Reports of the Academy of Sciences of the USSR

1. Volume 162, No. 5

MATHEMATICS

T. A. TIMAN

ON THE EXTENSION OF HÖLDER-CONTINUOUS FUNCTIONS

(Presented by Academician S. N. Bernstein on 12 XII 1964)

Let \(K\) be an arbitrary metric space, where the distance between any pair of elements \(x \in R,\ y \in R\) is determined by a certain nonnegative real function \(\rho(x,y)\) with the properties \(\rho(x,y)=0\) if and only if \(x=y\), and
\[ \rho(x,y)\leq \rho(x,z)+\rho(y,z). \]
A real function \(f(x)\) defined on \(R\) is called Hölder-continuous if, for some positive \(\alpha \leq 1\) and some \(C<\infty\), for any two elements \(x_1,x_2\in R\) for which \(\rho(x_1,x_2)\leq 1\), the inequality
\[ |f(x_1)-f(x_2)|\leq C\{\rho(x_1,x_2)\}^{\alpha} \tag{1} \]
holds.

Numerous problems of analysis are connected with the concept of Hölder continuity. In this note we wish to point out one addition to a known general theorem on the extension of continuous functions (see \((^1)\), Chap. VI, § 13), which refines this theorem for the case when the functions under consideration are Hölder-continuous.

Theorem. Whatever closed set of the metric space \(R\) is taken, and whatever bounded real Hölder-continuous function \(\varphi(x)\) is defined on it, it can be extended to the whole space \(R\) without increasing the maximum of its modulus, in such a way that the new function \(f(x)\) remains Hölder-continuous.

Let \(A\) and \(B\) be two closed subsets of the space \(R\), the distance \(d\) between them being positive. For any two real numbers \(a\) and \(b\) \((a<b)\) and for any positive \(\alpha\leq 1\), there exists a function \(F(x)\), defined in the whole space and such that \(a\leq F(x)\leq b\), \(F(x)=a\) for \(x\in A\), \(F(x)=b\) for \(x\in B\), and
\[ |F(x_1)-F(x_2)|\leq (b-a)\{\rho(x_1,x_2)/d\}^{\alpha}. \tag{2} \]

As in the consideration of all continuous functions, one constructs a sequence of functions \(\varphi_n(x)\), defined on the set \(P\), by the recurrence formula
\[ \varphi_n(x)=\varphi_{n-1}(x)-f_{n-1}(x)\quad (n\geq 1), \]
where \(\varphi_0(x)=\varphi(x)\); \(f_n(x)\) is the function indicated above, constructed for the sets \(A_n\) and \(B_n\), for which, respectively, \(\varphi_n(x)\geq \mu_n/3\) and \(\varphi_n(x)\leq -\mu_n/3\), with
\[ \mu_n=\max_{x\in P}|\varphi_n(x)| \]
and \(a=-\mu_n/3,\ b=\mu_n/3\). Passing to the consideration of Hölder-continuous functions, we introduce a sequence of distances \(d_n\) between the sets \(A_n\) and \(B_n\), and establish the following proposition, which plays the principal role here.

Lemma. If the function \(\varphi(x)\), for some positive \(\alpha\leq 1\) and some positive \(C_0\), satisfies, for distinct pairs of points \(x_1\in P,\ x_2\in P\), the condition
\[ |\varphi(x_1)-\varphi(x_2)|\leq C_0\{\rho(x_1,x_2)\}^{\alpha}, \tag{3} \]
then, whatever nonnegative integer value of \(n\) is taken, it is true

inequality

\[ C_0 d_n^{\alpha} \geq \frac{1}{2^{n-1}} \frac{\mu_n}{3}. \tag{4} \]

Inequality (4) is connected with the inequality

\[ \left|\varphi_n(x_1)-\varphi_n(x_2)\right| \leq 2^n C_0\{\rho(x_1,x_2)\}^{\alpha} \tag{5} \]

and, together with it, can be obtained by applying double induction.

The lemma given above makes it possible to show that the function \(f(x)=\sum_{n=0}^{\infty} f_n(x)\), which is an extension of the function \(\varphi(x)\), is Hölder continuous on the whole space \(R\), provided that \(\varphi(x)\) has this property on \(P\).

It should be noted that in the case when the space \(R\) is a finite-dimensional Euclidean space, the extension can be carried out in such a way that the exponent \(\alpha\) in the Hölder continuity condition does not decrease. For the one-dimensional case this fact is well known, and in the multidimensional case it was recently established in the work of E. M. Landis \((^2)\), where the constant \(C\) in condition (1) after extension depends essentially on the dimension of the space and, together with it, increases without bound.

Received
8 XII 1964

CITED LITERATURE

1. L. S. Aleksandrov, Introduction to the General Theory of Sets and Functions, 1948.
2. E. M. Landis, Uspekhi Mat. Nauk, 18, 1 (109), 3 (1963).