Journal of Mathematical Physics, Analysis, Geometry
2011, vol. 7, No. 1, pp. 3–18
Existence of Angular Boundary Values and
Cauchy–Green Formula
R.A. Aliyev
Baku State University
23, Z. Khalilov St., Baku, AZ 1148, Azerbaijan
E-mail:aliyevrashid@hotmail.ru, aliyevrashid@mail.ru
Received June 12, 2009
The aim of the paper is to find the conditions on the function F (z),
determined in the domain G and having in G continuous partial derivatives,
for validity of the following statements:
i) almost everywhere F (z) has the finite angular boundary values F (t)
on γ = ∂G;
ii) the boundary values F (t) are A-integrable on γ;
iii) the analog of the Cauchy–Green formula holds for F (z).
Key words: angular boundary values, A-integral, Cauchy–Green formula,
non-tangential maximal function, Cauchy integral.
Mathematics Subject Classification 2000: 30E25, 30E20.
Introduction
Let G be a bounded simply-connected domain of the complex plane C with
the boundary γ = ∂G. A set of functions satisfying the condition
‖f‖Lp(G)
def=


∫∫
G
|f(z)|pdxdy


1
p
< ∞, z = x+ iy, p ≥ 1,
is denoted by Lp(G).
For a function F , given in the domain G, on γ we determine the function
F ∗α(t) and put
F ∗α(t) = sup{|F (z)| : z ∈ Gt,α}, α ∈ (1,∞)
if the set Gt,α is not empty, and F ∗α(t) = 0 otherwise, where ρ(z, γ) is an Euclidean
distance from the point z to the curve γ, and Gt,α = {z ∈ G : |z− t| < αρ(z, γ)}.
The function F ∗α is a natural analog of a non-tangential maximal function (see
c© R.A. Aliyev, 2011
R.A. Aliyev
[1, Ch. 1, §5˙, p. 36]) for the case of functions F determined in the arbitrary
domains of the plane C.
Everywhere in the sequel, if there is no other restrictions onG, we will consider
γ as a Jordan rectifiable curve.
Let m be a Lebesgue measure on γ. A complex valued function f on γ that
is measurable with respect to the measure m is said to be A-integrable on γ if
m{t ∈ γ : |f(t)| > λ} = o
( 1
λ
)
, λ → +∞, (0.1)
and there exists a finite limit
(A)
∫
γ
f(t)dt def= lim
λ→+∞
∫
{t∈γ:|f(t)|≤λ}
f(t)dt. (0.2)
An attempt to determine an integral by means of limit (0.2) can hardly be
referred to any author. Without suitable restrictions this attempt meets an ob-
stacle because the integral determined by means of limit (0.2) has no additivity
property [2].
Titchmarch [2] showed that, when applied to the theory of trigonometric series
conjugate to the Fourier–Lebesgue series, this integration (determined through
the integral (0.2)) gives many natural results. Kolmogorov [3] showed that the
function conjugate to a summable function possesses the property (0.1). Titch-
march [2] noticed that it is the property (0.1) that guarantees the additivity of
A-integral.
The papers by P.L. Ulyanov [4, 6], Yu.S. Ochan [7], T.S. Salimov [8] were
devoted to the problem of application of A-integral to the theory of trigonometric
series and to the theory of boundary properties of analytic functions.
It is known [9, 10] that if the analytic function F (z) is a Cauchy-type integral
of some finite measure, then ν, then F (z) almost everywhere on γ has a finite
angular boundary value F (t). P.L. Ulyanov proved the following theorem.
Theorem A [6, Theorem 4]. Let a finite domain G be bounded by a contour
γ that satisfies the conditions of C (see the definition in [6]). Thereby, if
F (z) = 12pii
∫
γ
ϕ(t)
t− z dt (z ∈ G),
then
F (z) = 12pii(A)
∫
γ
F (t)
t− z dt (z ∈ G).
In other words, the Cauchy-type integral of absolutely continuous measure is
a Cauchy A-integral of its own boundary values.
4 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
Existence of Angular Boundary Values and Cauchy–Green Formula
The author of [11] gave an appropriate representation of the analytic function
F (z) in its own boundary values F (t) for a circle in the case of arbitrary finite
measures.
T.S. Salimov [8] considered a problem on the conditions on the analytic func-
tion F (z) determined in the domain G so that F (z) has almost everywhere the
finite angular boundary values F (t), the function F (t) is A-integrable, and F (z)
admits representation on the domain G by the Cauchy A-integral in its own
boundary values.
Theorem B [8, Theorem 2]. Let the function F (z) be analytic in G and for
some α ∈ (2,∞) m{t ∈ γ : F ∗α(t) > λ} = o
( 1
λ
) , λ → +∞. Then F has a finite
angular boundary value F (t) for almost all t ∈ γ and the following equalities are
valid:
a) (A) ∫
γ
F (t)dt = 0;
b) F (z) = 12pii(A)
∫
γ
F (t)
t− z dt, z ∈ G.
Theorem B is also proved for the case α ∈ (1, 2] under some additional con-
ditions set on the domain G (see [8], Theorem 9).
The present paper is devoted to finding the conditions on the arbitrary (not
necessarily analytic) functions F (z), determined in the domain G and having in
G continuous partial derivatives, for validity of the following statements:
i) almost everywhere F (z) has the angular boundary values F (t) on γ = ∂G
(see Theorem 1 in Sec. 1);
ii) the boundary values F (t) are A-integrable on γ (see item a) of the state-
ment of Theorem 2 in Sec. 2);
iii) the analog of the Cauchy–Green formula holds for F (z) (see item b) of the
statement of Theorem 2 in Sec. 2).
1. Existence of Angular Boundary Values
We will need the following known theorems.
Theorem C [12, Theorem 1.16]. If f = ∂g∂z ∈ L1(G), then
g(z) = Φ(z) + (TGf)(z), z ∈ G,
where Φ is analytic in G, (TGf)(z) = − 1pi
∫∫
G
f(ζ)
ζ − z dξdη, ζ = ξ + iη.
Theorem D [12, Theorem 1.19]. If f ∈ Lp(G) for some p ∈ (2,∞), then the
function TGf satisfies the condition
|(TGf)(z1)− (TGf)(z2)| ≤ Mp ‖f‖Lp(G) |z1 − z2|α, α = (p− 2)/p,
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 5
R.A. Aliyev
where z1 and z2 are arbitrary points of the plane C, and Mp is a constant depen-
dent only on p.
Theorem E [8, Lemma 6]. Let the function F be analytic in G and for some
α ∈ (1,∞) F ∗α(t) be finite for all t from the measurable set P ⊂ γ. Then F has
the finite angular boundary value F (t) for almost all t ∈ P .
Theorem 1. Let the function F be determined in the domain G and satisfy
the following conditions:
1) F is absolutely continuous in G, moreover, ∂F∂z ∈ Lp(G), z = x+ iy, forsome p ∈ (2,∞);
2) for some α ∈ (1,∞),
m{t ∈ γ : F ∗α(t) > λ} = o
( 1
λ
)
, λ → +∞.
Then, for m-almost all t ∈ γ there exists the finite angular boundary value F (t).
P r o o f. Assume
h(z) =
(
TG∂F∂ζ
)
(z) = − 1pi
∫∫
G
∂F
∂ζ
dξdη
ζ − z .
It follows from Theorem C that the function Ψ(z) = F (z) − h(z) is analytic in
G, and from Theorem D we get that the function h(z) is continuous in G. Hence
and from condition 2) of Theorem 1 it follows that the function Ψ(z) satisfies
all the conditions of Theorem E. Thus, Ψ(z) has a finite angular boundary value
Ψ(t) for almost all t ∈ γ. On the other hand, the function h(z) is continuous
in G, and therefore the function F (z) = Ψ(z) + h(z) also has the finite angular
boundary value F (t) for almost all t ∈ γ.
Theorem 1 is proved.
Corollary 1. If the function F (z) is bounded and has a bounded derivative
∂F
∂z in G, then there exists the finite angular boundary value F (t) for m-almostall t ∈ γ.
In fact, the second condition of Theorem 1 follows from the boundedness of
F (z), and the first condition follows from the boundedness of the derivative ∂F∂z .Therefore, F (z) has the finite angular boundary value F (t) for almost all t ∈ γ.
Remark 1. M.B. Balk [13] proved the following statement: if a polyana-
lytic function F (z), i.e. the function of the form F (z) = ϕ0(z) + zϕ1(z) + ... +
zn−1ϕn−1(z), where ϕi(z) are analytic functions, i = 0, n− 1, together with its
6 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
Existence of Angular Boundary Values and Cauchy–Green Formula
derivative ∂F∂z is bounded in G and there is an analytic arch Γ on the boundaryγ = ∂G, then there exists the finite angular boundary value F (t) almost every-
where on Γ. Corollary 1 shows that the existence of almost everywhere finite
angular boundary value holds for any bounded functions F (z) bounded by the
derivative ∂F∂z . Moreover, neither polyanaliticity nor the analytic arch Γ on thecurve γ is required.
2. Cauchy–Green Formula
In this section we prove the following
Theorem 2. Let the function F be determined in the domain G and satisfy
the following conditions:
1) F has the continuous partial derivatives ∂F∂x and
∂F
∂y in G, moreover,
∂F
∂z ∈ L1(G), z = x+ iy;2) for some α ∈ (1,∞),
m{t ∈ γ : F ∗α(t) > λ} = o
( 1
λ
)
, λ → +∞;
3) for m-almost all t ∈ γ there exists the finite angular boundary value F (t).
Then
a)
(A)
∫
γ
F (t)dt = 2i
∫∫
G
∂F
∂ζ dξdη, ζ = ξ + iη, (2.1)
where F (t) is a finite angular boundary value of the function F (z) as z → t ∈ γ;
b) for all z ∈ G,
F (z) = 12pii(A)
∫
γ
F (t)
t− z dt−
1
pi
∫∫
G
∂F
∂ζ ·
dξdη
ζ − z . (2.2)
Remark 2. If ∂F∂z ∈ Lp(G) for some p > 2, then the condition 3) followsfrom Theorem 1 and can be eliminated.
Remark 3. Since for the function F (z) analytic in the domain G the integral
in the right-hand side of (2.1) equals zero, then it follows from Theorems 1 and
2 that the statement of Theorem B holds for α ∈ (1,∞).
To prove Theorem 2 we will need the following known statements proved by
A.S. Bezikovich [14] and P.L. Ulyanov [6].
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 7
R.A. Aliyev
Theorem F ([14], Ch. 1, § 1, p. 13). Let A be a bounded set in Rn, and for
each x ∈ A there be given a closed Euclidean ball B(x, r(x)) centered in x and
of radius r(x). Then from the set {B(x, r(x)) : x ∈ A} we can choose at most a
denumerable set of balls {Bk} satisfying the following conditions:
i) A ⊂ ∪
k
Bk;
ii) none of the points from Rn is contained in more than θn balls from the set
{Bk}, where θn is a number dependent only on n;
iii) a set of the balls {Bk} can be divided into the ξn families of disjoint balls,
where ξn is a number dependent only on n.
Theorem G [6, Lemma 2]. Let the functions f(x) and ϕ(x) be determined
on the segment [a; b]. Thereby, if m{x ∈ [a, b] : |f(x)| > λ = o(1/λ), λ → +∞,
|ϕ(x)| ≤ D for x ∈ [a, b], where D is a positive constant, then
lim
λ→+∞



∫
{x∈[a;b]:|f(x)ϕ(x)|≤λ}
f(x)ϕ(x)dx−
∫
{x∈[a;b]:|f(x)|≤λ}
f(x)ϕ(x)dx



= 0.
2.1. Let E be an open set on γ = ∂G, E 6= ∅, E 6= γ, and α > 1 be a given
number. Denote P = γ\E,
r = α− 13α , β =
2α+ 1
α+ 2 , δ =
1
2 arccos
1
β , n =
[pi
δ
]
+ 1, (2.3)
where [x] is an entire part of a number x ∈ R.
Using Theorem F , from the system {B(t, rρ(t, P ))}t∈E we choose at most a
denumerable set of the circles {Bq}q∈Q (Bq = B(tq, rρ(tq, P )), q ∈ Q), such that
E ⊂ ∪
q∈Q
Bq (2.4)
and each point from C is overlapped by at most θ circles from {Bq}, and the
number θ is an absolute constant. As each point from E is overlapped by at most
θ circles from {Bq}, we get the estimation
∑
q∈Q
m(E ∩Bq) ≤ θmE.
Since the point tq (the center of the circle Bq) belongs to the set E, then the
measure of intersection of the set E with the circle Bq is smaller than the diameter
of the circle Bq, i.e. m(E ∩Bq) ≥ 2rρ(tq, P ), q ∈ Q, and thus we have
∑
q∈Q
ρ(tq, P ) ≤ θ2rmE. (2.5)
8 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
Existence of Angular Boundary Values and Cauchy–Green Formula
Let lk, k = 0, n− 1 be straight lines in the plane C given by the equations
x · sin
(pi
nk
)
− y · cos
(pi
nk
)
= 0, k = 0, n− 1.
For each point tq, q ∈ Q, divide the plane C into 2n sectors by the straight lines
parallel to lk, k = 0, n− 1, and crossing the point tq. Denote these sectors by
S(k)q , k = 1, 2n. Let
P (k)q = P ∩ S(k)q , k = 1, 2n.
Since the set P (k)q is closed, we can take a point t(k)q ∈ P (k)q , k = 1, 2n, such that
ρ
(
tq, P (k)q
)
= |tq − t(k)q |, k = 1, 2n.
Denoting
τ (k)q = tq +
tq − t(k)q
β2 − 1 , k = 1, 2n,
we set
Kq =
2n∩
k=1
B
(
τ (k)q ,
β
β2 − 1 |tq − t
(k)
q |
)
, q ∈ Q, (2.6)
and
G(E,α) = G\ ∪
q∈Q
Kq. (2.7)
Remark 4. In the paper [8], T.S. Salimov considered the set G(E,α) =
G\ ∪q∈Q B′q, where B′q is a circle centered at the point tq and of radius hρ(tq, P ),
h ∈ (0; 1). Unfortunately, this set is not suitable for obtaining the necessary result
for α ∈ (1; 2] for the domains with boundaries having external angles smaller
than 4 arccos α2 . Under such structure of the set G(E,α), if E is a sufficiently
small vicinity of the point t0 on γ = ∂G, with an external angle smaller than
4 arccos α2 , then for the point z0 on the internal bisectrix (and for the points z
in some proximity z0) at the distance rρ(t0, P ) from t0, neither the inequality
ρ(z, P ) < α · ρ(z, γ) (see Lemma 2.1 below) that provides inclusion z ∈ F ∗α(t0),
nor the inequality |F (z)| ≤ λ is fulfilled.
In the present paper, we consider the set G(E,α) = G\ ∪q∈Q Kq, where Kq
is determined by formula (2.6) that allows to establish Theorem B for the case of
the values α ∈ (1; 2).
Lemma 2.1. If z ∈ G(E,α), then ρ(z, P ) < αρ(z, γ).
P r o o f. Choose a point t ∈ γ such that ρ(z, γ) = |z − t|. In the case t ∈ P
we get ρ(z, P ) = |z − t| = ρ(z, γ) < αρ(z, γ). Consider the case t ∈ E. By (2.4),
t ∈ Bq for some q ∈ Q, hence for any k = 1, 2n
|tq − t| ≤ rρ(tq, P ) ≤ rρ(tq, P (k)q ) = r|tq − t(k)q |. (2.8)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 9
R.A. Aliyev
Further, in virtue of z 6∈ Kq (see (2.7)) there exists k0 ∈ {1, 2, . . . , 2n} such that
for
z 6∈ B
(
τ (k0)q ,
β
β2 − 1 |tq − t
(k0)q |
)
. (2.9)
It is easy to show that the geometric place of the points ξ, satisfying the inequality
|ξ − t(k0)q | ≥ β|ξ − tq|,
is the circle B
(
τ (k0)q , ββ2 − 1 |tq − t
(k0)q |
)
. Therefore, from (2.9) we get
|z − t(k0)q | < β|z − tq|. (2.10)
By virtue of the triangle inequality, from (2.8) and (2.10) we have
|z − tq| ≤ |z − t|+ |tq − t| ≤ |z − t|+ r|tq − t(k0)q |
≤ |z − t|+ r
{
|z − tq|+ |z − t(k0)q |
}
< |z − t|+ r(1 + β)|z − tq|,
whence
|z − tq| < 11− r(1 + β) |z − t|. (2.11)
Taking into account that |z − t| = ρ(z, γ) and ρ(z, P ) ≤ ρ(z, P (k0)q ) ≤ |z − t(k0)q |,
by (2.10), (2.11) and (2.3) we get
ρ(z, P ) ≤ |z − t(k0)q | < β|z − tq| <
β
1− r(1 + β)ρ(z, γ) = αρ(z, γ).
Lemma 2.2. Let t ∈ P and a tangent to γ exist at the point t. Then there
might be found a number ε > 0 such that for all the points z lying on the internal
normal to G at the point t and satisfying the inequality |z − t| < ε, the inclusion
z ∈ G(E,α) holds.
P r o o f. Choose ε1 > 0 such that if the point z lies on the internal normal
to G at the point t and |z− t| < ε1, then z ∈ G. Choose ε2 > 0 such that if τ ∈ γ
and |τ − t| < ε2, then the angle between the tangent at the point t and the ray
(tτ) is smaller than δ0.
Assume ε = min{ε1, β − 1β ε2}. Let the point z lie on the internal normal to G
at the point t and |z− t| < ε. Then |z− t| < ε1, and therefore z ∈ G. For proving
the lemma, by (2.7) it is enough to verify that z 6∈ Kq for all q ∈ Q. Since for
any q ∈ Q the plane C is divided into 2n sectors S(k)q , then there exists such a
number k1 ∈ {1, 2, . . . , 2n} that t ∈ S(k1)q . Denote an angle between the tangent
10 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
Existence of Angular Boundary Values and Cauchy–Green Formula
at the point t and the chord [t, tq] by ϕ1, and the angle between the straight lines
(ttq) and (tqt(k1)q ) by ϕ2. Two cases are possible.
1) Case |t− tq| < ε2. Here, by tq ∈ γ and t ∈ S(k1)q , there follows that ϕ1 < δ
and ϕ2 < 2pi2n < δ, respectively (see (2.3)). Let ` be a tangent to γ at the point
t. Since β = 2α+ 1α+ 2 < 2 and δ =
1
2 arccos
1
β <
pi
6 for any α > 1, we have
|z − τ (k1)q | ≥ np`
[
z; τ (k1)q
]
= np` [t; tq] + np`
[
tq; τ (k1)q
]
= |tq − t| · cosϕ1
+|τ (k1)q − tq| · cos(ϕ1 + ϕ2) > cos(2δ) ·
[
|tq − t|+ |τ (k1)q − tq|
]
,
where np`[z1; z2] is a length of the projection of the segment [z1, z2] on the straight
line `.
Taking into consideration that cos(2δ) = 1/β (see (2.3)), |tq− t| ≤ ρ(tq, P (k1)q )
= |tq− t(k1)q | by t ∈ S(k1)q , and |τ (k1)q − tq| = 1β2 − 1 |tq− t
(k1)q | by definition of τ (k)q ,
we get
|z − τ (k1)q | >
1
β
[
|tq − τ (k1)q |+
1
β2 − 1 |tq − t
(k1)q |
]
= ββ2 − 1 |tq − t
(k1)q |.
Hence it follows that z 6∈ B
(
τ (k1)q , ββ2 − 1 |tq − t
(k1)q |
)
and, consequently, z 6∈ Kq.
2) Case |t − tq| ≥ ε2. By the triangle inequality and the condition |z − t| <
ε ≤ β − 1β ε2 we have
|z − τ (k1)q | ≥ |t− τ (k1)q | − |t− z|
≥ |t− τ (k1)q | −
β − 1
β · ε2 ≥ |t− τ
(k1)q | −
β − 1
β |t− tq|. (2.12)
Denote by τ ′ a projection of the point τ (k1)q on the straight line (ttq). Since
ϕ2 < δ < pi6 , then
|t− τ (k1)q | ≥ |t− τ ′| = |t− tq|+ |tq − τ ′| = |t− tq|+ |τ (k1)q − tq| cosϕ2
> |t− tq|+ |τ (k1)q − tq| cos(2δ) = |t− tq|+
1
β |τ
(k1)q − tq|.
From the above and from (2.12) it follows that
|z − τ (k1)q | ≥
1
β
[
|t− tq|+ |τ (k1)q − tq|
]
.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 11
R.A. Aliyev
Further, taking into account that |tq − t| ≥ ρ(tq, P (k1)q ) = |tq − t(k1)q |, by t ∈ S(k1)q
and |τ (k1)q − tq| = 1β2 − 1 |tq − t
(k1)q | by the definition of τ (k)q , we get
|z − τ (k1)q | ≥
1
β
[
|tq − t(k1)q |+
1
β2 − 1 |tq − t
(k1)q |
]
= ββ2 − 1
[
|tq − t(k1)q |
]
and, consequently, z 6∈ Kq.
Lemma 2.2 is proved.
Lemma 2.3. G(E,α) is an open subset of the plane C.
P r o o f. Let z ∈ G(E,α). By (2.7), z ∈ G and therefore ρ(z, γ) > 0. At
first prove that for all q ∈ Q there holds the inclusion
Kq ⊂ B
(
tq, 1β − 1ρ(tq, P )
)
. (2.13)
In fact, for any point z ∈ Kq, by the definition of the set Kq, we have z ∈
B
(
τ (k)q , ββ2 − 1 |tq − t
(k)
q |
)
for each k = 1, 2n. Then, |z− τ (k)q | < ββ2 − 1 |tq − t
(k)
q |
and
|z − tq| = |z − τ (k)q + τ (k)q − tq| =
∣∣∣∣∣z − τ
(k)
q +
tq − t(k)q
β2 − 1
∣∣∣∣∣
≤ |z − τ (k)q |+
1
β2 − 1 · |tq − t
(k)
q | <
β
β2 − 1 · |tq − t
(k)
q |+
1
β2 − 1 · |tq − t
(k)
q |
= 1β − 1 · |tq − t
(k)
q | ≤
1
β − 1ρ(tq, P ),
whence inclusion (2.13) follows.
If some circle B
(
tq, 1β − 1ρ(tq, P )
)
intersects a circle B(z, ρ(z, γ)/2), then
by tq ∈ γ it follows that
1
β − 1ρ(tq, P ) + ρ(z, γ)/2 ≥ ρ(z, γ)
and thus ρ(tq, P ) ≥ β − 12 ρ(z, γ). The above and inclusion (2.4) givesthat the circle B(z, ρ(z, γ)/2) can intersect only a finite number of circles from{
B
(
tq, 1β − 1ρ(tq, P )
)}
q∈Q
. Taking also into account inequality (2.11), we get
that the circle B(z, ρ(z, γ)/2) can intersect only a finite number of sets from the
family {Kq}q∈Q. Since the sets Kq, q ∈ Q are closed, then for some δ > 0 the
12 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
Existence of Angular Boundary Values and Cauchy–Green Formula
circle B(z, δ) intersects none of the sets Kq, q ∈ Q. Then, by z ∈ G and (2.7) it
follows that some vicinity of the point z is contained in G(E,α).
Lemma 2.3. is proved.
Lemma 2.4. The following relations hold:
∂G(E,α) ∩ E = ∅, ∂G(E,α) ⊂ P ∪
(
∪
q∈Q
∂Kq
)
,
where P = γ\E.
P r o o f. It follows from the definition of the set G(E,α) (2.7) and Lemma
2.3 that
∂G(E,α) ⊂ γ ∪
(
∪
q∈Q
∂Kq
)
.
Therefore, it is enough to prove that ∂G(E,α) ∩ E = ∅. If t ∈ E, then by
(2.4), t ∈ Bq for some q ∈ Q. Subsequently, |t − tq| ≤ rρ(tq, P ). Since for any
k ∈ {1, 2, . . . , 2n} the inequality |tq − t(k)q | ≥ ρ(tq, P ) is fulfilled, and by (2.3),
r < (β + 1)−1, we have
|t− τ (k)q | ≤ |t− tq|+ |tq − τ (k)q | ≤ rρ(tq, P )
+ 1β2 − 1 |tq − t
(k)
q | ≤
[
r + 1β2 − 1
]
|tq − t(k)q | <
β
β2 − 1 |tq − t
(k)
q |.
From whence it follows that t is an internal point of the set Kq, and therefore
t 6∈ ∂G(E,α).
Lemma 2.4 is proved.
Let {Gj}j∈J be the connected components of the set G(E,α). From the
construction of G(E,α) it follows that ∂Gj is a Jordan curve for any j ∈ J .
Further, by Lemma 2.4 we have
∂Gj\γ ⊂ ∪q∈Q∂Kq, (2.14)
∂Gj ∩ γ ⊂ P, j ∈ J. (2.15)
The set Kq is an intersection of the 2n circles and therefore Kq is a convex set.
Since one of these circles is of radius ββ2 − 1ρ(tq, P ) and Kq is a subset of this
circle, then the length of the boundary Kq is not greater than the boundary of
the indicated circle, i.e. not greater than 2piββ2 − 1ρ(tq, P ). Therefore, from (2.14)
and (2.5) we get that ∂Gj is rectifiable, j ∈ J .
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 13
R.A. Aliyev
If at the point t ∈ P there exists a tangent γ, then by Lemma 2.2, t ∈ ∂Gj
for some j ∈ J and, consequently,
m
(
P\ ∪
j∈J
∂Gj
)
= 0. (2.16)
Lemma 2.5. If j1, j2 ∈ J and j1 6= j2, then the intersection ∂Gj1 ∩ ∂Gj2
contains at most one point.
P r o o f. By the construction of G(E,α), ∂Gj1 ∩ ∂Gj2 does not contain an
arch. Therefore, if this set contains at least two points, then the set C\(Gj1∪Gj2)
contains a bounded connected component D. Since ∂D ⊂ (∂Gj1 ∪ ∂Gj2) ⊂ G,
then from the coherence of G we get D ⊂ G. Furthermore, D ∩ G(E,α) 6= D,
otherwise the set Gj1∪Gj2∪D would be contained in some connected component
of the set G(E,α), that is impossible. Further, by D ⊂ G and D ∩G(E,α) 6= D,
we have D∩Kq 6= D for some q ∈ Q. Thus, taking into account the inclusion D ⊂
G, we get that the intersection of ∂D with the interior of Kq is not empty. The
last statement contradicts the inclusion ∂D ⊂ (∂Gj1 ∪ ∂Gj2) and the inequality
(2.5), and this proves the lemma.
2.2. Proof of item a) of Theorem 2.
Denote
Eλ = {t ∈ γ;F ∗α > λ}, Pλ = γ\Eλ.
If Eλ 6= ∅ for some λ > 0, then |F (z)| ≤ λ for all z ∈ G and the validity of items
a) and b) of Theorem 2 is established (see [12, pp. 28 and 42]).
Now, let Eλ 6= ∅ for all λ > 0. It follows from the definition of the function F ∗α
that Eλ is an open subset of γ. LetG(Eλ, α) be a set constructed in subsection 2.1,
{Gj}j∈J be the connected components of G(Eλ, α). On ∂Gj choose the positive
orientation (with respect to Gj). It follows from Lemma 2.1 and the definition
of the function F ∗α that |F (z)| ≤ λ for z ∈ G(Eλ, α). Therefore, |F (t)| ≤ λ for
almost all t ∈ ∂Gj , and the following equality is valid:
∫
∂Gj
F (t)dt = 2i
∫∫
Gj
∂F
∂ξ dxdy, ξ = x+ iy,
and
∣∣∣∣∣∣∣
∫
∂Gj∩γ
F (t)dt− 2i
∫∫
∂Gj
∂F
∂ξ dxdy
∣∣∣∣∣∣∣
=
∣∣∣∣∣∣∣
∫
∂Gj\γ
F (t)dt
∣∣∣∣∣∣∣
≤ λ
∫
∂Gj\γ
|dt|.
14 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
Existence of Angular Boundary Values and Cauchy–Green Formula
By summing up these inequalities and taking into account (2.14) and Lemma 2.6,
we get
∑
j∈J
∣∣∣∣∣∣∣
∫
∂Gj∩γ
F (t)dt− 2i
∫∫
Gj
∂F
∂ξ dxdy
∣∣∣∣∣∣∣
≤ λ
∑
q∈Q
∫
∂Kq
|dt|.
As the length of the boundary of the set Kq is not greater than 2piβ(β2 − 1)−1
×ρ(tq, P ), by using (2.5) and taking into account (2.3), we obtain
∑
j∈J
∣∣∣∣∣∣∣
∫
∂Gj∩γ
F (t)dt− 2i
∫∫
Gj
∂F
∂ξ dxdy
∣∣∣∣∣∣∣
≤ θ 3piβα(α− 1)(β2 − 1)λmEλ.
Further, by (2.15), (2.16) and Lemma 2.5, we have
∫
Pλ
F (t)dt =
∑
j∈J
∫
∂Gj∩γ
F (t)dt.
From the last two relations we get the estimation
∣∣∣∣∣∣∣
∫
Pλ
F (t)dt− 2i
∫∫
G(Eλ,α)
∂F
∂ξ dxdy
∣∣∣∣∣∣∣
≤ θ 3piβα(α− 1)(β2 − 1)λmEλ. (2.17)
Denote by P ′λ only a set of the points t ∈ γ having the angular boundary value
F (t), and |F (t)| ≤ λ. The conditions of the theorem imply that m(Pλ\P ′λ) = 0
and, therefore, ∫
P ′λ
F (t)dt =
∫
Pλ
F (t)dt+
∫
P ′λ\Pλ
F (t)dt.
On the other hand, we have
∣∣∣∣∣∣∣
∫
P ′λ\Pλ
F (t)dt
∣∣∣∣∣∣∣
≤ λm(P ′λ\Pλ) ≤ λm(γ\Pλ) = λmEλ. (2.18)
By estimations (2.17) and (2.18), we get
∣∣∣∣∣∣∣
∫
P ′λ
F (t)dt− 2i
∫∫
G
∂F
∂ξ dxdy
∣∣∣∣∣∣∣
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 15
R.A. Aliyev
≤
[
1 + θ 3piβα(α− 1)(β2 − 1)
]
λmEλ +
∫∫
G\G(Eλ,α)
∣∣∣∣
∂F
∂ξ
∣∣∣∣ dxdy.
Since the area of the domain G\G(Eλ, α) tends to zero as λ → +∞, then from
conditions 1) and 2) of Theorem 2 there follows the equality from item a) of
Theorem 2.
2.3. Proof of item b) of Theorem 2.
Fixing z ∈ G and using condition 2) of Theorem 2 choose λ0 > 0 such that
mEλ0 <
1
2 min{(β
2 − 1)ρ(z, γ),mγ}. (2.19)
Now, let an arbitrary λ > λ0. The set Eλ is open, Eλ 6= ∅, and by (2.19), Eλ 6= γ.
Using the construction from subsection 2.1 (for E = Eλ), we get a system of the
sets {Kq}q∈Q and the set G(Eλ, α). Let {Gj}j∈J be connected components of
G(Eλ, α). Choose the positive orientation on ∂Gj (with respect to Gj), j ∈ J .
At first prove that
ρ(z,Kq) > 12ρ(z, γ), q ∈ Q. (2.20)
Taking into account that Kq =
2n⋂
k=1
B
(
τ (k)q , ββ2 − 1 |tq − t
(k)
q |
)
, for any k ∈
{1, 2, . . . , 2n} we have
ρ(z,Kq) ≥ ρ
(
z,B
(
τ (k)q ,
β
β2 − 1 |tq − t
(k)
q |
))
≥ |z − τ (k)q | ≥ |z − tq| − |tq − τ (k)q | = |z − tq| −
1
β2 − 1 |tq − t
(k)
q |.
Let ρ(tq, Pλ) = |tq − t(k1)q |. Since |z − tq| ≥ ρ(z, γ) and ρ(z, Pλ) ≤ mEλ, then we
get
ρ(z,Kq) ≥ ρ(z, γ)− 1β2 − 1mEλ.
Noticing also that by (2.19) mEλ > β
2 − 1
2 ρ(z, γ), we prove the validity of theestimation (2.20).
From (2.7) and (2.20) we have z ∈ G(Eλ, α). Furthermore, it follows from
the definition of the set Pλ and Lemma 2.2 that |F (z′)| ≤ λ for z′ ∈ G(Eλ, α).
Consequently, in G(Eλ, α) the Cauchy–Green integral formula is applicable to
the function F . Thus, we have
F (z) = 12pii
∑
j∈J
∫
∂Gj
F (t)
t− z dt−
1
pi
∫∫
G(Eλ,α)
∂F
∂ξ
dxdy
ξ − z , ξ = x+ iy,
16 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
Existence of Angular Boundary Values and Cauchy–Green Formula
and ∣∣∣∣∣∣∣
F (z)− 12pii
∑
j∈J
∫
∂Gj∩γ
F (t)
t− z dt+
1
pi
∫∫
G
∂F
∂ξ
dxdy
ξ − z
∣∣∣∣∣∣∣
≤ 12pi
∑
j∈J
∫
∂Gj\γ
|F (t)|
|t− z| |dt|+
1
pi
∫∫
G\G(Eλ,α)
∣∣∣∣
∂F
∂ξ
∣∣∣∣
dxdy
|ξ − z|
≤ λ2pi
∑
j∈J
∫
∂Gj\γ
|dt|
|t− z| +
1
pi
∫∫
G\G(Eλ,α)
∣∣∣∣
∂F
∂ξ
∣∣∣∣
dxdy
|ξ − z| = 41 +42.
Estimate 41 and 42. Taking into account (2.14), Lemma 2.5 and inequality
(2.20), for 41 we have
41 ≤ λpiρ(z, γ)
∑
q∈Q
∫
∂Kq
|dt|.
Since the length of the boundary Kq is not greater than 2piβ(β2 − 1)−1ρ(tq, P ),
by inequality (2.5) we get
41 ≤ θ 3βα(α− 1)(β2 − 1)ρ(z, γ)λmEλ.
From the above and from condition 2) of Theorem 2 it follows that 41 tends to
zero as λ → +∞.
Further, by inequality (2.20), for 42 we have
42 ≤ 2piρ(z, γ)
∫∫
G\G(Eλ,α)
∣∣∣∣
∂F
∂ξ
∣∣∣∣ dxdy.
Since the area of the domain G\G(Eλ, α) tends to zero as λ → +∞, it follows
from condition 1) of Theorem 2 that 42 also tends to zero as λ → +∞.
Taking also into account that by (2.15), (2.16) and Lemma 2.5
∑
j∈J
∫
∂Gj∩γ
F (t)
t− z dt =
∫
Pλ
F (t)
t− z dt,
we get
F (z) = lim
λ→+∞
1
2pii
∫
Pλ
F (t)
t− z dt−
1
pi
∫∫
G
∂F
∂ξ
dxdy
ξ − z . (2.21)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 17
R.A. Aliyev
Since m(Pλ\P ′λ) = 0, then
∣∣∣∣∣∣∣
∫
P ′λ
F (t)
t− z dt−
∫
Pλ
F (t)
t− z dt
∣∣∣∣∣∣∣
=
∣∣∣∣∣∣∣
∫
P ′λ\Pλ
F (t)
t− z dt
∣∣∣∣∣∣∣
≤ 2λρ(z, γ)m(P
′
λ\Pλ) ≤
2λ
ρ(z, γ)mEλ.
The last expression tends to zero as λ → +∞ by condition 2) of Theorem 2.
Hence, in virtue of (2.21) and Theorem G there follows equality (2.2). The
theorem is proved.
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