ON THE DISTRIBUTIONS OF BOUNDARY

VALUES OF CAUCHY INTEGRALS.

Alexei G. Poltoratski

Abstract. We use new methods to give short proofs to some known results on the distributions of boundary values of Cauchy integrals. We also indicate some further generalizations.

Introduction.

Let φ be an analytic function on the unit disk 𝔻 such that |φ| ≤ 1. Then for every α ∈ 𝕋 = ∂𝔻 the function (α + φ)(α − φ)^{−1} has positive real part. Therefore, there is a unique positive measure μ_{α} on 𝕋 such that

Pμ_{α} = Re \frac{α + φ}{α − φ},

where Pμ_{α} denotes the Poisson integral:

Pμ_{α} = ∫_{𝕋} \frac{1 − |z|^{2}}{|ξ − z|^{2}} dμ_{α}(ξ).

The family M_{φ} = {μ_{α}}_{α∈𝕋} has many interesting properties. For instance, there is the following decomposition of the normalized Lebesgue measure m on 𝕋 into the integral of the measures μ_{α} , see [A1]:

(0.1)  ∫_{𝕋} μ_{α}dm(α) = m.

(This formula should be understood in the sense that any Lebesgue measurable set E ⊂ 𝕋 is μ_{α}-measurable for m-a.e. α and ∫_{𝕋} μ_{α}(E)dm(α) = m(E).)

Let now μ be a finite positive Borel measure on 𝕋. We can always find a family M_{φ} such that μ ∈ M_{φ}. After that, we can use the structure of the whole family, in particular the decomposition formula (0.1), to study the boundary behavior of the Cauchy integral

(Kμ)(z) = ∫_{𝕋} \frac{1}{1 − \overline{ξ}z} dμ(ξ),

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 30E20, 30D55. Secondary 42A50.

Key words and phrases. Cauchy integral, conjugate functions, singular measures.

or the conjugate Poisson integral

(Qμ)(z) = 2 Im(Kμ)(z) = ∫_{𝕋} \frac{2 Im(z\overline{ξ})}{|ξ − z|^{2}}dμ(ξ).

In this note we will apply this approach to give new short proofs to some known results on the distributions of boundary values of Cauchy integrals. We will also indicate some further generalizations.

The first result in this area is probably due to G. Boole who discovered, in 1857, the following formula in the case when μ is a finite positive linear combination of point masses:

(0.2)  m(\{Qμ > t\}) = m(\{Qμ < −t\}) = \frac{1}{π} arctan \frac{∥μ∥}{t} , t > 0

(the functions Kμ and Qμ are defined almost everywhere on 𝕋 by their nontangential boundary values).

Later on, this result was extended to the case of an arbitrary positive singular measure, see [T1], [T2], [D1]. We discuss the results of this type in Section 1.

In Section 2 we study the asymptotic behavior of m({|Qμ| > t}) as t → ∞ for arbitrary measures. The classical result of Kolmogorov states that if μ ≪ m (μ is absolutely continuous with respect to m) then

m({|Qμ| > t}) < C||μ||/t

(the exact constant C was obtained by Davis in [D2]) and therefore

(0.3)  μ ≪ m ⇒ m(\{|Qμ| > t\}) = o(\frac{1}{t}).

The case of an arbitrary measure was investigated by Vinogradov, Hruschev [V-H] and Goluzina [G]. In Section 2 we prove the Vinogradov-Hruschev result in the following refined form.

We denote by M the set of all finite complex Borel measures on 𝕋.

Theorem 1. Let μ ∈ M. Then

(i)  πtχ_{{|Kμ|>t}} · m ⟶^{∗−weakly}_{t→∞} |μ^{s}|.

Moreover, if μ is a real measure then

(ii)  πtχ_{{Qμ>t}} · m ⟶^{∗−weakly}_{t→∞} |μ^{s}|

and

(iii)  πtχ_{{Qμ<−t}} · m ⟶^{∗−weakly}_{t→∞} |μ^{s}|.

Here χ_{{Qμ>t}} is the characteristic function of the set {Qμ > t} ⊂ 𝕋 and |μ^{s}| is the variation of the singular component of μ.

In Section 3 we generalize (0.3) in another direction:

Theorem 2. Let μ, ν ∈ M. Then

ν^{s} ≪ μ^{s} ⇔ lim_{c→∞} lim_{t→∞} t · m({|Kν| > t} \ {|Kμ| > t/c}) = 0.

As was shown by Clark in [C], if φ is an inner function (|φ| = 1 a. e. on 𝕋) then M_{φ} is the system of the spectral measures of all unitary one dimensional perturbations of the model contraction with characteristic function φ. This connection with perturbation theory is even more transparent in the context of rank one perturbations of self-adjoint operators, see [Ar], [Do], [S-W] and [S].

In some problems on rank-one perturbations it is important to understand how the resolvent function ⟨(A−z)^{−1}ϕ, ϕ⟩ of a cyclic self-adjoint operator A depends on the choice of the vector ϕ. In terms of Cauchy integrals this amounts to the problem of comparing the integrals Kμ and Kν of two equivalent measures. Theorem 2 above gives a partial solution to this problem.

The author is grateful to N. Makarov for his help in the preparation of this paper and to D. Sarason for many useful comments.

1. Metric properties of conjugate functions.

In addition to (0.1) the family M_{φ} has the following properties, see [A1]. If μ_{α}^{s} and μ_{α}^{a.c.} are the singular and absolutely continuous components of μ_{α} and Σ = {ξ ∈ 𝕋 | |φ| = 1 }, then

(1.1)  χ_{Σ} · m = ∫_{𝕋} μ_{α}^{s}dm(α)

and

(1.2)  (1 − χ_{Σ}) · m = ∫_{𝕋} μ_{α}^{a.c.}dm(α).

An analogous result for the real line is contained in [S]. Instead of the measures μ_{α} Simon integrates the spectral measures of one-dimensional perturbations of a self-adjoint operator.

Let w : ℝ → 𝕋 be the mapping

w(λ) = \frac{λ + i}{λ − i}.

In this paper we will use (1.1) in the following form:

Lemma 1.1. Let φ be an analytic function in 𝔻, |φ| ≤ 1 and φ(0) ∈ ℝ. Let \{μ_{α}\}_{α∈𝕋} = M_{φ} and Λ = \{\frac{dμ_{1}}{dm} > 0\} . Then for any t > 0

(1.3)  χ_{({Qμ_{1}>t}\Λ)} · m = ∫_{w((t;∞))} μ_{α}^{s}dm(α);

in particular,

(1.4)  m({Qμ_{1} > t} \ Λ) = ∫_{w((t;∞))} ∥μ_{α}^{s}∥dm(α).

Proof. For each α ∈ 𝕋, the measure μ_{α}^{s} is concentrated on the set

Σ_{α} = \{ξ | lim_{\begin{matrix}z→ξ \\ ≯ \end{matrix}} φ = α\} .

Since the set Σ from (1.1) coincides up to a set of Lebesgue measure 0 with the set 𝕋 \ Λ and since

Qμ_{1} = Im \frac{1 + φ}{1 − φ},

we have that up to a set of Lebesgue measure 0

\{Qμ_{1} > t\} \ Λ = \{ξ | lim_{\begin{matrix}z→ξ \\ ≯ \end{matrix}} φ ∈ w((t; ∞))\} = ⋃_{α∈w((t;∞))} Σ_{α}.

That means that if we multiply both sides of formula (1.1) by χ_{{Qμ_{1}>t}\Λ}, we obtain (1.3). Integrating (1.3) over 𝕋 we obtain (1.4). ▲

Remark. If we dropped the condition φ(0) ∈ ℝ in the statement of the corollary, we would have to replace the set w((t; ∞)) in the formulas (1.2) and (1.3) with the set w((t + c; ∞)) where c = Im \frac{1+φ(0)}{1−φ(0)}.

Now we will give short proofs to some metric properties of conjugate functions of positive measures.

We denote by M_{+} the subset of M consisting of all nonnegative measures.

Theorem 1.2. Let μ ∈ M_{+}, t > 0 and Λ = \{\frac{dμ}{dm} > 0\}. Then

(i)  \frac{1}{π} arctan \frac{∥μ∥}{t} − m(Λ) ≤ m(\{Qμ > t\} \ Λ) ≤ \frac{1}{π} arctan \frac{∥μ∥}{t}

(ii)  \frac{1}{π} arctan \frac{∥μ∥}{t} − m(Λ) ≤ m(\{Qμ < −t\} \ Λ) ≤ \frac{1}{π} arctan \frac{∥μ∥}{t}.

Proof. Without loss of generality we can assume that ∥μ∥ = 1. Consider an analytic function φ such that φ(0) = 0 and

Pμ = Re \frac{1 + φ}{1 − φ}.

Since

∥μ_{α}∥ = \frac{α + φ(0)}{α − φ(0)} = 1,

we have :

(1.5)  ∫_{w([t;∞))} ∥μ_{α}^{s}∥dm(α) ≤ m(w([t; ∞))) = \frac{1}{π} ∫_{t}^{∞} \frac{dt}{1 + t^{2}} = \frac{1}{π} arctan \frac{1}{t}.

We also have :

(1.6,)  ∫_{w([t;∞))} ∥μ_{α}^{s}∥dm(α) = ∫_{w([t;∞))} 1 − ∥μ_{α}^{a.c.}∥dm(α) =

= \frac{1}{π} arctan \frac{1}{t} − ∫_{w([t;∞))} ∥μ_{α}^{a.c.}∥dm(α) ≥

≥ \frac{1}{π} arctan \frac{1}{t} − ∫_{𝕋} ∥μ_{α}^{a.c.}∥dm(α) = \frac{1}{π} arctan \frac{1}{t} − m(Λ)

because

∫_{𝕋} ∥μ_{α}^{a.c.}∥dm(α) = m(Λ)

by (1.2). Now, if we combine (1.4), (1.5) and (1.6), we obtain (i). Formula (ii) can be proven in the same way. ▲

Since for any two sets A and B

m(A) − m(B) ≤ m(A \ B) ≤ m(A),

Theorem 1.2 implies the following result of Tsereteli:

Corollary 1.3 ([T1], [T2]). Let μ ∈ M_{+}, t > 0. Then

(i)  |\frac{1}{π} arctan \frac{∥μ∥}{t} − m(\{Qμ > t\})| ≤ m(\{\frac{dμ}{dm} > 0\}).

(ii)  |\frac{1}{π} arctan \frac{∥μ∥}{t} − m(\{Qμ < −t\})| ≤ m(\{\frac{dμ}{dm} > 0\}).

In [T1] it is also shown that using this result one can prove the theorems of Riesz and Zygmund on metric properties of conjugate functions.

Both Theorem 1.2 and Corollary 1.3 imply Boole’s formula (0.2). Davis in [D1] proved (0.2) using the Brownian motion.

2. Reconstruction of the singular part of a measure.

Proof of Theorem 1. Let us first prove (ii) in three steps.

1) Let μ be a singular positive measure. Then for the measures from M_{φ}, where μ = μ_{1} ∈ M_{φ} we have

(2.1)  μ_{α} ⟶^{∗−weakly}_{α→1} μ_{1} = μ.

Indeed, the definition of M_{φ} implies that

(Kμ_{α})(z) -→_{α→1}(Kμ_{1})(z)

for any z ∈ 𝔻. But linear combinations of Cauchy kernels and their complex conjugates are dense in the space of continuous functions on 𝕋.

Now, formula (1.2) gives us

πtχ_{{Qμ>t}} · m = πt ∫_{w((t;∞))} μ_{α}dm(α).

Since πt ∼ \frac{1}{m(w((t;∞)))}, the right hand side of the last equation has the same limit (if any) as

(2.2)  \frac{1}{m(w((t; ∞)))} ∫_{w((t;∞))} μ_{α}dm(α).

But since

w(t) -→_{t→∞} w(∞) = 1,

the expression (2.2) is just the average over the interval “tending to the point 1”. Thus by (2.1) we obtain (ii).

2) Let μ be an arbitrary positive measure from M then the relation

({Qμ^{s} > (1 − ϵ)t} ∪ {Qμ^{a.c.} > ϵt}) ⊃
⊃ {Qμ > t} ⊃
⊃ ({Qμ^{s} > (1 − ϵ)t} ∩ {Qμ^{a.c.} > ϵt})

together with part 1) and formula (0.3) implies (ii).

(This part of the proof can also be obtained from the result of Vinogradov and Hruschev (see Corollary 2.1 below) or from the results of Tsereteli, mentioned in Section 1, via the following argument, suggested by the referee.

To prove (ii) for positive measures, let P_{z}(e^{iθ}) = \frac{1−|z|^{2}}{|e^{iθ}−z|^{2}} be the Poisson kernel. Start with the formula

πt · m({Qμ) > t}) -→_{t→∞} μ^{s}(𝕋).

Using linear fractional transformations it follows that

πt · ∫_{\{Qμ)>t\}} P_{z} \frac{dθ}{2π} -→_{t→∞} ∫ P_{z}dμ^{s}

for all z. The result then follows since linear combinations of the functions P_{z} are dense in C(𝕋). I am thankful to the referee for this remark.)

3) Now, let μ = μ_{+}−μ_{−} be a real measure, μ_{±} ∈ M_{+}, μ_{+} ⊥ μ_{−}. Our proposition easily follows from the previous part if μ_{+} and μ_{−} are concentrated on closed disjoint subsets of 𝕋, i.e., if there are closed subsets F_{+} and F_{−} of 𝕋 such that ∥μ_{±}∥ = μ_{±}(F_{±}), F_{+} ∩ F_{−} = ∅.

If now μ = μ_{+} − μ_{−} is an arbitrary real measure, consider disjoint closed subsets F_{+} and F_{−} of 𝕋 such that μ_{±}(F_{∓}) = 0, μ_{±}(F_{±}) = ∥μ_{±}∥ − ϵ^{2} where ϵ is a small positive constant. Let ν_{±} be the restriction of μ_{±} on F_{±}. Since

\{Q(μ − ν) > t\} ⊂ (\{Q(μ_{+} − ν_{+}) > \frac{t}{2}\} ∪ \{−Q(μ_{−} − ν_{−}) > \frac{t}{2}\}),

by (0.2) we have:

m(\{Q(μ − ν) > t\}) ≤ \frac{4ϵ^{2}}{πt}

(because measures μ_{+} − ν_{+} and μ_{−} − ν_{−} are positive). Thus, from the relation

(2.3)  ({Qν > (1 − ϵ)t} ∪ {Q(μ − ν) > ϵt}) ⊃
⊃ {Qμ > t} ⊃
⊃ ({Qν > (1 − ϵ)t} ∩ {Q(μ − ν) > ϵt})

we obtain:

(2.4)  πtχ_{{Qμ>t}} · m = πtχ_{{Qν>t}} · m + η,

where η ∈ M, ∥η∥ ≤ \frac{4ϵ}{πt}∥ν∥ + \frac{4ϵ}{πt} + o(\frac{1}{t}). Since

πtχ_{{Qν>t}} · m ⟶^{∗−weakly}_{t→∞} |ν|,

we obtain (ii).

4) To prove (iii) one should replace μ in (ii) by −μ.

To prove (i), let us notice that (ii) and (iii) imply (i) for real measures because for any such measure μ we have m({Pμ > t}) = o(1/t). Let us also notice that if μ is a complex measure then for any ϵ > 0 there exist real mutually singular measures μ_{1}, μ_{2}, ..., μ_{n} and real constants α_{1}, α_{2}, ..., α_{n} such that for ν = ∑_{k=1}^{n} e^{iα_{n}}μ_{n} we have ∥μ − ν∥ < ϵ. Since the measures μ_{k} are real and mutually singular , we can prove (i) for ν using the same argument as in part 3 (we just have to deal with μ_{1}, μ_{2}, ..., μ_{n} instead of μ_{+} and μ_{−}). Applying (ii) and (iii) to the real and imaginary parts of μ − ν we obtain : πtm({|K(μ − ν)| > t}) < 2ϵ. Now we can finish the proof using the estimates similar to the ones we used for μ and μ − ν from part 3. ▲

Corollary 2.1 ([V-H]). Let μ ∈ M. Then

πt · m({|Kμ| > t}) -→_{t→∞} ∥μ^{s}∥.

Remark. One can prove the following localized versions of (i), (ii) and (iii), cf. [G].

Let 0 ≤ θ_{1} < θ_{2} < 2π, a = e^{iθ_{1}}, b = e^{iθ_{2}}, and let I ⊂ 𝕋 be an open arc with the ends a and b : I = {e^{iθ} : θ_{1} < θ < θ_{2}}. Let μ ∈ M. Then

(i′)  πtχ_{({|Kμ|>t}∩I)} · m ⟶^{∗−weakly}_{t→∞} χ_{(I∪{a,b})} · |μ^{s}|.

If μ ∈ M_{+} then

(ii′)  πtχ_{({Qμ>t}∩I)} · m ⟶^{∗−weakly}_{t→∞} χ_{(I∪{a})} · |μ^{s}|

and

(iii′)  πtχ_{({Qμ<−t}∩I)} · m ⟶^{∗−weakly}_{t→∞} χ_{(I∪{b})} · |μ^{s}|.

3. Relations between Cauchy integrals.

Let us consider the following corollary of Theorem 1.

Corollary 3.1 ([T2]). Let μ ∈ M. If

lim_{t→∞} t · min(m({Qμ > t}), m({−Qμ > t})) = 0

then μ ≪ m.

This shows that the inverse of the statement (0.3) is also true. The situation is different when we replace m with an arbitrary measure from M_{+} (see Remark after the proof of Theorem 2 below). However, we still can obtain some necessary and sufficient condition for one measure to be absolutely continuous with respect to another measure (see Theorem 2 in the introduction).

Proof of Theorem 2. Put η(c) = |ν^{s}| − |cμ^{s}|. Let η_{+}(c) and η_{−}(c) be the positive and negative parts of η(c) (η(c) = η_{+}(c) − η_{−}(c), η_{±}(c) ∈ M_{+}, η_{+}(c) ⊥ η_{−}(c)).

By Theorem 1

πtχ_{\{|Kν|>t\}} · m − πtχ_{\{|Kμ|> \frac{t}{c}\}} · m ⟶^{∗−weakly}_{t→∞} η(c).

But

πtχ_{\{|Kν|>t\}} · m − πtχ_{\{|Kμ|> \frac{t}{c}\}} · m =
= πtχ_{\{|Kν|>t\}\\{|Kμ|> \frac{t}{c}\}} · m − πtχ_{\{|Kμ|> \frac{t}{c}\}\\{|Kν|>t\}} · m.

Thus

πtχ_{\{|Kν|>t\}\\{|Kμ|> \frac{t}{c}\}} · m ⟶^{∗−weakly}_{t→∞} η_{+}(c)

and, in particular,

πtm(\{|Kν| > t\} \ \{|Kμ| > \frac{t}{c}\}) -→_{t→∞} ||η_{+}(c)||.

It is left to notice that ||η_{+}(c)|| -→_{c→∞} 0 iff ν^{s} ≪ μ^{s}. ▲

Remark. The proof shows that if measures μ and ν were real then we could replace the sets \{|Kν| > t\} and \{|Kμ| > \frac{t}{c}\} from the statement of the theorem with \{Qν > t\} and \{Qμ > \frac{t}{c}\} or with \{Qν < −t\} and \{Qμ < −\frac{t}{c}\} respectively.

Remark. The fact that m(\{|Kν| > t\}) = o(\frac{1}{t}) as t → ∞ for any ν ≪ m can also be generalized in the following way:

Let μ, ν ∈ M, μ > 0. Then

(3.1)  ν^{s} ≪ μ^{s} ⇒ m(\{|\frac{Kν}{Kμ}| > t\}) = o(\frac{1}{t}) as t → ∞.

Proof. We can suppose that ∥μ∥ = 1. Theorem 2 suggests that for any ϵ > 0 there are positive c and T such that for any t > T

m(\{|Kν| > t\} \ \{|Kμ| > \frac{t}{c}\}) < \frac{ϵ}{t}.

Thus, for n = 1, 2, 3, ...

m(\{2^{n+1}T > |Kν| > 2^{n}T \} \ \{|Kμ| > \frac{2^{n}T}{c} \}) < \frac{ϵ}{2^{n}T} .

Since μ ≥ 0 and ∥μ∥ = 1, |Kμ| > 1/2 on 𝔻. Thus,

\{|\frac{Kν}{Kμ}| > t\} ⊂ \{|Kν| > t/2\}

for any t > 0. Hence, if n is big enough (such that 2^{n}T > c),

\{|\frac{Kν}{Kμ}| > 2^{n+1}T \} ⊂ ⋃_{k=n}^{∞}(\{2^{k+1}T > |Kν| > 2^{k}T \} \ \{|Kμ| > \frac{2^{k}T}{c} \}).

Hence

m(\{|\frac{Kν}{Kμ}| > 2^{n+1}T \}) ≤ ∑_{k=n}^{∞} \frac{ϵ}{2^{k}T} ≤ \frac{ϵ}{2^{n−1}T}

and because ϵ is arbitrary we obtain (3.1). ▲

The inverse of (3.1) is false. Even the weaker statement that m(\{|\frac{Kν}{Kμ}| > t\}) =

o(\frac{1}{t}) implies supp ν^{s} ⊂ supp μ^{s} is false, as is evident if

μ = \frac{1}{|ξ − 1|^{\frac{1}{2}}} m, ν = δ_{1}.

We can, however, prove the following:

Let f ∈ L^{1}(𝕋). Let

Σ_{f} = ⋂_{t>0} Clos {|Kf| > t} .

Let μ, ν ∈ M, μ = fm + μ^{s}. If

m(\{|\frac{Kν}{Kμ}| > t\}) = o(\frac{1}{t}) as t → ∞

then

supp ν^{s} ⊂ (Σ_{f} ∪ supp μ^{s}).

Proof. Suppose m(\{|\frac{Kν}{Kμ}| > t\}) = o(\frac{1}{t}) but |ν^{s}|(F_{r}) < ∥ν^{s}∥ for some r where F_{r} = Clos {|Kf| > r} ∪ supp μ^{s}. Then there exists an open set E such that |ν^{s}|(E) = δ > 0 and |μ| < T on E for some T > 0. But Theorem 1 implies that

m(E ∩ \{|Kν| > t\}) ∼ \frac{δ}{t} as t → ∞;

thus, m(\{|\frac{Kν}{Kμ}| > t\}) ≥ \frac{δ}{T ·t} + o(\frac{1}{t}) ≠ o(\frac{1}{t}). ▲

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California Institute of Technology, Dept. of Mathematics 253-37, Pasadena CA 91125, e-mail : aleksei@cco.caltech.edu