CC BY
74Probl. Anal. Issues Anal. Vol. 8 (26), No 1, 2019, pp. 65-71 65
DOI: 10.15393/j3.art.2019.5870
UDC 517.544
B.A. KATS, S.R. MlRONOVA, A. Yu. POGODINA
CAUCHY PROJECTORS ON NON-SMOOTH AND NON-RECTIFIABLE CURVES
Abstract. Let f (t) be defined on a closed Jordan curve r that divides the complex plane on two domains D+, D-, to € D-. Assume that it is representable as a difference f (t) = F+(t)-F-(t), t € r, where F±(t) are limits of a holomorphic in C \ r function F(z) for D± 9 z ^ t € r, F(to) = 0. The mappings f ^ F± are called Cauchy projectors.
Let Hv(r) be the space of functions satisfying on r the Holder condition with exponent v € (0,1]. It is well known that on any smooth (or piecewise-smooth) curve r the Cauchy projectors map Hv(r) onto itself for any v € (0,1), but for essentially non-smooth curves this proposition is not valid.
We will show that even for non-rectifiable curves the Cauchy projectors continuously map the intersection of all spaces Hv(r), 0 < v < 1 (considered as countably-normed Frechet space) onto itself.
Key words: Cauchy projectors, non-smooth curves, non-rectifiable curves
2010 Mathematical Subject Classification: 30E20
1. Introduction. Let r be a simple closed curve on the complex plane dividing it to two domains D+ and D-, to € D-. If r is rectifiable and a function f (t) is integrable over it, then the Cauchy type integral
C/(z):=¿ / f-* z € r. (1)
r
is defined. It represents an analytic in C \ r function that vanishes at the infinity. Let this function have limit values on r from both sides:
C±f (t) := lim Cf (z). (2)
D±3z^te r
(g) Petrozavodsk State University, 2019
These limits are called the Cauchy projectors.
The Cauchy type integral and the Cauchy projectors are of importance for the whole complex analysis. Their applications in Riemann-Hilbert boundary value problems are described in well-known monographs [1-3]. A contemporary results on this subject can be found in the survey [4].
We consider the Holder condition, i.e., the following restriction on a function f:
hv(f; r) := sup { lf : ti,2 e r, ti = t^ < rc, (3)
where the exponent v is positive, v e (0,1]. We denote by Hv(r) the set of all functions that satisfy this condition. It is a Banach space with the norm
|f ||v := sup{|f (t)| : t e r} + hv(f; r).
If r is a smooth curve and f e Hv(r), then the Cauchy projectors are representable as C±f = | (f ± Sf), where
Sf (t) := / ^
m J t — t r
is a singular integral with the Cauchy kernel.
The following result belongs to the base of the theory of singular integral equations [1,2]. Recall that a Ljapunov curve is a curve with Holder-continuous angle between its tangent and a fixed axis.
Theorem 1. ([5], p.199, item 4.5) Letacurve r be smooth. If f e Hv (r), then the singular integral operator S maps Hv (r), 0 < v < 1, onto itself. If r is a Ljapunov curve, then this mapping is bounded. If f e Hi(r), then S±f e Hv (r) for any v < 1.
Thus, the image of H1(r) is the set H1-0(r) := P| Hv(r). It is not a
v<1
Banach space, but any increasing sequence of exponents v1 < v2 < ■ ■ ■ < < vk < ... such that lim vk = 1 generates a family of semi-norms hvk (■; r)
in Hi-o(r).
In 1979, E. M. Dyn'kin [6] and T. Salimov [7] independently proved the following result for rectifiable (but not necessarily smooth) curves.
Theorem 2. Let a curve r be rectifiable. If f e Hv (r), v e ^, 1 , then
the boundary values C±f exist and belong to HM(r) for any ^ < 2v — 1, and this bound cannot be improved.
We see that on rectifiable curves the projectors C± do not map Hv(r) onto itself for 0 < v < 1, but they map H1-0(r) onto H1-0 (r).
In this paper, we study the action of the Cauchy projectors on coun-tably-normed space H1-0(r). In the sequel, we assume, without loss of generality, that the diameter of r is less than unit.
2. The Holder-Frechet space H1-0. Let A be a compact subset of the complex plane C with the diameter d < 1. We denote bt H1-0(A) the set of all functions on A that satisfy there the Holder condition with any exponent less than 1.
We consider a sequence v = {vk}£=0 of exponents such that 0 < v0 < < v1 < ■ ■ ■ < vk < ■ ■ ■ < 1, lim vk = 1, and norms
11/Ilk := sup{|f (t)| : t e A} + hvk(/; A), k = 0,1,... , (4)
and denote by H1-0(A; v) the set H1-0(A) equipped with these norms. As d < 1,
1/(t1) - /(t2)| > 1/(t1) - /(t2)| t t
> —r.---, t1,t2 e a,
|tl - Í2b+1 |tl - t
\Vk
i.e., the sequence of norms increases. Obviously, any two norms (4) are coordinated (see [8]), and H1-0(A; />) is a countably-normed Frechet space.
If r is a Ljapunov curve, then all mappings C±, S are bounded linear operators H1-0(r; £) M- H1-0(r;^). This fact follows from the proof of Theorem 1 .
There are many examples of compact integral operators in the space H1-0(r). For instance, the operator
Kf :=y f (t)fc(r,t)dr,
r
is compact (see [9,10]), if its kernel k(T, t) is continuous and satisfies the following condition of Holder type:
|k(T,t1) - k(T,t2)| ^ c(t)|t1 - t21, T,t1,t2 G r,
and the function c(t) is integrable over r. Then the singular integral equation
a(t)f (t) + b(t)S f (t) + Kf (t) = c(t), t G r,
with the desired function f and coefficients a,b,c G H1-0(r;^) keeps in the space H1-0(r; all its classic properties [1,2].
3. Non-smooth rectifiable curves. We consider non-smooth rectifiable curves r.
Theorem 3. If a simple closed curve r is rectifiable and a sequence V satisfies inequalities
1 1 + Vfc
vo > ^, vfc+i >—^—, k = 0,1,..., (5)
then the mappings C± are bounded linear operators
C± : Hi-o(r; V) ^ Hi-o(r; V).
Proof. Let f E H1-0(r; V). We apply the Whitney extension operator E0. As it is well-known (see, for instance, [11]), the obtained extension has the following properties:
- the function E0f is defined in the whole complex plane and E0f |r = f;
- if f E Hv(r), then Ef E Hv(C), and hv(E0f; C) = hv(f; r);
- the function E0f is differentiable with respect to real variables x and y in C \ r, and its first partial derivatives satisfy the estimate
hv (f; r) dist1-v (z, r)'
In the considered situation the last two properties are valid for any v < 1. We consider function u(z) that equals E0f (z) for z E D+ and to 0 for z E D-. The Cauchy type integral can be represented by means of the Stokes formula, as follows:
Cf (z)= u(z) - ¿ U tdS z E C \ r. (7)
D+
As the curve r is rectifiable, the function dist-p(z; r) is integrable in D+
du
for any p < 1 (see, for instance, [12]). Therefore, the derivative — is inte-
dz
grable in D+ with an arbitrarily large power p, and ||d«/dz||LP(D+) ^ Chv(f; r) for any v < 1 and p(1 — v) < 1. Here and in what follows the letter C stands for a positive value, which depends on v, p and r, but does not depend on f. Then we apply the well-known property of the integral operator
, ^ r^:= u mm.
D+
|VEf (z)| ^ ,. , z E r. (6)
(see, for instance, [13]): if 0 e Lp(D+), p > 2, then T0 e Hx_2 (C) and
p
h1-2(T0; C) ^ C |5u/5z|Lp(D+). Thus, ||C±/IIk+1 ^ C||/||k. □
The singular integral looses some of its properties at the points where r is not smooth. In this connection, we consider the equation
a(t)C+/(t) + b(t)C-/(t) + K/(t) = c(t), t e r,
with a,b, c e H1-0(r; v) and the same operator K. Clearly, it also keeps the classic properties of singular integral equations in the space H1-0(r; v).
4. Non-rectifiable curves. Let a simple closed curve r be not rectifiable. Then, the curvilinear integrals over that curve are undefined, and the formulas (1) and (2) loose their sense. However, we are able to define the Cauchy projectors in the following way.
We denote by dmh A and dm A the Hausdorff dimension and the upper Minkowskii dimension of a compact set A C C, correspondingly (see, for instance, [14]). Consider the jump problem, i. e., the problem of evaluation of a holomorphic in C \ r function $(z) that satisfies the conditions
(J1) $ has the limit values $±(t) from both sides at any point t e r, and these values are connected by the equality
$+ (t) - $-(t) = /(t), t e r; (8)
(J2) = 0.
If dmhr > 1, then a solution of this problem cannot be unique, because in this case there exists a non-trivial function that is continuous in C and holomorphic in C \ r (see [15]). However, the E. P. Dolzhenko theorem [15] states: if a function $ satisfies the Holder condition with the exponent ^ > dmhr — 1 in a domain D D r and is holomorphic in D \ r, then it is holomorphic in D. Therefore, we supply the formulation of the jump problem by the condition (J3) e HM(r), ^ > dmhr - 1,
and the problem (J1), (J2), (J3) cannot have more than one solution. If its unique solution exists, then we put
C±/ := ; (9)
we keep the notation C±, because on smooth curves both definitions of the Cauchy projectors coincide.
Theorem 4. If a simple closed non-rectifiable curve r satisfies the condition dmr < 2, and a sequence v satisfies inequalities
11—^ dmr + (2 — dmTW , ^ _ . .
vo > ^ dmr, vfc+1 >-^-LA, k = 0,1,... (10)
then the boundary values C± exist and are bounded linear operators from H1-0(r; v) into itself.
Proof. The solvability of the jump problem (J1), (J2), (J3) is studied in [12]. There it is proved that the function dist-p(z;T) is integrable in D+ for any p < 2 — dmr. The substitution of this estimate into the consideration of the previous section concludes the proof. □
Acknowledgment. The research of the first author is partially supported by the Russian Foundation for Basic Researches and the Government of Republic Tatarstan, grant 18-41-160003 r-a.
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Received July 28, 2018. In revised form,, December 21, 2018. Accepted December 24, 2018. Published online January 5, 2019.
Kazan Federal University 18 Kremlyovskaya str., Kazan 420008, Russia E-mail: [email protected]
