Lobachevskii Journal of Mathematics Vol. 7, 2000, 15–29

\begin{matrix}c \\ ◯ \end{matrix}B.A. Kats

B.A. Kats

THE CAUCHY INTEGRAL ALONG Φ-RECTIFIABLE CURVES

(submitted by A.M.Elizarov)

Abstract. The paper treats existence and boundary properties of the Cauchy integral over certain classes of non-rectifiable curves. These classes contain, in particular, known fractal curves: von Koch snowflakes, Weierstrass curves, lacunary wavelet trajectories and so on.

1. Introduction

Let Γ be a curve on the complex plane ℂ. The integral operator

C(Γ, f; z) = \frac{1}{2πi} ∫_{Γ} \frac{f(ζ)dζ}{ζ − z} ,  (1)

known as the Cauchy integral, is an important tool of the complex analysis. Various properties and applications of the integral (1) along rectifiable and piecewise smooth curves are described in a great body of publications.

In the present paper we study the Cauchy integral along non-rectifiable curves. This extension of the class of admissible curves seems to be of interest in connection with recent achievements of fractal analysis. In Sections 2,3,4 we are dealing with existence and properties of the integral ∫_{Γ} f(z)dg(z). Section 5 treats the existence of the integral (1). Section 6 contains the main results of the paper, namely, theorems on existence of boundary values of the Cauchy integral along non-rectifiable curves. We generalize for these curves certain known theorems on continuity of the integral (1) on Γ and describe some its boundary properties specific for non-rectifiable curves.

2. The Stieltjes integral over non-rectifiable curve

Let Γ be a simple arc on the plane ℂ with initial point a and end point b. As we call the point a the beginning and b the end, so we define an intrinsic ordering relation on the points of this curve: we write z_{1} ≺ z_{2} if z_{1,2} ∈ Γ and z_{1} precedes z_{2} on the curve Γ; the notation z_{1} ⪯ z_{2} means that either z_{1} ≺ z_{2} or z_{1} = z_{2}. Let τ = {z_{j}}_{j=0}^{j=n} be an ordered sequence of points of the curve Γ, i.e. a = z_{0} ≺ z_{1} ≺ z_{2} ≺ ⋯ ≺ z_{n} = b. We call it partition, and we set diam(τ) = max{|z_{j} − z_{j−1}| : j = 1, 2, …, n}. If, in addition, there are given points {w_{j}}_{j=1}^{j=n} such that z_{j−1} ⪯ w_{j} ⪯ z_{j}, j = 1, 2, …, n, then we say that the partition τ is pointed.

Definition 1. Let functions f(z) and g(z) be defined on a directed Jordan curve Γ. We call a value ∫_{Γ} fdg the Stieltjes integral of the function f with respect to g(z) along Γ if for any ε > 0 there exists δ > 0 such that

|∑_{j=1}^{n} f(w_{j})(g(z_{j}) − g(z_{j−1})) − ∫_{Γ} fdg| < ε

for any pointed partition satisfying the inequality diam(τ) < δ.

One can easily adapt the definition to closed directed Jordan curves, i.e. to the case a = b. In what follows we consider closed curves as oriented counterclockwise.

Let z = z(x) be a continuous one-to-one mapping of segment I = [0, 1] on the curve Γ, z(0) = a, z(1) = b. Clearly, ∑_{j=1}^{n} f(w_{j})(g(z_{j}) − g(z_{j−1})) co-

incide with the Riemann-Stieltjes sums for the customary Stieltjes integral ∫_{0}^{1} f(z(x))dg(z(x)). Therefore ∫_{Γ} fdg = ∫_{0}^{1} f(z(x))dg(z(x)), and these integrals exist (or do not exist) simultaneously. The homeomorphism z(x) : I ↦ Γ is not unique, but the value and the existence of the integral ∫_{0}^{1} f(z(x))dg(z(x)) do not depend on the choice of this mapping.

We are interested mainly in the integral ∫_{Γ} f(z)dz, the special case where g(z) is restriction of the function G(z) ≡ z onto Γ. In order to simplify notation we shall write in this situation g(z) = z.

The classical studies of the Stieltjes integral ∫_{0}^{1} fdg are based on assumption that the function g has finite variation on the segment I (see, for instance, [1]). But if z(x) has finite variation then its image Γ is rectifiable. Therefore we have to use here another results on existence of the Stieltjes integral.

One simple result of that kind is well known: the integral ∫_{0}^{1} fdg exists if g is continuous and f has bounded variation (see [1]). If g(z) = z then the continuity condition holds true for any curve Γ. The variation of the function f(z(x)) on the segment I does not depend on the homeomorphism z(x). Indeed, this variation equals to the least upper bound of the sums ∑_{j=1}^{n} |f(z_{j}) − f(z_{j−1})| taken over all ordered finite sequences {z_{1}, z_{2}, … } of points of the curve Γ. This upper bound can be considered as variation of the function f on the curve Γ. Let V (Γ) stand for the class of all functions of finite variation on Γ. Thus, there is valid

Proposition 1. If f ∈ V (Γ) and g is continuous, then the Stieltjes integral ∫_{Γ} fdg exists. In particular, the integral ∫_{Γ} fdz along arbitrary curve Γ exists if the integrand f has finite variation on this curve.

If the curve Γ is non-rectifiable, then the class V (Γ) does not contain even the function f(z) = z. But we can extend the class of integrands by restriction of the class of curves.

THE CAUCHY INTEGRAL ALONG Φ-RECTIFIABLE CURVES 17

Let Φ(x) be an increasing continuous real function defined for x ≥ 0 such that Φ(0) = 0. The value of the least upper bound

sup_{τ} ∑_{j=1}^{n} Φ(|f(z_{j}) − f(z_{j−1})|) = v_{Φ}(f; Γ),

taken over the set of all partitions τ = {z_{0}, z_{1}, …, z_{n}} of the curve Γ, is called the Φ-variation of f(z) on Γ. The class V_{Φ}(Γ) consists of all functions satisfying condition v_{Φ}(f; Γ) < ∞. If Φ(x) = x^{p}, p ≥ 1, then we denote this class by V_{p}(Γ) and the corresponding variation by v_{p}(f; Γ); it is called the p-variation.

For the segment I these classes were introduced by L.C. Young (see [2], [3]). He proved [3] that the Stieltjes integral ∫_{0}^{1} fdg exists if the functions f and g belong to the classes V_{Φ}(I) and V_{Ψ}(I) respectively, have not common discontinuities on I, and the inverse functions φ(x) = Φ^{−1}(x), ψ(x) = Ψ^{−1}(x) satisfy the condition

∑_{n=1}^{∞} φ(1/n)ψ(1/n) < ∞.  (2)

If Φ(x) = x^{p}, Ψ(x) = x^{q}, then the condition (2) reduces to the inequality

1/p + 1/q > 1.  (3)

The curvilinear version of this theorem is

Proposition 2. Let the Young series (2) converges. If the functions f and g belong to the classes V_{Φ}(Γ) and V_{Ψ}(Γ) respectively and have not common discontinuities on I, then the Stieltjes integral ∫_{Γ} fdg exists.

The function g(z) = z has not discontinuities. Its Φ-variation on Γ is

sup_{τ} ∑_{j=1}^{n} Φ(|z_{j} − z_{j−1}|) ≡ σ_{Φ}(Γ).  (4)

For Φ(x) ≡ x the value (4) is the length of Γ.

Definition 2. The value σ_{Φ}(Γ) is called the Φ-length of Γ. If this value is finite, then the curve Γ is said to be Φ-rectifiable.

We denote by R_{Φ} the class of all Φ-rectifiable curves. If Φ(x) = x^{p}, then we call it p-rectifiability and write R^{p} instead of R_{Φ}. The proposition 2 implies

Proposition 3. Let the Young series (2) converges. If Γ ∈ R_{Ψ} and f ∈ V_{Φ}(Γ), then the Stieltjes integral ∫_{Γ} f(z)dz exists. In particular, it exists if Γ ∈ R^{q}, f ∈ V_{p}(Γ) and inequality (3) holds true.

Note 1. We consider just the same definition of the integral ∫_{Γ} fdz along non-rectifiable curve as for rectifiable curves. On the other hand, there are publications where this integral is defined as a new object. One can find brief characteristics of these indirect definitions in [4].

A function f satisfies the Hölder condition with exponent ν on a set G ∈ ℂ if it is defined on this set and

sup\{\frac{|f(t′) − f(t′′)|}{|t′ − t′′|^{ν}} : t′, t′′ ∈ G, t′ ≠ t′′\} = h_{ν}(f; G) < ∞.

Let H_{ν}(G) stand for the class of all functions f satisfying this condition.

Corollary 1. Let Γ be q-rectifiable, f ∈ H_{α}(Γ) and g ∈ H_{β}(Γ). If α + β > q, then the integral ∫_{Γ} fdg exists. In particular, the integral ∫_{Γ} fdz exists if α > q − 1.

Proof. This corollary follows from Proposition 2 by virtue of inclusion H_{ν}(Γ) ⊂ V_{q/ν}(Γ), which is valid for any q−rectifiable curve Γ.

3. Φ-rectifiable curves

Let us consider certain properties of Φ-rectifiable curves.

Proposition 4. The curve Γ is Φ-rectifiable if and only if there exists one-to-one mapping z = ζ(x) : I → Γ and constant c > 0 such that

|ζ(x_{1}) − ζ(x_{2})| ≤ φ(c|x_{1} − x_{2}|)  (5)

for any x_{1,2} ∈ I.

Proof. Let σ_{Φ}(Γ) = c < ∞. We denote by Γ_{t} the part of Γ beginning at the point a and ending at a point t ∈ Γ. The function σ(t) ≡ σ_{Φ}(Γ_{t}) is a continuous one-to-one mapping of Γ onto segment [0, c] satisfying inequality σ(t_{1}) ≥ σ(t_{2}) + Φ(|t_{1} − t_{2}|) for any t_{1,2} ∈ Γ such that t_{2} ⪯ t_{1} (see [5]). Then the mapping ζ(x) = σ^{−1}(cx) is the desired one. The converse implication is obvious.

For Φ(x) = x^{p} inequality (5) means that ζ(x) satisfies on I the Hölder condition with exponent 1/p.

Obviously, R^{q} ⊂ R^{p} for q < p, and R^{1} is the class of all rectifiable (in usual sense) curves. In this connection we introduce rectifiability index 𝔯𝔢𝔠 Γ as the greatest lower bound of the set {p : Γ ∈ R^{p}}. We say that a curve Γ is a ν-Hölder image of I if there exists a homeomorphism ζ : I → Γ such that ζ ∈ H_{ν}(I). The proposition 4 implies that (𝔯𝔢𝔠 Γ)^{−1} is the least upper bound of the set {ν : Γ is the ν-Hölder image of I}. Thus, the union ∪_{p≥1}R^{p} consists of all possible Hölder images of the segment.

Let Y (x) be a real continuous function defined on the segment I,

Γ^{Y} = {x + iy : x ∈ I, y = Y (x)}.  (6)

The function ξ(x) = x + iY (x) maps I onto Γ^{Y}. If Y ∈ H_{ν}(I) then ξ belongs to the same class and, consequently, 𝔯𝔢𝔠 Γ^{Y} ≤ 1/ν. Let us show that this bound is attained on an extensive class of functions.

Proposition 5. Let function Y (x) satisfy inequalities

C^{−1} ≤ 2^{nν}( − Y (\frac{2k − 2}{2^{n}}) + 2Y (\frac{2k − 1}{2^{n}}) − Y (\frac{2k}{2^{n}} )) ≤ C  (7)

for n = n_{0}, n_{0} + 1, n_{0} + 2, …, k = 1, 2, …, 2^{n−1}, where C, n_{0} and ν are constants, n_{0} is a positive integer, C ≠ 0 and 0 < ν < 1. Then Y ∈ H_{ν}(I) and 𝔯𝔢𝔠 Γ^{Y} = 1/ν.

Proof. Let us expand Y in the Schauder series. The Schauder functions s_{m}(x) can be defined on the segment I as follows:

s_{0}(x) = 1, s_{1}(x) = x,

s_{2^{n}+k}(x) = \begin{cases}2^{n}(x − \frac{2k−2}{2^{n+1}}) for \frac{2k−2}{2^{n+1}} < x < \frac{2k−1}{2^{n+1}}, \\ 2^{n}(\frac{2k}{2^{n+1}} − x) for \frac{2k−1}{2^{n+1}} < x < \frac{2k}{2^{n+1}} , \\ 0 for all other x ∈ I \end{cases}

for n = 0, 1, 2, …, k = 1, 2, …, 2^{n}. Any continuous function Y (x) is representable as the Schauder series Y (x) = ∑_{j=0}^{∞} b_{j}s_{j}(x), where b_{0} = Y (0), b_{1} = Y (1) − Y (0), and b_{j} = −Y (\frac{2k−2}{2^{n+1}}) + 2Y (\frac{2k−1}{2^{n+1}}) − Y (\frac{2k−1}{2^{n+1}}) for j = 2^{n} + k, n = 0, 1, …, k = 1, 2, …, 2^{n}. This series converges on I uniformly (see [5]; the Schauder functions in this work are defined with another normalizing factors). The condition (7) means that

C^{−1} ≤ j^{ν}b_{j} ≤ C  (8)

for all sufficiently large j. The standard estimations (see, for example, considerations of the Weierstrass functions in the book [6]) show that under this condition Y ∈ H_{ν}(I). Consequently, 𝔯𝔢𝔠 Γ^{Y} ≤ 1/ν. In order to prove the inverse inequality for the rectifiability index of the curve Γ^{Y} we have to find a lower bound for the sums S_{k} = ∑_{j=0}^{2^{k}−1} |Y ((j + 1)2^{−k}) − Y (j2^{−k})|^{p}. Let us divide the segment I into 2^{k} equal segments I_{j,k} = [j2^{−k}, (j+1)2^{−k}]. The collection of the Schauder functions s_{2^{n}+l}, l = 1, …, 2^{n}, contains only one function such that it does not vanish at the end points of the segment I_{j,k}. The number m = m(j, n) of this function can be determined in the following way. We divide j by 2^{n} with remainder and denote the quotient by q. Then m(j, n) = 2^{n} + q + 1 and

Y ((j+1)2^{−k})−Y (j2^{−k}) = b_{1}2^{−k}+∑_{n=0}^{k−1} b_{m(j,n)}(s_{m(j,n)}((j + 1)2^{−k}) − s_{m(j,n)}(j2^{−k})).

The function s_{m(j,n)} is linear on the segment I_{j,k} and its angular coefficient is equal to ±2^{n}. The sign here can be determined in terms of binary representation j = 2^{k−1}d_{0}(j)+2^{k−2}d_{1}(j)+⋯+2d_{k−2}(j)+d_{k−1}(j), where every digit d_{i}(j) is either zero or unit. The sign of the angular coefficient of the function s_{m(j,n)} on the segment I_{j,k} is plus for d_{n}(j) = 0 and minus for d_{n}(j) = 1. Therefore s_{m(j,n)}((j + 1)2^{−k}) − s_{m(j,n)}(j2^{−k}) = (−1)^{d_{n}(j)}2^{n−k} and Y ((j + 1)2^{−k}) − Y (j2^{−k}) = 2^{−k}(b_{1} + B_{j,k}), where

B_{j,k} = ∑_{n=0}^{k−1} (−1)^{d_{n}(j)}2^{n}b_{m(j,n)}.

Hence, S_{k} = ∑_{j=0}^{2^{k}−1} |2^{−k}(b_{1} + B_{j,k})|^{p}. Let us put 2^{nν}b_{m(j,n)} = c_{j,n}. By virtue of (7) these values are bounded and separated from zero. We can assume without loss of generality that C > 0. Then c_{j,n} > 0 for any j and n. Obviously,

B_{j,k} = ∑_{n=0}^{k−1} (−1)^{d_{n}(j)}2^{n(1−ν)}c_{j,n}.

In other words, B_{j,k} is the sum of k first terms of disturbed geometric progression with ratio Q = 2^{1−ν} > 1. The disturbation consists in multiplication of every term by the bounded and separated from zero positive factor c_{j,n} and by the sign factor (−1)^{d_{n}(j)}.

Lemma 1. Let the double sequences c_{j,n} and ε_{j,n} satisfy the following conditions:

i. there exists positive constant c such that c^{−1} ≤ c_{j,n} ≤ c;

ii. for j = 0, 1, …, 2^{k} − 1 the vector e_{j} = {ε_{j,0}, ε_{j,1}, …, ε_{j,k−1}} ∈ ℝ^{k} runs through all 2^{k} distinct k-dimensional vectors with coordinates ±1.

Then for any Q > 1 there exist an integer M = M(Q, c) and positive r = r(Q, c), such that for k > M the collection of 2^{k} values

B_{j,k} = ∑_{n=0}^{k−1} ε_{j,n}c_{j,n}Q^{n}, j = 0, 1, …, 2^{k} − 1,

contains no less than 2^{k−M} numbers satisfying inequality |B_{j,k}| ≥ rQ^{k}.

Proof of Lemma. Let m last coordinates of the vector e_{j} have the common sign. There exist 2^{k−m+1} such vectors, and for every one we have |B_{j,k}| ≥ ∑_{n=k−m}^{k−1} c^{−1}Q^{n} − ∑_{n=0}^{k−m−1} cQ^{n} = c^{−1} \frac{Q^{k}−Q^{k−m}}{Q−1} − c\frac{Q^{k−m}−1}{Q−1} ≥ Q^{k} \frac{1−(1+c^{2})Q^{−m}}{c(Q−1)} . The value 1 − (1 + c^{2})Q^{−m} tends to 1 as m → ∞. Let M(Q, c) be the minimal integer m such that this value exceeds 1/2. Then the statement of Lemma is valid for r(Q, c) = 1/2c(Q − 1).

Now we are able to conclude the proof of Proposition 5. For j = 0, 1, …, 2^{k} − 1 the vector e_{j} = {(−1)^{d_{0}(j)}, (−1)^{d_{1}(j)}, …, (−1)^{d_{k−1}(j)}} runs through all 2^{k} various vectors with coordinates ±1, and Lemma 1 is applicable. By virtue of the lemma there exist such integer M and positive r that for k > M the inequality |B_{j,k}| ≥ r2^{k(1−ν)} is valid for at least 2^{k−M} various values j. Hence, S_{k} ≥ 2^{k−M−kp}(r2^{k(1−ν)} − |b_{1}|)^{p} ≥ C′2^{k(1−pν)}, where C′ > 0 does not depend on k. Therefore the p-variation of function Y on the segment I cannot be finite for pν < 1 and 𝔯𝔢𝔠 Γ^{Y} ≥ 1/ν, which completes the proof.

Note 2. The condition (7) holds for any sum Y (x) = ∑_{j=0}^{∞} b_{j}s_{j}(x) with coefficients b_{j} satisfying inequality (8). Therefore this condition describes a wide class of curves, which includes well known fractal curves. For instance, it contains the Weierstrass curve (i. e. graph of the Weierstrass series; see, for instance, [6] and [7]) and graphs (trajectories) of lacunary wavelet series (see [8]).

Another class of interesting examples is classical fractal curves. We consider so-called von Koch snowflake with rotation angle α ∈ (0, \frac{π}{2}) (see, for example, [7], where this curve is considered for α = π/3 as the triangular von Koch curve). It is the Haussdorf limit of following sequence of polygonal lines (prefractals):

F_{1} consists of four segments connecting consequtively the points 0 (the beginning), l_{1} = \frac{1}{2(1+cos α)}, \frac{1}{2} + il_{1} sin α, 1 − l_{1} and 1 (the end); obviously, all these segments have the same length l_{1};

F_{n}, n ≥ 2, is the result of replacement of each segment of F_{1} by polygonal line, which is similar to F_{n−1} with similarity coefficient l_{1}; the polygon F_{n} consists of 4^{n} segments of length l_{n} = l_{1}^{n} each.

The limit curve F_{∞} contains all corner points of prefractals F_{n}. Therefore it can be p-rectifiable only if the sequence 4^{n}l_{n}^{p}, n = 1, 2, …, is bounded, i. e. if p ≥ \frac{log 4}{log 1/l_{1}} . Easy estimations show that F_{∞} really is \frac{log 4}{log 1/l_{1}} -rectifiable. Thus,

𝔯𝔢𝔠 F_{∞} = \frac{log 4}{log (2(1 + cos α))}.

The right side of this equality is equal to the well known fractal dimension of the curve F_{∞}. The upper and lower metric dimensions of a compact set G ⊂ ℂ are the limits

lim sup_{k→∞} k^{−1} log_{2} N_{k}(G) = 𝔇𝔪 G, lim inf_{k→∞} k^{−1} log_{2} N_{k}(G) = 𝔡𝔪 G,

where N_{k}(G) stand for the minimal quantity of disks of radius 2^{−k} necessary for covering of G. For the von Koch curve we have 𝔯𝔢𝔠 F_{∞} = 𝔇𝔪 F_{∞} = 𝔡𝔪 F_{∞}. But in general the rectifiability index is not equal to any fractal dimension. For example, if coefficients of the Schauder series Y (x) = ∑_{j=0}^{∞} b_{j}s_{j}(x) satisfy the condition (8), then 𝔯𝔢𝔠 Γ^{Y} = 1/ν by Proposition 5, but 𝔇𝔪 Γ^{Y} = 2 − ν by Z. Ciesielski theorem [9].

Proposition 6. 𝔇𝔪 Γ ≤ 𝔯𝔢𝔠 Γ.

Proof. We consider the following special partition of Γ. Let z_{0} = a. We define z_{1} as the minimal (with respect to the ordering ≺) point satisfying conditions |z_{1} − z_{0}| = 2^{−k} and z_{0} ≺ z_{1}; if this point does not exist then we put z_{1} = b. Analogously, z_{2} is the minimal point of the set {z ∈ Γ : |z_{2}−z_{1}| = 2^{−k}, z_{1} ≺ z_{2}}, and z_{2} = b if this set is empty, and so on. As a result we obtain a partition τ = {z_{j}}_{j=1}^{m} such that |z_{j} − z_{j−1}| = 2^{−k} for j = 1, …, m − 1, and |z_{m} − z_{m−1}| ≤ 2^{−k}. If Γ is p-rectifiable, then (m − 1)2^{−kp} < ∑_{j=1}^{m} |z_{j} − z_{j−1}|^{p} ≤ σ, where σ is the p-length of Γ. Hence, m < 1 + σ2^{kp}. On the other hand, any arc of Γ whose initial point is z_{j−1} and end point is z_{j}, j = 1, …, m, is contained in a disk of radius 2^{−k}, and these disks cover Γ. Thus, N_{k}(Γ) < c + c2^{kp}, where constant c does not depend on k, and 𝔇𝔪 Γ ≤ p by definition of the upper metric dimension. Proposition is proved.

4. A non-rectifiable version of integral calculus

First we consider few customary properties of integral which are violated for the integral along non-rectifiable curve.

Let a curve Γ be not rectifiable. Then the functional f → ∫_{Γ} fdz cannot be bounded with respect to the uniform norm. Indeed, if | ∫_{Γ} fdz| ≤ C sup{|f(z)| : z ∈ Γ} for certain C > 0 and any continuous function f ∈ V (Γ), then Γ must be rectifiable by virtue of F. Riesz theorem. On the other hand, one can obtain certain bounds for this integral in terms of v_{Φ}(f; Γ) from corresponding results of [2], [3], [5].

Another broken property is the Newton - Leibniz formula. Let f(z) be an analytic function in a neighborhood of curve Γ including the curve itself. Obviously, the integral ∫_{Γ} df(z) exists and equals f(b) − f(a). But the integral ∫_{Γ} f′(z)dz can be divergent. For example, let f(z) = z^{2}/2, f′(z) = z. We consider an arbitrary partition τ = {z_{0}, z_{1}, …, z_{n}} of the curve Γ pointed in two different ways: w_{j} = z_{j}, j = 1, 2, …, n, and w_{j} = z_{j−1}, j = 1, 2, …, n. The difference of corresponding integral sums for integral ∫_{Γ} zdz is r(τ) = ∑_{j=1}^{n} (z_{j} − z_{j−1})^{2}. If this integral exists, then |r(τ)| → 0 if diam τ → 0. But this condition is broken for curves Γ^{Y} satisfying inequalities (7) with ν ≤ 1/2 (see proof of Proposition 5). The following result shows that the Newton - Leibniz formula is valid if the curve is q-rectifiable for q < 2.

Proposition 7. Let Γ ∈ R^{q}, f ∈ V_{p}(Γ) and the Young condition (3) holds true. If a function g(z) in a neighborhood of Γ has partial derivatives satisfying the Lipschiz condition (i. e. the Hölder condition with exponent ν = 1) and q < 2, then

∫_{Γ} fdg = ∫_{Γ} f \frac{∂g}{∂z} dz + f \frac{∂g}{∂\overline{z}} d\overline{z},  (9)

and both integrals exist.

Proof. If Γ ∈ R^{q} and h(z) ∈ H_{1}(Γ), then h ∈ V_{q}(Γ). Therefore, the derivatives \frac{∂g}{∂z} and \frac{∂g}{∂\overline{z}} belong to V_{q}(Γ). If f ∈ V_{p}(Γ) and h ∈ V_{q}(Γ), then these functions are bounded, and, obviously, |f(t_{1})h(t_{1})−f(t_{2})h(t_{2})| ≤ c_{1}|f(t_{1})−f(t_{2})|+ c_{2}|h(t_{1}) − h(t_{2})| ≤ c_{3} max{|f(t_{1}) − f(t_{2})|, |h(t_{1}) − h(t_{2})|}, where constants c with various subscripts do not depend on t_{1,2} ∈ Γ. Consequently, fh ∈ V_{r}(Γ), where r = max(p, q). By assumption 1/r + 1/q > 1, and the integrals in both sides of (9) exist by Proposition 3. Let us prove the equality of these integrals. We consider partition τ = {z_{0}, z_{1}, …, z_{n}} with points w_{j} = z_{j−1} and denote by S_{1} and S_{2} the corresponding Stieltjes sums for integrals from the left and right sides of (9). Let λ(j) stand for straightlinear segment connecting the points z_{j−1} and z_{j}. If diam τ is sufficiently small, then all these segments are situated in the neighborhood of Γ, where g is differentiable. Obviously,

|S_{1} − S_{2}| ≤ ∑_{j=1}^{n} |f(z_{j−1})| |∫_{λ(j)} (\frac{∂g}{∂z} (z) − \frac{∂g}{∂z} (z_{j−1}))dz + (\frac{∂g}{∂\overline{z}} (z) − \frac{∂g}{∂\overline{z}} (z_{j−1}))d\overline{z}|,

and under assumptions of the proposition we have

|S_{1} − S_{2}| ≤ C ∑_{j=1}^{n} |z_{j} − z_{j−1}|^{2} ≤ C′(diam τ)^{2−q},

where C and C′ do not depend on τ. This estimation concludes the proof.

If Γ is q-rectifiable, then by virtue of Proposition 4 there exists one-to-one mapping ζ : I → Γ satisfying the Hölder condition with exponent 1/q. Let ξ = {x_{0}, x_{1}, …, x_{n}} be a partition of the segment I, i. e. 0 = x_{0} < x_{1} < ⋯ < x_{n−1} < x_{n} = 1. We consider the function ζ_{ξ} : I → ℂ, which is linear on the segments [x_{j−1}, x_{j}], j = 1, 2, …, n, and satisfies equalities ζ_{ξ}(x_{j}) = ζ(x_{j}) for j = 0, 1, …, n. The image Γ_{ξ} = ζ_{ξ}(I) is a polygonal line. It is well known (see, for example, [5]) that the interpolation mapping ζ → ζ_{ξ} is bounded in the Hölder spaces, i. e. ζ_{ξ} ∈ H_{1/q}(I) and h_{1/q}(ζ_{ξ}; I) ≤ h_{1/q}(ζ; I). If we have a sequence of partitions ξ^{1} ⊂ ξ^{2} ⊂ ξ^{3} … such that diam ξ^{n} → 0, then the functions ζ_{n} uniformly converge to ζ. If the functions f ∈ H_{α}(Γ) and g ∈ H_{β}(Γ) are restrictions F and G to Γ, which are defined in a neighborhood B of Γ and satisfy the Hölder condition with exponents α and β, respectively, in its closure \overline{B}, then the sequences of functions f_{n}(x) = F (ζ_{ξ^{n}}(x)) and g_{n}(x) = G(ζ_{ξ^{n}}(x)) uniformly converge to f(ζ(x)) and g(ζ(x)) respectively, and the sequences of generalized variations v_{q/α}(f_{n}; I) and v_{q/β}(g_{n}; I) are bounded. Therefore for α + β > q we have

∫_{Γ} fdg = lim_{n→∞} ∫_{Γ_{n}} F dG  (10)

by virtue of theorem on term-by-term integration (see [3], [5]); here Γ_{n} stands for Γ_{ξ^{n}}. In this connection we must note that any function f satisfying the Hölder condition on Γ admits continuation onto the whole complex plane keeping validity of this condition and its exponent (the Whitney continuation; see, for instance, [10]).

The equality (10) is also valid in certain situations, where the curve Γ is not q-rectifiable and the function g does not satisfy the Hölder condition. For example, let Γ = Γ^{Y} (see (6)) for some continuous function Y. Then we put Γ_{n} = Γ^{Y_{n}} for piecewise-linear function Y_{n} = Y_{ξ^{n}}. If the function f(z) is defined on Γ, then it is continuable into the strip B = I × ℝ by F (x + iy) = f(x + iY (x)), x ∈ I, y ∈ ℝ. If f ∈ V (Γ), then F |_{Γ_{n}} ∈ V (Γ_{n}) for any n, and all these functions have the same variation. Now (10) follows from classical theorems on term-by-term Stieltjes’ integration. Obviously, the same construction works if Γ consists of several rotated curves of the form (6). We call the curves Γ graph-like. The described above continuation operator f → F can be easily modified for these curves, and we call it vertical continuation. Proposition 8. The equality (10) is valid for certain sequence of polygonal lines Γ_{n} if one of following two assumptions holds true:

i. Γ ∈ R^{q}, F ∈ H_{α}(\overline{B}), f = F |_{Γ}, G ∈ H_{β}(\overline{B}), g = G|_{Γ}, where B is any domain containing Γ, and α + β > q;

ii. Γ is a graph-like curve, g is continuous on Γ, f ∈ V (Γ), F is the vertical continuation of f and G is an arbitrary continuation of g.

Now we consider a closed curve Γ which is a boundary of domain D. Let one of assumptions of the preceding proposition holds true, and, in addition, the continuations are differentiable. The polygonal approximations Γ_{n} can be chosen among simple curves. If the curve Γ_{n} is a boundary of domain D_{n}, n = 1, 2, …, then by the Stokes’ formula we obtain

∫_{Γ_{n}} fdg = ∫_{Γ_{n}} f \frac{∂g}{∂z} dz + f \frac{∂g}{∂\overline{z}} d\overline{z} = ^{∫∫} _{D_{n}} (\frac{∂F}{∂z} \frac{∂G}{∂\overline{z}} − \frac{∂F}{∂\overline{z}} \frac{∂G}{∂z})dzd\overline{z}.

Consequently, the Stokes’ formula for non-rectifiable curves

∫_{Γ} fdg = ^{∫∫} _{D} (\frac{∂F}{∂z} \frac{∂G}{∂\overline{z}} − \frac{∂F}{∂\overline{z}} \frac{∂G}{∂z})dzd\overline{z}  (11)

is valid if the double integral in its right side exists. Let us set [F, G] = \frac{∂F}{∂z} \frac{∂G}{∂\overline{z}} − \frac{∂F}{∂\overline{z}} \frac{∂G}{∂z}. Proposition 9. Let Γ be a closed curve bounding domain D ⊂ B. If F and G are continuations into B of continuous functions f and g on Γ such that there is valid one of assumptions (i), (ii) of Proposition 8, F and G are differentiable in B \ Γ, and [F, G] is integrable in \overline{B}, then the Stokes formula (11) is valid. Corollary 2. Let a closed curve Γ be q-rectifiable and bound domain D. If f ∈ H_{α}(Γ), f ∈ H_{β}(Γ), α + β > q, and F, G are the Whitney’s continuations of f and g respectively, then the Stokes formula (11) is valid.

Proof. Let F be the Whitney continuation of function f ∈ H_{α}(Γ). It satisfies the Hölder condition with the same exponent α in the whole complex plane. In ℂ \ Γ it has partial derivatives of any order (see [10]), and the first derivatives are integrable near Γ in any degree p less than (2 − 𝔇𝔪 Γ)/(1 − α) (see [11]). Therefore these derivatives are integrable in D under condition α > 𝔇𝔪 Γ − 1. But, by assumption, α > q − 1, and this condition holds by Proposition 6. Analogously, let G be the Whitney continuation of function g ∈ H_{β}(Γ). Its first derivatives are integrable, and the products \frac{∂F}{∂z} \frac{∂G}{∂\overline{z}}, \frac{∂F}{∂\overline{z}} \frac{∂G}{∂z} are integrable in D if

1 > \frac{1 − α}{2 − 𝔇𝔪 Γ} + \frac{1 − β}{2 − 𝔇𝔪 Γ} = \frac{2 − (α + β)}{2 − 𝔇𝔪 Γ},

i. e. if α + β > 𝔇𝔪 Γ. But the last inequality follows from condition Γ ∈ R^{q}, α + β > q by virtue of Proposition 6.

Note 3. One can consider the equality (11) as definition of its left side. This approach allows to study the integral ∫_{Γ} fdg for f and g with large sets of singularities. Their extensions into D can be constructed, for instance, in terms of the Dirichlet problem; see [12], where this problem is solved for von Koch snowflake.

5. Existence of the Cauchy integral

Now let us consider the Cauchy integral (1). In order to prove its convergence one can use Propositions 1 and 3. But their immediate application implies restrictive assumptions on the function f. Indeed, Proposition 1 yields inclusion f(·)/(· − z) ∈ V (Γ) for arbitrarily fixed z ∉ Γ. But this inclusion means rectifiability of Γ even for f(ζ) ≡ ζ.. Proposition 3 brings similar difficulties. In order to avoid this obstacle we consider the integral

ℭ(Γ, f; z) = \frac{1}{2πi} ∫_{Γ} f(ζ)d log (ζ − z),  (12)

which we also call the Cauchy integral. If Γ is an open arc, then the branch of logarithm is selected by means of cut of the complex plane which connects the points z and ∞ and has not common points with Γ; if Γ is a closed curve and z lies in it, then we must decompose Γ into union of open arcs. If Γ ∈ R_{Ψ} and z ∉ Γ, then log (ζ − z) belongs to the class V_{Ψ}(Γ) as function of variable ζ.

Proposition 10. The Cauchy integral (12) exists if one of the following two assumptions holds true:

i. f ∈ V (Γ);

ii. f ∈ V_{Φ}(Γ), Γ ∈ R_{Ψ}, and the Young series (2) converges.

The existence of integral (12) under assumption (i) follows from Proposition 1, and under assumption (ii) from Proposition 3. Now Proposition 7 implies

Corollary 3. Let Γ ∈ R^{q}, q < 2, and f ∈ V_{p}(Γ). If the Young condition (3) holds true, then the integral (1) exists and equals integral (12).

Thus, the formula (12) defines the Cauchy integral for wider classes of curves and functions than the initial formula (1).

6. Boundary values of the Cauchy integral.

First we obtain the Green formula for Cauchy integral (12).

Lemma 2. Let Γ be a closed curve bounding domain D ⊂ B, a function f is defined on Γ, and F is its continuation into the domain B, D ⊂ B. If there one of the assumptions

i. Γ ∈ R^{q}, F ∈ H_{α}(\overline{B}), where α > q − 1, and distributional derivatives of the function F are integrable in \overline{B},

ii. Γ is graph-like curve, g is continuous on Γ, f ∈ V (Γ), F is vertical continuations of f and its distributional derivatives are integrable in \overline{B}, holds true, then the Green formula

ℭ(Γ, f; z) = F (z)χ(z) − \frac{1}{2πi} ^{∫∫} _{D} \frac{∂F}{∂\overline{ζ}} \frac{dζd\overline{ζ}}{ζ − z}  (13)

is valid for any z ∉ Γ. Here χ(z) is the characteristic function of the set D, i. e. χ(z) = 1 for z ∈ D and χ(z) = 0 for z ∈ ℂ \ D.

Proof. If z ∈ D^{−} ≡ ℂ \ D, then the equality

\frac{1}{2πi} ∫_{Γ} f(ζ)d log (ζ − z) = −\frac{1}{2πi} ^{∫∫} _{D} \frac{∂F}{∂\overline{ζ}} \frac{dζd\overline{ζ}}{ζ − z}

is the Stokes formula (11) for G(ζ) = log (ζ − z). If z ∈ D, then we have to apply the same formula for the domain D \ {ζ : |ζ − z| ≤ ε}. The passage to the limit, as ε → 0, gives the equality

\frac{1}{2πi} ∫_{Γ} f(ζ)d log (ζ − z) = F (z) − \frac{1}{2πi} ^{∫∫} _{D} \frac{∂F}{∂\overline{ζ}} \frac{dζd\overline{ζ}}{ζ − z} .

These equalities together are equivalent to (13).

Properties of potential operator ϕ(ζ) ↦ T ϕ ≡ ^{∫∫} \frac{ϕ(ζ)dζd\overline{ζ}}{ζ−z} are well known (see, for instance, [13]). The function T ϕ is continuous in the whole complex plane if ϕ has compact support and is integrable with degree p > 2. Thus, the Cauchy integral (12) has continuous boundary values on Γ from D and from D^{−} if first partial derivatives of continuation F are integrable in D with that degree.

We consider first the situation where the assumption (ii) holds true. It suffices to study the domain D^{Y} = {z = x + iy : x ∈ I, 0 < y < Y (x)} for a positive continuous function Y. If a function f is defined on Γ^{Y} , then its vertical continuation F (x + iy) = f(x + iY (x)) has restriction f^{♭} ≡ F |_{I}, i. e. f^{♭}(x) = f(x + iY (x)). Let us suppose that f^{♭} ∈ H_{1}(I). Then f ∈ V (Γ), F has essentially bounded derivatives in D^{Y} and, consequently, the Cauchy integral (12) along boundary ∂D^{Y} of domain D^{Y} is continuous in closures of D^{Y} and ℂ\D^{Y} . But ∂D^{Y} \Γ^{Y} is a polygonal line, and properties of the Cauchy integral along piecewise smooth curve are well known (see, for instance, [14], [15].) As a result, we obtain asymptotic formulas

ℭ(Γ, f; z) = −\frac{f(a)}{2πi} log (z − a) + r(z), z → a,  (14.1)

and

ℭ(Γ, f; z) = \frac{f(b)}{2πi} log (z − b) + r(z), z → b,  (14.2)

where a is the initial point of Γ^{Y}, b is its end point, and the remainder r(z) iz continuous at the points a and b.

In order to prove the same result for general graph-like curves we need a definition of appropriate class of functions f. A graph-like curve Γ is a union of rotated graphs Γ_{1}, Γ_{2}, …, Γ_{m}. Here Γ_{j} = {z = (x + iy)e^{iθ_{j}} : a_{j} < x < b_{j}, y = Y_{j}(x)}, where [a_{j}, b_{j}] ≡ I_{j} is a segment of real axis, and θ_{j} is the angle of rotation, j = 1, 2, …, m. If a function f is defined on Γ, then it generates m functions f_{j}^{♭}(x) ≡ f|_{Γ_{j}}(e^{iθ_{j}}(x + iY_{j}(x))), x ∈ I_{j}.

Definition 3. The class ℌ_{1}(Γ) consists of all continuous on Γ functions f such that f_{j}^{♭} ∈ H_{1}(I_{j}), j = 1, 2, …, m, for a decomposition of Γ into union of graphs.

By summation of formulas (14.1) and (14.2) for all curves Γ_{j} we obtain

Theorem 1. Let Γ be a graph-like curve, f ∈ ℌ_{1}(Γ). Then the Cauchy integral (12) has continuous limit values ℭ^{+}(Γ, f; t) and ℭ^{−}(Γ, f; t) from the left and from the right respectively at any point t ∈ Γ, excluding, possibly, its initial point a and end point b. If the curve is not closed, i. e. it has an initial point a and an end point b, then near these points there are valid asymptotic equalities (14.1) and (14.2) with continuous at these points remainder r(z). At all other points of the curve Γ the boundary values ℭ^{±} are related by equality

ℭ^{+}(Γ, f; t) − ℭ^{−}(Γ, f; t) = f(t).  (15)

Note 4. The author supposes that this result is valid for an arbitrary (i. e. not graph-like) closed curve if f ∈ V (Γ) ∩ H_{1}(Γ). In general this class is wider than ℌ_{1}(Γ).

Now let us consider the situation (i) of Lemma 2. If F is the Whitney continuation of function f ∈ H_{α}(Γ), then its first derivatives are integrable with exponent p > 2 under condition α > 𝔇𝔪 Γ/2 (see [11]). In addition, here the Cauchy integrals (1) and (12) both exist and are equal each other. Thus, there is valid

Theorem 2. Let Γ be a closed q-rectifiable curve, f ∈ H_{α}(Γ), α > q − 1 and α > 𝔇𝔪 Γ/2. Then the Cauchy integral (1) exists and has continuous limit values C^{+}(Γ, f; t) and C^{−}(Γ, f; t) from the left and from the right respectively at any point t ∈ Γ, and these boundary values are related by equality

C^{+}(Γ, f; t) − C^{−}(Γ, f; t) = f(t).  (16)

One can obtain an analog of the result of Theorem 1 concerning nonclosed curves under additional assumption that Γ is a part of boundary of a domain D such that the complementary curve ∂D \ Γ is piecewise smooth.

If Γ is rectifiable, then all assumptions of Theorem 2 reduce to inequality α > 1/2, i. e. it turns into the known E.M. Dynkin’s result (see [16]).

The following example shows that the restriction α > 𝔇𝔪 Γ/2 cannot be omitted. Fix a value P > 1. We put d_{n} = 2^{−n−[nP]}, where [·] means the integer part, x_{n,j} = 2^{−n} + jd_{n}, and G_{n,j} = {z = x + iy : x_{2j−1,n} ≤ x ≤ x_{2j,n}, 0 ≤ y ≤ 2^{−n}} for j = 1, …, 2^{[nP]−1}, n = 1, 2, …. In addition, we set G^{∗} = ∪_{n=1}^{∞}∪_{j=1}^{2^{[nP]−1}}G_{n,j}, G = {z = x + iy : −1 ≤ x ≤ 1, −1 ≤ y ≤ 0} and \overline{D} = G^{∗} ∪ G. Let Γ be a boundary of domain D. Easy estimations show that 𝔯𝔢𝔠 Γ = P. The upper metric dimension 𝔇𝔪 Γ is calculated in [11]. It equals to 2P/(P + 1). Hence, under condition P < P^{∗} ≡ (\sqrt{5} + 1)/2 we have 𝔇𝔪 Γ/2 > 𝔯𝔢𝔠 Γ − 1. In what follows we suppose that P ∈ (1, P^{∗}).

Then we have to construct the integrand f. We fix a value α ∈ (P − 1, \frac{P}{P +1}] and set f^{♭}(x_{2j−1,n}) = d_{n}^{α}, f^{♭}(x_{2j,n}) = 0 for j = 1, 2, …, 2^{[nP]−1}, n = 1, 2, …. On all segments [x_{j−1,n}, x_{j,n}] the function f^{♭}(x) is linear, and f^{♭}(x) = 0 for −1 ≤ x ≤ 0. As is known, this function satisfies the Hölder condition with exponent α on the segment [−1, 1] (see, for instance, [16].) Then we set F (x+iy) = f^{♭}(x) for −1 ≤ x ≤ 1 and f(z) = F |_{Γ}(z). Obviously, f ∈ H_{α}(Γ).

All assumptions of Lemma 2 are valid here. Therefore the Cauchy integral is representable in the form (13). Consequently,

C(Γ, f; z) = C(∂G, F |_{∂G}; z) + F (z)χ^{∗}(z) − \frac{1}{2πi} ^{∫∫} _{G^{∗}} \frac{∂F}{∂\overline{ζ}} \frac{dζd\overline{ζ}}{ζ − z} ,

where χ^{∗} is characteristic function of the set G^{∗}. Hence, the boundary values C^{±}(Γ, f; 0) exist if and only if the series

Ξ(z) = ^{∫∫} _{G^{∗}} \frac{∂F}{∂\overline{ζ}} \frac{dζd\overline{ζ}}{ζ − z} = − ∑_{n=1}^{∞} ∑_{j=1}^{2^{[nP]−1}} \frac{d_{n}^{α−1}}{2} ^{∫∫} _{G_{n,j}} \frac{dζd\overline{ζ}}{ζ − z}

have finite limits for G^{∗} ∋ z → 0 and G^{−} ∋ z → 0, G^{−} = ℂ \ \overline{G}^{∗}. But immediate evaluation gives

|Ξ(re^{iα})| > \begin{cases}Cr^{α(P +1)−P} , α < \frac{P}{P +1}, \\ C log \frac{1}{r}, α = \frac{P}{P +1}, \end{cases}

where |α − π| ≤ α_{0} < π/4, and C > 0 does not depend on r.

Thus, for any values Δ and α satisfying inequalities 0 < α ≤ Δ/2 and 1 < Δ < \sqrt{5} − 1 there exist a curve Γ and a function f ∈ H_{α}(Γ) such that 𝔇𝔪 Γ = Δ and all assumptions of Theorem 2 (excluding inequality α < 𝔇𝔪 Γ/2) hold true, but the Cauchy integral has not boundary values at certain point of the curve Γ. Probably, the upper bound Δ < \sqrt{5}−1 here is not exact.

One can replace the rectangles G_{n,j} by trapezoids keeping validity of all these considerations. Hence, the condition α > 𝔇𝔪 Γ/2 cannot be omitted in the class of graph-like curves, too.

The following result shows that the equality (15) can be true in a certain generalized sense even if the boundary values ℭ^{±} itself do not exist.

Theorem 3. Let Γ = Γ^{Y} be a Ψ-rectifiable curve of the form (6), f ∈ V_{Φ}(Γ). If the Young condition (2) holds true, then

lim_{h→+0} (ℭ(f, Γ; z + ih) − ℭ(f, Γ; z − ih)) = f(z)  (17)

for any inner point z of the curve Γ, where function f is continuous.

Proof. Obviously, for h > 0 we have

ℭ(f, Γ; z + ih) − ℭ(f, Γ; z − ih) = ∫_{Γ} f(ζ)dK_{h}(ζ − z),

where

K_{h}(ζ) = \frac{1}{2πi} log \frac{ζ − ih}{ζ + ih},

and the single-valued branch of logarithm is defined in domain ℂ\{ζ = x+iy : x = 0, |y| ≥ h}. If z is an inner point of Γ, then for Γ ∋ ζ ≠ z

lim_{h→+0} K_{h}(ζ − z) = H(ζ − z) = \begin{cases}−\frac{1}{2} if ζ ≺ z, \\ +\frac{1}{2} if z ≺ ζ, \end{cases}

i. e. pointwise limit of the kernel K_{h}(ζ) is the Heaviside function. Immediate calculation shows that Ψ-variations of functions K_{h}(ζ) are uniformly bounded for h → 0. Consequently, by virtue of theorem on term-by-term integration from [5] (see item 4.15) we obtain

lim_{h→+0} ∫_{Γ} f(ζ)dK_{h}(ζ − z) = ∫_{Γ} f(ζ)dH(ζ − z) = f(z).

Theorem is proved.

Note 5. Analogous consideration (using the classical Helly theorem instead of theorem from [5]) proves the equality (17) for rectifiable Γ and continuous f. In this connection we have to note that the boundary values C^{±} in general exist only if Γ is piecewise smooth and f satisfies the Hölder condition.

The relations (15) and (16) show that the Cauchy integral along non-rectifiable curve can be applied for resolvation of the boundary value problems in domains with non-rectifiable boundaries in just the same way as for piecewise smooth domains (see [14], [15]). The relation (17) can be used for resolvation of that problems in certain generalized sense.

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[16] E.M. Dynkin, Smoothness of the Cauchy type integral, Zapiski nauchn. sem. Leningr. dep. mathem. inst. AN USSR 92(1979), pp.115–133

Russia, Tatarstan, Kazan, 420043, Zelenaya street, 1,Kazan State Academy of Architecture and Constructions E-mail address: kats@ksaba.kcn.ru

Received July, 5, 2000