On the boundary behavior of conjugate harmonic functions
It is proved that if a harmonic function u on the unit disk D in C has angular limits on a measurable set E of the unit circle, then its conjugate harmonic function v in D also has (finite !) angular limits a.e. on E and both boundary functions are measurable on E. The result is extended to arbitrar...
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| Опубліковано в: : | Праці Інституту прикладної математики і механіки НАН України |
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Інститут прикладної математики і механіки НАН України
2017
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| Цитувати: | On the boundary behavior of conjugate harmonic functions / V.I. Ryazanov // Праці Інституту прикладної математики і механіки НАН України. — Слов’янськ: ІПММ НАН України, 2017. — Т. 31. — С. 117-123. — Бібліогр.: 19 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859906520465014784 |
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| author | Ryazanov, V.I. |
| author_facet | Ryazanov, V.I. |
| citation_txt | On the boundary behavior of conjugate harmonic functions / V.I. Ryazanov // Праці Інституту прикладної математики і механіки НАН України. — Слов’янськ: ІПММ НАН України, 2017. — Т. 31. — С. 117-123. — Бібліогр.: 19 назв. — англ. |
| collection | DSpace DC |
| container_title | Праці Інституту прикладної математики і механіки НАН України |
| description | It is proved that if a harmonic function u on the unit disk D in C has angular limits on a measurable set E of the unit circle, then its conjugate harmonic function v in D also has (finite !) angular limits a.e. on E and both boundary functions are measurable on E. The result is extended to arbitrary Jordan domains with rectifiable boundaries in terms of angular limits and of the natural parameter. This result is essentially based on the Fatou theorem on angular limits of bounded analytic functions and on the construction of Luzin and Priwalow to their uniqueness theorem for analytic and meromorphic functions. The result will have interesting applications to the study of the various Stieltjes integrals in the theory of harmonic and analytic functions and, in particular, of the Hilbert–Stieltjes inyegral.
Доказывается, что если гармоническая функция u, заданная в единичном круге D комплексной плоскости C, имеет угловые пределы на измеримом множестве E единичной окружности, то ее сопряженная гармоническая функция v в D также имеет угловые пределы п.в. на E и обе граничные функции п.в. конечны и измеримы на E. Затем этот результат распространяется на произвольные жордановы области со спрямляемыми границами в терминах угловых пределов относительно естественного параметра. Результат существенно основывается на теореме Фату об угловых пределах ограниченных аналитических функций и конструкции Лузина и Привалова к их теореме единственности для аналитических и мероморфных функций. Результат будет иметь интересные приложения к изучению различных интегралов Стилтьеса в теории гармонических и аналитических функций и, в частности, интеграла Гильберта–Стилтьеса.
Доводиться, що якщо гармонiйна функцiя u, що задана в одиничному колi D комплексної площинi C, має кутовi межi на вимiрної множинi E одиничного кола, то її сполучена гармонiйна функцiя v в D також має кутовi межi п.в. на E i обидвi граничнi функцiї п.в. кiнцевi та вимiрнi на E. Потiм цей результат поширюється на довiльнi жорданова областi з границями, що спрямляються в термiнах кутових меж щодо природного параметра. Результат iстотно ґрунтується на теоремi Фату про кутовi межи обмежених аналiтичних функцiй та конструкцiї Лузiна i Привалова до їх теоремi єдиностi для аналiтичних i мероморфних функцiй. Результат буде мати цiкавi додатки до вивчення рiзних iнтегралiв Стiлтьєса в теорiї гармонiйних i аналiтичних функцiй i, зокрема, iнтеграла Гiльберта–Стiлтьєса.
|
| first_indexed | 2025-12-07T16:00:34Z |
| format | Article |
| fulltext |
ISSN 1683-4720 Працi IПММ НАН України. 2017. Том 31
УДК 517.54
c⃝2017. V. I. Ryazanov
ON THE BOUNDARY BEHAVIOR OF CONJUGATE
HARMONIC FUNCTIONS
It is proved that if a harmonic function u on the unit disk D in C has angular limits on a measurable
set E of the unit circle, then its conjugate harmonic function v in D also has (finite !) angular limits a.e.
on E and both boundary functions are measurable on E. The result is extended to arbitrary Jordan
domains with rectifiable boundaries in terms of angular limits and of the natural parameter. This
result is essentially based on the Fatou theorem on angular limits of bounded analytic functions and
on the construction of Luzin and Priwalow to their uniqueness theorem for analytic and meromorphic
functions. The result will have interesting applications to the study of the various Stieltjes integrals in
the theory of harmonic and analytic functions and, in particular, of the Hilbert–Stieltjes inyegral.
Key words: correlation, boundary behavior, conjugate harmonic functions, rectifiable Jordan curves,
angular limits, boundary value problems.
2010 Mathematics Subject Classification: Primary 30C62, 31A05, 31A20, 31A25, 31B25; Secon-
dary 30E25, 31C05, 34M50, 35F45, 35Q15.
1. Introduction.
First of all, recall that a path in D := {z ∈ C : |z| < 1} terminating at ζ = eiϑ ∈ ∂D
is called nontangential at ζ if its part in a neighborhood of ζ lies inside of an angle in
D with the vertex at ζ. Hence limits along all nontangential paths at ζ are also named
angular at ζ. The latter is a traditional tool of the geometric function theory, see e.g.
monographs [1]–[6]. Note that every closed rectifiable Jordan curve has a tangent a.e.
with respect to the natural parameter and the angular limit has the same sense at its
points with a tangent.
It is known the very delicate fact due to Lusin that harmonic functions in the unit
circle with continuous (even absolutely continuous !) boundary data can have conjugate
harmonic functions whose boundary data are not continuous functions, furthemore,
they can even be even not essentially bounded in neighborhoods of each point of the
unit circle, see e.g. Theorem VIII.13.1 in [7]. Thus, a correlation between boundary
data of conjugate harmonic functions is not a simple matter, see also I.E in [3].
Denote by hp, p ∈ (0,∞), the class of all harmonic functions u in D with
sup
r∈(0,1)
2π∫
0
|u(reiϑ)|p dϑ
1
p
< ∞ .
It is clear that hp ⊆ hp
′ for all p > p′ and, in particular, hp ⊆ h1 for all p > 1.
Remark 1. It is important that every function in the class h1 has a.e. nontangential
boundary limits, see e.g. Corollary IX.2.2 in [8].
117
V. I. Ryazanov
It is also known that a harmonic function u in D can be represented as the Poisson
integral
u(reiϑ) =
1
2π
2π∫
0
1− r2
1− 2r cos(ϑ− t) + r2
φ(t) dt (1.1)
with a function φ ∈ Lp(0, 2π), p > 1, if and only if u ∈ hp, see e.g. Theorem IX.2.3
in [8]. Thus, u(z) → φ(ϑ) as z → eiϑ along any nontangential path for a.e. ϑ, see e.g.
Corollary IX.1.1 in [8]. Moreover, u(z) → φ(ϑ0) as z → eiϑ0 at points ϑ0 of continuity
of the function φ, see e.g. Theorem IX.1.1 in [8].
Note also that v ∈ hp whenever u ∈ hp for all p > 1 by the M. Riesz theorem, see
[9], see also Theorem IX.2.4 in [8]. Generally speaking, this fact is not trivial but it
follows immediately for p = 2 from the Parseval equality, see e.g. the proof of Theorem
IX.2.4 in [8]. The case u ∈ h1 is more complicated.
The correlation of the boundary behavior of conjugate harmonic functions outside
the classes hp was not investigated at all. This is just the subject of the present article.
2. The case of the unit disk with respect to the arc length.
Here we apply in a certain part a construction of Luzin–Priwalow from the proof
of their theorem on the boundary uniqueness for analytic functions, see [10], see also
[3], Section III.D.1, and [6], Section IV.2.5.
Theorem 1. Let u : D → R be a harmonic function that has angular limits on a
measurable set E of the unit circle ∂D. Then its conjugate harmonic functions v have
(finite !) angular limits a.e. on E and both boundary functions are measurable on E.
Remark 2. By the Luzin–Priwalow uniqueness theorem for meromorphic functions
u as well as v cannot have infinite angular limits on a subset of D of a positive measure,
see Section IV.2.5 in [6].
Proof. By Remark 2 we may consider that angular limits of u are finite everywhere
on the set E. Moreover, the measurable set E admits a countable exhaustion by measure
of the arc length with its closed subsets, see e.g. Theorem III(6.6) in [11], and hence
with no loss of generality we may also consider that E is compact, see e.g. Proposition
I.9.3 in [12].
Following [3], Section III.D.1, we set, for ζ ∈ ∂D,
Sζ =
{
z ∈ D : |z| > 1√
2
, | arg (ζ − z) | < π
4
}
(2.1)
and
D =
∪
ζ∈E
Sζ ∪D∗ (2.2)
where
D∗ =
{
z ∈ C : |z| ≤ 1√
2
}
.
118
On the boundary behavior of conjugate harmonic functions
It is easy geometrically to see that ∂D contains E and it is a rectifiable Jordan curve
because ∂D\E is open set and hence it consists of a countable collection of arcs of ∂D,
see the corresponding illustrations in [3], Section III.D.1.
By the construction, the radii of D to every ζ ∈ E belong to D and the well defined
real-valued function φ(ζ) := lim
n→∞
φn(ζ), φn(ζ) := u(rnζ), n = 1, 2, . . ., with arbitrary
sequence rn → 1− 0 as n → ∞, is measurable, see e.g. Corollary 2.3.10 in [13]. Thus,
by the known Egorov theorem, see e.g. Theorem 2.3.7 in [13], with no loss of generality
we may assume that φn → φ uniformly on E and that φ is continuous on E, see e.g.
Section 7.2 in [14].
Let us consider the sequence of the functions
ψn(ζ) := sup
z∈Sζ∩Dn
ζ
|u(z)− φ(ζ)| , ζ ∈ E , (2.3)
where Dn
ζ = {z ∈ C : |z − ζ| < εn} with εn ↘ 0 as n → ∞. First of all, ψn(ζ) → 0
as n → ∞ for every ζ ∈ E. Moreover, the functions ψn(ζ) are measurable again by
Corollary 2.3.10 in [13] because of ψn(ζ) = lim
m→∞
ψmn(ζ) asm→ ∞ where the functions
ψmn(ζ) := max
z∈Sζ∩Rmn
ζ
|u(z)− φ(ζ)| , Rmnζ := Dn
ζ \Dn+m
ζ , ζ ∈ E , (2.4)
are continuous. Indeed, ψmn(ζ) coincide with the Hausdorff distance between the compact
sets u(Sζ ∩ Rmnζ ) and {φ(ζ)}, see e.g. Theorem 2.21.VII in [15], and any distance
is continuous with respect to its variables, recall that both functions u and φ are
continuous.
Again by the Egorov theorem with no loss of generality we may consider that ψn → 0
uniformly on E. The latter implies that the restriction U of the harmonic function u to
the domain D is bounded. Indeed, let us assume that there exists a sequence of points
zn ∈ D such that |u(zn)| ≥ n, n = 1, 2, . . .. With no loss of generality we may consider
that zn → ζ ∈ E because the function u is bounded on the compact subsets of D and
by the construction E = ∂D ∩ ∂D and E is compact. Moreover, by the construction
of D, we also may consider that zn ∈ Sζn , ζn ∈ E, n = 1, 2, . . . and that ζn → ζ as
n→ ∞. Consequently, it should be that u(zn) → φ(ζ) because ψn(ζn) → 0 as ζn → ζ,
see e.g. Theorem 7.1(2) and Proposition 7.1 in [14]. The latter conclusion contradicts
the above assumption.
Further, by the construction the domain D is simply connected and hence by the
Riemann theorem there exists a conformal mapping w = ω(z) of D onto D, see e.g.
Theorem II.2.1 in [8]. Note that the function U∗ := U ◦ ω−1 is a bounded harmonic
function in D and there exists its conjugate harmonic function V∗ in D, i.e. F :=
U∗ + i V∗ is an analytic function in D. Let N be a positive number that is greater
than sup
w∈D
|U∗(w)| = sup
z∈D
|U(z)|. Then the analytic function g(w) := F (w)/(N −F (w)),
w ∈ D, is bounded. Thus, by the Fatou theorem, see e.g. Corollary III.A in [3], g has
finite angular limits as w → W for a.e. W ∈ ∂D. By Remark 2 these limits cannot be
119
V. I. Ryazanov
equal to 1 on a subset of ∂D of a positive measure. Consequently, the function F (w)
has also (finite!) angular limits as w →W for a.e. W ∈ ∂D.
Let us consider the analytic function f = F ◦ ω given in the domain D. By the
construction Re f = U = u|D and hence V := Im f is its conjugate harmonic function
in D. By the standard uniqueness theorem for analytic functions, we have that V = v|D
where v is a conjugate harmonic function for u in D. Recall that the latter is unique
up to an additive constant. Thus, it remains to prove that the function f(z) has (finite
!) angular limits as z → ζ for a.e. ζ ∈ E. For this goal, note that the rectifiable curve
∂D has tangent a.e. with respect to its natural parameter. It is clear that tangents at
points ζ ∈ E to ∂D (where they exist !) coincide with the corresponding tangents at ζ
to ∂D.
By the Caratheodory theorem ω can be extended to a homeomorphism of D onto
D and, since ∂D is rectifiable, by the theorem of F. and M. Riesz length ω−1(E) = 0
whenever E ⊂ ∂D with length E = 0, see e.g. Theorems II.C.1 and II.D.2 in [3]. By the
Lindelöf theorem, see e.g. Theorem II.C.2 in [3], if ∂D has a tangent at a point ζ, then
arg [ω(ζ)− ω(z)]− arg [ζ − z] → const as z → ζ .
In other words, the conformal images of sectors in D with a vertex at ζ ∈ ∂D is
asymptotically the same as sectors in D with a vertex at w = ω(ζ) ∈ ∂D up to
the corresponding shifts and rotations. Consequently, nontangential paths in D are
transformed under ω−1 into nontangential paths in D and inversely at the corresponding
points of ∂D and ∂D.
Thus, in particular, v(z) has finite angular limits φ∗(ζ) for a.e. ζ ∈ E. Moreover, the
function φ∗ : E → R is measurable because φ∗(ζ) = lim
n→∞
vn(ζ) where vn(ζ) := v(rnζ),
n = 1, 2, . . ., with rn → 1− 0 as n→ ∞, see e.g. Corollary 2.3.10 in [13]. 2
In particular, we have the following consequence of Theorem 1.
Corollary 1. Let u : D → R be a harmonic function that has angular limits a.e.
on the unit circle ∂D. Then its conjugate harmonic functions v in D also have angular
limits a.e. on ∂D and both boundary functions are measurable.
By Remark 1 we have also the next consequence of Theorem 1.
Corollary 2. Let u : D → R be a harmonic function in the class h1. Then its
conjugate harmonic functions v : D → R have (finite !) angular limits v(z) → φ(ζ) as
z → ζ for a.e. ζ ∈ ∂D.
3. The case of rectifiable Jordan domains.
Theorem 2. Let D be a Jordan domain in C with a rectifiable boundary and u :
D → R be a harmonic function that has angular limits on a measurable set E of ∂D
with respect to the natural parameter. Then its conjugate harmonic functions v : D → R
also have (finite !) angular limits a.e. on E with respect to the natural parameter and
both boundary functions are measurable on E with respect to this parameter.
120
On the boundary behavior of conjugate harmonic functions
Proof. Again by the Riemann theorem there exists a conformal mapping w = ω(z)
of D onto D and by the Caratheodory theorem ω can be extended to a homeomorphism
of D onto D. As known, a rectifiable curves have tangent a.e. with respect to the natural
parameter. Hence ∂D has a tangent at every point ζ of the set E except its subset E
with length E = 0. By the Lindelöf theorem, for every ζ ∈ E \ E ,
arg [ω(ζ)− ω(z)]− arg [ζ − z] → const as z → ζ .
Thus, the harmonic function u∗ := u ◦ ω−1 given in D has angular limits φ∗(w) at all
points w of the set E∗ := ω(E \ E) ⊆ ∂D. Consequently, by Theorem 1 its conjugate
harmonic function v∗ : D → R has (finite !) angular limits ψ∗(w) at a.e. point w ∈ E∗
and the boundary functions φ∗ : E∗ → R and ψ∗ : E∗ → R are measurable. The
harmonic function v := v∗ ◦ω is conjugate for u because the function f := f∗ ◦ω, where
f∗ := u∗ + v∗, is analytic. Finally, by theorems of Lindelöf and F. and M. Riesz v has
(finite !) angular limits ψ(ζ) = ψ∗(ω(ζ)) at a.e. point ζ ∈ E.
The boundary functions φ = φ∗◦ω and ψ = ψ∗◦ω of u and v on E, correspondingly,
are measurable functions on E because φ(ζ) = lim
n→∞
φn(ζ) for all ζ ∈ E and ψ(ζ) =
lim
n→∞
ψn(ζ) for a.e. ζ ∈ E, where the functions φn(ζ) := u∗(rnω(ζ)) and ψn(ζ) :=
v∗(rnω(ζ)) with rn → 1− 0 as n→ ∞ are continuous, see e.g. Corollary 2.3.10 in [13].
�
Corollary 3. Let D be a Jordan domain in C with a rectifiable boundary and
u : D → R be a harmonic function that has angular limits a.e. on ∂D with respect to
the natural parameter. Then its conjugate harmonic functions v : D → R also have
(finite !) angular limits a.e. on ∂D and both boundary functions are measurable on E
with respect to the natural parameter.
Remark 3. These results can be extended to domains whose boundaries consist of
a finite number of mutually disjoint rectifiable Jordan curves (through splitting into a
finite collection of Jordan’s domains !).
The established facts can be applied to various boundary value problems for harmonic
and analytic functions in the plane, see e.g. [16]–[19].
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ArXiv 1711.02717v7 [math.CV], 18 pp.
В. И. Рязанов
О граничном поведении сопряженных гармонических функций.
Доказывается, что если гармоническая функция u, заданная в единичном круге D комплексной
плоскости C, имеет угловые пределы на измеримом множестве E единичной окружности, то
ее сопряженная гармоническая функция v в D также имеет угловые пределы п.в. на E и обе
граничные функции п.в. конечны и измеримы на E. Затем этот результат распространяется на
произвольные жордановы области со спрямляемыми границами в терминах угловых пределов
относительно естественного параметра. Результат существенно основывается на теореме Фату об
угловых пределах ограниченных аналитических функций и конструкции Лузина и Привалова к
их теореме единственности для аналитических и мероморфных функций. Результат будет иметь
интересные приложения к изучению различных интегралов Стилтьеса в теории гармонических
и аналитических функций и, в частности, интеграла Гильберта–Стилтьеса.
Ключевые слова: корреляция, граничное поведение, сопряженные гармонические функции,
спрямляемые жордановы кривые, угловые пределы, краевые задачи.
В. I. Рязанов
Про граничну поведiнку пов’язаних гармонiйних функцiй.
Доводиться, що якщо гармонiйна функцiя u, що задана в одиничному колi D комплексної площинi
C, має кутовi межi на вимiрної множинi E одиничного кола, то її сполучена гармонiйна функцiя
v в D також має кутовi межi п.в. на E i обидвi граничнi функцiї п.в. кiнцевi та вимiрнi на E.
Потiм цей результат поширюється на довiльнi жорданова областi з границями, що спрямляються
в термiнах кутових меж щодо природного параметра. Результат iстотно ґрунтується на теоремi
Фату про кутовi межи обмежених аналiтичних функцiй та конструкцiї Лузiна i Привалова до їх
122
On the boundary behavior of conjugate harmonic functions
теоремi єдиностi для аналiтичних i мероморфних функцiй. Результат буде мати цiкавi додатки
до вивчення рiзних iнтегралiв Стiлтьєса в теорiї гармонiйних i аналiтичних функцiй i, зокрема,
iнтеграла Гiльберта–Стiлтьєса.
Ключовi слова: кореляцiя, гранична поведiнка, пов’язанi гармонiйнi функцiї, жордановi кривi,
що спрямляються, кутовi границi, крайовi задачi.
Institute of Applied Mathematics and Mechanics,
National Academy of Sciences of Ukraine, Slavyansk, Ukraine
vl.ryazanov1@gmail.com
Получено 07.11.17
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| id | nasplib_isofts_kiev_ua-123456789-145115 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1683-4720 |
| language | English |
| last_indexed | 2025-12-07T16:00:34Z |
| publishDate | 2017 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Ryazanov, V.I. 2019-01-15T18:48:12Z 2019-01-15T18:48:12Z 2017 On the boundary behavior of conjugate harmonic functions / V.I. Ryazanov // Праці Інституту прикладної математики і механіки НАН України. — Слов’янськ: ІПММ НАН України, 2017. — Т. 31. — С. 117-123. — Бібліогр.: 19 назв. — англ. 1683-4720 2010 Mathematics Subject Classification: Primary 30C62, 31A05, 31A20, 31A25, 31B25; Secondary 30E25, 31C05, 34M50, 35F45, 35Q15. https://nasplib.isofts.kiev.ua/handle/123456789/145115 517.54 It is proved that if a harmonic function u on the unit disk D in C has angular limits on a measurable set E of the unit circle, then its conjugate harmonic function v in D also has (finite !) angular limits a.e. on E and both boundary functions are measurable on E. The result is extended to arbitrary Jordan domains with rectifiable boundaries in terms of angular limits and of the natural parameter. This result is essentially based on the Fatou theorem on angular limits of bounded analytic functions and on the construction of Luzin and Priwalow to their uniqueness theorem for analytic and meromorphic functions. The result will have interesting applications to the study of the various Stieltjes integrals in the theory of harmonic and analytic functions and, in particular, of the Hilbert–Stieltjes inyegral. Доказывается, что если гармоническая функция u, заданная в единичном круге D комплексной плоскости C, имеет угловые пределы на измеримом множестве E единичной окружности, то ее сопряженная гармоническая функция v в D также имеет угловые пределы п.в. на E и обе граничные функции п.в. конечны и измеримы на E. Затем этот результат распространяется на произвольные жордановы области со спрямляемыми границами в терминах угловых пределов относительно естественного параметра. Результат существенно основывается на теореме Фату об угловых пределах ограниченных аналитических функций и конструкции Лузина и Привалова к их теореме единственности для аналитических и мероморфных функций. Результат будет иметь интересные приложения к изучению различных интегралов Стилтьеса в теории гармонических и аналитических функций и, в частности, интеграла Гильберта–Стилтьеса. Доводиться, що якщо гармонiйна функцiя u, що задана в одиничному колi D комплексної площинi C, має кутовi межi на вимiрної множинi E одиничного кола, то її сполучена гармонiйна функцiя v в D також має кутовi межi п.в. на E i обидвi граничнi функцiї п.в. кiнцевi та вимiрнi на E. Потiм цей результат поширюється на довiльнi жорданова областi з границями, що спрямляються в термiнах кутових меж щодо природного параметра. Результат iстотно ґрунтується на теоремi Фату про кутовi межи обмежених аналiтичних функцiй та конструкцiї Лузiна i Привалова до їх теоремi єдиностi для аналiтичних i мероморфних функцiй. Результат буде мати цiкавi додатки до вивчення рiзних iнтегралiв Стiлтьєса в теорiї гармонiйних i аналiтичних функцiй i, зокрема, iнтеграла Гiльберта–Стiлтьєса. en Інститут прикладної математики і механіки НАН України Праці Інституту прикладної математики і механіки НАН України On the boundary behavior of conjugate harmonic functions О граничном поведении сопряженных гармонических функций Про граничну поведiнку пов’язаних гармонiйних функцiй Article published earlier |
| spellingShingle | On the boundary behavior of conjugate harmonic functions Ryazanov, V.I. |
| title | On the boundary behavior of conjugate harmonic functions |
| title_alt | О граничном поведении сопряженных гармонических функций Про граничну поведiнку пов’язаних гармонiйних функцiй |
| title_full | On the boundary behavior of conjugate harmonic functions |
| title_fullStr | On the boundary behavior of conjugate harmonic functions |
| title_full_unstemmed | On the boundary behavior of conjugate harmonic functions |
| title_short | On the boundary behavior of conjugate harmonic functions |
| title_sort | on the boundary behavior of conjugate harmonic functions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/145115 |
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