ASYMPTOTIC BEHAVIOR OF ARGUMENTS OF CAUCHY INTEGRALS

ALEXEI POLTORATSKI

Abstract. We study asymptotic behavior of Cauchy integrals of measures on the real line. We describe the sets of measures whose Hilbert transform (conjugate Poisson integral) can grow faster than the Poisson transform and vice versa.

To my teacher Viktor Petrovich Havin on the occasion of his 75th birthday

1. Introduction

In this note we continue the discussion of asymptotic properties of Cauchy integrals started in [9]. Our main goal is to identify the classes of measures whose Poisson transform grows faster (or slower) than the conjugate Poisson transform (or the Hilbert transform) near a large set of points. Besides pure curiosity, such questions are motivated by their frequent appearance in complex function theoretic proofs. The idea to estimate the singular Cauchy (Hilbert) transform with a much more manageable Poisson transform lies in the foundation of complex analysis. Such estimates have produced many of the classical results: from Harnack’s Lemma to Fatou’s Theorem. The questions considered here may be viewed as an attempt to identify those measures whose Cauchy integrals allow (disallow) such estimates. Such problems have many interesting connections, like the connection with planar Cauchy transforms via a problem by P. Mattila, that was solved in [8], or the connection with mathematical physics via reflectionless measures that are discussed in the next section.

Let M(ℝ) be the space of all complex measures μ on the real line satisfying

∫_{ℝ} \frac{d|μ|(t)}{1 + t^{2}} < ∞.

1991 Mathematics Subject Classification. 30E20.

Key words and phrases. Hilbert transform, Cauchy integral, Poisson integral, boundary behavior.

The author is supported in part by N.S.F. grant DMS 0800300.

We will denote by P μ and Qμ the Poisson and the conjugate Poisson integrals of μ ∈ M(ℝ) in the upper half-plane ℂ_{+} respectively:

P μ(x + iy) = \frac{1}{π} ∫_{ℝ} \frac{y}{(x − t)^{2} + y^{2}} dμ(t)

and

Qμ(x + iy) = \frac{1}{π} ∫_{ℝ} \frac{x − t}{(x − t)^{2} + y^{2}} + \frac{t}{1 + t^{2}} dμ(t).

We denote by Cμ the Cauchy integral in ℂ_{+}:

Cμ = i(P μ + iQμ) = \frac{1}{π} ∫_{ℝ} \frac{1}{t − z} − \frac{t}{1 + t^{2}} dμ(x).

We say that a measure μ ∈ M(ℝ) belongs to Class N, (N = I, II, III or IV) if μ-almost every x satisfies the corresponding condition below. The conditions are:

(I) P μ(z) = o(Qμ(z)) as z ⟶_{∢} x;

(II) Qμ(z) = o(P μ(z)) as z ⟶_{∢} x;

(III) P μ(z) = O(Qμ(z)) as z ⟶_{∢} x;

(IV) Qμ(z) = O(P μ(z)) as z ⟶_{∢} x.

We also discuss these conditions with the non-tangential convergence z ⟶_{∢} x replaced with radial z = x + iy, y → 0+. The corresponding classes of measures will be denoted by N_{r}.

The word ”argument” in the title is justified by the fact that, if μ is a real measure, each of the relations (I)-(IV) can be interpreted as a statement about the behavior of arg Cμ. From this point of view real measures from Class II, for instance, are such that arg Cμ tends to ±π at μ-a.e point, etc.

A study of these classes was started in [9] where some results on their radial versions were obtained. Many interesting open problems, especially those related to Class II, were also discussed. One of the open problems was later solved in [8], see section 2 below.

In this note we provide a short review of the new results and connections that appeared since [9]. One of such connections concerns so-called reflectionless measures that recently became an object of active study in complex analysis and mathematical physics, see [1, 2, 7, 12, 14]. We discuss reflectionless measures in section 2. We also give answers to some of the remaining questions and provide several new examples of measures from Classes I-IV.

Let us start the discussion of Classes I-IV with a remark that the subset of each class consisting of singular measures is spectral. We say that a set of measures is spectral if together with any measure μ it contains the whole equivalence class of μ, i.e. all measures fμ where f ∈ L^{1}(|μ|). This immediately follows from

Theorem 1.1. [10] Let μ, ν ∈ M(ℝ), μ = fν + η where f ∈ L^{1}(|ν|) and η ⊥ ν. Then the limit

lim_{z⟶_{∢}x} \frac{Cμ(x)}{Cν(x)}

exists ν-a.e. It is equal to f(x) ν_{s}-a.e.

Theorem 1.1 also takes care of other questions of similar nature. For instance, the fact that Class II does not contain any singular measures (see below) implies, by Theorem 1.1, that it does not contain any measures with non-trivial singular parts, etc.

Now let us pass to a more detailed discussion of each class. As was proved in [9], Classes I and I_{r} are trivial. Theorem 2.7 [9] actually gives a slightly more general result where non-tangential convergence is replaced with convergence along a non-tangential path. Similar versions of some of the results discussed below would also be interesting.

2. Class II

Singular measures. As was proved in [8] Class II does not contain any singular measures. This follows from a more general result on Riesz transforms in ℝ^{n}.

For each μ ∈ M(ℝ^{n−1}) the Riesz transform Rμ(x) is defined on the half-space ℝ_{+}^{n} = {(x_{1}, ...x_{n})|x_{n} > 0} as

Rμ(x) =< R_{1}μ(x), ..., R_{n}μ(x) >,

R_{i}μ(x) = ∫_{ℝ^{n−1}} \frac{x_{i} − y_{i}}{|x − y|^{n}} dμ(y), i = 1, 2, ..., n.

It is well known, that Riesz transforms can be viewed as generalizations of the Poisson integral (the n-th transform R_{n}) and the conjugate Poisson integral (all other transforms R_{1}, ..., R_{n−1}) to higher dimensions.

If x ∈ ℝ^{n−1} and 0 < ϕ < π/2 we denote by Γ_{x}^{ϕ} the truncated cone:

Γ_{x}^{ϕ} = {y =< y_{1}, ..., y_{n} >∈ ℝ_{+}^{n} | y_{n}/|y − x| > sin ϕ, y_{n} < 1}.

As usual we write y ⟶_{∢} x if y → x and there exists ϕ = ϕ(x) such

that y ∈ Γ_{x}^{ϕ}.

Theorem 2.1. [8] Let μ ∈ M(ℝ^{n−1}) and let Σ ⊂ ℝ^{n−1} be such that

| < R_{1}μ(y), ..., R_{n−1}μ(y) > | = o(R_{n}μ(y))  (1)

as y ⟶_{∢} x for μ-a.e. x ∈ Σ. Then the restriction of μ on Σ is

absolutely continuous with respect to the Lebesgue measure in ℝ^{n−1}.

It would be interesting to see what other problems considered here can be generalized to the settings of Riesz transforms.

Class II_{r} does not contain any singular continuous measures, but contains all discrete measures:

Theorem 2.2. [8] Let μ ∈ M(ℝ) and let Σ be a Borel subset of ℝ. Then the following conditions are equivalent:

1) Qμ(x + iy) = o(P μ(x + iy)) as y → 0+ for μ_{s}-a.e. x ∈ Σ;

2) the restriction of μ_{s} on Σ is discrete.

This result answers a question raised in [9]. It also solves a problem by P. Mattila and some related problems in the theory of planar Cauchy transforms, see [8]. For an alternative proof and further developments see also [16].

Absolutely continuous measures. Absolutely continuous measures from Class II are called reflectionless, see [12]. For absolutely continuous measures non-tangential and radial convergence are much closer related. In particular, for any N = I, ..., IV, the set of absolutely continuous measures belonging to Class N coincides with the analogous subset in Class N_{r}.

A more general definition of a reflectionless measure pertains to finite complex measures in the plane ℂ. In [5] a finite complex measure μ in ℂ is called reflectionless if it has no point masses and its planar Cauchy transform, defined as the convolution of μ with \frac{1}{z} in the sense of principal value, converges to 0 at μ-almost every point in ℂ.

If μ happens to lie on the real line, this definition is equivalent to our definition with a slightly different kernel for the conjugate Poisson transform: without the term \frac{t}{1+t^{2}} dμ(t) under the integral. The two definitions are, of course, closely related. In particular, the two different conjugate Poisson transforms coincide for measures such that

∫ \frac{t}{1 + t^{2}} dμ(t) = 0

which is true, for instance, for all even measures.

Theorem 2.1 above implies that reflectionless measures on the line cannot have non-trivial singular parts. However, many absolutely continuous measures on the line are reflectionless. Examples are given immediately by Theorem 2.3. [5] Let E be a compact set of positive length, E ⊂ ℝ. Let ω be a harmonic measure of ℂ \ E with pole at infinity. Suppose that ω is absolutely continuous. Then ω is reflectionless.

Perhaps the simplest example of a reflectionless measure is the harmonic measure on [−1, 1]: μ = \frac{1}{π}(1 − x^{2})^{−1/2}dx.

A description of the set of reflectionless measures on R seems to be an interesting open problem. In [5] a partial description of the class of positive reflectionless measures on ℝ is given using geometric properties of conformal maps. Here we provide a full description of this set in terms of the arguments of the Cauchy transforms of the measures. A description of all complex reflectionless measures, even in the line case, is yet to be found.

Let us mention that positive reflectionless measures on the line naturally appear in Mathematical Physics as spectral measures of Jacobi matrices. In addition to the measures defined as above, in such problems one is also interested in measures reflectionless on a fixed set, i.e. those measures for which the condition (II) holds a.e. on a fixed subset of the line. This connection brings up a number of interesting questions on reflectionless measures, see [12, 14, 7, 2]

The study of reflectionless measures on the line also seems like a logical first step towards obtaining a description of all reflectionless measures in the plane. It was proved in [17] (see also [5] for an alternative proof), that a reflectionless measure in the plane cannot be absolutely continuous with respect to the two-dimensional Lebesgue measure. The results of [8] imply that a reflectionless measure may not have Hausdorff dimension less than 1. Examples of reflectionless measures on the real line give measures of dimension 1.

Do there exist planar reflectionless measures of Hausdorff dimensions between 1 and 2? This seems to be another intriguing open problem. First significant progress in this direction was recently obtained by F. Nazarov and A. Volberg [6]. It was proved that each planar reflectionless measure corresponds to a Hausdorff gauge function that tends to zero strictly slower than x^{2}.

Our goal for the rest of this section is to provide a workable description of the set of all positive reflectionless measures on the line that could help answer some of the most basic questions about this set: Does Theorem 2.3 give all possible examples of positive reflectionless measures on ℝ? Is every set of positive Lebesgue measure a minimal support for a reflectionless measure? Is such a measure unique?

As we will see shortly, not all reflectionless measures on the line are harmonic measures. Every set of positive Lebesgue measure is indeed a minimal support of a reflectionless measure. Such a measure is unique if and only if the set is an interval.

Let μ ∈ M(ℝ). We say that a Borel set E supports μ (E is a support of μ) if |μ|(E^{c}) = 0. If μ is absolutely continuous, we say that E is a minimal support of μ if E is a support of μ and for any set G ⊂ E, |G| > 0, |μ|(G) > 0.

Notice that our definition does not require a support to be closed. Minimal supports of absolutely continuous measures are unique up to sets of Lebesgue measure zero.

Denote by M(ℝ̂) the space of complex measures on R̂: if μ ∈ M(ℝ̂) then

μ = ν + cδ_{∞},

where ν ∈ M(ℝ), c ∈ ℝ and δ_{∞} is the unit point mass at infinity. Then the Cauchy integral of μ, Cμ, is defined as

Cμ = Cν + cz.

The norm in M(ℝ̂) is defined as

||μ|| = ∫ \frac{dμ(x)}{1 + x^{2}} + |c|.

Consider the principal branch of logarithm, log z, defined in ℂ \ (−∞, 0], that takes values in the strip |ℑz| < π. If μ is a positive measure from M(ℝ̂) then Cμ has positive imaginary part in ℂ_{+}. Therefore log Cμ is a well defined function in ℂ_{+} with ℑ log Cμ taking values in (0, π). Denote

u_{μ}(x) = ℑ log Cμ(x)  (2)

for x ∈ ℝ. I.e. u_{μ} is equal to the boundary values of arg Cμ on the real line, that exist almost everywhere on ℝ. Here arg stands for the principal branch of the argument in ℂ \ (−∞, 0] taking values in (−π, π). Then u_{μ} is a bounded function on ℝ, 0 ≤ u_{μ} ≤ π. Denote by Π the Poisson measure on the real line, \frac{1}{π}Π = \frac{dx}{1+x^{2}} . The Poisson integral P u_{μ} is equal to ℑ log Cμ(x) in ℂ_{+}. In particular

P u_{μ}(i) = ∫_{ℝ} udΠ = π/2

because our choice of the conjugate Poisson kernel implies Qμ(i) = 0. Sometimes it is more convenient to rewrite (2) as

exp Cu_{μ}(x) = \frac{1}{||μ||}Cμ(x).  (3)

Moreover, for any bounded function u on ℝ, 0 ≤ u ≤ π, ∫_{ℝ} udΠ = π/2, one can solve (3) backwards to obtain a unique probability measure μ ∈ M(ℝ̂) satisfying

exp Cu_{μ}(x) = Cμ(x).  (4)

Indeed, the left hand side is an analytic function in ℂ_{+} with a positive imaginary part and, by the representation theorem, it must be equal to a Cauchy integral of a positive measure μ ∈ M(ℝ̂), up to a real constant. The constant is zero because the left hand side is imaginary at i. Thus one can consider a one-to-one correspondence between bounded functions u on ℝ, taking values on [0, π], satisfying ∫ udΠ = π/2 and probability measures on ℝ̂.

In perturbation theory u_{μ} appears as the so-called Krein-Lifshits shift function corresponding to the self-adjoint operator with the spectral measure μ. The relation u_{μ} ↔ μ described above is well studied in the theory of rank-one perturbations, see for instance [3, 11, 15]. Here we use some of the well-known properties of this relation to establish the following:

Lemma 2.4. Let E be a Borel subset of ℝ. A probability measure μ is a reflectionless measure minimally supported on E if and only if the function u_{μ} defined by (4) satisfies

1) u_{μ}(x) = π/2 for almost every x ∈ E and u_{μ}(x) equals either 0 or π for almost every x ∉ E;

2) ∫_{ℝ} u(x)dΠ(x) = π/2;

3) Π({x ∈ ℝ | Qu_{μ}(x) > t}) = o(e^{−t}) as t → ∞;

4) either ∫_{1}^{∞} \frac{u_{μ}(x)dx}{x} = ∞ or ∫_{−∞}^{−1} \frac{π−u_{μ}(x)dx}{x} = −∞.

Remark 2.5. If E is a simple set the third and fourth conditions can be made much more explicit. For instance, if E is a union of finitely many intervals, these conditions are equivalent to the condition that u_{μ} does not make any jumps ”up” from 0 to π on the compliment of E, including at infinity.

The conjugate Poisson transform can be replaced with the Hilbert transform in this statement.

As usual let us denote by μ_{ac} the absolutely continuous part and by μ_{s} the singular part of μ.

Proof. The Poisson integral of μ has non-zero limits a.e. on any minimal support of μ_{ac} and zero limits a.e. outside of that support. Therefore the argument of the Cauchy integral of μ is equal to 0 or π outside of a minimal support of μ and is strictly between 0 and π a.e. on it. The argument of Cμ is equal to π/2 a.e. on E iff Qμ is zero a.e on that set. Hence the first condition is equivalent to the condition that E is a minimal support of μ_{ac} and that Qμ is zero a.e on E.

The second condition is equivalent to the condition that μ is a probability measure.

The third condition can be rewritten as

Π(\{|Cμ| > t\}) = o (\frac{1}{t})

as t → ∞. This is equivalent to the statement that μ is absolutely continuous on ℝ.

Finally, the fourth condition means that μ does not have a point mass at infinity.

Corollary 2.6. Let E ⊂ ℝ be a Borel subset. Then there exists a reflectionless probability measure μ minimally supported on E. Such a measure is unique iff E is an interval.

Proof. WLOG we can assume that E ≠ ℝ. To prove existence, notice that the function u = \frac{π}{2}χ_{E} satisfies conditions 1, 3 and 4 of the lemma. I is left to ”correct” it to satisfy 2 (for u we obviously have ∫_{ℝ} u(x)dΠ(x) < π/2).

Let u_{a}, a ∈ ℝ be defined as

u_{a}(x) = \begin{cases}π/2 if x ∈ E \\ π if x ∉ E, x > a \\ 0 if x ∉ E, x < a \end{cases} .

Obviously, one can choose a = a_{0} so that u_{a_{0}} satisfies 2. It will also satisfy 1 and 4. Moreover, if a_{0} is a point of non-zero lower density for E, then u_{a_{0}} will still satisfy 2. Thus if a_{0} is such a point we are done: μ can be chosen so that u_{μ} = u_{a_{0}}.

If a_{0} is not a point of non-zero lower density for E then the construction could be finished in the following way. Let b be any point of non-zero lower density for E. Then there are two possibilities:

a) ∫_{ℝ} u_{b}(x)dΠ(x) < π/2

or

b) ∫_{ℝ} u_{b}(x)dΠ(x) > π/2.

Suppose we are in the case a. Then one can choose a point a < b so that the function u_{ab} defined a.e. on the line as

u_{ab}(x) = \begin{cases}π/2 if x ∈ E \\ π if x ∉ E, x < a or x > b \\ 0 if x ∉ E, a < x < b \end{cases}

satisfies the second condition of the lemma. Since the jump at a is a negative jump, it will still satisfy 3 and the rest of the conditions.

Similarly, if we are in the case b, then one can choose a point c > b and the function u_{bc} defined as

u_{bc}(x) = \begin{cases}π/2 if x ∈ E \\ 0 if x ∉ E, x < b or x > c \\ π if x ∉ E, b < x < c \end{cases}

to satisfy the conditions of the lemma.

To prove uniqueness for the case when E =< a, b > is an interval, notice that if ∫_{ℝ} u_{b}(x)dΠ(x) < π/2 then the only function satisfying conditions 1-4 is the function v_{c} defined as

v_{c}(x) = \begin{cases}π/2 if a < x < b \\ 0 if c < x < a \\ π if x < c or b < x \end{cases}

where c is chosen so that v_{c} satisfies 2. Similarly, if ∫_{ℝ} u_{b}(x)dΠ(x) > π/2, then one can choose c > b and define v_{c} as

v_{c}(x) = \begin{cases}π/2 if a < x < b \\ 0 if x < a or x > c \\ π if b < x < c \end{cases} .

From our construction above one can see that when E is a more complicated set there exist infinitely many functions satisfying the conditions of the lemma. We discuss the case of two intervals below and leave the details of this part of the proof to the reader.

Example 2.7. Suppose that E = [−2, −1] ∪ [1, 2]. Elementary calculations show that the set of all functions satisfying the conditions of lemma 2.4 can be described as follows. Define u_{c}, c ∈ [0, 1/2] as

u_{c}(x) = \begin{cases}π/2 if x ∈ (−2, −1) ∪ (1, 2) \\ 0 if x < −2 or x ∈ (c, 1) or x ≥ 1/c \\ π if x ∈ (−1, c) or x ∈ (2, 1/c) \end{cases}

(if c = 0 then 1/c should be replaced with +∞). For c ∈ [−1/2, 0), u_{c} is defined as

u_{c}(x) = \begin{cases}π/2 if x ∈ (−2, −1) ∪ (1, 2) \\ 0 if 1/c < x < −2 or x ∈ (c, 1) \\ π if x < 1/c or x ∈ (−1, c) or x > 2 \end{cases} .

Except for such u_{c}, c ∈ [−1/2, 1/2] there are no functions satisfying the conditions 1 and 2, that have no jumps from 0 to π at any finite points or at infinity (which, as we remarked earlier, is equivalent to satisfying the conditions 3 and 4). Thus by lemma 2.4 all probability reflectionless measures on [−2, −1]∪[1, 2] form a one-parameter family μ_{c}, c ∈ [−1/2, 1/2] defined via 4 as u_{μ_{c}} = u_{c}.

A description of general complex reflectionless measures on ℝ as well as numerous problems on reflectionless measures in the plane (some of them mentioned above) remain open.

3. Class III

This class obviously contains all absolutely continuous measures from M(ℝ). Does it contain any singular measures? Theorem 3.1 below shows that the answer is negative is the size of the sector is large enough.

As before, if x ∈ ℝ and 0 < ϕ < π, denote by Γ_{x}^{ϕ} the sector

Γ_{x}^{ϕ} = \{u + iv ∈ ℂ_{+} | − ϕ/2 < arctan \frac{v}{u − x} < ϕ/2\}.

Theorem 3.1. Let μ ∈ M(ℝ) be a real measure. Define the set E ⊂ ℝ as the set of all x ∈ ℝ for which there exist ϕ > π/2 and ϵ > 0 such that Qμ(z) ≠ 0 for all z ∈ Γ_{x}^{ϕ}, ℑz < ϵ. Then |μ_{s}|(E) = 0.

Proof. The proof utilizes some of the methods from [8].

Suppose that |μ_{s}|(E) > 0. Then there exists a set

F ⊂ E, |μ_{s}|(F) > 0

such that there exist uniform ϕ > π/2 and ϵ > 0 such that Qμ(z) ≠ 0 for all z ∈ Γ_{x}^{ϕ}, ℑz < ϵ for any x ∈ F. Note that since Qμ is a continuous real function in Γ_{x}^{ϕ}, WLOG one can assume that Qμ(z) > 0 for all z ∈ Γ_{x}^{ϕ}, ℑz < ϵ for any x ∈ F. Since P μ/P |μ| tends to 1 or −1 at μ-a.e. x, we can also assume that P μ(z) > 0 for all z ∈ Γ_{x}^{ϕ}, ℑz < ϵ for any x ∈ F. Denote

Γ = ∪_{x∈F} Γ_{x}^{ϕ} ∩ {ℑz < ϵ}.

Then the function QμP μ = ℑ(Cμ)^{2} is a positive harmonic function in Γ. Therefore it is summable on the boundary ∂Γ of Γ with respect to the harmonic measure. Hence its harmonic conjugate function

(P μ)^{2} − (Qμ)^{2}

belongs to L^{p} for any p < 1 with respect to the harmonic measure on ∂Γ. Since

|(Qμ)^{2}| ≤ (|P μQμ| + |(P μ)^{2} − (Qμ)^{2}|),

(Qμ)^{2} belongs to L^{p} for any p < 1 with respect to the harmonic measure on ∂Γ. Therefore Qμ is in L^{p} for any p < 2 with respect to the harmonic measure on ∂Γ.

Denote by Ω the lower part of the boundary ∂Γ:

Ω = ∂Γ ∩ {ℑz < ϵ/2}.

Since Γ is a Lipschitz domain, the harmonic measure on ∂Γ is absolutely continuous with respect to the arclength: it is equal to wds for some density w. Since ϕ > π/2, on the lower part of the boundary the density satisfies

∫_{Ω} w^{−q}ds < ∞

for some q > 1.

Choose 1 < r < 2 < s so that 1/r + 1/s = 1, r < 2 and s/r < q. Then

∫_{Ω} |Qμ|ds ≤ (∫_{Ω} |Qμ|^{r}wds)^{1/r} (∫_{Ω} w^{−s/r})^{1/s} < ∞.

Hence Qμ is summable with respect to the arclength on Ω. Since Qμ is bounded in {ℑz > ϵ/2}, this implies that Qμ is summable with respect to the arc length on the whole boundary ∂Γ. By lemma 3.1 from [8], this can only happen when |μ_{s}|(F) = 0.

Since for any μ ∈ M(ℝ) the Poisson integral P μ tends to infinity non-tangentially at μ_{s}-a.e. point, Theorem 3.1 implies that for large sectors P μ(z) cannot be equal to O(Qμ(z)) as z → x, z ∈ Γ_{x}^{ϕ} for a nonzero set of x with respect to μ_{s}. Can ϕ > π/2 be replaced with ϕ > 0? in this statement? Furthermore, can nontangential convergence be strengthened to radial? The methods that allowed a similar transition in [8] do not seem to apply here.

To end this section let us remark, that together with results of [8], Theorem 3.1 shows that for any μ ∈ M(ℝ), |Qμ| oscillates wildly in μ_{s}-a.e. sector Γ_{x}^{ϕ} from 0 to a value comparable to |P μ| that tends to infinity. Once again, it would be interesting to see if such oscillation happens on μ_{s}-a.e. vertical line as well.

4. Class IV

All measures from M(ℝ) that do not have a singular continuous part, including all absolutely continuous measures or all pure point measures, belong to this class. Moreover, measures whose singular continuous part is ”reasonable,” like the standard Cantor measure for example, do belong to this class as well. Nonetheless, this class does not contain the whole M(ℝ) and even the larger set, Class IV_{r} does not include some of the measures.

As was mentioned in [9], the existence of a measure that does not belong to this class follows from a construction of an inner function given by W. Rudin in [13]. This counterexample is quite implicit and does not allow one to study the resulting measure. The existence of the measure follows from the fact that almost every measure from the Clark family corresponding to the Rudin’s inner function must not satisfy (IV).

Here we give a simpler example of a singular continuous measure not belonging to Class IV. Our construction, once again, utilizes the argument of the Cauchy integral of the measure, the Krein function u_{μ}. In terms of this function, the example is explicit.

For our construction we will need the following

Lemma 4.1. [4] A probability measure μ ∈ M(ℝ) has a point mass at x iff

∫_{x−1}^{x+1}(πχ_{(x;x+1)} − u_{μ}(y))\frac{dy}{y − x} < ∞.

Example 4.2. Our goal is to construct the Krein function u on the real line such that the probability measure μ corresponding to u (i.e. such that u = u_{μ}) does not belong to Class IV.

Denote I = [−1, 1]. For ϵ > 0 consider the intervals

J_{n} = (1 − 2^{−n}, 1 − 2^{−n−1} − ϵ2^{−n−1})

for n = 0, 1, 2, ... and

J_{n} = (−1 + 2^{n−1} + ϵ2^{n}, −1 + 2^{n})

for n = −1, −2, −3, .... Put

I^{ϵ} = I \ ∪J_{n}.

For an arbitrary interval L = [a, b] ”rescale and shift” the above set and denote

L^{ϵ} = \{x | x = \frac{b − a}{2}(t + 1) + a, t ∈ I^{ϵ}\}.

The closed set L^{ϵ} occupies a small portion of L and is thin (has density ϵ) near the endpoints of L. Because of that property one can show that for any ε > 0 there exist ϵ, δ > 0 such that for any x ∈ L

1 − ε < P χ_{L_{ϵ}}(x + iδ dist(x, ℝ \ L)) < 1.  (5)

Now we are ready to define the function u. First let us define it on [0, 1]. Choose a sequence of small positive numbers ϵ_{n} → 0. Start with the unit interval I_{0,1} = [0, 1]. Consider the set I_{0,1}^{ϵ_{0}} defined as above. Define u to be π on [0, 1] \ I_{0,1}^{ϵ_{0}}.

Next, consider the disjoint closed intervals I_{1,n}, n = 1, 2, 3, ..., that form I_{0,1}^{ϵ_{0}}:

I_{0,1}^{ϵ_{0}} = ∪I_{1,n}.

On each of these intervals consider the set I_{1,n}^{ϵ_{1}}. Define u as 0 on each of I_{1,n} \ I_{1,n}^{ϵ_{1}}.

On the k-th step consider the disjoint closed intervals I_{k,n} that compose the remainder of [0, 1] where u is not yet defined. On each of them consider I_{k,n}^{ϵ_{k}}. If k is even, define u as π on all I_{k,n} \ I_{k,n}^{ϵ_{k}}, if k is odd, define u as 0.

Since | ∪_{n,k} I_{k,n}^{ϵ_{k}}| = 1, the limit function u will be defined a.e. on the unit interval. Assume that ϵ_{1} < 1/2. Then outside of [0, 1] define u as π on (1, ∞) ∪ (−∞, c) and as 0 on (c, 0), where the negative constant c is chosen so that

∫ udΠ = 0

.

Denote by C the Cantor set ∩_{n,k}I_{k,n}^{ϵ_{k}}. Let μ be a probability measure on ℝ such that u_{μ} = u. One can easily show that μ is supported on C. The set C is a closed set that is equal to [0, 1] minus the union of countably many complimentary open intervals. The measure does not have point masses at the endpoints of the complimentary intervals of C. This can be seen from the Lemma 4.1 and the definition of the set I^{ϵ} at the beginning of our construction.

Next, notice that

lim sup_{z⟶_{∢}x} P u(z) = π and lim inf_{z⟶_{∢}x} P u(z) = 0

for every x ∈ C that is not an end point of a complimentary interval. This also follows from the construction of L^{ϵ}, see (5). Hence, by (3),

lim sup_{z⟶_{∢}x} arg Cμ(z) = π and lim inf_{z⟶_{∢}x} arg Cμ(z) = 0

for μ-a.e. x. Therefore the condition (IV) is not satisfied and μ does not belong to Class IV.

It would be interesting to describe Class IV and to understand its relations with such properties of the measure as dimension, symmetry, smoothness, etc.

References

[1] F. Gesztesy and P. Yuditskii, Spectral properties of a class of reflectionless Schrödinger operators, J. Funct. Anal. 241 (2006), 486–527.

[2] F. Gesztesy and M. Zinchenko, Local spectral properties of reflectionless Jacobi, CMV, and Schrödinger operators, preprint,

[3] D. Hundertmark and B. Simon, An optimal Lp -bound on the Krein spectral shift function, J. d’Anal. Math, 87 (1), 2002, 199–208.

[4] M. Martin and M. Putinar, Lectures on Hyponormal Operators, Operator Theory: Advances and Applications, 39, Birkhäuser Verlag, Basel, 1989.

[5] M. Melnikov, A. Poltoratski, and A. Volberg, Uniqueness theorems for Cauchy integrals, to appear in Publ. Math.

[6] F. Nazarov and A. Volberg, A remark on reflectionless measures, preprint [7] F. Nazarov, A. Volberg, and P. Yuditskii, Reflectionless measures with a point

mass and singular continuous component, preprint.

[8] P. W. Jones and A. Poltoratski, Asymptotic Behavior of Cauchy Transforms, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 99–120.

[9] A. Poltoratski, Images of non-tangential sectors under Cauchy transforms, J. d’Anal. Math., 89 (2003), 385–395.

[10] A. Poltoratski, On the boundary behavior of pseudocontinuable functions, St. Petersburg Math. J.,5 (1994), 389–406.

[11] A. Poltoratski, The Krein spectral shift and rank one perturbations of spectra, St. Petersburg Math. J., 10 (1999), pp 833-859.

[12] A. Poltoratski and C. Remling, Reflectionless Herglotz functions and Jacobi matrices, to appear in Comm. Math. Phys.

[13] W. Rudin, Inner function images of radii, Proc. Camb. Phil. Soc., 85 (1979), 357-372.

[14] M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of characterautomorphic functions, J. Geom. Anal. 7 (1997), 387–435.

[15] B. Simon, Spectral analysis of rank one perturbations and applications, CRM Proc. and Lect. Notes, 8, 1995.

[16] X. Tolsa, Growth Estimates for Cauchy Integrals of Measures and Rectifiability, Geom. and Func. Anal., 17 (2), 2007, 605–643.

[17] X. Tolsa, J. Verdera, May the Cauchy transform of a non-trivial finite measure vanish on the support of the measure? Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 479–494.

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

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