THE BOUNDARY BEHAVIOR OF

PSEUDOCONTINUABLE FUNCTIONS

A. G. Poltoratskiĭ

24/APR/1992

Abstract. We investigate the functions belonging to an arbitrary invariant subspace of the backward shift operator, namely, their boundary properties relative to the spectral measures of one-dimensional perturbations of the compression of the shift to this subspace. As an application, we solve certain well-known problems about multiplication and division for Cauchy-Stieltjes type integrals.

Introduction

According to a well-known theorem of D. Clark [1], each fixed closed nontrivial invariant subspace of the backward shift operator on the Hardy class H^{2} gives rise to a specific unitary operator identifying this invariant subspace with the L^{2}-space of a certain singular measure. In [2] and [3], A. B. Aleksandrov studied further properties of such measures and those of the identification mapping regarded as an operator on L^{p} with p ≠ 2. The main purpose of the present paper is to shed some light upon the mechanism of action of the above operator. Also, the paper contains some generalizations and applications of the results pertaining to the study of Clark’s operator.

Let H^{p} (0 < p < ∞) denote the Hardy class on the unit disk 𝔻, 𝔻 =^{def} { z ∈ ℂ | |z| < 1 }, i.e., H^{p} is the space of all functions f holomorphic in 𝔻 for which

∥f∥_{p} =^{def} sup_{o<r<1} ( ∫_{𝕋} |f(rξ)|^{p} dm(ξ))^{1/p} < +∞,

where 𝕋 is the unit circle, 𝕋 =^{def} { z ∈ ℂ | |z| = 1 }, and m stands for normalized Lebesgue measure on 𝕋. For p = +∞ the Hardy class H^{p} consists of the functions f that are analytic and bounded in 𝔻:

∥f∥_{∞} =^{def} sup_{z∈𝔻} |f(z)| < +∞.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 30D55, Secondary 30E20, 47B37.

Key words and phrases. Invariant subspaces of the backward shift operator, Clark theorem, de Branges spaces, multiplication and division for Cauchy–Stieltjes type integrals.

Since any H^{p}-function has boundary values almost everywhere on 𝕋, one can treat H^{p} as a closed subspace of L^{p}(m) (for more details see [4] or [5]). A function θ, θ ∈ H^{∞}, is called inner if |θ(ξ)| = 1 for m-almost all ξ in 𝕋. We set

H_{0}^{p} =^{def} { f ∈ H^{p} | f(0) = 0 },
H_{−}^{p} =^{def} \{ f ∈ L^{p} | \overline{f} ∈ H_{0}^{p} \}.

With each inner function θ we associate the subspaces θ^{∗}(H^{p}), p ≥ 1:

θ^{∗}(H^{p}) =^{def} \{ f ∈ H^{p} | f\overline{θ} ∈ H_{−}^{p} \}.

Each function lying in the space θ^{∗}(H^{p}) is known to possess the so-called “pseudocontinuation” (see, e.g., [6]). This is reflected by the title of the paper. Furthermore, the spaces θ^{∗}(H^{p}) are precisely the closed invariant subspaces of the backward shift operator S^{∗}: H^{p} ⟶ H^{p}, (S^{∗}f)(z) = (f(z) − f(0))/z.

By C(𝕋) we denote the space of all continuous functions on the circle 𝕋. The dual of C(𝕋) will be naturally identified with the space M(𝕋) of all finite regular complex measures on 𝕋. By M_{+}(𝕋) we denote the subset of M(𝕋) consisting of the nonnegative measures. The space L^{1}(m) is identified with the set of all measures in M(𝕋) absolutely continuous with respect to m.

With a measure μ ∈ M(𝕋) we associate the following functions defined on ℂ \ 𝕋:

μ_{+}(z) = ∫_{𝕋} \frac{1}{1 − z\overline{ξ}} dμ(ξ),
P μ(z) = ∫_{𝕋} \frac{1 − |z|^{2}}{|z − ξ|^{2}} dμ(ξ),
Qμ(z) = ∫_{𝕋} \frac{2 Im(z\overline{ξ})}{|z − ξ|^{2}} dμ(ξ).

In other words, μ_{+}, P μ, and Qμ are the convolutions of the measure μ with the Cauchy kernel, the Poisson kernel, and the conjugate Poisson kernel, respectively.

To each inner function θ we relate the family of measures {σ_{α}}_{α∈𝕋} in M_{+}(𝕋) such that

P σ_{α} = Re \frac{α + θ}{α − θ} .

Conversely, for each singular measure σ in M_{+}(𝕋) there exists an inner function θ such that σ = σ_{1} for this θ. Thus, the correspondence θ ↔ σ_{1} (as well as any correspondence θ ↔ σ_{α}, α ∈ 𝕋) provides a parametrization of the set of inner functions by means of singular measures from M_{+}(𝕋).

It is not hard to see that the measure σ_{α} is supported on the set of points at which the radial limits of θ are equal to α. It follows that the measures σ_{α} are singular with respect to Lebesgue measure, and also pairwise singular. Besides, we have the following theorem due to A. B. Aleksandrov.

Theorem 1 ([2]). The equality

m = ∫_{𝕋} σ_{α} dm(α),

is valid, if understood in the following “weak” sense:

∫_{𝕋} f dm = ∫_{𝕋}(∫_{𝕋} f dσ_{α}) dm(α)

for every f ∈ C(𝕋).

As it is shown in [2], this latter relation actually holds for every nonnegative Lebesgue measurable function f. In particular, any such function is σ_{α}-measurable for almost all α.

The measures σ_{α} were introduced by Clark [1] who was concerned with the completeness of the family of functions k_{λ} = (1 − \overline{θ(λ)}θ(z))/(1 − \overline{λ}z) in the space θ^{∗}(H^{2}) (this question is important, in particular, by the reason that, as a special case, it contains the classical problem on the completeness of collections of exponentials e^{iγt} in L^{2}([0, 1]). Sufficient conditions for completeness can be derived from a well-known theorem of B. Sz.-Nagy on biorthogonal systems (cf. [8]), provided that θ^{∗}(H^{2}) possesses a convenient orthogonal basis. In particular, such a basis exists if the operator Y_{α}: θ^{∗}(H^{2}) ⟶ θ^{∗}(H^{2}), α ∈ 𝕋, defined by

Y_{α}f = \{ \begin{matrix}zf & if f ⊥ S^{∗}θ, \\ αk_{0} & if f = S^{∗}θ \end{matrix}

has purely point spectrum. In this case the eigenvectors of Y_{α} are precisely the functions k_{λ} with |λ| = 1. The operators Y_{α} were studied in [1], and it was shown that σ_{α}’s are the spectral measures for these operators.

The following theorem was also proved.

Theorem 2 ([1]). Consider the operator U_{α} originally defined on the linear hull of the functions k_{λ} with |λ| < 1 and mapping this linear hull into the space L^{0}(σ_{α}) of all σ_{α}- measurable functions (mod 0) according to the rule

(U_{α}k_{λ})(z) = (1 − \overline{θ(λ)}α)/(1 − \overline{λ}z).

After an appropriate extension, U_{α} becomes a unitary operator mapping θ^{∗}(H^{2}) onto L^{2}(σ_{α}).

Let λ_{a} be the conformal automorphism of the unit disk that sends a to 0. Now the

operator given by f ↦ f · ^{\frac{\sqrt{1−|a|^{2}}}{1−\overline{a}θ}} takes θ^{∗}(H^{p}) onto (λ_{a} ∘ θ)^{∗}(H^{p}); in the case p = 2 this operator is an isometry. Furthermore, letting {σ′_{α}}_{α∈𝕋} be the family of measures corresponding to the inner function λ_{a} ∘ θ, we have σ′_{β} = \frac{σ_{α}}{|λ′_{a}(α)|} for β = λ_{a}(α). This observation will enable us to confine ourselves to the case where θ(0) = 0. Under this assumption, all the σ_{α}’s are probability measures, and the formulas become much simpler. In particular, the operator U_{α}^{∗}: L^{2}(σ_{α}) ⟶ θ^{∗}(H^{2}), which is the adjoint of the unitary operator U_{α} mapping θ^{∗}(H^{2}) onto L^{2}(σ_{α}), is now given by

U_{α}^{∗}f = \frac{(fσ_{α})_{+}}{(σ_{α})_{+}} = (1 − \overline{α}θ)(fσ_{α})_{+}.

In [3], A. B. Aleksandrov studied the behavior of the map U_{α}: θ^{∗}(H^{p}) ⟶ L^{p}(σ_{α}) for p ≠ 2. His results were obtained as corollaries to the following general theorem.

Theorem 3 ([3]). Let μ ∈ M_{+}(𝕋). With this μ we associate the map

V_{μ}f =^{def} (f_{μ})_{+}/μ_{+} (f ∈ L^{1}(μ)).

The operator V_{μ} is of weak type (1, 1), and it is a continuous map from L^{p}(μ) to L^{p}(m) (1 < p ≤ 2). Moreover, the norm of V_{μ} can be estimated in terms of p only.

Further properties of the map V_{μ} will be considered in §2. Since for 1 < p ≤ 2 we have U_{α}^{−1} = V_{σ_{α}} (recall that we assume θ(0) = 0), Theorem 3 implies the following statements.

Corollary 1 ([3]). Let 1 < p ≤ 2. Then

U_{α}^{−1}(L^{p}(σ_{α})) ⊂ θ^{∗}(H^{p}).

Corollary 2 ([3]). Let 2 < p ≤ +∞. Then

U_{α}(θ^{∗}(H^{p})) ⊂ L^{p}(σ_{α}).

In [3], A. B. Aleksandrov also showed that though U_{α}(θ^{∗}(H^{p})) and L^{p}(σ_{α}) coincide for p = 2 by Clark’s theorem, this is no longer valid for p ≠ 2 except for some degenerated cases.

Theorem 4 ([3]). Let μ be a singular measure in M_{+}(𝕋), p > 2. Suppose that V_{μ}(C(𝕋)) ⊂ L^{p}. Then μ is discrete.

Corollary ([3]). Let θ be an inner function and let α ∈ 𝕋. Suppose that U_{α}^{−1}(L^{p}(σ_{α})) = θ^{∗}(H^{p}) for some p ∈ (1, 2) or U_{α}(θ^{∗}(H^{p})) = L^{p}(σ_{α}) for some p ∈ (2, +∞]. Then σ_{α} is a discrete measure.

Now the following question arises: how does the operator U_{α} act on functions that do not belong to the linear hull of the family {k_{λ}}_{λ∈𝔻}? In Clark’s paper [1] it was shown that if a function f in θ^{∗}(H^{2}) extends analytically across an arc of the circle 𝕋, then we have f = U_{α}f almost everywhere with respect to σ_{α} on that arc. In A. B. Aleksandrov’s paper [3] it was mentioned that the relation f = U_{α}f also holds a.e. with respect to σ_{α} in the case where f ∈ C(𝕋) ∩ θ^{∗}(H^{2}). In §1 of the present paper we obtain some results showing that the relation f = U_{α}f remains true for the rest of the functions in θ^{∗}(H^{p}); moreover, it can be understood “literally”.

In [7], D. Sarason proved that, given a probability measure μ in M_{+}(𝕋), the map V_{μ} is an isometry of the space H^{2}(μ) (the latter is the closure in L^{2}(μ) of the set of all polynomials of z) onto the so-called de Branges space ℌ(f) (the notation is the same as in [7]), where f = 1 − 1/μ_{+}. So, in a sense, the map V_{μ} is a generalization of the operator U_{α}^{∗}. In §2 we extend the results of §1 to the case of V_{μ}, where μ is an arbitrary measure from M(𝕋).

Finally, in §3 we apply the results obtained in the previous sections to some old problems concerning division and multiplication for Cauchy–Stieltjes-type integrals.

The author expresses his deep gratitude to A. B. Aleksandrov for having posed the problems treated here, and also for his kind help in preparing the manuscript.

§1. Convergence of functions lying in θ^{∗}(H^{p}) to their boundary values

We start with some propositions showing that for p ≥ 2 the image U_{α}f of a function f ∈ θ^{∗}(H^{p}) under Clark’s operator U_{α} coincides a.e. [σ_{α}] with the boundary values (introduced in a certain way) of f on 𝕋.

Theorem 1.1. Suppose that 2 ≤ p < +∞, f ∈ θ^{∗}(H^{p}), and let f have the expansion f(z) = ∑_{n≥0} a_{n}z^{n} for z in 𝔻. The following statements hold.

1) The partial sums of the series ∑ a_{n}z^{n} are bounded in the L^{p}(σ_{α})-norm by C∥f∥_{H^{p}}, where the constant C depends only on p.

2) The series ∑ a_{n}z^{n} converges in L^{p}(σ_{α}) to the function U_{α}f.

Proof. We note that the k-th partial sum of the series ∑ a_{n}z^{n} coincides a.e. [σ_{α}] with U_{α}f −z^{k+1}U_{α}S^{∗k}f. This fact is obvious if f ∈ C(𝕋), because the map U_{α} takes continuous functions to themselves. But if f ∉ C(𝕋), one can choose a sequence {f_{n}}_{n≥0}, f_{n} ∈ C(𝕋) ∩ θ^{∗}(H^{p}), for which f_{n} → f in H^{p} and apply a limit argument. By Theorem 3 and its Corollary 2, there is a constant C_{0} = C_{0}(p) such that

∥z^{k+1}U_{α}S^{∗k}f∥_{L^{p}(σ_{α})} = ∥U_{α}S^{∗k}f∥_{L^{p}(σ_{α})} ≤ C_{0}∥S^{∗k}f∥_{H^{p}} ⟶ 0. □

Corollary. Let f ∈ θ^{∗}(H^{p}), p ≥ 2, f_{r}(z) = f(rz), 0 < r < 1. Then 1) f_{r} ⟶ U_{α}f in L^{p}(σ_{α}) as r → 1^{−}.

2) ∥f_{r}∥_{L^{p}(σ_{α})} ≤ C∥f∥_{H^{p}}, where C = C(p).

Proof. It suffices to apply the Abel–Poisson summation method to the power series of f and then to use Theorem 1.1. □

Remark 1. In A. B. Aleksandrov’s paper [3] it was shown that the inclusion map of θ^{∗}(H^{p}) into θ^{∗}(H^{q}) is compact if p > q ≥ 1. It can be seen from the proof of Theorem 1.1 that this statement yields an estimate, in terms of p and θ only, for the rate of convergence of the series ∑ a_{n}z^{n} = f(z) (f ∈ θ^{∗}(H^{p})) and of the functions f_{r} in L^{p}(σ_{α}).

Remark 2. The restriction p ≥ 2 cannot be dropped in the statements of the theorem and its corollary. This can be seen from the following example.

Let us take an inner function θ with the following properties:

1) \frac{θ(z)−θ(1)}{z−1} ∈ H^{2} and

2) \frac{θ(z)−θ(1)}{z−1} ∉ H^{p} for every p > 2.

Such a function can be constructed, e.g., with the help of Theorem 3.3 in [3]. In the same paper it was pointed out that 1) implies σ_{α}({1}) ≠ 0 for α = θ(1), and 2) implies that the point evaluation at 1 is a discontinuous functional on θ^{∗}(H^{p}) for p < 2. Therefore, for any p < 2 there is a function f ∈ θ^{∗}(H^{p}) for which the limit

lim_{r→1^{−}} f(r)

does not exist. So the functions f_{r} also do not converge as r → 1^{−}. Nor does the power series of f converge in L^{p}(σ_{α}).

The above example also shows that for p < 2 the functions f_{r} do not necessarily converge a.e. [σ_{α}]. However, soon we shall see that they do converge for p ≥ 2. In order to treat the problem of convergence almost everywhere, we have now some preparatory work to be done.

We say that z tends to ξ, ξ ∈ 𝕋, nontangentially (and write z →_{≮} ξ) if z tends to ξ from

inside the region Δ_{ξ}^{φ},

Δ_{ξ}^{φ} =^{def} 𝔻 ∩ \{ z | | arg(1 − z\overline{ξ})| < φ \}, where φ ∈ (0, \frac{π}{2} )

(from now on we denote by arg the principal branch of the argument, so its values lie in (−π, π]).

Let μ ∈ M(𝕋), f ∈ L^{1}(μ). A point ξ ∈ 𝕋 will be called a Lebesgue point of f with respect to the measure μ if

\frac{1}{2h} ∫_{I(ξ,h)} |f(ξ) − f(z)| dμ(z) ⟶ 0 as h → 0,

where I(ξ, h) is the subarc of 𝕋 centered at ξ whose length is 2h.

Lemma 1.2. Let μ ∈ M(𝕋), f ∈ L^{1}(μ). Then for |μ|-almost all ξ ∈ 𝕋

P (fμ)/P μ ⟶_{z→_{≮}ξ} f(ξ).

Proof. First we consider the case μ ≥ 0. Let ξ be a Lebesgue point of f with respect to the measure μ. We may assume f ≥ 0 and f(ξ) = 0. Standard arguments involving the definition of a Lebesgue point and the properties of the Poisson kernel show that in this case

(P (fμ)/P μ)(rξ) ⟶_{r→1^{−}} 0.

The passage from a radius to a sector will require some estimates similar to those that occur in Harnack’s lemma. We apply these estimates in the same fashion as it is done in the well-known proof of a generalized version of Fatou’s theorem (see, e.g., [4, Chapter I, Section D]). Note that for any z ∈ 𝔻 and ξ ∈ 𝕋 we have

(1)  \frac{1 − |z|}{1 + |z|} ≤ \frac{1 − |z|^{2}}{|z − ξ|^{2}} ≤ \frac{1 + |z|}{1 − |z|}.

Integrating (1) with respect to an arbitrary measure from M_{+}(𝕋), we obtain for an arbitrary nonnegative harmonic function u on 𝔻

(2)  u(0)\frac{1 − |z|}{1 + |z|} ≤ u(z) ≤ \frac{1 + |z|}{1 − |z|}u(0)  (z ∈ 𝔻).

Further, let z tend to ξ from inside the sector Δ_{ξ}^{φ_{1}}, φ_{1} ∈ (0, π/2). We choose φ_{2} between φ_{1} and π/2. Let z_{0} be a point on the radius terminating in ξ (z_{0} = rξ, 0 ≤ r < 1). Denote by 𝔻_{1} and 𝔻_{2} the disks centered at z_{0} and inscribed into Δ_{ξ}^{φ_{1}} and Δ_{ξ}^{φ_{2}}, respectively. The ratio γ (γ < 1) of the radii of 𝔻_{1} and 𝔻_{2} does not depend on z_{0}. Since the restrictions of P (fμ) and P μ to 𝔻_{2} are positive harmonic functions, it follows from (2) that for all z in 𝔻_{1}

0 ≤ (P (fμ)/P μ)(z) ≤ \frac{\frac{1+γ}{1−γ} (P (fμ))}{\frac{1−γ}{1+γ} (P μ)}(z_{0}) ≤ (\frac{1 + γ}{1 − γ} )^{2}(\frac{P (fμ)}{P μ})(z_{0}).

Letting z_{0} tend to ξ along the radius and applying Lebesgue’s theorem we come to the desired conclusion.

Now let μ be an arbitrary measure from M(𝕋), and let g be a μ-measurable function such that μ = g|μ|. The statement of the lemma now follows from the relation

P (fμ)/P μ = \frac{P (fg|μ|)}{P (|μ|)} · \frac{P |μ|}{P (g|μ|)},

since |g| = 1 ≠ 0 a.e. [μ]. □

Lemma 1.3. Let μ and ν be measures in M(𝕋), μ ⊥ ν. Then for ν-almost all ξ ∈ 𝕋 we have

(P μ/P ν)(z) ⟶_{z→_{≮}ξ} 0.

Proof. We consider the case where ν ≥ 0 and μ ≥ 0. Let f be a function defined on 𝕋 such that f = 0 a.e. [μ] and f = 1 a.e. [ν]. Now by Lemma 1.2 we have for ν-almost all ξ

lim_{z→_{≮}ξ} \frac{P (f(μ + ν))}{P (μ + ν)} = lim_{z→_{≮}ξ} \frac{P ν}{P (μ + ν)} = 1.

Consequently,

lim_{z→_{≮}ξ} \frac{P μ}{P ν} = lim_{z→_{≮}ξ} \frac{P (μ + ν)}{P ν} − 1 = 0

for ν-almost all ξ.

The passage to the case of arbitrary μ and ν is similar to that in the previous proof. □

Consider the space θ^{∗}(H_{0}^{p}) =^{def} { f ∈ θ^{∗}(H^{p}) | f(0) = 0 }. For f ∈ θ^{∗}(H^{p}) we denote the function θ\overline{f} by f̃. The map f ↦ f̃ is an involution on the space θ^{∗}(H_{0}^{p}). In what follows a function f in θ^{∗}(H^{p}) will be called a Hermitian element if f̃ = f.

Some of the proofs of the theorems below rely on the following nice property of Hermitian elements: if f ∈ θ^{∗}(H^{p}), p ≥ 2, and f = f̃, then arg U_{α}f = \frac{arg α}{2} a.e. [σ_{α}]. In particular, the function U_{1}f is real a.e. [σ_{1}], and U_{−1}f is purely imaginary a.e. [σ_{−1}].

Theorem 1.4. Let f ∈ θ^{∗}(H^{p}), p ≥ 2, α ∈ 𝕋. Then for σ_{α}-almost all ξ ∈ 𝕋

f(z) ⟶_{z→_{≮}ξ} (U_{α}f)(ξ).

Proof. Assuming that f(0) = 0, we represent f as the sum

f = \frac{f + f̃}{2} + \frac{f − f̃}{2}.

Since both f + f̃ and i(f − f̃) are Hermitian elements, it suffices to prove the theorem in the case of Hermitian f.

In what follows we write f instead of U_{α}f if this leads to no confusion.

Now let f be a Hermitian element, α = 1, and let ξ be such that

(A) P (fσ_{1})/P σ_{1} ⟶ f(ξ) as z →_{≮} ξ,

(B) (P (f^{2}σ_{−1})/P σ_{1})(z) ⟶ 0 as z →_{≮} ξ,

(C) (1 − θ(z)) ⟶ 0 as z →_{≮} ξ.

(Lemmas 1.2, 1.3 and the definition of σ_{1} show that σ_{1}-almost all ξ’s enjoy the above properties.)

We note that

P (fσ_{1}) = 2 Re(fσ_{1})_{+} = 2 Re(\frac{1 + θ}{1 − θ} (fσ_{−1})_{+})

= 2(Re(\frac{1 + θ}{1 − θ} ) Re((fσ_{−1})_{+}) − Im(\frac{1 + θ}{1 − θ} ) Im((fσ_{−1})_{+}))

= i(P σ_{1}Q(fσ_{−1}) − Qσ_{1}P (fσ_{−1})),

because f is a Hermitian element. Therefore, using condition (A) we get

(1)  i(Q(fσ_{−1}) − \frac{Qσ_{1} · P (fσ_{−1})}{P σ_{1}}) ⟶_{z→_{≮}ξ} f(ξ).

Next, by Hölder’s inequality we have

|P (fσ_{−1})| ≤ \sqrt{P (σ_{−1})|P (f^{2}σ_{−1})|}.

In view of condition (B) imposed on ξ, it follows that

(2)  |P (fσ_{−1})| = o(\sqrt{P σ_{−1}P σ_{1}}) = o(\frac{1 − |θ|^{2}}{|1 − θ^{2}|}).

Now we are able to estimate the second summand on the left-hand side of (1):

\frac{Qσ_{1} · P (fσ_{−1})}{P σ_{1}} = o(Im(\frac{1 + θ}{1 − θ} ) · \frac{|1 − θ|^{2}}{|1 − θ^{2}|}) = o(\frac{Im(1 − θ)}{|1 − θ^{2}|}).

¿From condition (C), imposed on the point ξ at the beginning of the proof, we derive Im(1 − θ)/|1 − θ^{2}| = O(1) as →_{≮} ξ, and so (1) yields

iQ(fσ_{−1})(z) ⟶_{z→_{≮}ξ} f(ξ).

Furthermore, (2) implies P (fσ_{−1}) = o(1). It remains to notice that ∫ f dσ_{−1} = f(0) = 0, and so

\begin{aligned}f(z) & = (1 + θ(z)) · (fσ_{−1})_{+}(z) \\ & = (1 + θ(z))(\frac{1}{2}P (fσ_{−1}) + \frac{i}{2}Q(fσ_{−1}))(z) ⟶_{z→_{≮}ξ} f(ξ). □ \end{aligned}

The rest of this section is devoted to the properties of maximal functions corresponding to elements of the space θ^{∗}(H^{p}) with p ≥ 2.

Let f ∈ L^{1}(μ), μ ∈ M(𝕋). By f_{μ}^{M} we denote the Hardy–Littlewood maximal function:

f_{μ}^{M}(ξ) =^{def} sup_{ε>0} \frac{1}{μ(I(ξ, ε))} ∫_{I(ξ,ε)} |f| dμ,

where ξ ∈ 𝕋 and I(ξ, ε) is the subarc of 𝕋 of length 2ε and centered at ξ.

In what follows we make use of the so-called Luzin–Privalov “Ice-cream cone construction” (cf. [4]). Let ξ ∈ 𝕋 and φ ∈ (0, π/2). Into the sector Δ_{ξ}^{φ} we inscribe a circle 𝕋_{φ} centered at the origin. The two points at which 𝕋_{φ} is tangent to the rays { z | arg(1 − zξ) = ±φ } divide the circle 𝕋_{φ} into two arcs. We denote by Γ_{ξ}^{φ} the open region bounded by the shorter arc and the rays \{ z | arg(1 − z\overline{ξ}) = ±φ \}. In other words, of the three parts into which 𝕋_{φ} divides Δ_{ξ}^{φ}, Γ_{ξ}^{φ} is the one adjacent to the point ξ.

Let E ⊂ 𝕋. By Γ_{E}^{φ} we denote the set

⋃_{ξ∈E} Γ_{ξ}^{φ}.

For a function f ∈ H^{p} we introduce its nontangential maximal function. Letting φ ∈ (0, π/2) we set

f_{φ}^{∗}(ξ) =^{def} sup_{z∈Γ_{ξ}^{φ}} |f(z)|.

Before proceeding with the “maximal theorem” 1.6, we establish an auxiliary statement.

Lemma 1.5. Let μ, ν ∈ M_{+}(𝕋), f ∈ L^{1}(μ + ν), φ ∈ (0, π/2). Then there is a positive constant C such that for ν-almost all ξ ∈ 𝕋 and for all z in Γ_{ξ}^{φ} we have

1) |P (fν)| ≤ Cf_{ν}^{M}P ν,

2) |P (fμ)| ≤ Cf_{μ+ν}^{M}P (μ + ν).

Proof. We may assume that f ≥ 0. A standard argument shows that inequality 1) holds for all points z on the radius terminating at a Lebesgue point of f with respect to ν. In order to extend inequality 1) from the radius to the sector Γ_{ξ}^{φ}, one should reproduce the argument that was used in the proof of Lemma 1.2 for similar purposes.

To deduce 2) from 1), one should substitute μ + ν for ν in 1) and recall that both f and ν are positive. □

Now everything is ready to prove the maximal theorem.

Theorem 1.6. Assume that α ∈ 𝕋, φ ∈ (0, π/2), f ∈ θ^{∗}(H^{p}), p ≥ 2. If p > 2, then f_{φ}^{∗} ∈ L^{p}(σ_{α}). If p = 2, then f_{φ}^{∗} belongs to the “weak L^{2}”, i.e.,

σ_{α}(\{ ξ ∈ 𝕋 | f_{φ}^{∗}(ξ) > λ \}) < \frac{c}{λ^{2}} ,

where c is a constant.

Proof. The crucial estimates will be similar to the estimates in the proof of the pointwise convergence.

Since (h + g)_{φ}^{∗} ≤ h_{φ}^{∗} + g_{φ}^{∗}, it follows that we can once again confine ourselves to the case where f is a Hermitian element.

Now, by Lemma 1.5

|P (fσ_{1})/P σ_{1}| ≤ Cf_{σ_{1}}^{M}.

¿From this we derive (like we did in the proof of Theorem 1.4) that

(1)  |Q(fσ_{−1}) − Qσ_{1}P (fσ_{−1})/P σ_{1}| ≤ Cf_{σ_{1}}^{M}.

Hölder’s inequality gives

|P (fσ_{−1})| ≤ \sqrt{P (σ_{−1})|P (f^{2}σ_{−1})|}.

Furthermore, by Lemma 1.5

|P (f^{2}σ_{−1})| ≤ C(f^{2})_{σ_{1}+σ_{−1}}^{M} · P (σ_{1} + σ_{−1})

whence

\begin{aligned}|P (fσ_{−1})| & ≤ \sqrt{C(f^{2})_{σ_{1}+σ_{−1}}^{M} · P (σ_{1} + σ_{−1}) · P (σ_{−1})} \\ & ≤ C_{1} \frac{1 − |θ|^{2}}{|1 − θ^{2}||1 + θ|}\sqrt{(f^{2})_{σ_{1}+σ_{−1}}^{M}}. \end{aligned} (2)

Next, we use (2) to estimate the second summand on the left-hand side of (1):

\begin{aligned}|Qσ_{1} · P (fσ_{−1})/P σ_{1}| & ≤ C_{1}|Im(\frac{1 + θ}{1 − θ} )| · \frac{|1 − θ|^{2}}{|1 − θ^{2}||1 + θ|}\sqrt{(f^{2})_{σ_{1}+σ_{−1}}^{M}} \\ & ≤ \frac{C_{2}}{|1 + θ|} · \sqrt{(f^{2})_{σ_{1}+σ_{−1}}^{M}}. \end{aligned} (3)

Since f is a Hermitian element, we have f(0) = ∫ f dσ_{−1} = 0, and so (1), (2), and (3) together yield

|f(z)| = \frac{1}{2}|(1 + θ(z))(P (fσ_{−1}) + iQ(fσ_{−1}))(z)| ≤ Cf_{σ_{1}}^{M} + \frac{C_{1} + C_{2}}{2} \sqrt{(f^{2})_{σ_{1}+σ_{−1}}^{M}}.

The fact that f belongs to both L^{p}(σ_{1}) and L^{p}(σ_{−1}), combined with the well-known Hardy–Littlewood maximal theorem, implies that in the case p > 2 the two summands in the last expression belong to L^{p}(σ_{1}), whereas in the case p = 2 the first summand is in L^{2}(σ_{1}) and the second one is in the “weak L^{2}(σ_{1})”. □

Remark. From the proof one sees that the norm of the maximal function f_{φ}^{∗} can be majorized by the quantity C∥f∥_{H^{p}}, where C depends only on p and φ.

§2. Boundary properties of functions lying in the range of the operator V_{μ}

In this section we generalize the results of §1 to the case of an arbitrary measure μ ∈ M(𝕋) instead of the singular measure σ_{α}. The space θ^{∗}(H^{p}) will now be replaced by the image of L^{p}(μ) under the map V_{μ}: L^{p}(μ) ⟶ H^{p}(𝔻).

Let R denote the operator from L^{p}(μ) to L^{p}(μ) defined by the formula Rf = \overline{z} · (f − ∫_{𝕋} f d\frac{μ}{∥μ∥}).

Lemma 2.1. Let μ ∈ M_{+}(𝕋), f ∈ L^{1}(μ), F = V_{μ}f, and let ∑_{n≥0} a_{n}z^{n} be the Fourier series of F. Then for each k ∈ ℤ, k ≥ 0 we have

(1)  ∑_{n=0}^{k} a_{n}z^{n} = f − z^{k+1}R^{k+1}f

a.e. [μ] on 𝕋.

Proof. We may assume that μ is a probability measure. Set g = f − ∫ f dμ. We note that ∫ g dμ = 0. We also have

(\frac{g(ξ)}{ξ} · μ)_{+}(z) = ∫ \frac{1}{1 − z\overline{ξ}} \frac{g(ξ)}{ξ} dμ(ξ)

= (∫ \frac{1}{1 − z\overline{ξ}} g(ξ) dμ(ξ) − ∫ \frac{1 − z\overline{ξ}}{1 − z\overline{ξ}} g(ξ) dμ(ξ))/z

= \frac{g(μ)_{+}(z)}{z}.

Consequently,

V_{μ}Rf = \frac{((Rf)_{μ})_{+}}{μ_{+}} = \frac{(\frac{g(ξ)}{ξ}μ)_{+}}{μ_{+}} = (V_{μ}f − (V_{μ}f)(0))/z = S^{∗}V_{μ}f.

Thus, for all k ∈ ℤ, k ≥ 0 we have V_{μ}R^{k}f = S^{∗k}F. Now we prove the statement of the lemma by induction. For k = 0, (1) obviously holds, since F (0) = ∫_{𝕋} f dμ. Further, if

∑_{n=0}^{k} a_{n}z^{n} = f − z^{k+1}R^{k+1}f a.e. [μ] on 𝕋,

then

∑_{n=0}^{k+1} a_{n}z^{n} − ∑_{n=0}^{k} a_{n}z^{n} + a_{k+1}z^{k+1} = (f − z^{k+1}R^{k+1}f) + z^{k+1}((S^{∗k}V_{μ}f)(0)
= f − z^{k+1}(R^{k+1}f − (V_{μ}R^{k+1}f)(0))
= f − z^{k+1}(R^{k+1}f − ∫ R^{k+1}f dμ)
= f − z^{k+2}R^{k+1}f. □

Lemma 2.2. Let μ ∈ M_{+}(𝕋), f ∈ L^{2}(μ), F = V_{μ}f, and let ∑_{n≥0} a_{n}z^{n} be the Fourier series of F. Then for each k ∈ ℤ, k ≥ 0, we have

∥ ∑_{n=0}^{k} a_{n}z^{n}∥_{L^{2}(μ)} ≤ C∥f∥_{L^{2}(μ)},

where C is an absolute constant.

Proof. It is sufficient to prove the lemma for an arbitrary continuous function f.

Consider a sequence of singular measures {μ_{n}}_{n>0}, μ_{n} ∈ M_{+}(𝕋), converging to μ in the weak^{∗} topology of the space M(𝕋). Since f is continuous, we may deal with the sequence of functions F_{n} = V_{μ_{n}}f. Let S_{k} denote the k-th partial sum of the Fourier series of F, and let S_{k}^{n} stand for the similar sum associated with F_{n}. By Theorem 1.1,

(1)  ∥S_{k}^{n}∥_{L^{2}(μ_{n})} ≤ C∥f∥_{L^{2}(μ_{n})}.

We note that F_{n} → F uniformly on compact subsets of 𝔻, and so S_{k}^{n} → S_{k} uniformly on 𝕋. It remains to let n → ∞ in (1). □

Remark. Clark’s theorem and the proof of Theorem 1.1 together imply that the constant C occurring in the statement of Lemma 2.2 can be chosen to equal 2.

Lemma 2.3. Suppose that a measure μ satisfies one of the following two conditions: (A) μ = m + fm, where f ≥ 0, f ∈ L^{1}(m); (B) μ = m + ν, where ν is a singular probability measure in M_{+}(𝕋). Then for each function g in L^{2}(μ) the Fourier series of V_{μ}g converges in L^{2}(μ).

Proof. In view of Lemma 2.2 and the Banach–Steinhaus theorem, it suffices to find a dense subset of L^{2}(μ) such that the statement holds for its elements.

Let μ satisfy (A). We shall prove that the power series of 1/μ_{+} converges in L^{2}(μ). Set μ_{r} = μ_{+}(rz), 0 < r < 1. We have

\begin{aligned}∫ |S^{∗n} \frac{1}{μ_{r}} |^{2} Re μ_{r} dm & = \frac{1}{2} ∫ S^{∗n} \frac{1}{μ_{r}} · \overline{S}^{∗n} \frac{1}{μ_{r}} (μ_{r} + \overline{μ}_{r}) dm \\ & = \frac{1}{2}(∫ \frac{z^{n}μ_{r}}{\overline{μ}_{r}} S^{∗n} \frac{1}{μ_{r}} dm + ∫ \frac{\overline{z^{n}μ_{r}}}{μ_{r}} \overline{S^{∗n} \frac{1}{μ_{r}}} dm) \\ & = Re ∫ S^{∗n} \frac{1}{μ_{r}} · \frac{z^{n}μ_{r}}{\overline{μ}_{r}} dm. \end{aligned} (1)

Now, since 1/μ_{+} ∈ H^{∞}, the functions |S^{∗n} \frac{1}{μ_{r}} |^{2} are bounded by a constant independent

of r, and they converge to |S^{∗n} \frac{1}{μ_{r}} |^{2} as r → 1^{−} a.e. [m]. Thus, by the Lebesgue dominated

convergence theorem, |S^{∗n} \frac{1}{μ_{r}} |^{2} tends to |S^{∗n} \frac{1}{μ_{+}} |^{2} as r → 1^{−} in the weak^{∗} topology of L^{∞}(m). Besides, Re μ_{r} ⟶ Re μ_{+} in L^{1}(m). Consequently,

(2)  ∫ |S^{∗n} \frac{1}{μ_{r}} |^{2} Re μ_{r} dm ⟶_{r→1−} ∫ |S^{∗n} \frac{1}{μ_{r}} |^{2} Re μ_{r} dm.

Since μ_{r} ⟶_{r→1^{−}} μ_{+} a.e. [m] and 1/μ_{+} ∈ H^{∞}, we have

(3)  Re ∫ S^{∗n} \frac{1}{μ_{r}} · \frac{z^{n}μ_{r}}{\overline{μ}_{r}} dm ⟶_{r→1−} Re ∫ S^{∗n} \frac{1}{μ_{+}} · \frac{z^{n}μ_{+}}{\overline{μ}_{+}} dm.

Now (1), (2), (3), and the properties of S^{∗} yield

∫ |S^{∗n} \frac{1}{μ_{+}} |^{2} Re μ_{+} dm = Re ∫ S^{∗n} \frac{1}{μ_{+}} \frac{z^{n}μ_{+}}{\overline{μ}_{+}} dm ⟶_{n→∞} 0.

Therefore, the power series of 1/μ_{+} converges in L^{2}(μ). We set h = P/f + 1, where P is a polynomial in z and \overline{z}. Then V_{μ}h = P_{1} + P_{2}/μ_{+}, where P_{1} and P_{2} are polynomials in z. Thus, the series of V_{μ}h converges in L^{2}(μ). In view of the remark made at the beginning of the proof, it remains to observe that the functions h of the above form are dense in L^{2}(μ).

Let μ satisfy (B). By θ we denote the inner function for which ν_{+} = 1/(1 − θ). Let P_{1} be a polynomial in z, P_{2} a polynomial in \overline{z}, and h_{0} a function in L^{2}(ν). Consider a function h in L^{2}(μ) defined by

h = \{ \begin{matrix}h_{0} & a.e. [ν], \\ (2 − θ)P_{1} + P_{2} + V_{ν}h_{0} & a.e. [m]. \end{matrix}

We have

V_{μ}h = \frac{(1 − θ)((2 − θ)P_{1} + V_{ν}h_{0} + V_{ν}h_{0}/(1 − θ))}{2 − θ} = (1 − θ)P_{1} + V_{ν}h_{0}.

By Theorem 1.1, the Fourier series of θ and V_{ν}h_{0} converge in L^{2}(ν); they also converge in L^{2}(m) because both θ and V_{ν}h_{0} are in H^{2}. Finally, the functions h under consideration form a dense subset of L^{2}(μ). □

Lemma 2.4. Let μ ∈ M_{+}(𝕋). Then the Fourier series of 1/μ_{+} converges in L^{2}(μ).

Proof. The measure μ can be written in the form μ = fm + σ, where f ≥ 0, f ∈ L^{1}(m), and σ is a singular measure in M_{+}(𝕋). We may assume that σ is a probability measure. Now we shall show that the function 1/μ_{+} lies both in the range of V_{(f+1)m} and in the range of V_{σ+m}.

Since |μ_{+}| ≥ 1/2, the function 1/μ_{+} is bounded. By Theorem 3, (fm)_{+}/μ_{+} is in H^{2}. Consequently,

\frac{(fm)_{+} + 1}{μ_{+}(f + 1)} ∈ L^{2}((f + 1)m),

and so

V_{(f+1)m}(\frac{(fm)_{+} + 1}{μ_{+}(f + 1)}) = \frac{((f + 1)m)_{+}}{μ_{+}} /((f + 1)m)_{+} = \frac{1}{μ_{+}} .

Further, consider a function h in L^{2}(σ + m) such that h = 0 a.e. [σ] and h = (σ_{+} + 1)/μ_{+} a.e. [m]. (We note that h ∈ L^{2}(m) because 1/μ_{+} is bounded and σ_{+}/μ_{+} ∈ H^{2} by Theorem 3). We have

V_{σ+m}(h) = \frac{(σ_{+} + 1)}{μ_{+}}/(σ_{+} + 1) = \frac{1}{μ_{+}} .

Now the desired conclusion follows from Lemma 2.3. □

The next theorem is a generalization of Theorem 1.1 in the case p = 2.

Theorem 2.5. Let μ ∈ M_{+}(𝕋) and f ∈ L^{2}(μ). Then the Fourier series of V_{μ}f converges in L^{2}(μ).

Proof. By Lemma 2.2 and the Banach–Steinhaus theorem, it again suffices to find a dense subset of L^{2}(μ) for which the statement is valid.

First we show that if the series of V_{μ}f converges in L^{2}(μ), then the same is true for V_{μ}(zf) and V_{μ}(\overline{z}f). Computations similar to those in the proof of Lemma 2.1 show that

(V_{μ}(zf))(ξ) = ∫_{𝕋} zf(z) dμ(z)/μ_{+}(ξ) + ξ(V_{μ}f)(ξ),
(V_{μ}(\overline{z}f))(ξ) = \overline{ξ}((V_{μ}f)(ξ) − ∫_{𝕋} f dμ/m_{+}(ξ)).

The convergence of the series of V_{μ}(zf) and V_{μ}(\overline{z}f) now follows by Lemma 2.4.

Since for c ∈ ℂ one has V_{μ}c = c, the polynomials in z and \overline{z} form a dense subset with the required property. □

Corollary. Under the assumptions of the theorem, we set

F_{r} = (V_{μ}f)(rz), r ∈ [0, 1).

Then the F_{r}’s converge in L^{2}(μ) as r → 1^{−}.

Remark 1. Clearly, in the theorem (respectively, in its corollary) it is impossible to guarantee that the series converges to f (respectively, that the functions F_{r} converge to f).

However, the sum of the series and the limit of F_{r} do coincide with f almost everywhere with respect to the singular component of the measure μ. This is a consequence of Theorem 2.7 below.

Remark 2. In the statement of the theorem, one cannot replace L^{2} by L^{p}, p ≠ 2. This can be seen from the following example.

Let σ be a singular probability measure in M_{+}(𝕋), and let p > 2. If σ is not discrete, then the corollary to Theorem 4 implies the existence of a function f in L^{p}(σ) such that V_{σ}f ∉ H^{p}. Consider the measure μ = m + σ and the function g ∈ L^{p}(μ) which equals 0 a.e. [m] and equals f a.e. [σ]. We denote by θ the inner function that equals 1 − 1/σ_{+}. Then V_{μ}g = V_{σ}f · (2 − θ) ∉ H^{p}. Consequently, the Fourier series of V_{μ}g does not converge in L^{p}(m), so not in L^{p}(μ) either. Moreover, the L^{p}(μ)-norms of the partial sums S_{n} of that series tend to infinity. By Lemma 2.1, S_{n} = g − z^{n+1}R^{n}g a.e. [μ] on 𝕋. Therefore, ∥R^{n}∥ → ∞ as n → ∞.

Consider the adjoint operator R^{∗}: L^{q}(μ) ⟶ L^{q}(μ), where q = \frac{p}{p−1}. From the definition of R one easily derives that R^{∗} is given by the formula R^{∗}f = zf − ∫ zf d\frac{μ}{∥μ∥}. Since ∥R^{∗n}∥ ⟶ ∞ as n → ∞, the Banach–Steinhaus theorem implies that for some function h ∈ L^{q}(μ) the norms ∥R^{∗n}h∥_{L^{q}(μ)} are not uniformly bounded. Now for the powers of R and R^{∗} we have the relation R^{∗n}f = (\overline{zR^{n}(\overline{zf})}) holding for all f ∈ L^{q}(μ). It follows that the norms ∥R^{n}(\overline{zh})∥_{L^{q}(μ)} are not bounded, and so the Fourier series of V_{μ}(\overline{zh}) does not converge in L^{q}(μ).

Now we are going to focus on the problem concerning nontangential limits of functions lying in the range of the map V_{μ}. This problem was treated in Section 1 for singular measures μ ∈ M_{+}(𝕋). In the proof of Theorem 2.7 below we reduce the general case to that special setting with the help of the following statement.

Lemma 2.6. Assume that I is an open subarc of the circle 𝕋, J = 𝕋 \ I, μ ∈ M_{+}(𝕋), μ(J) = 0, f ∈ L^{∞}(μ), φ ∈ (0, π/2). Then for each ε > 0 there exists a discrete measure μ_{0} and a function f_{0} ∈ L^{∞}(μ_{0}) such that ∥μ_{0}∥ ≤ ∥μ∥, ∥f_{0}∥_{L^{∞}(μ_{0})} ≤ ∥f∥_{L^{∞}(μ)}, and for all z in Γ_{J}^{φ} one has

(1)  |μ_{+}(z) − (μ_{0})_{+}(z)| < ε,
(2)  |(fμ)_{+}(z) − (f_{0}μ_{0})_{+}(z)| < ε.

Proof. We may assume that ∂I = {e^{iα}, e^{−iα}}, where α ∈ (0, π] (if α = 0, then μ itself is discrete). We shall show how to partition the arc I into suitable subarcs I_{k}, k ∈ ℤ, so that the desired measure μ_{0} could be obtained by replacing μ with a point mass on each of these subarcs.

For each ε_{0} > 0 one can find a sequence {α_{n}}_{n≥0} of numbers in [0, α) with the following properties:

1) α_{0} = 0,

2) α_{n+1} > α_{n} (n = 0, 1, 2, …),

3) μ({e^{iα_{n}}}) = 0 (n = 0, 1, 2, …),

4) α_{n} ⟶ α as n → ∞, 5) for each k ∈ ℕ

sup_{z∈Γ_{J}^{φ}} |\frac{1}{1 − ze^{−iα_{k−1}}} − \frac{1}{1 − ze^{−iα_{k}}} | < ε_{0}.

We construct such a sequence for a suitable sufficiently small ε_{0} (the choice of ε_{0} will be specified at the end of the proof). Denote by I_{k}, k ∈ ℕ, the arc between e^{iα_{k−1}} and e^{iα_{k}} contained in I. By I_{−k} we denote the arc symmetric to I_{k} with respect to the real axis: I_{−k} = \{ ξ | \overline{ξ} ∈ I_{k} \}. Now let μ_{0} be the measure having the point mass μ(I_{k}) at each e^{iα_{k}} and the point mass μ(I_{−k}) at each e^{−iα_{k}}, k ∈ ℕ. We define the function f_{0} by letting it equal \frac{1}{μ(I_{k})} ∫_{I_{k}} f dμ at e^{iα_{k}} and \frac{1}{μ(I_{−k})} ∫_{I_{−k}} f dμ at e^{−iα_{k}} (k ∈ ℕ).

The conditions ∥μ_{0}∥ ≤ ∥μ∥ and ∥f∥ ≤ ∥f_{0}∥ are easily verified. Further, for z ∈ Γ_{J}^{φ} we have

|μ_{+}(z) − (μ_{0})_{+}(z)| ≤ ∑_{k∈ℤ\\{0\}} |μ(I_{k})| · sup_{ξ∈I_{k}} |\frac{1}{1 − z\overline{ξ}} − \frac{1}{1 − ze^{−iα_{k}}} | ≤ C∥μ∥ε_{0},

where the constant C depends only on φ. Thus, taking ε_{0} smaller than ε/C∥μ∥, we get (1). Writing the same estimates for fμ and f_{0}μ_{0} instead of μ and μ_{0}, we arrive at (2). □

Theorem 2.7. Let μ ∈ M(𝕋) and f ∈ L^{1}(μ). Denote by F the function V_{μ}f, which is meromorphic in 𝔻. Then for μ-almost all ξ ∈ 𝕋 the limit

lim_{z→_{≮}ξ} F (z)

exists, and it equals f(ξ) almost everywhere with respect to the singular component of the measure μ.

Proof. A) First we consider the case where μ ≥ 0 and f ∈ L^{∞}(μ). We denote the singular component of μ by σ. Since μ ≥ 0, Theorem 3 yields F ∈ H^{2}, and so the nontangential limits of F exist a.e. [m]. It remains to prove that they also exist and equal f a.e. [σ]. Let a subset E ⊂ 𝕋 be such that σ(E) = ∥σ∥, m(E) = 0. Since σ is a regular measure, there exist closed sets E_{1}, E_{2}, … with ⋃_{n∈ℕ} E_{n} = E. Let n be a fixed positive integer. The set 𝕋 \ E_{n} can be written as the union of a countable collection of pairwise disjoint open arcs I_{1}, I_{2}, . . .. By Lemma 2.6, for every ε > 0 and φ ∈ (0, π/2) one can find a discrete measure μ_{k}^{ε} supported on I_{k} and a function f_{k}^{ε} ∈ L^{∞}(μ_{k}^{ε}) such that ∥μ_{k}^{ε}∥ ≤ μ(I_{k}), ∥f_{k}^{ε}∥_{L^{∞}(μ_{k}^{ε})} ≤ ∥f∥_{L^{∞}(μ)} and for every z ∈ Γ_{𝕋\I_{k}}^{φ} one has

| ∫_{I_{k}} \frac{1}{1 − z\overline{ξ}} dμ − ∫_{I_{k}} \frac{1}{1 − z\overline{ξ}} dμ_{k}^{ε}| < \frac{ε}{2^{k}}

and

| ∫_{I_{k}} \frac{1}{1 − z\overline{ξ}} f dμ − ∫_{I_{k}} \frac{1}{1 − z\overline{ξ}} f_{k}^{ε} dμ_{k}^{ε}| < \frac{ε}{2^{k}} .

Let χ_{E_{n}} denote the characteristic function of the set E_{n}. We set μ_{ε} = ∑_{k∈ℕ} μ_{k}^{ε} +χ_{E_{n}} ·σ. Let a function f_{ε} be equal to f_{k}^{ε} a.e. [μ_{k}^{ε}] for all k ∈ ℕ, and to f a.e. [σ] on E_{n}. Then we have ∥μ_{ε}∥ ≤ ∥μ∥, ∥f_{ε}∥_{L^{∞}(μ_{ε})} ≤ ∥f∥_{L^{∞}(μ)}. Setting F_{ε} = V_{μ_{ε}}f_{ε}, we see that the following relations hold on Γ_{E_{n}}^{φ}:

\begin{aligned}|F − F_{ε}| & ≤ |\frac{(fμ)_{+}}{μ_{+}} − \frac{(f_{ε}μ_{ε})}{μ_{+}} | + |\frac{(f_{ε}μ_{ε})_{+}}{μ_{+}} − \frac{(f_{ε}μ_{ε})_{+}}{(μ_{ε})_{+}} | \\ & = \frac{1}{|μ_{+}|}(|(f_{μ})_{+} − (f_{ε}μ_{ε})_{+}| + |F_{ε}||(μ_{ε})_{+} − μ_{+}|) \\ & ≤ \frac{2ε}{∥μ∥}(1 + |F_{ε}|). \end{aligned} (1)

By Theorem 1.6, the maximal function (F_{ε})_{φ}^{∗} is finite a.e. [μ_{ε}], and consequently also a.e. [μ] on E_{n}. From (1) it follows that F_{φ}^{∗} is also finite a.e. [μ] on E_{n}. Let {ε_{k}}_{k≥0} be a sequence of positive numbers tending to zero. For each ε_{k}, we construct a measure μ_{ε_{k}} and functions f_{ε_{k}} and F_{ε_{k}} in the same way as μ_{ε}, f_{ε}, and F_{ε} were constructed for ε. Let a point ξ in E_{n} be such that for all k ∈ ℕ

lim_{z→_{≮}ξ} F_{ε_{k}}(z) = f(ξ)

and F_{φ}^{∗} < +∞ ( μ-almost all points in E_{n} enjoy these properties). Then for each z ∈ Γ_{ξ}^{φ} we have

(2) \begin{aligned}|F (z) − F_{ε_{k}}(z)| & ≤ |\frac{(fμ)_{+}}{μ_{+}} − \frac{(fμ)_{+}}{(μ_{ε_{k}})_{+}} | + |\frac{(fμ)_{+}}{(μ_{ε_{k}})_{+}} − \frac{(f_{ε_{k}}μ_{ε_{k}})_{+}}{(μ_{ε_{k}})_{+}} | \\ & = |F (z)||\frac{1}{(μ_{ε_{k}})_{+}} ||μ_{+} − (μ_{ε_{k}})_{+}| + \frac{1}{|(μ_{ε_{k}})_{+}|}|(fμ)_{+} − (f_{ε_{k}}μ_{ε_{k}})_{+}| \\ & ≤ F_{φ}^{∗}(ξ)\frac{4ε_{k}}{∥μ∥}. \end{aligned}

Since ε_{k} → 0 and all F_{ε_{k}} tend to f(ξ) along Γ_{ξ}^{φ}, the Stokes–Seidel theorem and (2) together imply that F also tends to f(ξ) along the set Γ_{ξ}^{φ}. Thus,

lim_{z→_{≮}ξ} F (z) = f(ξ)

for σ-almost all ξ ∈ E_{n}. Recalling that ⋃ E_{n} = E, we come to the desired conclusion.

B) Now let μ ∈ M_{+}(𝕋), f ∈ L^{1}(μ), and let σ be the singular part of μ. We may assume that f ≥ 1. By A), the limit

(3)  lim_{z→_{≮}ξ}(V_{fμ}1/f)(z)

exists a.e. [fμ] (and consequently a.e. [μ]), its value being equal to 1/f(ξ) a.e. [fσ] (and consequently a.e. [σ]). Moreover, the function V_{fμ}1/f belongs to H^{2}. Therefore, the limit (3) is nonzero a.e. [m] on 𝕋. Since f ∈ L^{1}(μ), one has 1/f ≠ 0 a.e. [σ]. Consequently, the limit (3) is also nonzero a.e. [σ], and it is equal to 1/f(ξ) for σ-almost all ξ. The statement of the theorem now follows from the relation

V_{μ}f = (V_{fμ}1/f)^{−1}.

C) Finally, let μ ∈ M(𝕋) and f ∈ L^{1}(μ). Let a function g be such that g|μ| = μ. Then |g| = 1 a.e. [μ]; the desired conclusion is now implied by B) and the fact that

V_{μ}f = V_{|μ|}fg/V_{|μ|g}. □

Corollary 1. Assume that μ, ν ∈ M(𝕋), ν is singular, μ ⊥ ν. Then

μ_{+} = o(ν_{+}) as z →_{≮} ξ

for ν-almost all ξ.

Proof. Consider a function f in L^{∞}(μ + ν) such that f = 1 a.e. [ν] and f = 0 a.e. [μ]. Now, by Theorem 2.7,

V_{μ+ν}f = \frac{ν_{+}}{μ_{+} + ν_{+}} ⟶_{z→_{≮}ξ} 1

for ν-almost all ξ. □

Corollary 1′. Let f ∈ H^{1} and let θ be an inner function. Then

(1 − θ)f ⟶ 0 as z →_{≮} ξ

for σ_{1}-almost all ξ.

Proof. Set fm = μ, σ_{1} = ν and apply Corollary 1. □

Thus, the set { ξ ∈ 𝕋 | (1 − θ)f ⟶_{z→_{≮}ξ} 0 } is nonempty, whenever f ∈ H^{1} and θ is a

nonconstant inner function.

Remark 1. Since Clark’s operator U_{α}: θ^{∗}(H^{p}) ⟶ L^{p}(σ_{α}) is unbounded for p ∈ (1; 2), the theorems of Section 1 deal with the case p ≥ 2 only. Theorem 2.7 enables us to generalize Theorem 1.4 in the following way.

Let f ∈ θ^{∗}(H^{p}) and U_{α}f ∈ L^{1}(σ_{α}) (i.e., f = V_{σ_{α}}g for some g ∈ L^{1}(σ_{α})). Then for σ_{α}-almost all ξ one has

lim_{z→_{≮}ξ} f(z) = (U_{α}f)(ξ).

Remark 2. As it was mentioned in the Introduction, for μ ∈ M_{+}(𝕋) the range of the map V_{μ} contains de Branges’s space ℌ(f) with f = 1 − \frac{∥μ∥}{μ_{+}}. Thus, the theorems proved in this section also describe the boundary behavior of functions in ℌ(f) with respect to the measure μ.

§3. Applications to multiplication and division problems for Cauchy–Stieltjes-type integrals

In this section we apply the above results to some well-known problems. The questions treated here were discussed earlier in [9, 10, 11].

Let θ be an inner function. It is well known that θ possesses a natural extension to 𝔻_{−} = ℂ \ (𝔻 ∪ 𝕋) as a meromorphic function. Given f ∈ H^{∞}, we say that f has a pseudocontinuation equal to θg, where θ is inner and g is analytic and bounded on 𝔻_{−} with g(∞) = 0, provided that for m-almost all ξ one has

lim_{r→1^{−}} f(rξ) = lim_{r→1^{+}}(θg)(rξ).

We define the Hardy class H^{p}(𝔻_{−}) of the region 𝔻_{−} as the set of all functions f analytic in 𝔻_{−} and satisfying

sup_{r>1} ∫_{𝕋} |f(rξ)|^{p} dm(ξ) < +∞.

The following theorem is due to A. B. Aleksandrov.

Theorem 3.1 ([12, 13]). Assume that f ∈ H^{∞}(𝔻) and f has a pseudocontinuation equal to θg. Let μ ∈ M(𝕋) and θμ_{+} ∈ H^{p}(𝔻_{−}) for some p > 0. Then there is a measure ν such that

fμ_{+} = ν_{+}

on ℂ \ 𝕋 (it is meant that f = θg on ℂ \ (𝔻 ∪ 𝕋)).

Now the following question arises: what is the relationship between the measures μ and ν? More precisely, when is it possible to let ν equal fμ? Some results on this matter can be found in [9], [10], and [11]. Theorem 2.7 enables us to prove the following proposition.

Theorem 3.2. Under the assumptions of the preceding theorem, for μ-almost all ξ the limit

lim_{z→_{≮}ξ} f(z) = f(ξ)

exists. The measure ν can be chosen to equal fμ:

fμ_{+} = (fμ)_{+}.

Proof. From I. I. Privalov’s results [14] it follows that the absolutely continuous component ν_{a} of the measure ν equals fμ_{a} ( μ_{a} being the absolutely continuous component of the measure μ). It was shown in M. G. Goluzina’s paper [9] that ν is absolutely continuous with respect to μ. Thus, ν equals gμ, where g ∈ L^{1}(μ) and g = f a.e. [m] on 𝕋.

It remains to prove that f has angular limits almost everywhere with respect to μ_{s} (which is the singular component of μ) and these limits equal the values of g a.e. [μ_{s}].

By Theorem 2.7, the function ν_{+}/μ_{+} has angular limits a.e. [μ]; moreover, for μ_{s}-almost all ξ

(\frac{ν_{+}}{μ_{+}} )(z) = f(z) ⟶_{z→_{≮}ξ} g(ξ). □

Another problem is related to the following theorem, due to S. A. Vinogradov.

Theorem 3.3 ([15]). Let μ ∈ M(𝕋), let θ be an inner function, and let μ_{+}/θ ∈ H^{p} for some p > 0. Then there is a measure ν ∈ M(𝕋) such that μ_{+}/θ = ν_{+}.

Once again, the question of relationship between μ and ν arises. One of the ways to construct the measure ν for given μ and θ was pointed out in [10]. On the other hand, applying Theorem 2.7 we get the following result.

Theorem 3.4. Under the assumptions of the preceding theorem, the function θ has nontangential boundary values of modulus one a.e. [μ] on the circle 𝕋. The measure ν can be chosen to equal μ\overline{θ}.

Proof. It was shown in [10] that ν can be chosen to equal Iμ, where I is a function of modulus one a.e. [μ]. Now the desired conclusion follows from Theorem 3.2. □

References

1. D. N. Clark, One dimensional perturbations of restricted shifts, J. Analyse Math. 25 (1972), 169–191.

2. A. B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), no. 5, 490–503; English transl. in J. Soviet Math..

3. A. B. Aleksandrov, Inner functions and related spaces of pseudocontinuable functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 170, 1989, pp. 7–33.

4. P. Koosis, Lectures on H^{p} spaces, Cambridge Univ. Press, London, 1980.

5. J. B. Garnett, Bounded analytic functions, Academic Press, New York and London, 1981.

6. N. K. Nikol′skii, Treatise on the shift operator, Grundlehren Math. Wiss., vol. 273, Springer-Verlag, Berlin and New York, 1986.

7. D. Sarason, Function theory and de Branges’s spaces, Operator Theory and Operator Algebras, Proc. Symp. Pure Math., AMS, Providence, 1990.

8. F. Riesz and B. Sz.-Nagy, Leçons d’analyse fonctionnelle, Gauthier-Villars, Paris, 1968.

9. M. G. Goluzina, On the multiplication and division of Cauchy–Stieltjes-type integrals, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1981), no. 4, 8–15.

10. , Several remarks on the factorization of Cauchy–Stieltjes-type integrals, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1982), no. 1, 107–110.

11. S. A. Vinogradov, M. G. Goluzina, and V. P. Khavin, Multipliers and divisors of Cauchy–Stieltjestype integrals.

12. A. B. Aleksandrov, Hardy classes H^{p} with p ∈ (0, 1), Summary of Ph.D. Thesis, 1979.

13. , A-integrability of boundary values of harmonic functions, Mat. Zametki 30 (1981), no. 1, 59–72.

14. I. I. Privalov, Randeigenschaften analytischer Funktionen, Deutscher Verlag der Wissenschaften, Berlin, 1956.

15. S. A. Vinogradov, Properties of multipliers of Cauchy–Stieltjes-type integrals and some factorization problems for analytic functions, Proceedings of 7th Winter School (Drogobych, 1974), Moscow (1974), 5–39.

St. Petersburg State University, Department of Mathematics and Mechanics, Bibliotechnaya sq. 2, Staryĭ Petergof, St. Petersburg, 198904, Russia

Current address: Department of Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA

E-mail address: Aleksei@cco.caltech.edu (Aleksei Poltoratskii)