} is complete in L^{2}(−a, a)}.

(A family is complete if finite linear combinations of its elements are dense in the corresponding space.) In [1]-[2], Beurling and Malliavin established a formula for R(Λ) in terms a certain density of Λ at infinity.

If Λ ⊂ ℝ, then the effective (or Beurling-Malliavin) density D_{eff}(Λ) is the supremum of the set of numbers a ≥ 0 such that there is a collection of disjoint intervals {l_{j}} satisfying the following two conditions:

∑_{j} \frac{|l_{j}|^{2}}{1 + d_{j}^{2}} = ∞, d_{j} := dist(0, l_{j}),

and

∀j, #(Λ ∩ l_{j}) ≥ a|l_{j}|.

Beurling-Malliavin’s ”Second Theorem” (BM2 for short) states that if Λ ⊂ ℝ, then

R(Λ) = πD_{eff}(Λ).

This formula extends to the general case Λ ⊂ C as follows. If Λ satisfies the Blaschke condition

(B)  ∑_{λ∈Λ} |ℑ λ^{−1}| < ∞,

then

R(Λ) = πD_{eff}(Λ^{∗}),

The first author is supported by N.S.F. Grant No. 0201893 .

The second author is supported by N.S.F. Grant No. 0500852.

where

Λ^{∗} = {λ^{∗} : λ ∈ Λ, ℜλ ≠ 0} , λ^{∗} := [ℜ λ^{−1}]^{−1} ;

otherwise

Λ ∉ (B)  ⇒  R(Λ) = ∞.

The Beurling-Malliavin theorem crowned a long search for a solution of the completeness problem, see [29], [21], [32], [20]. We refer to [31] for historical information; let us only mention that one of the earliest results of the theory was the estimate

R(Λ) ≤ πD(Λ),  (Λ ⊂ ℝ),  (1.1)

where D(Λ) is the usual upper density of Λ at infinity.

The Beurling-Malliavin theory also comprises their ”First Theorem” (BM1), a result of considerable independent interest and (so far) a necessary step in the proof of BM2. A detailed exposition of BM theory (including clarification and further improvements of the argument) is presented in the monographs [17], [18], [9], see also [14], [6]. New applications and new approaches to various parts of the theory have been suggested; see [3], [10], [24] for some recent developments; also see [15] for a modern overview of the completeness problem for exponential systems.

In this paper we generalize BM theory to many other families of special functions. We state our results in the language of Toeplitz kernels referring to our paper [23] for a detailed explanation of how results of this type are related to the completeness problem for families of solutions of general Sturm-Liouville problems.

1.2. Toeplitz kernels. The completeness radius problem can be restated in terms of Toeplitz operators as follows. Recall that the Toeplitz operator T_{U} with a symbol U ∈ L^{∞}(ℝ) is the map

T_{U} : H^{2} → H^{2}, F ↦ P_{+}(UF),

where P_{+} is the orthogonal projection in L^{2}(ℝ) onto the Hardy space H^{2} = H^{2}(C_{+}) in the upper halfplane C_{+} = {ℑz > 0}. By duality and the definition of the classical Fourier transform,

f(t) ↦ f̂(z) = ∫ e^{izt}f(t)dt,

the exponential family E_{Λ} is complete in L^{2}(−a, a) if and only if there is a non-trivial function F in the Paley-Wiener space

PW_{a} = {f̂: f ∈ L^{2}(−a, a)}

such that F = 0 on Λ. According to Paley-Wiener’s theorem, the Fourier transform isometrically identifies L^{2}(0, ∞) with H^{2}(C_{+}), and therefore

PW_{a} = e^{−iaz} [H^{2} ⊖ e^{2aiz}H^{2}] .

The subspace H^{2} ⊖ e^{2aiz}H^{2} is the so called model space of the inner function e^{2aiz}. More generally, one defines model spaces

K_{Θ} = H^{2} ⊖ ΘH^{2}

for arbitrary inner functions Θ. The elements of K_{Θ} are analytic functions in C_{+} but if Θ has a meromorphic extention to the whole complex plane, (we call such Θ’s meromorphic inner functions), then the elements of K_{Θ} are defined as functions in C. The completeness problem for exponentials is exactly the problem of describing the sets of uniqueness for the model space of e^{2aiz}.

Suppose now that Λ is a subset of C_{+} satisfying the Blaschke condition, and let B_{Λ} be the corresponding Blaschke product. A simple argument shows that Λ is a set of uniqueness for K_{Θ} if and only if the Toeplitz operator with the symbol U = B_{Λ}Θ̄ has a trivial kernel. In particular, we obtain the formula

R(Λ) = inf {a : ker T_{B_{Λ}e^{−2aiz}} ≠ 0} .

There is a similar statement in the general case Λ ⊂ C, see [23], Section 3.1. For example, if Λ ⊂ ℝ, then

R(Λ) = inf {a : ker T_{J_{Λ}e^{−2aiz}} ≠ 0} ,

where J_{Λ} denotes some/any meromorphic inner function J such that Λ is precisely the level set {J = 1}.

We should mention that the idea of the Toeplitz operator approach in the study of exponential systems was introduced in the series of papers [30], [26], [12]. This approach has been particularly successful for the interpolation and sampling theory in Paley-Wiener spaces, see [22], [28], [33].

We will use the following notation for kernels of Toeplitz operators (or Toeplitz kernels in H^{2}):

N[U] = ker T_{U}.

(For example, N[Θ̄] = K_{Θ} if Θ is an inner function.) We will also consider Toeplitz kernels in the Smirnov-Nevanlinna class N^{+} = N^{+}(C_{+}),

N^{+}[U] = {F ∈ N^{+} ∩ L_{loc}^{1}(ℝ) : ŪF̄ ∈ N^{+}},

and in the Hardy spaces H^{p} = H^{p}(C_{+}),

N^{p}[U] = N^{+}[U] ∩ L^{p}(ℝ), (0 < p ≤ ∞).

See [16], [7], [27] for general references concerning the Hardy-Nevanlinna theory.

1.3. Generalization of Beurling-Malliavin theory. A natural way to generalize the completeness radius problem (and the BM2 theorem) is to ask about the exact value of the infimum

inf {a : ker T_{JS̄^{a}} ≠ 0}  (1.2)

for arbitrary meromorphic inner functions J and S. We will give an answer in the case where the argument of S has a power law type behavior,

(arg S)′(x) ≍ |x|^{κ}, x → ±∞,

with κ ≥ 0. (We call the case κ ≥ 0 super-exponential to underline the relation to the classical case S(x) = e^{iax}. In Section 1.5 below we will comment on the sub-exponential case κ < 0.)

As explained in [23], the computation of the ”radius” (1.2) has some immediate consequences for the theory of Sturm-Liouville (SL) operators. Roughly speaking, the case of SL operators with eigenvalues

λ_{n} ≍ n^{ν} belongs to the theory with parameter

κ = \frac{2}{ν} − 1 ≥ 0.

If κ > 0, the SL operators are singular in contrast to the BM case S(x) = e^{iax}, which applies to regular operators. In addition to the completeness problem for systems of solutions of SL equations, cf [11], the generalized BM theory applies to certain problems of spectral theory as well as the theory of (Weyl-Titchmarsh) Fourier transforms associated with SL operators and the corresponding (de Branges) spaces of entire functions.

To state our results, we need to introduce the notion of BM intervals. Let γ be a continuous function ℝ → ℝ such that γ(∓∞) = ±∞. i.e.

lim_{x→−∞} γ(x) = +∞,  lim_{x→+∞} γ(x) = −∞.

The family BM(γ) is defined as the collection of the components of the open set

{x : γ(x) ≠ max_{[x,+∞)} γ} .

For an interval l ∈ BM(γ), we denote its length by |l| or simply by l, and we denote the distance to the origin by d = d(l).

1.4. Super-exponential case. If κ ≥ 0, then we say that γ is (κ)-almost decreasing if

γ(∓∞) = ±∞,  ∑_{l∈BM(γ)} d^{κ−2}l^{2} < ∞,  (1.3)

where the sum is taken over intervals satisfying d(l) ≥ 1.

The standard terminology in the classical κ = 0 case is the following: the family BM(γ) is short if γ is almost decreasing; otherwise we say that BM(γ) is long.

Theorem A. Let κ ≥ 0, and let U = e^{iγ} and S = e^{iσ} be smooth unimodular functions on ℝ such that

γ′(x) ≳ −|x|^{κ}, σ′(x) ≳ |x|^{κ},  (x → ∞).  (1.4)

(i) If γ is not (κ)-almost decreasing, then N^{+}[US^{ϵ}] = 0 for all ϵ > 0.

(ii) If γ is (κ)-almost decreasing, then N^{p}[US̄^{ϵ}] ≠ 0 for all ϵ > 0 and all p < \frac{1}{3}.

Here and throughout the paper the notation f(x) ≳ g(x) means that f(x) ≥ cg(x) for some c > 0 and all x such that |x| ≫ 1.

Given two unimodular functions U and S as in Theorem A, we can consider the family of symbols

US̄^{a} = e^{iγ_{a}},  γ_{a} = γ − aσ,  (a ∈ ℝ).

If a_{1} > a and if γ_{a} is decreasing near ∞, then γ_{a_{1}} is also decreasing. It is not difficult to see that the same is true for almost decreasing functions, so we can define the transition parameter

c ≡ c(U, S; κ) = inf{a : γ_{a} is (κ)-almost descreasing} ∈ (−∞, +∞].

Corollary. Let U = e^{iγ} and S = e^{iσ} be such that

γ′(x) ≳ −|x|^{κ}, σ′(x) ≍ |x|^{κ}, (x → ∞),

and let c = c(U, S; κ). Then for all p < 1/3 we have

N^{p}[US̄^{a}] = 0 (a < c), N^{p}[US̄^{a}] ≠ 0 (a > c).

Indeed, if a < c then γ_{a+ϵ} is not almost decreasing for some ϵ > 0, and we have

N^{p}[US̄^{a}] ⊂ N^{+}[US̄^{a+ϵ}S^{ϵ}] = 0

by Theorem A, which can be applied because γ′_{a}(x) ≳ −|x|^{κ} for all a’s. Similarly, if a > c, then γ_{a−ϵ} is almost decreasing for some ϵ > 0, and we have

N^{p}[US̄^{a}] = N^{p}[US̄^{a−ϵ}S̄^{ϵ}] ≠ 0.

In the special case where U is an inner function, we can extend the statement of he corollary to all values of p, in particular p = 2.

Theorem B. Let J be a meromorphic inner function, and suppose that a unimodular function S satisfies

(arg S)′(x) ≍ |x|^{κ}, x → ∞.

Denote c = c(J, S; κ). Then for all p ≤ ∞ we have

N^{p}[JS̄^{a}] = 0 (a < c), N^{p}[JS̄^{a}] ≠ 0 (a > c).

1.5. Sub-exponential case. It is easy to see that the statement of Theorem A (and Theorem B) can not be extended to the case κ < 0. For example, the functions

σ(x) = 2 sign(x) |x|^{1/4}, γ(x) = 2(1 + \sqrt{2}) 1_{ℝ_{−}}(x) |x|^{1/4}

satisfy the conditions (1.4) with κ = −3/4, and of course γ(+∞) ≠ −∞. Nevertheless, for U = e^{iγ} and S = e^{iσ} we have

N^{∞}[US] ≠ 0,

because

US = f̄/f, f(z) = exp{−(1 + i)z^{1/4}} ∈ H^{∞}(C_{+}).

(Also note that the sum ∑ d^{κ−2}l^{2} in (1.3) is always finite if κ < 0.)

The Beurling-Malliavin theory extends to the sub-exponential case in a different fashion. For κ ∈ (−1, 0] we consider the weighted (non-linear) Smirnov-Nevanlinna classes

N_{κ}^{+} = \{F ∈ N^{+} : log |F | ∈ L^{1} (ℝ, \frac{1}{1 + |x|^{2+κ}} )\} ,

and define the corresponding Toeplitz kernels as follows:

N_{κ}^{+}[U] = N^{+}[U] ∩ N_{κ}^{+}, N_{κ}^{p}[U] = N^{p}[U] ∩ N_{κ}^{+}.

Theorem C. Let κ ∈ (−1, 0], and let U = e^{iγ} and S = e^{iσ} be smooth unimodular functions such that

γ′(x) ≳ −|x|^{κ}, σ′(x) ≳ |x|^{κ} (x → ∞).

(i) If the family BM(γ) is long, then N_{κ}^{+}[US^{ϵ}] = 0 for all ϵ > 0.

(ii) If the family BM(γ) is short, then N_{κ}^{p}[US̄^{ϵ}] ≠ 0 for all ϵ > 0 and all p < \frac{1}{3}.

One can also state a theorem similar to Theorem B. Applications of these results to Volterra operators, see [8], and higher order differential operators will be discussed in a separate paper.

1.6. Hilbert transform. The main tool in the proof of the theorems stated above is the one-dimensional Hilbert transform. Let Π denote the Poisson measure on ℝ,

dΠ(t) = \frac{dt}{1 + t^{2}} .

If h ∈ L_{Π}^{1} ≡ L^{1}(ℝ, Π) is a real-valued function, and if Sh denotes its Schwarz integral,

Sh(z) = \frac{1}{πi} ∫_{ℝ} [\frac{1}{t − z} − \frac{t}{1 + t^{2}} ] h(t) dt,  (z ∈ C_{+}),  (1.5)

then h̃, the Hilbert transform of h, is defined a.e. on ℝ as the angular limit of ℑ[Sh]. Alternatively, h̃ can be defined as a singular integral:

h̃(x) = \frac{1}{π} v.p. ∫ [\frac{1}{x − t} + \frac{t}{1 + t^{2}} ] h(t)dt, (x ∈ ℝ).

(As a general rule we identify Nevanlinna class functions in the halfplane C_{+} with their angular boundary values on ℝ; e.g. we may write Sh = h + ih̃.)

The relevance of the Hilbert transform in the theory of Toeplitz kernels can be explained by the following simple observation, see [23], Section 2.

Suppose γ : ℝ → ℝ is a smooth function. Then N^{+} [e^{iγ}] ≠ 0 if and only if

γ = −α + h̃  (1.6)

for some smooth increasing function α and some h ∈ L_{Π}^{1}. There is a similar criterion for Toeplitz kernels in Hardy spaces: N^{p} [e^{iγ}] ≠ 0 if and only if γ admits a representation (1.6) with α being the argument of some inner function and with h ∈ L_{Π}^{1} such that e^{−h} ∈ L^{p/2}(ℝ).

For further references, we recall some properties of the Hilbert transform. We denote by L_{Π}^{(1,∞)} the usual weak L^{1}-space with respect to the Poisson measure. Kolmogorov’s theorem states that

h̃ ∈ L_{Π}^{o(1,∞)},

where L_{Π}^{o(1,∞)} stands for the ”little o” subspace of L_{Π}^{(1,∞)}, i.e.

Π\{|h̃| > A\} = o (\frac{1}{A}) ,  A → ∞.  (1.7)

For bounded functions we have the following (Smirnov-Kolmogorov) estimate :

∥h∥_{∞} < \frac{π}{2}  ⇒  eh̃ ∈ L_{Π}^{1}.

Together with the criterion (1.6), this implies

∥γ∥_{∞} < \frac{π}{p}  ⇒  N^{p}[b̄^{2/p}e^{iγ}] ≠ 0,  (1.8)

where b is the Blaschke factor

b(z) = \frac{i − z}{i + z} ,  z ∈ C_{+}.  (1.9)

1.7. The structure of BM theory. In the remaining sections of the paper we prove Theorems A and B. We closely follow all the steps in our presentation of the classical Beurling-Malliavin theory in [23], combining them with certain estimates of the Hilbert transform, which we derive in Section 2. To make the proof selfcontained, in several places we had to repeat the argument outlined in [23]. To avoid further repetitions we decided to omit the proof of Theorem C because the reasoning in the sub-exponential case is quite similar. The proof of Theorems A and B is organized as follows.

(1) Upper density estimate: γ(±∞) ≠ ∓∞ implies N^{+}[US^{ϵ}] = 0; Section 3.1. This statement is analogous to the estimate (1.1).

(2) Effective density estimate: ∑ d^{κ−2}l^{2} = ∞ implies N^{+}[US^{ϵ}] = 0; Sections 3.2-3.4. Together with (1) this generalizes the estimate R(Λ) ≤ πD_{eff}(Λ) in BM2.

(3) Little multiplier theorem: if γ is almost decreasing, then N^{+}[US̄^{ϵ}] = 0; Section 4.

(4) BM multiplier theorem: if the weighted Dirichlet norm of log W is finite, then W belongs to some Hardy space up to a factor from N^{+}[S̄^{ϵ}]; Sections 5.1-5.2.

(5) A version of BM1: the logarithm of any outer function in N^{+}-kernel has a finite weighted Dirichlet norm. This is used to show that non-triviality of N^{+}-kernels implies non-triviality of N^{p}-kernels for symbols involving inner functions; Sections 5.3-5.4.

(6) L^{p}-multipliers: approximation by inner functions and multiplying the elements of N^{p}-kernels down to H^{∞}; Section 6.

2. One-sided Lipschitz condition for the Hilbert transform

In this section we discuss various consequences of the weighted one-sided Lipschitz condition

h̃′(x) ≲ |x|^{κ}, x → ∞,

for smooth real-valued functions h ∈ L_{Π}^{1}. (Here and elsewhere h̃′ means (h̃)′.)

2.1. Application of Kolmogorov’s theorem.

Lemma 1. If κ ≥ 0 and h ∈ L_{Π}^{1}, then

h̃′(x) ≲ x^{κ} ⇒ h̃(x) = o (x^{κ+1}) as x → +∞.

Proof: By Kolmogorov’s theorem, we have

h̃ ∈ L_{Π}^{o(1,∞)}.  (2.1)

If x_{∗} ≫ 1 and h̃(x_{∗}) ≥ cx_{∗}^{κ+1} for some c > 0, then for all x such that 1 ≪ x ≤ x_{∗} we have

h̃(x) ≥ h̃(x_{∗}) − a ∫_{x}^{x_{∗}} t^{κ} dt ≥ cx_{∗}^{κ+1} − a(x_{∗}^{κ+1} − x^{κ+1}),

and it follows that

h̃(x) ≳ x_{∗}^{κ+1} ≥ x_{∗}

for all x in some interval (x_{∗∗}, x_{∗}) of length ≍ x_{∗}. Since Π(x_{∗∗}, x_{∗}) ≍ 1/x_{∗}, this contradicts (2.1), see (1.7).

If h̃(x_{∗}) ≤ −cx_{∗}^{κ+1} for some c > 0, then by a similar argument we have

h̃(x) ≲ −x_{∗}^{κ+1} ≤ x_{∗}

for all x in some interval (x_{∗}, x_{∗∗}) of length ≍ x_{∗}, which again contradicts (2.1). □

We will also need the following version of this lemma.

Lemma 2. Let h ∈ L_{Π}^{1}, κ ≥ 0, and a ∈ ℝ. If

h̃′(x) + ax^{−1}h̃(x) ≤ x^{κ},  x ≫ 1,  (2.2)

then

h̃(x) = o (x^{κ+1}) and h̃′(x) ≤ x^{κ} + o (x^{κ}) as x → +∞.

Proof: Suppose we have h̃(x_{∗}) ≥ cx_{∗}^{κ+1} for some x_{∗} ≫ 1. Let x_{1} be the smallest positive number such that h̃(x_{1}) = cx_{∗}^{κ+1}, so we have 1 ≪ x_{1} ≤ x_{∗} and h̃ ≤ cx_{∗}^{κ+1} on (0, x_{1}). Together with (2.2), this implies

h̃′(x) ≲ \frac{x_{∗}^{κ+1}}{x_{1}} , x ∈ (\frac{x_{1}}{2} , x_{1}) .

Arguing as in the previous proof, we see that h̃ ≳ x_{∗}^{κ+1} ≥ x_{1} on some interval of length ≍ x_{1}, which contradicts the weak L^{1}-estimate (1.7). The argument in the case h̃(x_{∗}) ≤ −cx_{∗}^{κ+1} is similar. □

2.2. Hilbert transform in weighted L^{1}-spaces.

Lemma 3. If κ ∈ [−1, 0) and h ∈ L^{1} (|x|^{−2−κ}), then

h̃′(x) ≲ x^{κ} ⇒ h̃(x) = o (x^{κ+1}) , x → +∞.

Proof: If κ ∈ (−1, 0), then the weight |x|^{−2−κ} satisfies the Muckenhoupt (A_{1}) condition at infinity, and therefore we have

h ∈ L^{1} (|x|^{−2−κ}) ⇒ h̃ ∈ L^{o(1,∞)} (|x|^{−2−κ}) ,

see [13]. (One can also give an elementary proof for this particular weight.) We then argue as in the proof of Lemma 1. For example, if x_{∗} ≫ 1 and h̃(x_{∗}) ≥ cx_{∗}^{κ+1}, then h̃ ≳ x_{∗}^{κ+1} on some interval [x_{∗∗}, x_{∗}] of length ≍ x_{∗}. The weighted length of this interval is ≍ x_{∗}^{−1−κ}, which contradicts the weak L^{1}-estimate.

If κ = −1, then we consider the function

h_{1}(x) = xh(x) ∈ L_{Π}^{1}.

Since h̃_{1}(x) = xh̃(x), we have

h̃′_{1}(x) = h̃(x) + xh̃′(x) ≤ x^{−1}h̃_{1}(x) + O(1), x → +∞.

By Lemma 2, we get h̃_{1}(x) = o(x) and therefore h̃(x) = o(1). □

2.3. Persistence of 1-sided Lipschitz condition.

Lemma 4. Let f ∈ L_{Π}^{1}, 0 ∉ supp f, and let

0 ≤ α ≤ β,  or  0 ≤ β < α < 2.

Denote

g(x) = |x|^{−α}f(x).

Then

f̃′(x) ≤ (1 + o(1)) |x|^{β} ⇒ g̃′(x) ≤ (1 + o(1)) |x|^{β−α},  (2.3)

and

x^{−1}f̃′(x) ≤ (1 + o(1)) |x|^{β−1} ⇒ x^{−1}g̃′(x) ≤ (1 + o(1)) |x|^{β−α−1}. (2.4)

Proof: We will prove (2.3) for x → +∞. The proof of the other cases is similar. Since the statement is trivial for α = 0, we will assume α > 0.

It is clear that we can modify f on any finite interval, so we will assume that f(x) = x^{N} near the origin for some N ≫ 1. If we specify f̃(0) = 0, then

|x|^{−α}f̃(x) ∈ L_{Π}^{1}.  (2.5)

Indeed, by Lemma 1 we have f̃ = O(|x|^{N}), and therefore

|x|^{−α}|f̃| = |x|^{−α} |f̃|^{α/N} |f̃|^{1−(α/N)} ≲ |f̃|^{1−(α/N)} ∈ L_{Π}^{1}

by Kolmogorov’s estimate (1.7).

Consider the analytic function

u(z) + iũ(z) := z^{−α}(f + if̃)(z), z ∈ C^{+},

where z^{−α} denotes the branch positive on ℝ_{+}. Note that

u(x) = g(x), ũ(x) = |x|^{−α}f̃(x) for x ∈ ℝ_{+},

and

u(x) = |x|^{−α} [f(x) cos απ + f̃(x) sin απ] for x ∈ ℝ_{−}.

By (2.5),

g − u ∈ L_{Π}^{1}, g − u = 0 on ℝ_{+},

so if we define

δ(x) = g̃(x) − |x|^{−α}f̃(x),

then δ = g̃ − ũ on ℝ_{+}, and we have the following representation for the derivative:

δ′(x) = { \atop (g − u)}′(x) = ∫_{−∞}^{0} \frac{c_{1}f(t) + c_{2}f̃(t)}{(t − x)^{2}} \frac{dt}{|t|^{α}} , (x > 0).

By the dominated convergence theorem

δ′(x) = o(1), x → +∞,

in particular

δ′(x) = o (x^{β−α}) if β ≥ α.

In the case 0 ≤ β < α < 2, we consider the integrals involving f and f̃ separately. We have

∫_{−∞}^{−1} \frac{|f(t)|}{(t − x)^{2}} \frac{dt}{|t|^{α}} ≤ ∫_{−∞}^{−x} \frac{1}{|t|^{α}} \frac{|f(t)| dt}{|t|^{2}} + \frac{1}{x^{α}} ∫_{−x}^{−1} \frac{|t|^{2−α}}{x^{2−α}} \frac{|f(t)| dt}{|t|^{2}}

≤ \frac{1}{x^{α}} ∫_{−∞}^{−x} \frac{|f(t)| dt}{|t|^{2}} + \frac{1}{x^{α}} ∫_{−x}^{−1} \frac{|f(t)| dt}{|t|^{2}} = o (x^{−α}) = o (x^{β−α}) .

Since β ≥ 0, by Lemma 1 we have

f̃(t) = o (|t|^{1+β}) ,

and since

∫_{−∞}^{−1} \frac{|t|^{1+β−α}}{(t − x)^{2}} dt ≤ ∫_{−∞}^{−x} |t|^{β−α−1}dt + \frac{1}{x^{2}} ∫_{−x}^{−1} |t|^{1+β−α}dt ≍ x^{β−α},

we have

∫_{−∞}^{−1} \frac{|f̃(t)|}{(t − x)^{2}} \frac{dt}{|t|^{α}} = o (x^{β−α}) .

It follows that in all cases we have

δ′(x) = o (x^{β−α}) , x → +∞,

and therefore

g̃′(x) = x^{−α}f̃′(x) − αx^{−α−1}f̃(x) + δ′(x) ≤ x^{β−α} + o (x^{β−α}) .
□

2.4. A converse. We will also need a converse of (2.3). We state it only for the range of parameters that will be used later.

Lemma 5. Let w ∈ L_{Π}^{1}, 0 ∉ supp w, and

0 < α ≤ β,  or  1 ≤ α ≤ min(2, β + 1).

Denote

h(x) = |x|^{−α}w(x).

Then

h̃′(x) ≤ (1 + o(1)) |x|^{β−α}  ⇒  w̃′(x) ≤ (1 + o(1)) |x|^{β}.

Proof: (a) The case β ≥ α. Let n be an even integer such that

α_{1} := n − α ∈ [0, 2).

Define

g(x) := x^{−n}w(x) = |x|^{−α_{1}}h(x).

Since α_{1} < 2 and β_{1} := β − α ≥ 0, we can apply Lemma 4 to f = h and g and obtain the estimate

g̃′(x) ≤ |x|^{β_{1}−α_{1}} + ⋯ = |x|^{β−n} + ….

Since w̃(x) = x^{n}g̃(x), we have

w̃′(x) = nx^{n−1}g̃(x) + x^{n}g̃′(x) ≤ nx^{−1}w̃(x) + |x|^{β} + …,

and by Lemma 2,

w̃′(x) ≤ (1 + o(1)) |x|^{β}.

(b) The case α ∈ [1, 2] and β − α ∈ [−1, 0]. Note that this implies β ≥ 0. Define the functions

g(x) = x^{−1}w(x), f(x) = xh(x),

so

g(x) = |x|^{−α_{1}}f(x), α_{1} = 2 − α ∈ [0, 1].

Let us show that

x^{−1}f̃′ ≤ |x|^{β_{1}−1} + …,  β_{1} := β − α + 1.  (2.6)

Since

h ∈ L^{1} (\frac{1}{|x|^{2−α}} ) ⊂ L^{1} (\frac{1}{|x|^{2+κ}} ) , κ := β − α,

by Lemma 3 we have

h̃(x) = o(|x|^{κ+1}),

and since f̃(x) = xh̃(x), we obtain (2.6):

x^{−1}f̃′(x) = h̃′(x) + x^{−1}h̃(x) ≤ |x|^{κ} + o(|x|^{κ}).

We can now apply Lemma 4 with parameters α_{1} and β_{1}. (Note that f ∈ L_{Π}^{1} and the parameters are admissible.) By (2.4) we get the estimate

x^{−1}g̃′(x) ≤ |x|^{β_{1}−α_{1}−1} + ⋯ = |x|^{β−2} + …,

and from w̃(x) = xg̃(x) we derive

w̃′(x) = g̃(x) + xg̃′(x) ≤ x^{−1}w̃(x) + |x|^{β} + ….

Applying Lemma 2 we conclude the proof. □

3. Triviality of Toeplitz kernels

In this section we prove the first part of Theorem A, which gives a sufficient condition for the triviality of a Toeplitz kernel. Let us fix κ ≥ 0 and consider two unimodular functions U = e^{iγ} and S = e^{iσ} on ℝ satisfying

γ′(x) ≳ −|x|^{κ},  σ′(x) ≳ |x|^{κ},  (x → ∞).  (3.1)

3.1. Upper density estimate.

Proposition. If N^{+}[US^{ϵ}] ≠ 0 for some ϵ > 0, then γ(∓∞) = ±∞.

Proof: If N^{+}[US^{ϵ}] ≠ 0, then by the basic criterion (1.6) we have

γ + ϵσ + α = h̃, α′ ≥ 0, h ∈ L_{Π}^{1}.

Therefore,

h̃′(x) ≳ γ′(x) ≳ −|x|^{κ},

and h̃(x) = o (|x|^{κ+1}) by Lemma 1. It follows that

\frac{γ(x)}{x} ≲ \frac{γ(x)}{x} + \frac{α(x)}{x} = −ϵ\frac{σ(x)}{x} + o (|x|^{κ}) ≲ −|x|^{κ},

which implies γ(∓∞) = ±∞. □

3.2. Effective density estimate. Let c > 0 be a fixed constant. For an interval l ⊂ ℝ we denote by l′ and l′′ the intervals of length c|l| adjacent to l from the left and from the right respectively, and we define

Δ_{l}^{∗}[γ] = inf_{l′′} γ − sup_{l′} γ.

Lemma. Let ϵ > 0 and suppose

γ(∓) = ±∞,  ∑_{l∈BM(γ)} d^{κ−2}l^{2} = ∞.  (3.2)

Then there is a constant c > 0 and there is a collection of disjoint intervals {l_{n}} in [1, +∞) or in (−∞, −1] such that

∑ d_{n}^{κ−2}l_{n}^{2} = ∞,  10l_{n} ≤ d_{n},  mult{5l_{n}} < ∞,  (3.3)

and

Δ_{l_{n}}^{∗}[arg(US^{ϵ})] ≥ cd_{n}^{κ}l_{n}.  (3.4)

Here 5l is the notation for the interval of length 5|l| concentric with l, and mult{·} is the multiplicity of the covering.

Proof: Suppose the sum (3.2) over BM intervals in ℝ_{+} is infinite. If there are infinitely many BM intervals l = (ã_{n}, b_{n}) in ℝ_{+} satisfying 10|l| > d, then we set

l_{n} = (a_{n}, b_{n}), a_{n} := \frac{10}{11} b_{n};

otherwise we simply enumerate BM intervals such that 10|l| ≤ d. In any case, we get a collection of intervals l_{n} = (a_{n}, b_{n}) satisfying the first two conditions in (3.3) and also the inequality

γ(b_{n}) ≥ γ(a_{n}).

By (3.1), the latter implies that the intervals also satisfy (3.4) for some c > 0. Finally, we take a subfamily {l_{n_{k}}} such that {5l_{n_{k}}} is a subcover of ⋃ 5l_{n} of finite multiplicity and observe that we still have the divergence of the series ∑ d^{κ−2}l^{2}. Indeed, if ⋃ l_{j} ⊂ 5l, then

d_{j} ≍ d, ∑ l_{j}^{2} ≲ l^{2},

and so

∑_{j} d_{j}^{κ−2}l_{j}^{2} ≲ d^{κ−2}l^{2}.

□

The following proposition completes the proof of the first part of Theorem A.

Proposition. Suppose γ′(x) ≳ −|x|^{κ} and suppose there is a collection {l} of disjoint intervals in [1, +∞) such that

∀l, Δ_{l}^{∗}[γ] ≥ cld^{κ},

and

∑ d^{κ−2}l^{2} = ∞,  10l ≤ d,  mult{5l} < ∞,

then N^{+} [e^{iγ}] = 0.

3.3. Proof of the proposition. The statement corresponds to the so-called Beurling’s lemma in the classical BM theory. There are several versions of the proof of Beurling’s lemma, e.g. Koosis [17] applies the Beurling-Tsuji estimate of harmonic measure, Nazarov [25] uses the Bellman function, and Kargaev’s proof [19] is based on PDE techniques. We suggest yet another approach.

According to the criterion (1.6), we have to exclude the possibility

γ + α = h̃, α ↑, h ∈ L_{Π}^{1}.

Denote by h_{l} the restriction of h to the interval 5l. We say that l is of type I if

d^{κ−2}l^{2} ≤ C∥h_{l}∥_{Π},  (3.5)

where C is a sufficiently large constant; otherwise we call l an interval of type II. Clearly, we have

∑_{l∈I} d^{κ−2}l^{2} < ∞,

and to get a contradiction we need to show

∑_{l∈II} d^{κ−2}l^{2} < ∞  (3.6)

Consider the 2D Hilbert transform

H(z) = ∫_{ℝ} \frac{h^{−}(t) dt}{(t − z)^{2}}, (z ∈ C_{+}).

where h^{−} = max{0, −h}.

Lemma. If l is of type II, then

|H(z)| ≳ d^{κ}, ∀z ∈ Q_{l} := {x + iy : x ∈ l, l < y < 2l}.

We prove this lemma in the next subsection, and we now explain how the lemma implies (3.6). Denote

ψ = ∑_{l∈II} d^{κ}l · 1_{l}.

We have

∑_{l∈II} d^{κ−2}l^{2} ≍ ∫_{1}^{∞} \frac{ψ(t) dt}{t^{2}} = \frac{8}{3} ∫_{1}^{∞} \frac{dA}{A^{3}} ∫_{A/2}^{A} ψ(t) dt.  (3.7)

For every A > 1 let

H_{A}(z) = ∫_{−CA}^{CA} \frac{h^{−}(t) dt}{(t − z)^{2}},

where C > 0 is a large constant, and let II(A) denote the set of all intervals l ∈ II intersecting (A/2, A). If l ∈ II(A) and z ∈ Q_{l}, then

|H(z) − H_{A}(z)| ≤ ∫_{|t|>CA} \frac{h^{−}(t) dt}{|t − z|^{2}} ≍ ∫_{|t|>CA} \frac{h^{−}(t) dt}{1 + t^{2}} ≪ 1,

so by the lemma we have

|H_{A}| ≳ A^{κ} on ⋃_{l∈II(A)} Q_{l}.

Applying the weak-L^{1} estimate for the 2D Hilbert transform, see [4], we get

∑_{l∈II(A)} l^{2} ≤ Area(|H_{A}| ≳ A^{κ}) ≲ A^{−κ} ∫_{−CA}^{CA} h^{−}(t) dt,

and therefore

∫_{A/2}^{A} ψ(t) dt ≲ A^{κ} ∑_{l∈II(A)} l^{2} ≲ ∫_{−CA}^{CA} h^{−}(t) dt,

Combining this with (3.7), we conclude

∑_{l∈II} d^{κ−2}l^{2} ≲ ∫_{1}^{∞} \frac{dA}{A^{3}} ∫_{−CA}^{CA} h^{−}(t) dt ≲ ∥h^{−}∥_{Π},

which proves (3.6)

3.4. Proof of the lemma. Since α in the representation h̃ = γ + α is increasing, we have

Δ_{l}^{∗}[h̃] ≳ ld^{κ}.  (3.8)

On the other hand, for intervals of type II we have

Δ_{l}^{∗}[h̃_{l}] ≪ ld^{κ}.  (3.9)

Indeed, if Δ_{l}^{∗}[h̃_{l}] ≳ ld^{κ} , then |h̃_{l}| ≳ ld^{κ} on either l′ or l′′. Applying the weak type inequality with A ≍ ld^{κ}, we get

d^{−2}l ≲ Π{|h̃_{l}| > A} ≲ A^{−1}∥h_{l}∥_{Π},

which contradicts the definition of type II.

Denote f ≡ f_{l} = 1_{ℝ\5l} · h, so h̃ = h̃_{l} + f̃_{l}. From (3.8)-(3.9) we conclude that there are points a ∈ l′ and b ∈ l′′ such that

f̃(b) − f̃(a) ≥ \frac{c}{2}ld^{κ}.  (3.10)

Represent f = f^{+} − f^{−} with f^{+} = max{f, 0}, and note that the functions f̃_{±} := (f^{±})˜ are decreasing on [a, b]:

f̃′_{±}(x) = −\frac{1}{π} ∫_{ℝ\(5l)} \frac{f^{±}(t) dt}{(t − x)^{2}} < 0, (x ∈ 5l).

From (3.10) it then follows that

f̃_{−}(a) − f̃_{−}(b) ≳ ld^{κ},

so there is a point x_{∗} ∈ (a, b) such that

\frac{1}{π} ∫ \frac{f^{−}(t)dt}{(t − x_{∗})^{2}} = −f̃′_{−}(x_{∗}) = \frac{f̃_{−}(a) − f̃_{−}(b)}{b − a} ≳ d^{κ}.

Observe that if z ∈ Q_{l} and t ∈ ℝ \ 5l, then

ℜ [\frac{1}{(t − z)^{2}} ] ≍ \frac{1}{(t − x_{∗})^{2}} ,

and we have

|∫ \frac{f^{−}(t)dt}{(t − z)^{2}} | ≥ ℜ ∫ \frac{f^{−}(t)dt}{(t − z)^{2}} ≍ ∫ \frac{f^{−}(t)dt}{(t − x_{∗})^{2}} ≳ d^{κ}.

It follows that

|H(z)| ≥ |∫ \frac{f_{l}^{−}(t)dt}{(t − z)^{2}} | − |∫ \frac{h_{l}^{−}(t)dt}{(t − z)^{2}} | ≳ d^{κ},

because

|∫ \frac{h_{l}^{−}(t)dt}{(t − z)^{2}} | ≲ \frac{d^{2}}{l^{2}} ∥h_{l}∥_{Π} ≪ d^{κ}

provided that the constant C in (3.5) is large enough.

4. Non-triviality of Toeplitz kernels in Smirnov-Nevanlinna class

4.1. A version of the ”little multiplier” theorem. In this section we prove the following statement. Let κ ≥ 0 and suppose that U = e^{iγ}, S = e^{iσ} satisfy conditions (1.4) of Theorem A.

Proposition. If γ is almost (κ) decreasing, then N^{+}[US̄^{ϵ}] ≠ 0 for all ϵ > 0.

Proof: By assumption we have

∑_{l∈BM(γ)} d^{κ−2}l^{2} < ∞.  (4.1)

Recall that the BM intervals of γ are the components of the open set {γ^{∗} ≠ γ}, where γ^{∗}(x) = max γ[x, +∞). Denote

f = γ^{∗} − γ,

so f = 0 outside the union of BM intervals, and f′(x) ≲ |x|^{κ} on BM intervals. By (4.1) we have l ≲ d, and therefore

0 ≤ f ≲ ld^{κ}  on l,  (4.2)

Together with (4.1) this implies

f ∈ L_{Π}^{1}.

The estimate (4.2) also shows that we can assume l ≥ d^{−κ} for all BM intervals; otherwise we can eliminate short intervals by adding a bounded function to γ (this will not affect the N^{+}-kernel). In particular, we will assume that BM intervals don’t cluster to a finite point. The non-triviality of N^{+}[US̄^{ϵ}] is a consequence of the following statement which will be verified in the next two subsections.

Lemma. For any ϵ > 0, there is a function β such that

f + β ∈ L̃_{Π}^{1}, β′(x) ≤ ϵ|x|^{κ} for |x| ≫ 1.

Indeed, if for instance σ′(x) ≥ |x|^{κ} near ±∞, then we can write

γ − ϵσ = −(f + β) + (β − ϵσ) + γ^{∗}.

The first term in the RHS is in L̃_{Π}^{1}, and the last two terms are decreasing near infinities, so we can apply the basic criterion (1.6). □

4.2. Proof of the lemma. We will construct disjoint intervals l_{n} such that they cover all BM intervals and satisfy the following two conditions:

∑_{n} d_{n}^{κ−2}l_{n}^{2} < ∞,  (4.3)

and

∀n ∃ϵ_{n} ∈ [0, ϵ],  ∫_{l_{n}} \frac{f(x) − ϵ_{n}|x|^{κ}T_{n}(x)}{1 + x^{2}} dx = 0,  (4.4)

where T_{n} is the ”tent” function of the interval l_{n},

T_{n}(x) = dist(x, ℝ \ l_{n}).

Let us show that the existence of such intervals l_{n} implies the statement of the lemma. Define

β(x) = − ∑_{n} ϵ_{n}|x|^{κ}T_{n}(x),

and

g(x) = f(x) − ∑ ϵ_{n}|x|^{κ}T_{n}(x).

Clearly, we have

|β′(x)| ≲ ϵ|x|^{κ},

and all we need is to check g ∈ L̃_{Π}^{1}.

Let us show that g belongs to the real Hardy space H_{Π}^{1}(ℝ). We can represent g as follows:

g = ∑ g_{n} = ∑ λ_{n} \frac{g_{n}}{λ_{n}} := ∑ λ_{n}A_{n},

where

g_{n} = g · 1_{l_{n}}, λ_{n} = Π(l_{n}) ∥g_{n}∥_{∞}.

The functions A_{n} = λ_{n}^{−1}g_{n} are ”atoms”:

∫ A_{n} dΠ = \frac{1}{λ_{n}} ∫_{l_{n}} g dΠ = 0 by (4.4),

and

∥A_{n}∥_{∞} = \frac{∥g_{n}∥_{∞}}{λ_{n}} = \frac{1}{Π(l_{n})}.

Since

∥g_{n}∥_{∞} ≲ d_{n}^{κ}l_{n},

(use l_{n} ≲ d_{n} and (4.2) for the BM intervals covered by l_{n}), we have

∑ λ_{n} ≲ ∑ \frac{l_{n}}{d_{n}^{2}} d_{n}^{κ}l_{n} < ∞ by (4.3).

It follows that ∑ λ_{n}A_{n} ∈ H_{Π}^{1}(ℝ), see [5].

4.3. Construction of intervals l_{n}. Let us assume that all BM intervals l lie in [1, +∞). In the general case we will need to apply the procedure described below to BM intervals in (−∞, −1] and in [1, +∞) separately.

We construct our intervals l_{1}, l_{2}, … by induction. The left endpoint a_{1} of l_{1} will be the left endpoint of the leftmost BM interval. Suppose the left endpoint a_{n} of l_{n} has been constructed so that a_{n} is also the left endpoint of some BM interval l = (a_{(l)}, b_{(l)}), i.e. a_{n} = a_{(l)}. Consider the function

F (b) = ∫_{a_{n}}^{b} [f − ϵ|x|^{κ}T_{(a_{n},b)}] dΠ,

where T_{(a_{n},b)}(·) = dist( · , {a_{n}, b}) is the tent function. We define b_{n}, the right endpoint of l_{n}, as the nearest point in the complement of BM intervals at which F is non-positive,

b_{n} = min{b ≥ b_{(l)} : f(b) = 0, F (b) ≤ 0}.

Since f ∈ L_{Π}^{1}, we have F (+∞) = −∞ and so b_{n} < ∞. Finally, we define a_{n+1} as the leftmost endpoint of BM intervals not covered by l_{1} ∪ ⋯ ∪ l_{n}. (Recall that we assumed that there are no finite cluster points.)

It is clear from the construction that the intervals l_{n} cover all BM intervals. We also get (4.4) by defining ϵ_{n} from the equation

∫_{a_{n}}^{b_{n}} [f − ϵ_{n}|x|^{κ}T_{(a_{n},b_{n})}] dΠ = 0;

clearly we have 0 < ϵ_{n} ≤ ϵ. In remains to verify (4.3). We have three types of intervals l_{n}:

(a) F (b_{n}) < 0 but there is a BM interval l ⊂ l_{n} such that |l| ≍ |l_{n}|,

(b) F (b_{n}) = 0,

(c) other intervals.

Property (4.3) is obvious for the collection of intervals of type (a): we have l ≪ d (except for finitely many l’s) and therefore d ≍ d_{n} and d^{κ−2}l^{2} ≍ d_{n}^{κ−2}l_{n}^{2}.

To prove (4.3) for the collection of intervals of type (b), we note that ϵ_{n} = ϵ if l_{n} ∈(b), and since

∫_{a}^{b} x^{κ}T_{(a,b)}(x) \frac{dx}{x^{2}} ≥ \frac{a^{κ}}{b^{2}} ∫_{a}^{b} T_{(a,b)} ≳ \frac{a^{κ}}{b^{2}} (b − a)^{2},

we have

∑_{(b)} \frac{d_{n}^{κ}l_{n}^{2}}{(d_{n} + l_{n})^{2}} ≲ ∑_{(b)} ∫_{a_{n}}^{b_{n}} |x|^{κ}T_{(a_{n},b_{n})} dΠ = \frac{1}{ϵ} ∫_{⋃_{(b)} l_{n}} f dΠ < ∞.

Since d_{n} → ∞, it follows that there are only finitely many intervals l_{n} ∈ (b) satisfying d_{n} ≤ l_{n}, so the last estimate implies

∑_{(b)} d_{n}^{κ−2}l_{n}^{2} < ∞.

The argument for intervals of type (c) is the same if we can show that if l_{n} ∈(c), then ϵ_{n} > ϵ/2, i.e.

∫_{a_{n}}^{b_{n}} [f − \frac{ϵ}{2}|x|^{κ}T_{(a_{n},b_{n})}] dΠ > 0.  (4.5)

Since l_{n} is not of type (b), we have F (b_{n}) < 0 and by construction, b_{n} is the right endpoint of some BM interval l = (c, b_{n}). Note that |l| ≪ |l_{n}| because l_{n} is not of type (a). Since f > 0 on l, we have

∫_{a_{n}}^{b_{n}} (f − \frac{ϵ}{2}x^{κ}T_{(a_{n},b_{n})}) dΠ > ∫_{a_{n}}^{c} (f − ϵx^{κ}T_{(a_{n},c)}) dΠ +

+ [ϵ ∫_{a_{n}}^{c} x^{κ}T_{(a_{n},c)} dΠ − \frac{ϵ}{2} ∫_{a_{n}}^{b_{n}} x^{κ}T_{(a_{n},b_{n})} dΠ] .

The first term in the RHS is equal to F (c) and therefore positive by construction. Since |l| ≪ |l_{n}|, the second term in the RHS is also positive, and we get (4.5)

5. Multiplier theorem

5.1. BM multipliers. Let S be a unimodular function and let 0 < p ≤ ∞. If w ∈ L_{Π}^{1} is a real function, then we write

w ∈ M_{p}(S)

if the outer function

W = e^{w+iw̃}

satisfies the following condition:

∀ϵ > 0,  ∃G ∈ N^{+}[S̄^{ϵ}],  W G ∈ H^{p}.

In other words, w ∈ M_{p}(S) if the corresponding outer function belongs to H^{p} up to an arbitrarily small (compared to S) factor.

We can restate this property in terms of Toeplitz kernels.

Lemma. w ∈ M_{p}(S) iff

∀ϵ > 0, N^{p} [S̄^{ϵ} \frac{W̄}{W} ] ≠ 0.

Proof: ⇒ Let G ∈ N^{+}[S̄^{ϵ}] be such that F := GW ∈ H^{p}. Then

F ∈ N^{p} [S̄^{ϵ} \frac{W̄}{W} ] ,

and the Toeplitz kernel is non-trivial. Indeed,

S̄^{ϵ} \frac{W̄}{W} F = (S̄^{ϵ}G)W̄ ∈ N^{−} ∩ L^{p} = H̄^{p}.

⇐ If F is in the Toeplitz kernel, i.e. F ∈ H^{p} and F S̄^{ϵ}W̄ /W ∈ H̄^{p}, then we define G = F/W ∈ N^{+}. Since

S̄^{ϵ}G = S̄^{ϵ} \frac{W̄}{W} F \frac{1}{W̄} ∈ N^{−},

we have G ∈ N^{+}[S̄^{ϵ}] and W G ∈ H^{p}. □

Corollary. Suppose (arg S)′ ≳ |x|^{κ}. If a real function w_{0} ∈ L_{Π}^{1} satisfies the following condition:

∀ϵ > 0, ∃w ∈ L_{Π}^{1}, w ≥ w_{0}, w̃′ > −ϵ|x|^{κ} + o (|x|^{κ}) ,

then w_{0} ∈ M_{p}(S) for all p < 1.

Proof: Without loss of generality, (arg S)′ ≥ 2|x|^{κ} for |x| ≫ 1. We have

− arg [S̄^{2ϵ} \frac{W̄_{0}}{W_{0}} ] = 2 (ϵ arg S + w̃) + 2(w̃_{0} − w̃) := α + g̃,

where

α′ ≥ ϵ|x|^{κ}, (|x| ≫ 1),

and

g ∈ L_{Π}^{1}, g ≤ 0.

For sufficiently large N, the function α + arg b^{N}, where b is the Blaschke factor (1.9), is monotone increasing on ℝ, and therefore there is an inner function Φ, not a finite Blaschke product, such that

α + arg b^{N} = arg Φ + δ, ∥δ∥_{∞} ≤ π.

Clearly, N^{∞} [e^{−ig̃}] ≠ 0, i.e.

N^{∞} [e^{iδ} b̄^{N} Φ S̄^{2ϵ} \frac{W̄_{0}}{W_{0}} ] ≠ 0.

By (1.8), we also have

N^{p} [e^{−iδ} b̄^{N}] ≠ 0,

provided that p < 1 and N is sufficiently large, and of course

N^{∞} [b^{2N} Φ̄] ≠ 0.

It follows that

N^{p} [S̄^{2ϵ} \frac{W̄_{0}}{W_{0}} ] ≠ 0.

□

The main result of this section is the following version of the Beurling-Malliavin multiplier theorem.

Theorem. Suppose (arg S)′ ≳ |x|^{κ}, and let w_{0} ∈ L_{Π}^{1} be a real function. Then

|x|^{− \frac{2+κ}{2}} w_{0}(x) ∈ D(ℝ, ∞)  ⇒  w_{0} ∈ M_{p}(S), (∀p < 1).

Here the notation f ∈ D(ℝ, ∞) means that there is a neighborhood of infinity where f coincides with some function from the Dirichlet space D(ℝ). Recall that the Hilbert space D(ℝ) consists of functions h ∈ L_{Π}^{1} such that the harmonic extension u = u(z) of h to C_{+} has a finite gradient norm,

∥h∥_{D}^{2} ≡ ∥u∥_{∇}^{2} = ∫_{C_{+}} |∇u|^{2} dA < ∞,

(dA is the area measure). If h ∈ D(ℝ) is a smooth function, then we also have

∥h∥_{D}^{2} = ∫_{ℝ} h̄ h̃′ dx.

In the next two subsections we use some ideas from the proof of Theorem 64 in [6].

5.2. Proof of the multiplier theorem. It is clear that we can assume that the function w_{0} + iw̃_{0} is analytic and has a zero of sufficiently large multiplicity at the origin; in particular

h_{0}(x) := |x|^{− \frac{2+κ}{2}} w_{0}(x) ∈ D(ℝ).

Let us fix ϵ > 0. According to the last corollary we need to construct w such that

(i) w ∈ L_{Π}^{1},  (ii) w ≥ w_{0},  (iii) w̃′ > −ϵ|x|^{κ} + o (|x|^{κ}) .

We define

w(x) = |x|^{\frac{2+κ}{2}} h(x),

where h is a solution of the following extremal problem:

min\{I(h) : h ≥ h_{0}\}, I(h) := ∥h∥_{D}^{2} + ϵ ∫ |x|^{\frac{2+κ}{2}} |h(x)| dΠ(x).

The existence of a solution follows from the usual argument: the set

A = {h : ∥h∥_{D} ≤ I(h_{0}), h ≥ h_{0} a.e.} ⊂ D(ℝ)

is bounded, closed, and convex in D(ℝ), therefore it is weakly compact. Let I_{0} denote the minimum of I(h) over A. Then there is a sequence of functions h_{n} ∈ A such that I(h_{n}) → I_{0} and h_{n} weakly converge to some function g ∈ A. It is then routine to see that

I_{0} ≤ I(g) ≤ lim inf I(h_{n}) = I_{0},

so g is a solution of the extremal problem.

By construction, w satisfies (i) and (ii). To prove (iii) we first note that

h̃′(x) ≥ −ϵ|x|^{\frac{κ−2}{2}} .  (5.1)

Indeed, by the extremality of h we have

∥ϕ∥_{D}^{2} + 2 ∫ ϕh̃′ + ϵ ∫ (|h + ϕ| − |h|)(x) \frac{|x|^{\frac{2+κ}{2}}}{1 + x^{2}} dx = I(h + ϕ) − I(h) ≥ 0

for all smooth test functions ϕ = ϕ(x) ≥ 0. (The integral ∫ ϕh̃′ has to be interpreted in the sense of the theory of distributions.) Since

\frac{ϕ(x)}{x^{2}} ≥ \frac{|h(x) + ϕ(x)| − |h(x)|}{1 + x^{2}},

we conclude

∥ϕ∥_{D}^{2} + 2 ∫ ϕ(x) [h̃′(x) + ϵ|x|^{\frac{κ−2}{2}} ] dx ≥ 0

Replacing ϕ(x) with δϕ(x) and letting δ → 0, we get

∫ ϕ(x) [h̃′(x) + ϵ|x|^{\frac{κ−2}{2}} ] dx ≥ 0

for all ϕ ≥ 0, which proves (5.1)

To derive (iii) from (5.1) we apply Lemma 5 in Section 2 with

α = 1 + \frac{κ}{2},  β = κ,  β − α = \frac{κ}{2} − 1.

The parameters α and β are admissible because for κ ≥ 2 we have α ≤ β, and if 0 ≤ κ ≤ 2 then 1 ≤ α ≤ 2 and α ≤ β + 1.

5.3. Multipliers and one-sided Lipschitz condition.

Proposition. If w ∈ L_{Π}^{1}, w ≥ 0, and w̃′ ≲ |x|^{κ}, then

|x|^{− \frac{2+κ}{2}} w(x) ∈ D(ℝ, ∞).

Proof: We will assume that the function w_{0} + iw̃_{0} is analytic and has a zero of sufficiently large multiplicity at the origin. Let u = u(z) be the harmonic extention of |x|^{− \frac{2+κ}{2}} w(x) to the upper half plane C_{+}, and let v = ũ. We need to show that the gradient norm of u + iv in C_{+} is finite,

∥u + iv∥_{∇}^{2} = lim_{r→∞} ∫_{∂D(r)} udv < ∞,

where D(r) is the semidisc {|z| < r} ∩ C_{+}.

We first prove that the integrals over ∂D(r)∩ℝ are uniformly bounded from above. Applying Lemma 4 in Section 2 with (admissible) parameters

α = 1 + \frac{κ}{2},  β = κ,  β − α = \frac{κ}{2} − 1

to the functions f = w and g = u, we see that

v′(x) ≲ |x|^{\frac{κ−2}{2}} , x ∈ ℝ.

Since u > 0 we have

∫_{∂D(r)∩ℝ} udv = ∫_{−r}^{r} v′u ≲ ∫_{−r}^{r} |x|^{\frac{κ−2}{2}} |x|^{− \frac{2+κ}{2}} w(x) dx ≲ ∥w∥_{Π} < ∞.

To finish the proof of the proposition it remains to show that the integrals

∫_{∂D(r)\ℝ} u dv = rI′(r), I(r) := \frac{1}{2} ∫_{0}^{π} u^{2} (re^{iθ}) dθ,

don’t tend to +∞ as r → ∞. In fact, it is enough to show

I(r) ↛ ∞,

because if rI′(r) → +∞, then I′(r) ≥ 1/r for all r ≫ 1, and we have I(r) → ∞. Since κ ≥ 0, we can apply the following lemma.

Lemma. If u ∈ L^{1}(1 + |x|^{−1}), then I(r) ↛ ∞.

Proof: We will prove an equivalent statement for functions in the unit disc D. Let f = u + iũ be an analytic function in D such that

\frac{u(ζ)}{1 − |ζ|} ∈ L^{1}(∂D).

Define

h(z) = \frac{1 + z}{1 − z} u(z), z ∈ D,

and denote by h^{∗}(ζ), ζ ∈ ∂D, the angular maximal function. By Hardy-Littlewood maximal theorem,

h^{∗} ∈ L_{weak}^{1}(∂D).  (5.2)

Let us show that as ϵ → 0,

\frac{1}{ϵ} ∫_{C_{ϵ}} |f(z)|^{2}|dz| ↛ ∞, C_{ϵ} := \{|1 − z| = ϵ\} ∩ D.

We have

\frac{1}{ϵ} ∫_{C_{ϵ}} |f|^{2} = ϵ ∫_{C_{ϵ}} |h|^{2} ≲ [ϵh^{∗}(ζ)]^{2} + [ϵh^{∗}(ζ̄)]^{2} ,

where ζ ∈ ∂D, |1 − ζ| = ϵ. The RHS can not tend to infinity because otherwise for all small ϵ, we would have

h^{∗}(ζ) + h^{∗}(ζ̄) ≫ \frac{1}{ϵ}

on an interval of length ϵ, which would contradict (5.2). □

5.4. A version of BM1.

Proposition. Suppose (arg S)′ ≳ |x|^{κ} and let Θ be a meromorphic inner function satisfying |Θ′| ≲ |x|^{κ}. Then

W ∈ N^{+}[Θ̄]  ⇒  log |W | ∈ M_{p}(S), (∀p < 1).

Proof: We have W Θ̄ = H̄ for some H ∈ N^{+}. Define

W_{1} = W H + Θ,

and let W_{1}^{e} be the outer part of W_{1}. From the identity

Θ̄^{2}W_{1} = Θ̄W Θ̄H + Θ̄ = H̄W̄ + Θ̄ = W̄_{1},

we deduce

|W_{1}| = |W W̄ Θ + Θ| = 1 + |W |^{2} ≥ 1,

and

|W_{1}^{e}| ≥ 1,  |W | ≤ |W_{1}^{e}|,  (arg W_{1}^{e})′ ≲ |x|^{κ}.

By Proposition 5.3 and the multiplier theorem, we have log |W_{1}| ∈ M_{p}(S) and therefore

log |W | ∈ M_{p}(S).
□

Corollary. Let S and Θ be as above. Then for any meromorphic inner function J and any p < 1, we have

N^{+}[Θ̄J] ≠ 0  ⇒  ∀ϵ, N^{p}[S̄^{ϵ}Θ̄J] ≠ 0.

Proof: Take an outer function W ∈ N^{+}[Θ̄J]. Then W ∈ N^{+}[Θ̄], and by the last proposition,

∃G ∈ N^{+}[S̄^{ϵ}]. W G ∈ H^{p}.

It then follows that

W G ∈ N^{+}[S̄^{ϵ}Θ̄J] ∩ H^{p} = N^{p}[S̄^{ϵ}Θ̄J].
□

6. Non-triviality of Toeplitz kernels in Hardy spaces

In this final section we finish the proof of Theorems A and B.

6.1. Approximation by inner functions. It is well known that given any two intertwining discrete sets A = {a_{n}} and B = {b_{n}} of real numbers, ...a_{n} < b_{n} < a_{n+1}..., there exists a meromorphic inner function Θ such that

{Θ = 1} = A,  {Θ = −1} = B.  (6.1)

Indeed, the sequences A, B determine the set

E = {ℑΘ > 0} ∩ ℝ = ∪(a_{n}, b_{n}),

and we can define Θ in C_{+} by the (Krein’s shift) formula

\frac{1}{πi} log \frac{Θ + 1}{Θ − 1} = Su + ic,  u := 1_{E} − \frac{1}{2}, c ∈ ℝ,  (6.2)

where Su is the Schwarz integral (1.5), so ℜ[Su] is the Poisson extension of u to the halfplane. (Note that u is the boundary function of the expression in the LHS of (6.2), provided that Θ is an inner function with level sets A and B, and in fact Krein’s shift formula parametrizes all such inner functions.)

An immediate consequence of this construction is the following statement: for any increasing, continuous function σ : ℝ → ℝ, there exists a meromorphic inner function Θ = e^{iθ} such that

∥θ − σ∥_{∞} ≤ π.

We will need the following version of this statement.

Lemma. If σ′(x) ≍ |x|^{κ}, then there is a meromorphic inner function Θ = e^{iθ} such that

θ − σ ∈ L^{∞}(ℝ), θ′(x) ≍ |x|^{κ}.

Proof: We can assume that σ is strictly increasing on ℝ. Define the intertwining sequences A = {a_{n}} and B = {b_{n}} by the equations

σ(a_{n}) = 2πn,  b_{n} = \frac{a_{n} + a_{n+1}}{2} ,  (n ∈ Z),

so we have

a_{n} ≍ (sign n) |n|^{\frac{1}{1+κ}} ,

and

δ_{n} := b_{n} − a_{n} ≍ |a_{n}|^{−κ}.

Let Θ be an inner function satisfying (6.1),

∥θ − σ∥_{∞} ≤ 2π,

and let μ_{1}, μ_{−1} be the corresponding (Aleksandrov-Clark’s) measures defined by the Herglotz representation

\frac{1 + Θ}{1 − Θ} = Sμ_{1} + const, \frac{1 − Θ}{1 + Θ} = Sμ_{−1} + const.

The measures μ_{1}, μ_{−1} have the following form:

μ_{1} = ∑ α_{n}δ_{a_{n}}, μ_{−1} = ∑ β_{n}δ_{b_{n}}

for some positive numbers α_{n}, β_{n}. (It is easy to see that μ_{±1}{∞} = 0 though we don’t actually need this fact.) We claim that

α_{n} ≍ δ_{n},  β_{n} ≍ δ_{n}.  (6.3)

The estimate θ′(x) ≍ |x|^{κ} easily follows from (6.3). Since

|Θ′| ≍ |1 − Θ^{2}| |(Sμ_{1})′| , |Θ′| ≍ |1 + Θ^{2}| |(Sμ_{−1})′| ,

we have

θ′(x) ≍ min \{∑ \frac{α_{n}}{(x − a_{n})^{2}} , ∑ \frac{β_{n}}{(x − b_{n})^{2}} \} , (x ∈ ℝ).

It follows that if x ∈ (a_{m}, a_{m+1}), then by (6.3)

θ′(x) ≍ ∫_{|t−x|≳δ_{m}} \frac{dt}{(x − t)^{2}} ≍ δ_{m}^{−1} ≍ |a_{m}|^{κ} ≍ |x|^{κ}.

Proof of (6.3). We will explain the estimate for α_{n}’s; the proof for β_{n}’s is similar. According to (6.2), we have

\frac{1 − Θ}{1 + Θ} = const e^{Ku},

where u = 1_{E} − 1/2,

E = ⋃_{k=−∞}^{∞} (a_{k}, b_{k}),

and Ku is the improper integral

Ku(z) = ∫ \frac{u(t) dt}{t − z}, (z ∈ C_{+}).

By construction,

α_{n} = const Res_{a_{n}}e^{Ku}.

Denote

g_{n}(z) = exp \{∫_{b_{n−1}}^{b_{n}} \frac{u(t) dt}{t − z}\} = \frac{\sqrt{(b_{n} − z)(b_{n−1} − z)}}{a_{n} − z},

and

A_{n} = exp \{∫_{ℝ\(b_{n−1},b_{n})} \frac{u(t) dt}{t − a_{n}}\} ,

so

Res_{a_{n}}e^{Ku} = A_{n} Res_{a_{n}}g_{n}, |Res_{a_{n}}g_{n}| ≍ δ_{n}.

It remains to show that A_{n} = e^{O(1)}. This can be done as follows.

For j > n we have

∫_{a_{j}}^{a_{j+1}} \frac{u(t) dt}{t − a_{n}} = log \frac{b_{j} − a_{n}}{a_{j} − a_{n}} − log \frac{a_{j+1} − a_{n}}{b_{j} − a_{n}}

= log (1 + \frac{δ_{j}}{a_{j} − a_{n}} ) − log (1 + \frac{δ_{j}}{b_{j} − a_{n}} )

= \frac{δ_{j}}{a_{j} − a_{n}} − \frac{δ_{j}}{b_{j} − a_{n}} + O (\frac{δ_{j}^{2}}{(a_{j} − a_{n})^{2}} ) = O (\frac{δ_{j}^{2}}{(a_{j} − a_{n})^{2}} ) ,

where we used the relation log(1 + x) = x + O(x^{2}) for 0 < x ≲ 1. Since

∑_{j=n+1}^{∞} \frac{δ_{j}^{2}}{(a_{j} − a_{n})^{2}} ≍ ∑_{j=n+1}^{∞} \frac{δ_{j}}{a_{j}^{κ}(a_{j} − a_{n})^{2}}

≍ ∫_{a_{n}+δ_{n}}^{∞} \frac{dt}{t^{κ}(t − a_{n})^{2}} = ∫_{a_{n}+δ_{n}}^{2a_{n}} + ∫_{2a_{n}}^{∞}

≲ \frac{1}{a_{n}^{κ}} ∫_{a_{n}+δ_{n}}^{∞} \frac{dt}{(t − a_{n})^{2}} + ∫_{a_{n}}^{∞} \frac{dt}{t^{2+κ}}

≍ \frac{1}{a_{n}^{κ}} \frac{1}{δ_{n}} + \frac{1}{a_{n}^{1+κ}} = O(1),

we get

∫_{b_{n}}^{∞} \frac{u(t) dt}{t − a_{n}} = O(1).

A similar estimate holds for the integral over (−∞, b_{n−1}), and we have A_{n} = e^{O(1)}. □

6.2. Proof of Theorem A. The first part of the theorem was established in Section 3. The second part states that if γ is almost decreasing and ϵ > 0, then

N^{p}[US̄^{2ϵ}] ≠ 0,  (p < 1/3).  (6.4)

By Lemma 6.1 there exists an inner function Θ satisfying

(arg Θ)′ ≍ |x|^{κ}.

We will assume that U^{2}Θ has an increasing argument (otherwise we can replace Θ with Θ^{n} for a large integer n). We will also assume that SΘ̄ has an increasing, unbounded argument (otherwise we replace S with a large power). By Proposition 4.1 we have

N^{+}[UΘ^{1−ϵ}Θ̄] ≠ 0.  (6.5)

Since the argument of UΘ^{1−ϵ} is increasing, there is an inner function J such that

UΘ^{1−ϵ} = XJ, ∥ arg X∥_{∞} ≤ π.

From (6.5) we have N^{+}[JΘ̄] ≠ 0, and so by Corollary 5.4

N^{p}[JΘ̄S̄^{ϵ}] ≠ 0,  (p < 1).  (6.6)

Note that

US̄^{2ϵ} = (UΘ^{1−ϵ}Θ̄S̄^{ϵ}) (Θ^{ϵ}S̄^{ϵ}) = X (JΘ̄S̄^{ϵ}) (Θ^{ϵ}S̄^{ϵ}) .

Since the argument of S^{ϵ}Θ̄^{ϵ} is increasing and unbounded, we can find an infinite Blaschke product Ψ such that

Θ^{ϵ}S̄^{ϵ} = Y Ψ̄, ∥ arg Y ∥_{∞} ≤ π.

Thus the symbol US̄^{2ϵ} has the following representation:

US̄^{2ϵ} = (JΘ̄S̄^{ϵ}) (XY Ψ̄), ∥ arg XY ∥_{∞} ≤ 2π,

and by (1.8) we have

N^{p}[XY Ψ̄] ≠ 0,  (p < 1/2).  (6.7)

Combining (6.6) and (6.7), we get (6.4) by Hölder’s inequality.

Remark. It is clear from the proof that if γ is almost decreasing, then the N^{p}-kernels are infinite dimensional.

6.3. Proof of Theorem B. Recall that J is a meromorphic inner function, S = e^{iσ} with σ′(x) ≍ |x|^{κ}, and c = c(J, S; κ). Applying Theorem A (or rather its corollary) to U = J we conclude that if a < c then N^{+}[JS̄^{a}] = 0 and therefore N^{p}[JS̄^{a}] = 0 for all p > 0. On the other hand, if a > c, then N^{p}[JS̄^{a}] ≠ 0 for some p > 0, and in fact the kernel is infinite dimensional, as we just mentioned. The following proposition completes the proof.

A unimodular function S is called tempered if ∃n, S′(x) = O (|x|^{n}) as x → ∞.

Proposition. If S is a tempered unimodular function, then for any meromorphic inner function J and any p > 0,

dim N^{p}[JS̄] = ∞  ⇒  dim N^{∞}[JS̄] = ∞.

Proof: First of all we observe that the statement is true if S is a tempered inner function, S = Θ. By Carleson’s type embedding theorem [34], all elements in N^{p}[JΘ̄] have at most polynomial growth at infinity, see details in [23], Section 4.1. Since the kernel is infinite dimensional, it contains functions with many zeros in C_{+}. Dividing such functions by appropriate polynomials we obtain functions in N^{∞}[JΘ̄].

Let now S be an arbitrary tempered unimodular function. By Lemma 6.1 we can find a tempered inner function Θ and a bounded real-valued function χ such that

S = ΘX̄, X = e^{2iχ}.

By the previous observation, we have

dim N^{∞}[JΘ̄] = ∞,  (6.8)

and it remains to show that

∃n, N^{∞}[Xb̄^{n}] ≠ 0,  (6.9)

(b is the Blaschke factor (1.9)). Indeed, combining (6.8) and (6.9) we conclude that the kernel

N^{∞}[JS̄b̄^{n}] = N^{∞}[JΘ̄ Xb̄^{n}]

is infinite dimensional, which allows us to get rid of b^{n}.

To prove (6.9), consider the outer function

H = e^{χ̃−iχ}, so X = \frac{H̄}{H} .

We will have

(z + i)^{−n}H(z) ∈ N^{∞}[Xb̄^{n}], (n ≫ 1)

if we can show that h := |H| = eχ̃ has at most polynomial growth at infinity. Without loss of generality, we can assume that the L^{∞}-norm of χ is so small that h ∈ L_{Π}^{2}. We have

h(x) − h(0) ≤ ∫_{0}^{x} |h′| = ∫_{0}^{x} h |χ̃′| ≲ ∥h∥_{L_{Π}^{2}} (1 + x^{2})^{1/2} (∫_{0}^{x} |χ̃′|^{2})^{1/2} .(6.10)

Since |χ′(t)| ≲ |t|^{n} by construction, for each x > 0 we can represent χ as the sum of two smooth functions,

χ = χ_{1} + χ_{2},

such that

∥χ′_{1}∥_{L^{2}} ≲ |x|^{n},  ∥χ_{2}∥_{L^{∞}} ≍ 1,  χ_{2} = 0 on (−2x, 2x).

(For example, take χ_{1} = ϕχ, where ϕ is a smooth ”bump” function such that ϕ is equal to 1 on (−2x, 2x) and 0 on ℝ \ (−3x, 3x).) Then we have

∥χ̃′_{1}∥_{L^{2}} ≲ |x|^{n}, |χ̃′_{2}| ≲ 1 on (0, x),

and so (6.10) shows that h has at most polynomial growth. □

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California Institute of Technology, Department of Mathematics, Pasadena, CA 91125, USA

   E-mail address: makarov@its.caltech.edu

Texas A& M University, Department of Mathematics, College Station, TX 77843, USA

   E-mail address: alexeip@math.tamu.edu