MATHEMATICS

JOSEF KRÁL

ON ANGULAR LIMIT VALUES OF INTEGRALS OF CAUCHY TYPE

(Presented by Academician V. I. Smirnov on 6 XI 1963)

In the present note some necessary and sufficient conditions are indicated for the existence of angular limit values of integrals of Cauchy type at a prescribed point of a curve, under the assumption that the corresponding densities of the integrals belong to certain classes of functions on the curve. For simplicity we shall confine ourselves to the case where (C) is a simple rectifiable oriented arc in the Euclidean plane (E_2), which we identify with the set of complex numbers. For (M \subset C), denote by (\lambda M) the length (linear measure) of the set (M) (see ((^5))). We agree that (\varphi) will always denote some finite (in general complex-valued) function, integrable ((\lambda)) on (C). If (\eta \in C), then, by definition, we put

[
\Phi_\eta(z)=\int_C \frac{\varphi(\xi)-\varphi(\eta)}{\xi-z}\,d\xi,\qquad z\in E_2\setminus C.
]

If it is known that (\varphi) assumes on (C) only real values, then we also put

[
\Phi_\eta^1(z)=\operatorname{Re}\Phi_\eta(z),\qquad
\Phi_\eta^2(z)=\operatorname{Im}\Phi_\eta(z).
]

Let (p) be a bounded, nonnegative, lower semicontinuous function defined on (C). We shall investigate the behavior of the integrals (\Phi_\eta(z)), (\Phi_\eta^k(z)) as (z) approaches (\eta) along nontangential paths, under the assumption that (\varphi) satisfies one of the conditions

[
\varphi(\xi)-\varphi(\eta)=O(p(\xi)),\qquad \xi\to\eta;
\tag{1}
]

[
\varphi(\xi)-\varphi(\eta)=o(p(\xi)),\qquad \xi\to\eta.
\tag{2}
]

For this we shall need the quantities (v_R^k(\eta;p)), (V_R^k(\eta;p)), to whose definition we now proceed.

If (a\leq b) and (R>0), then we put

[
S_R(\eta; a,b)={\eta+re^{iy};\ 0<r<R,\ a\leq y\leq b}.
]

For real (x) and (R>0), denote by (\Gamma_R(x;\eta,p)) the sum

[
\sum_\xi p(\xi),\qquad \xi\in C\cap S_R(\eta; x,x)
]

(of course, (\Gamma_R(x;\eta,p)=+\infty) if (p(\xi)>0) for an uncountable set of points (\xi\in C\cap S_R(\eta; x,x))). Analogously we put

[
\gamma_R(x;\eta,p)=\sum_\xi |\xi-\eta|\,p(\xi),\qquad \xi\in C\cap S_R(\eta; x,x).
]

For each (x>0) we also put

[
v(x;\eta,p)=\sum_\xi p(\xi),\qquad \xi\in C,\quad |\xi-\eta|=x.
]

The functions (\Gamma_R(x;\ldots)), (\gamma_R(x;\ldots)), (v(x;\ldots)) are Lebesgue measurable with respect to (x). Put

[
v_R(\eta;p)=\int_0^{2\pi}\gamma_R(x;\eta,p)\,dx,\qquad
v_R^2(\eta;p)=\int_0^R v(x;\eta,p)\,dx,
]

[
V_R^1(\eta;p)=\int_0^R x^{-1}v(x;\eta,p)\,dx,\qquad
V_R^2(\eta;p)=\int_0^{2\pi}\Gamma_R(x;\eta,p)\,dx.
]

Obviously, the quantities (v_R^k(\ldots)), (V_R^k(\ldots)) do not decrease as (R) increases. We are now in a position to formulate the following theorems, in which we always assume that (\eta\in C), (R>0), (a<b), and (C) does not intersect (S_R(\eta;a,b)). In Theorems 1 and 3 we also assume that (C) does not intersect (S_R(\eta;a+\pi,b+\pi)).

Theorem 1. Let (a<x0). If, for every real function (\varphi) satisfying the inequality

[
|\varphi(\xi)-\varphi(\eta)|\leq Kp(\xi),\qquad \xi\in C,
\tag{3}
]

the relation
[
\limsup_{r\to0+}\left|\Phi_\eta^k(\eta+re^{ix})\right|<\infty
]
holds, then necessarily

[
V_\infty^k(\eta;p)<\infty,
\tag{4(_k)}
]

[
\sup_{r>0} r^{-1}v_r^k(\eta;p)<\infty.
\tag{5}
]

The following converse of this theorem is also valid:

Theorem 2. If (5) holds and, for some (\rho>0),

[
V_\rho^k(\eta;p)<\infty,
\tag{6(_k)}
]

then, for every real function (\varphi) satisfying condition (1), the corresponding function (\Phi_\eta^k) is bounded on (S_r(\eta;a_1,b_1)), where (a<a_1\leq b_1<b), (0<r<R); all functions (\Phi_\eta^k) corresponding to functions (\varphi) satisfying inequality (3) (with one and the same constant (K)) are uniformly bounded on (S_r(\eta;a_1,b_1)). If, instead of (1), one requires (2), then (\Phi_\eta^k(z)) extends continuously to the point (\eta) along the set (S_r(\eta;a_1,b_1)). If (5) is replaced by the condition

[
v_\rho^k(\eta;p)=o(\rho),\qquad \rho\to0+,
]

then all functions (\Phi_\eta^k) corresponding to functions (\varphi) satisfying (3) are equicontinuous on (S_r(\eta;a_1,b_1)).

From these theorems the following propositions follow:

Theorem 3. Let (a<x<b). If
[
\limsup_{r\to0+}\left|\Phi_\eta(\eta+re^{ix})\right|<\infty
]
for every function (\varphi) satisfying (3), then (4(_1)) and (4(_2)) hold.

Theorem 4. If, for some (\rho>0), (6(_1)) and (6(_2)) hold, then, for every function (\varphi) satisfying condition (1), the relation

[
\int_C \frac{\varphi(\xi)-\varphi(\eta)}{\xi-\eta}\,d\xi
=
\lim_{z\to\eta}\Phi_\eta(z),\qquad
z\in S_R(\eta;a_1,b_1),
]

holds, where (a<a_1\leq b_1<b). All functions (\Phi_\eta) corresponding to functions (\varphi) satisfying (3) (with one and the same constant (K)) are equicontinuous on (S_r(\eta;a_1,b_1)), where (0<r<R).

If the function (p) has the form (p(\zeta)=h(|\zeta-\eta|)), where (h) is a function of one real variable, then conditions ((4_1)), ((4_2)) in Theorem 3 and conditions ((6_1)), ((6_2)) in Theorem 4 can be replaced by the single condition

[
\int_0^\infty \rho^{-1} h(\rho)\, dm(\rho)<\infty,
]

where (m(\rho)=\lambda{\zeta;\ \zeta\in C,\ |\zeta-\eta|<\rho}).

The proofs of these theorems use methods from the theory of functions of a real variable connected with Banach’s theorem on the total variation of a continuous function.

The quantities (V_R^2(\eta;p)), (v_R^2(\eta;p)) for the particular case (p\equiv 1) were introduced in ((^2)). The quantity corresponding to (V_\infty^2(\eta;1)) in three-dimensional space was considered in ((^1)). For references to studies on integrals of Cauchy type, see the monographs ((^3,^4)) and the survey article ((^6)).

Karlov University
Prague, Czechoslovakia

Received
4 XI 1963

REFERENCES

(^1) N. D. Burago, V. G. Maz’ya, V. D. Sapozhnikova, DAN, 147, No. 3, 523 (1962).
(^2) J. Král, Commentationes Math. Univ. Carolinae, 3, No. 1, 3 (1962).
(^3) N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
(^4) I. I. Privalov, Boundary Properties of Analytic Functions, Moscow, 1950.
(^5) S. Saks, Theory of the Integral, N. Y., 1937.
(^6) G. Ts. Tumarkin, S. Ya. Khavinson, in Mathematics in the USSR over 40 Years, 1917–1957, Moscow, 1959.