MATHEMATICS

K. I. BABENKO

ON SOME CLASSES OF SPACES OF INFINITELY DIFFERENTIABLE FUNCTIONS

(Presented by Academician I. M. Vinogradov on 22 II 1960)

In the book (¹) various spaces of type \(S\) are introduced and considered. We shall examine some spaces of type \(S\) from the point of view of the stock of functions in these spaces.

Let the space \(S_{l_k}^{m_n}\) consist of infinitely differentiable functions \(\varphi(x)\) satisfying the conditions

\[ \left|x^k\varphi^{(n)}(x)\right| \leq A^k B^n l_k m_n,\qquad k,n=0,1,2,\ldots;\ -\infty<x<\infty, \tag{1} \]

where \(A\) and \(B\) are constants depending only on the function \(\varphi\). Below we shall always assume that the sequences \(\{l_k\}\) and \(\{m_n\}\) are nondecreasing and \(l_0=m_0=1\).

Introduce the functions

\[ L(x)=\sup_k \frac{|x|^k}{l_k},\qquad M(x)=\sup_n \frac{|x|^n}{m_n}. \tag{2} \]

Conditions (1) can be written in the equivalent form

\[ \left|\varphi^{(n)}(x)\right|\leq B^n \frac{m_n}{L(x/A)},\qquad n=0,1,2,\ldots \tag{3} \]

If the dual space is denoted by \(\widetilde{S}_{l_k}^{m_n}\), then, as shown in paper (²), \(\widetilde{S}_{l_k}^{m_n}\in S_{m_k}^{\,l_{n+2}}\), and consequently

\[ \left|\widetilde{\varphi}^{(n)}(x)\right| \leq A^n\frac{l_{n+2}}{M(x/B_1)}. \tag{4} \]

In paper (¹) it is shown that, for \(|z|\geq 1,\ z=x+iy\), the inequalities

\[ |\varphi(z)|\leq \frac{B|z|^2}{A^2 L(|z|/4A)} \int_0^\infty \frac{\operatorname{ch} Byt}{M(t)}\,dt, \tag{5} \]

\[ |\widetilde{\varphi}(z)|\leq \frac{A}{M(|z|/B)} \int_0^\infty \frac{\operatorname{ch} Ayt}{L(t)}\,dt. \tag{6} \]

hold.

Simultaneously with the spaces \(S_{l_k}^{m_n}\) and \(\widetilde{S}_{l_k}^{m_n}\), we shall consider the space \(S_l^m\), consisting of entire functions \(f(z)\) subject to the condition

\[ |f(z)|\leq \exp\{\,l(A|y|)-m(B|z|)\,\}, \tag{7} \]

where \(z=x+iy\); \(A\) and \(B\) are constants depending on \(f\). We shall assume that the functions \(l(x)\) and \(m(x)\) satisfy the condition

\[ xl'(x)\uparrow\infty,\qquad xm'(x)\uparrow\infty. \]

It is easy to give a necessary and sufficient condition for the nontriviality of the space \(S_l^m\) under the assumption that one of the functions \(l,m\) satisfies certain regularity requirements on its growth. A condition for the triviality of the space \(S_l^m\) is contained in the paper \((^3)\). We shall give here a result of M. M. Dzhrbashyan in the form of the following lemma.

Lemma 1. If for every \(\theta>0\)

\[ \lim_{x\to\infty}\left[\frac{l(\theta x)}{x}-\int_1^x \frac{m(u)}{u^2}\,du\right]=-\infty, \]

then the space \(S_l^m\) is empty.

Let us consider the question of the nontriviality of the space \(S_l^m\). Since, by assumption, \(xm'(x)\uparrow\infty\), we may assume that

\[ m(x)=\int_0^x \frac{N(u)}{u}\,du, \]

where \(N(x)\) is a nondecreasing function taking only integer values. Denote the discontinuity points of the function \(N(x)\), arranged in increasing order, by \(\rho_1,\rho_2,\ldots,\rho_n,\ldots\), where each point is repeated as many times as there are units in the corresponding jump.

Lemma 2. If the function \(m(x)\) is such that, for sufficiently large \(k\), as \(x\to\infty\)

\[ x^{-k}\int_0^x \frac{m(u)}{u^2}\,du\downarrow 0, \tag{8} \]

and if

\[ \lim_{x\to\infty}\left[\frac{l(\theta x)}{x}-\int_1^x \frac{m(u)}{u^2}\,du\right]>-\infty, \]

then the space \(S_l^m\) is nontrivial.

Proof. Let \(p\) be an integer. Consider the entire function

\[ f(x)=\frac{2}{C_{2p}^p}\sum_{j=1}^p(-1)^{j-1}C_{2p}^{p+j}\frac{\sin jx}{j}. \]

It is not difficult to verify that

\[ f'(x)=1-\frac{2^{2p}}{C_{2p}^p}\sin^{2p}\frac{x}{2}. \]

If \(x_0,\ 0<x_0<\pi\), is a zero of the function \(f'(x)\), then \(f(x_0)=\max |f(x)|=M\).

Consider the function \(g(x)=\frac{1}{M}f(Mx)\). It is clear that \(g(x)\) satisfies the conditions

\[ g'(0)=1,\qquad |g(x+iy)|\le e^{p|y|}. \tag{9} \]

It is not difficult to verify that in the domain \(G_{\eta,\delta}\), defined by the inequalities
\(|\arg z|\le\delta,\ |\arg z-\pi|\le\delta,\ 0\le |z|\le 1+\eta\), the relations

\[ \left|\frac{g(z)}{z}\right|\le 1,\qquad |g(z)|\le 1 \tag{10} \]

hold.

The quantities \(\delta\) and \(\eta>0\) are sufficiently small and depend on \(p\). Choose \(p\) so that \(2p>k+1\), and consider the function

\[ f(z)=\prod_{m=1}^{\infty}\left(\frac{\rho_m g(z/\rho_m)}{z}\right)^2. \]

In view of (8) and the choice of the number \(p\), the infinite product will converge. Let us estimate \(|f(z)|\). Suppose that one of the conditions \(|\arg z|\leqslant \delta\), \(|\arg z-\pi|\leqslant \delta\) is satisfied. Define \(n\) from the inequalities \(\rho_n \leqslant |z|<\rho_{n+1}\). Then

\[ |f(z)|= \left| \frac{\prod_{m=1}^{n} g^2(z/\rho_m)} {\left(z^n/\rho_1\rho_2\ldots\rho_m\right)^2} \prod_{n+1}^{\infty} \frac{g^2(z/\rho_m)\rho_m^2}{z^2} \right| < \left| \prod_{1}^{n} g^2\left(\frac{z}{\rho_m}\right) \right|e^{-2m(|z|)}, \]

since for \(m\geqslant n+1\), \(z/\rho_m\in G_{\eta,\delta}\). Using (9) and (10), we obtain

\[ |f(z)|<\exp\left[ 2\rho y\sum_{\rho_m\leqslant \frac{|z|}{1+\eta}}\frac{1}{\rho_m} -2m(|z|) \right]. \]

But

\[ \sum_{\rho_m\leqslant \frac{|z|}{1+\eta}}\frac{1}{\rho_m} < A_\eta\int_{1}^{y}\frac{m(t)}{t^2}\,dt + \frac{A_\eta}{y}\, m\left(\frac{|z|}{1+\eta/2}\right). \]

Therefore

\[ |f(z)|\leqslant \exp\left\{ 2\rho A_\eta y\int_{1}^{y}\frac{m(u)}{u^2}\,du -m(|z|)+C \right\}. \tag{11} \]

Let us now consider the case when \(\delta<\arg z<\pi-\delta\) or \(\pi+\delta<\arg z<2\pi-\delta\). Since
\(\log \dfrac{g(z)}{z}=O(z^{2p})\) for \(|z|\leqslant 1\), it follows that

\[ |f(z)|\leqslant \exp\left\{ 2\rho y\sum_{1}^{n}\frac{1}{\rho_m} + C_1|z|^{2p}\sum_{n+1}^{\infty}\frac{1}{\rho_m^{2p}} -2m(|z|) \right\}. \]

But

\[ \sum_{n+1}^{\infty}\frac{1}{\rho_m^{2p}} < (2p+1)^3 \int_{|z|}^{\infty}\frac{dt}{t^{2p}} \int_{1}^{t}\frac{m(u)}{u^2}\,du. \]

Applying (8), we obtain

\[ |f(z)|\leqslant \exp\left\{ C_2(z)\int_{1}^{2|z|}\frac{m(u)}{u^2}\,du -m(|z|) \right\}. \tag{12} \]

From inequalities (11) and (12) it follows that, for any \(z\), the inequality

\[ |f(z)|< \exp\left\{ \sqrt{\theta}\,|y| \int_{1}^{\sqrt{\theta}|y|}\frac{m(u)}{u^2}\,du -m(|z|)+C \right\} \]

holds, where \(\theta\) is a constant depending on \(p\). By the condition of the lemma,

\[ |f(z)|<\exp\{l(\theta |y|)-m(|z|)+D\}, \]

where \(D\) is a constant. Thus the space \(S_l^m\) is nontrivial.

Theorem 1. Let \(m(x)\) satisfy condition (8). In order that the space \(S_l^m\) be nontrivial, it is necessary and sufficient that, for at least one \(\theta>0\),

\[ \lim_{x\to\infty} \left[ \frac{l(\theta x)}{x} - \int_{1}^{x}\frac{m(u)}{u^2}\,du \right] >-\infty . \]

Let the function \(L(x)\) be such that \(\dfrac{d}{dx}\log L(x)\uparrow\infty\). Then, if we denote by \(\psi(x)\) the function inverse to \(\dfrac{d}{dx}\log L(x)\), and set

\[ l(x)=\int_{0}^{x}\psi(t)\,dt, \tag{13} \]

we immediately obtain that the question of nontriviality of the space \(S_{l_k}^{m_{r}}\) is equivalent to the question of nontriviality of the space \(S_l^m\), where \(l(x)\) is defined by (13), and \(m(x)=\log M(x)\).

Consider the case when \(M(x)\) is a function of infinite order, i.e.,
\[ \varlimsup_{x\to\infty}\frac{\log\log M(x)}{\log x}=\infty . \tag{14} \]
If (14) holds, we shall require that
\[ \varliminf_{x\to\infty}\frac{\log\log M(x)}{\log x}>1 . \tag{15} \]
Then the function
\[ \int_{1}^{\infty}\frac{\operatorname{ch} xt}{M(t)}\,dt \]
is of finite order, and for the nontriviality of the class it is necessary that \(L(x)\) also be of finite order. We shall say that the functions \(L(x)\) and \(M(x)\) satisfy condition \((\alpha)\) if the following relations are fulfilled:

1. If \(M(x)\) is of finite order, then there always exists a \(k\) such that
   \[ x^{-k}\int_{0}^{x}\frac{\log M(u)}{u^{2}}\,du\downarrow 0, \tag{16} \]
   and the function \(L(x)\) satisfies the condition
   \[ \frac{d}{dx}\log L(x)\uparrow\infty . \]

2. If \(M(x)\) satisfies (14), then (15) also holds, and item 1 holds for the functions \(L(x)\) and \(M(x)\).

From Theorem 1 it follows

Theorem 2. If \(L(x)\) and \(M(x)\) satisfy condition \((\alpha)\), then the space \(S_{l_k}^{m_n}\) is nontrivial if and only if, for at least one \(\theta>0\),
\[ \varliminf\left[\frac{l(\theta x)}{x}-\int_{0}^{x}\frac{\log M(u)}{u^{2}}\,du\right]>-\infty, \]
where \(l(x)\) is defined by formula (13).

Finally, the conditions for nontriviality of the space \(S_{l_k}^{m_n}\) can be formulated in the following interesting way. Let \(\mu(x)\) be the function inverse to the function
\[ \int_{0}^{x}\frac{\log M(u)}{u^{2}}\,du . \]

Theorem 3. If for every \(\theta>0\)
\[ \lim_{x\to\infty}\frac{L(\theta x)}{M[|\mu(x)|]}=\infty, \tag{17} \]
then the space \(S_{l_k}^{m_n}\) is trivial. If the functions \(L(x)\), \(M(x)\) satisfy condition \((\alpha)\) and for at least one \(\theta>0\)
\[ \lim_{x\to\infty}\frac{L(\theta x)}{M(\mu(x))}<\infty, \tag{18} \]
then \(S_{l_k}^{m_n}\) is nontrivial.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
11 II 1960

REFERENCES

1. I. M. Gel'fand, G. E. Shilov, Spaces of Basic Generalized Functions, Moscow, 1958.
2. K. I. Babeshko, Transactions of the Moscow Mathematical Society, 5, 523 (1956).
3. M. M. Dzhrbashyan, Izv. Acad. Sci. Armenian SSR, Ser. Phys.-Math. Sci., 10, No. 6, 7 (1957).