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c Pleiades Publishing, Ltd., 2017. ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2017, Vol. 38, No. 3, pp. 456–465. 

Jump Problem for Differentiable Functions with Applications D. B. Katz* and B. A. Kats** (Submitted by A. M. Elizarov) Kazan (Volga Region) Federal University, ul. Kremlevskaya 18, Kazan, Russia Received June 24, 2016

Abstract—We solve a boundary value problem on non-rectifiable contours for differentiable functions, and show that it gives us a tool for solving a number of problems for analytical and generalized analytical functions. DOI: 10.1134/S1995080217030143 Keywords and phrases: Non-rectifiable curve, generalized curvilinear integral, jump problem.

Dedicated to Professor Leonid Aleksandrovitch Aksentiev INTRODUCTION We consider a closed Jordan curve Γ on the complex plane C dividing it onto domains D + and D − , ∞ ∈ D − . Generally speaking, this curve is not rectifiable. We study first a problem on evaluation of differentiable in C \ Γ function φ(z) such that it has limit values φ± (t) from domains D ± at any point t ∈ Γ, and these limit values are related by equality φ+ (t) − φ− (t) = j(t),

t ∈ Γ,

(1)

where j(t) is given function (the jump). Clearly, solution of the problem cannot be unique. In particular, we can multiply any its solution by a smooth function with compact support equaling to 1 in a neighborhood of Γ, and obtain a solution with compact support. Below we consider only solutions with compact supports. A solution φ of this problem generates distribution  ∂(φω) ∞ dz dz, (2) C (C)  ω → − ∂z C

and we will see that under certain additional assumptions the jump j(t) determines this distribution uniquely. The distribution (2) is a generalization of the curvilinear integration for non-rectifiable curves. Indeed, if Γ would be rectifiable, then     ∂(φω) ∂(φω) dz dz, dz dz φ+ ω dt = − φ− ω dt = ∂z ∂z Γ

D+

Γ

by Stokes formula, and, consequently, 

 jω dt = −

Γ * **

C

E-mail: [email protected] E-mail: [email protected]

456

∂(φω) dz dz. ∂z

D−

JUMP PROBLEM FOR DIFFERENTIABLE FUNCTIONS

457

In general case the right hand part of the last equality can be used as a definition of the left one. We will apply this generalized integration as a tool for solving of certain boundary value problems. In the section 1 we consider the problem (1) and related distribution (2). Then we solve the Riemann problem for analytical (section 2) and β-analytical (section 3) functions by means of the generalized integrations (2). 1. MARCINKIEWICZ EXPONENTS AND GENERALIZED INTEGRATIONS The concept of Marcinkiewicz exponents is introduced by D.B. Katz [1]. Here we offer its local version. In what follows Γ has zero plane Jordan measure, i.e., domains D ± are measurable. Definition 1. Inner and outer Marcinkiewicz exponents of curve Γ at its point t are m± (Γ; t) := sup{p : lim Ip± (t; r) < ∞}, r→0

where Ip± (t; r)

 =

dx dy , dist (x + iy, Γ) p

B ± (t;r)

r > 0, p > 0, B(t; r) := {z : |z − t| < r}, B ± (t; r) := B(t; r) ∩ D ± . The Marcinkiewicz exponent equals to m(Γ; t) := max{m+ (Γ; t), m− (Γ; t)}. Clearly, these values are invariant with respect to parallel transfer, rotation and similarity transforms. The name “Marcinkiewicz exponents” is connected with Marcinkiewicz’s characterization of sophisticated sets in terms of certain integrals over their complements (see, for instance, [2]). Theorem 1. The Marcinkiewicz exponents of any plane curve satisfy inequalities 2 − dmΓ ≤ m± (Γ; t) ≤ 1, where dmΓ is upper Minkowski dimension of Γ (see, for instance, [3]). If a curve Γ is rectifiable in a neighborhood of a point t ∈ Γ, then m± (Γ; t) = 1. Proof. We call to our mind one of definitions of the Minkowski dimension. We consider all non-overlapping squares of the form Q = {z = x + iy : mε ≤ x < (m + 1)ε, nε ≤ y < (n + 1)ε}, where ε > 0, m, n ∈ Z. Let N (Γ; ε) be number of squares such that Q ∩ Γ = ∅. Then the upper Minkowski dimension of Γ is ln N (Γ; ε) . dmΓ := lim sup − ln ε ε→0 This value is known also as box-counting dimension, Kolmogorov dimension, upper metric dimension and so on. It does not exceed 2 for any plane set; the Minkowski dimension of any rectifiable curve equals to 1. Then we consider the Whitney partition of C \ Γ (see, for instance, [2]). It consists of non-overlapping dyadic squares Q (i.e., squares vith vertices at points 2−n m and with sides 2−n , m, n ∈ Z) such that diamQ ≤ dist(Q, Γ) ≤ cdiamQ, where c is positive constant independent on Q; in what follows we keep this notation for various positive constants. Clearly, for a square Q with side 2−n from this partition  2−2n dxdy ≤ c , (dist(z, Γ))p (2−n )p Q

and Ip± (t; r)

≤c

∞ 

wn · 2n(p−2) ,

n=1

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where wn is a number of squares with side 2−n in the Whitney partition. For any d > dmΓ we have wn ≤ c2nd for sufficiently large n (see, for instance, [4]). Therefore, integrals Ip± (t; r) converge if ∞ n(p−2+d) < ∞, i.e., for p ≥ 2 − dmΓ. The inequalities m± (Γ; t) ≥ 2 − dmΓ are proved. n=1 2 There exist curves Γ such that {t ∈ Γ : m± (Γ; t) > 2 − dmΓ} = ∅. A reader can find examples of that curves in [5]. Then we will prove that I1+ (t; r) = ∞. Any segment λ ⊂ B + (t; r) can be maid horizontal by means of rotation. We consider that the rotation transform is maid, and λ = {x + iy : x1 ≤ x ≤ x2 , y = y0 }, x1 < x2 . Moreover, we can choose the segment (and the rotation) so that λ lies higher set Γ ∩ B(t; r), and put δ(x) = max{y < y0 : x + iy ∈ Γ} for x1 < x < x2 . This function is upper semi-continuous, and, consequently, measurable. Let Δ = {x + iy ∈ B + (t; r) : x1 < x < x2 , δ(x) < y < y0 }. Clearly,  I1+ (t; r) =

dxdy ≥ dist(z, Γ)

B + (t;r)



dxdy ≥ dist(z, Γ)

x2 dx x1

Δ

y0

dy , y − δ(x)

δ(x)

but the last integral diverges. Hence, m+ (Γ; t) ≤ 1. The proof of inequality m− (Γ; t) ≤ 1 is analogous. Let Γ contain a rectifiable arc γ, t ∈ γ. As known, any rectifiable arc has Minkowski dimension equaling 1 (see, for instance, [3]). Obviously, m(Γ; t) = m(γ; t) and 1 = 2 − dmγ ≤ m(γ; t) ≤ 1. Thus, m(Γ; t) = 1. Theorem is proved. ¨ We introduce a local version of the Holder condition. Definition 2. Let v(t) be a real function defined on Γ, 0 < ν ≤ v(t) < 1, t ∈ Γ. A defined on Γ function f (t) belongs to class Hvloc (Γ) if for any point t ∈ Γ there exists r = r(t) > 0 such that restriction of f on Γ ∩ B(t; r) satisfies the Holder ¨ condition with exponent v(t). ¨ class consisting of defined on set A ⊂ C functions f By Hμ (A) we denote customary Holder satisfying condition   |f (t) − f (t )|   : t, t ∈ A, t = t < ∞. sup |t − t |μ Theorem 2. Let j ∈ Hvloc (Γ). Then the boundary value problem (1) has a solution such that its derivatives are integrable in D + and D − in any degree p satisfying inequality p
1. For any point t ∈ Γ there exists r = + (t; r(t)) < r(t) > 0 such that f |Γ∩B(t;kr(t)) ∈ Hv(t) (Γ ∩ B(t; kr(t)), and at least one of conditions Im(t)

− (t; r(t)) < ∞. The disks B(t, r(t)) cover Γ, and we can select its finite covering by disks Bj = ∞, Im(t) B(tj , r(tj )), j = 1, 2, . . . , n. Let function σj (z) ∈ C ∞ (C) equals to 1 in Bj , zero in C \ B(tj , kr(tj )), and it is positive in ring B(tj , kr(tj )) \ Bj . Then sum S(z) = nj=1 σj (z) is enclosed between 1 and   n in domain nj=1 Bj := B ⊃ Γ, vanishes outside of domain B := nj=1 B(tj , kr(tj )), and is positive in B \ B. Let function ω(z) ∈ C0∞ have support inside domain B and be equal to 1 in B. We put sj (z) := σj (z)ω(z)/S(z) for z ∈ B and sj (z) := 0 outside of this domain. Thus, there is built a system  of functions sj (z) ∈ C0∞ (C) such that suppsj ⊂ B(tj , kr(tj ) and nj=1 sj (t) = 1 for t ∈ Γ.

Now we define on union of set Γj := Γ ∩ B(tj , kr(tj )) and circle Cj bounding disk B(tj , kr(tj )) function fj equaling to f sj on Γj and to zero on Cj . Clearly, fj ∈ Hv(tj ) (Cj ∪ Γj ), and we can apply to it the Whitney extension operator (see, for instance, [2]). By virtue of known properties of this operator we obtain a function fjw (z) such that —it is defined on the whole complex plane and coincides with fj on Cj ∪ Γj ; ¨ —it satisfies the Holder conditon with exponent v(tj ) in the whole complex plane; LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 38 No. 3 2017

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—it has continuous partial derivatives in C \ Γj , and |∇fjw (z)| ≤

c dist

1−v(tj )

(z, Γ)

.

In addition, outside of B(tj , kr(tj )) function fjw (z) is identical zero, i.e., this extension coincides with  f sj not only on Γj , but on the whole curve Γ, and nj=1 fjw (t) = f (t) for t ∈ Γ. + (t; r(t)) < ∞. Then we put ψj (z) := fjw (z)χ+ (z), where Assume that at point tj we have Im(t j)

χ+ (z) is characteristic function of domain D + . Clearly, ψj+ (t) − ψj− (t) = f (t)sj (t), and ∇ψj is integrable with degree p =

m(tj ) 1−v(tj ) .

t ∈ Γ,

− If Im(t (t; r(t)) < ∞, then we put ψj (z) := fjw (z) × j)

(χ+ (z) − 1), and it has the same properties.  Obviously, sum φ := nj=1 ψj satisfies relation (1). The value m(t) can be arbitrarily close to m(Γ; t), what concludes the proof. This theorem implies integrability of solution φ for v(t) > 1 − m(Γ; t),

t ∈ Γ.

Hence, distribution (2) is defined under condition (4). We temporarily denote this distribution i.e., φ

 ω(z) dz := − C

∂(φω) dz dz, ∂z

φ

(4) ·dz,

ω ∈ C ∞ (C).

Now we will show that under certain additional restrictions the distribution depends only on curve Γ and jump j. Theorem 3. If solutions φ1 and φ2 of the jump problem (1) have integrable derivatives and satisfy the Holder ¨ condition with exponent (5) μ > dmH Γ − 1   φ φ in neighborhoods of Γ in D + and D − , then 1 ·dz = 2 ·dz. Here dmH stands for the Hausdorff dimension. Proof. Let us cite first definition of the Hausdorff dimension. Let E be a bounded set on the complex plane. Its λ-dimensional Hausdorff r-content is defined as n  n 

λ λ rk : E ⊂ B(xk , rk ), xk ∈ E, 0 < rk ≤ r , Hr (E) := inf k=1

λ-dimensional Hausdorff measure equals to

k=1

Hλ (E)

:= lim Hrλ (E), and Hausdorff dimension is r↓0

dmH (E) := inf{λ ≥ 0 : Hλ (E) = 0}. As known (see, for instance, [3]), Hausdorff dimension of any set does not exceed its Minkowski dimension, and for rectifiable curves it equals to 1. We fix α ≥ dmH Γ such that Hα (Γ) = 0; then for  any αε > 0 we can cover Γ by a family of disks Bj = B(zj , rj ), j = 1, 2, . . . , such that rj < ε and j>0 rj < ε. We denote Λ boundary of the union ¨ condition with B = ∪j>0 Bj , and φ = φ1 − φ2 . Clearly, φ is continuous on Γ and satisfies the Holder exponent μ in its neighborhood. Then for any ω ∈ C ∞ (C) we have φ     1 φ2 ∂(φω) . ω dz − ω dz ≤ dzdz + φωdz B ∂z Λ

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The first term of the right side vanishes for ε → 0 because Γ has null plane measure. We will prove that the second term has null limit, too, what will conclude the proof. We assume without loss of generality that the family of covering disks is finite, and that none of them are covered by the union of the others. Let us enumerate these disks in decreasing order of their radiuses.   Then we denote Δ1 = B1 , Δ2 = B2 \ Δ1 , Δ3 = B3 \ 2k=1 Δk , Δ4 = B4 \ 3k=1 Δk , and so on. As a result, we represent B as union of finite number of non-overlapping simply connected domains Δk ⊂ Bk such that boundary λk = ∂Δk consists of circular arcs, and its length does not exceed 2πrk . Clearly,   φω dz = φω dz, Λ

and



 φω dz = λk

We have

k λ k

 (φ(z) − φ(zk )ω(z) dz.

φ(zk )ω(z) dz + λk

λk



 φ(zk )ω(z) dz = −

φ(zk ) Δk

λk

∂ω dz dz, ∂z

and due to boundedness of function φ ∂ω ∂z   φ(z )ω(z) dz ≤ cmes2 B, k k λk where mes2 stands for the plane Lebesgue measure, i.e., this sum vanishes for ε → 0. Finally,    μ+1  (φ(z) − φ(zk )ω(z) dz ≤ c rk ≤ c rkα k k k λk if μ > α − 1, and this sum also vanishes for ε → 0. Theorem is proved. ¨ Thus, if we restrict ourself by solutions of the problem (1) satisfying near Γ the Holder condition with exponent μ > dmH Γ − 1, then all these solutions generate one distribution. It depends on j only;  (S) we call it Stokes integral and denote Γ j(t) · dt. It is a simplification of the concept of integral over non-rectifiable curve introduced in [6]. Obviously, support of this distribution belongs to Γ. ¨ The solution from the proof of Theorem 2 satisfies in one-sided neighborhoods of Γ the Holder condition with exponent ν = min{v(t) : t ∈ Γ}.  (S) We introduce distribution Γ j(t) · dt by relation  (S) ∂(φω) dz dz. j(t)ω(t) dt = − ∂z C

Γ

Its uniqueness features are the same. 2. THE RIEMANN BOUNDARY VALUE PROBLEM The Riemann boundary value problem for analytic functions is well known; see, for instance, [7]. This is problem of evaluation of analytic in CΓ function Φ(z) satisfying relation Φ+ (t) = G(t)Φ− (t) + g(t),

t ∈ Γ,

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 38 No. 3 2017

(6)

JUMP PROBLEM FOR DIFFERENTIABLE FUNCTIONS

461

where G, g are defined on Γ given functions, and Φ± (t) are limit values of Φ at a point t ∈ Γ from domains D ± correspondingly. If G ≡ 1, then this problem turns into the jump problem for analytical functions. The Riemann problem is reducible to the jump problem by means of factorization technique. In the classical results on this problem the curve Γ is assumed piecewise-smooth, and the main tool is the Cauchy type integral  g(t) dt 1 , z ∈ Γ. Φ(z) = 2πi t−z Γ

Φ± ,

and Φ+ (t) − Φ− (t) = f (t) for any t ∈ Γ (the PlemeljFor g ∈ Hμ (Γ) this integral has limit values Sokhotski formula). Hence, the Cauchy type integral solves the jump problem for analytic functions. If curve Γ is not rectifiable, then the integral over it is undefined. The first solution of the problem (6) on non-rectifiable curve (see [4] and recent survey [8]) was obtained without application of curvilinear integration. Here we solve this problem by means of the Stokes integral introduced in the previous section. We begin from the jump problem on non-rectifiable curves for analytic functions. Let g ∈ Hvloc (Γ), where v(t) satisfies conditions (4) and (7) v(t) > dmH Γ − 1, t ∈ Γ. Then the jump problem for differentiable functions (1) with j(t) ≡ g(t) has a solution φ(z) satisfying ¨ near Γ the Holder condition with exponent greater than dmH Γ − 1, and it generates the Stokes integral  (S) Γ g(t) · dt. We apply it to a function 1 − ω1 (t) , z ∈ Γ, t−z where ω1 ∈ C0∞ (C) equals to 1 in a neighborhood of z and suppω1 ∩ Γ = ∅. Obviously, ωz (t) ∈ C ∞ (C), 1 in a neighborhood of Γ and vanishes in a neighborhood of z. As it coincides with the Cauchy kernel t−z a result, we obtain Stokes–Cauchy integral ωz (t) =

1 Φ(z) = 2πi

(S) Γ

1 g(t) dt = t−z 2πi

(S) g(t)ωz (t) dt,

z ∈ Γ.

(8)

Γ

Let B be sufficiently large disk containing Γ and suppφ, and B(z, r) be a small disk such that Γ ∩ B(z, r) = ∅ and suppω1 ⊂ B(z, r). Then by definition of the Stokes integral 1 2πi

(S) Γ

1 g(t) dt =− t−z 2πi

 C

1 ∂(φωz ) dz dz = − ∂z 2πi



B\B(z,r)

for any sufficiently small r > 0. We assume that derivative product

∂φ 1 ∂ζ ζ−z

1 ∂φ dζ dζ − 2πi ∂ζ ζ − z

∂φ ∂ζ

C

We transform the integral over B(z, r) by the Stokes formula, and obtain   ∂(φωz ) φ(ζ) dζ dζ dζ = − . ζ−z ∂ζ |ζ−z|=r

B(z,r)

Hence,

 lim

r→0 B(z,r)

∂(φωz ) dζ dζ ∂ζ

B(z,r)

is integrable in B wth degree p > 2. Then

is integrable there as function of ζ, and   ∂φ dζ dζ ∂φ dζ dζ = . lim r→0 ∂ζ ζ − z ∂ζ ζ − z B\B(z,r)



∂(φωz ) dζ dζ = −2πiφ(z). ∂ζ

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Thus, there is valid Cauchy–Green formula for the Stokes integral 1 2πi

(S) Γ

1 g(t) dt = φ(z) − t−z 2πi

 C

∂φ dζ dζ . ∂ζ ζ − z

(9)

is integrable with degree p > 2, then it The last integral term is well known, see, for instance, [9]. If ∂φ ∂ζ is continuous in the whole complex plane, and function (8) satisfies boundary value condition (6) with G ≡ 1, i.e., the jump problem for analytical functions. The relation (3) means, that derivative

∂φ ∂ζ

is integrable with degree p > 2 under retriction

1 (10) v(t) > 1 − m(Γ; t), t ∈ Γ. 2 Thus, inequality (10) is sufficient condition for solvability of the jump problem for analytic functions. It is better than known condition g ∈ Hν (Γ), ν > 12 dmΓ (see [4]) if {t ∈ Γ : m± (Γ; t) > 2 − dmΓ} = ∅. Let us consider uniqueness of the solution. If the curve Γ is rectifiable, then uniqueness of solution of the jump problem for analytic functions follows from two fundamental results – Liouville and Painleve´ theorems. According the Painleve´ theorem (see, for instance, [10]), any function, which is continuous in domain D ⊃ Γ and analytic in D \ Γ is analytic in D, too. Hence, difference of two solutions is analytic in C. Then it is constant by virtue of the Liouville theorem (see, for instance, [11]). If we solve the jump problem in the class of functions vanishing at ∞, then this constant equals to zero. If Γ is not rectifiable, then we have to replace the Painleve´ theorem in this consideration by the E.P. Dolzhenko theorem [12]. It asserts that a function F ∈ Hμ (D), which is analytic in in D \ Γ (here, as above, D is domain containing Γ), is analytic in D if the exponent μ satisfies inequality (5). Thus, the condition (5) defines uniqueness class for the jump problem on a non-rectifiable curve. We return to features of the last term of representation (9). If

∂φ ∂ζ

is integrable with degree p > 2,

¨ then it satisfies the Holder condition with exponent μ = 1 − 2/p in the whole complex plane (see [9]). According (3), we obtain μ 0, then general solution is Φ(z) = Φ0 (z) + X(z)P (z), where P (z) is arbitrary algebraic polynomial of degree κ; —if κ = 0, then Φ0 (z) is a unique solution; —if κ < 0, then the problem has a unique solutions under −κ solvability conditions. 3. RIEMANN BOUNDARY VALUE PROBLEM FOR β-ANALYTIC FUNCTIONS One of intrinsic and important for applications of Cauchy-Riemann equations is Beltrami equation ∂φ = μ∂φ, |μ(z)| < 1. As usually,



 ∂ 1 ∂ ∂ 1 ∂ +i , ∂ := −i . ∂ := 2 ∂x ∂y 2 ∂x ∂y For μ ≡ 0 it turns into the Cauchy–Riemann system, and its solutions are analytic functions. In 1985 A.B. Tungatarov [13] established that in the case μ(z) = β zz , 0 ≤ β < 1, the solutions of the Beltrami equation admit integral representation analogous to the Cauchy integral. These solutions are called β-analytic functions. That functions and β−analytic analog of the Cauchy integral were studied by R. Abreu-Blaya, J. Bory-Reyes, D. Pena-Pena and J.-M. Vilaire (see [14, 15] and references in the last paper). In particular, they investigated the Riemann boundary value problem for β-analytic functions on nonrectifiable curve. Here we improve the results of paper [15] in terms of the Marcinkiewicz exponent and the Stokes integral. If 0 ∈ D + and Γ is rectifiable, then the role of the Cauchy type integral for β-analytic functions plays function   β f (ζ) dζ ζf (ζ) dζ 1 , (14) + Φ(z) = θ 2(1 − β)πi ζ − z|z/ζ| 2(1 − β)πi ζ(ζ − z|z/ζ|θ ) Γ

Γ

where θ = 2β/(1 − β) (see [13, 14]). If curve Γ is not rectifiable, then we replace both integrals in this  (S)  (S) equality by Stokes ones, i.e., we apply distributions Γ f (t) · dt and Γ f (t) · dt to function ωz (ζ) =

(1 − ω1 (ζ))(1 − ω2 (ζ)) ζ − z|z/ζ|θ

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and ζωz (ζ)/ζ. Here ω1,2 ∈ C ∞ (C) are functions with sufficiently small compact supports equaling to 1 in neighborhoods of 0 and z correspondingly. In just the same way as the formula (9), we prove that under assumption f ∈ Hvloc (Γ) and restriction (10)   ζ ∂φ ∂φ dζ dζ 1 −β , (15) Φ(z) = φ(z) − 2(1 − β)πi ∂ζ ζ ∂ζ ζ − z|z/ζ|θ C

where φ is differentiable solution of the jump problem (1) with j ≡ f . Then we repeat the considerations of the previous section and conclude, that for G ≡ 1, f ∈ Hvloc (Γ), the boundary value problem is solvable in β-analytic functions under restriction (10). This restrictionis, generally speaking, weaker than the solvability condition from the paper [15]. In order to establish uniqueness of solution we need β-analytic generalizations for the Liouville and E.P. Dolzhenko theorems. That generalizations are proved in the paper [14]. As a result, we obtain the following β-analytic versions of Theorems 4 and 5. Theorem 6. Let G ≡ 1 and g ∈ Hvloc (Γ), where exponent v(t) satisfies condition (10). Then function 1 Φ(z) = 2(1 − β)πi

(S) Γ

β g(ζ) dζ + θ ζ − z|z/ζ| 2(1 − β)πi

(S) Γ

ζg(ζ) dζ , ζ(ζ − z|z/ζ|θ )

(16)

is unique vanishing at infinity point solution of the jump problem (6) in the class of β-analytic in C \ Γ functions satisfying in one-sided neighborhoods of Γ the Holder ¨ condition with exponent μ if 

2(1 − v(t)) 1 + β , t ∈ Γ. (17) dmH Γ − 1 < μ < min{v(t), 1 − m(Γ; t) 1−β Theorem 7. Let G, g ∈ Hvloc (Γ), exponents v(t) and μ satisfy conditions (10) and (17), G(t) = 0 for t ∈ Γ. Then dimension of set of vanishing at infinity point solutions of the Riemann boundary value problem in the class of β-analytic in C \ Γ functions satisfying in one-sided neighborhoods of Γ the Holder ¨ condition with exponent μ is determined by the winding number κ of G: —if κ > 0, then general solution is Φ(z) = Φ0 (z) + X(z)P (z|z|θ ), where Φ0 and X are certain β-analytic functions, and P (z) is arbitrary algebraic polynomial of degree κ; —if κ = 0, then solution is unique; —if κ < 0, then the problem has a unique solutions under −κ solvability conditions. 4. ACKNOWLEDGMENTS The authors are supported by Russian Foundation for Basic Research, grant no. 17-01-00018. REFERENCES 1. D. B. Katz, “The Marcinkiewicz exponent with applications,” in Proceedings of the 9th International Society for Analysis, its Applications, and Computations ISAAC Congress, Krakow, Aug. 5–9, 2013 (Springer, 2014), pp. 106–107. 2. E. M. Stein, Singular Integrals and Differential Properties of Functions (Princeton Univ. Press, Princeton, 1970). 3. K. J. Falconer, Fractal Geometry, 3rd ed. (Wiley, Chichester, UK, 2014). 4. B. A. Kats, “Riemann problem on a closed Jordan curve,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 68–80 (1984). 5. D. B. Katz, “New metric characteristics of non-rectifiable curves with applications,” Sib. J. Math. (in press). 6. R. Abreu-Blaya, J. Bory-Reyes, and B. A. Kats, “Integration over non-rectifiable curves and Riemann boundary value problems,” J. Math. Anal. Appl. 380, 177–187 (2011). 7. F. D. Gakhov, Boundary Value Problems (Nauka, Moscow, 1988) [in Russian]. 8. B. A. Kats, “The Riemann boundary value problem on non-rectifiable curves and related questions,” Complex Var. Elliptic Equ. 59, 1053–1069 (2014). LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 38 No. 3 2017

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9. I. N. Vekua, Generalized Analytical Functions (Nauka, Moscow, 1988) [in Russian]. 10. A. I. Markushevich, Selected Chapters of Theory of Analytic Functions (Nauka, Moscow, 1976) [in Russian]. 11. A. I. Markushevich, Theory of Functions of a Complex Variable (AMS Chelsea, Providence, RI, 2011). 12. E. P. Dolzhenko, “On ’rasing’ of singularities of analytic functions,” Usp. Mat. Nauk 18 (4), 135–142 (1963). 13. A. B. Tungatarov, “Properties of certain integral operator in classes of summable functions,” Izv. AN Kazakh. SSR, Ser. Fiz. Mat. 132 (5), 58–62 (1985). 14. R. Abreu-Blaya, J. Bory-Reyes and D. Pena-Pena, “On the jump problem for β-analytic functions,” Complex Var. Elliptic Equ. 51, 763–775 (2006). 15. R. Abreu-Blaya, J. Bory-Reyes and J.-M. Vilaire, “The Riemann boundary value problem for β-analytic functions over D-summable closed curves,” Int. J. Pure Appl. Math. 75, 441–453 (2012).

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