S. G. Krein

On the Concept of a Normal Scale of Spaces

(Presented by Academician N. N. Bogolyubov, 28 I 1960)

In the note (¹) we introduced for consideration the concept of an analytic scale of spaces as a family of Banach spaces obtained by a special construction. The norms in these spaces depend analytically on the parameter. In the present article we consider a more general definition of a scale of spaces, in which, instead of analyticity, only logarithmic convexity is required of the dependence of the norms on the parameter.

1°. Definition. A family of Banach spaces (E_\alpha) ((\alpha_0 \leqslant \alpha \leqslant \beta_0)) with norms (|x|_\alpha) is called a normal scale of spaces if the following conditions are satisfied:

1. For (\beta > \alpha), the set (E_\beta) is contained in the space (E_\alpha), is dense in it, and

[
|x|\alpha \leqslant |x|\beta
\quad \text{for } x \in E_\beta .
\tag{1}
]

1. If (\alpha_0 \leqslant \alpha \leqslant \beta \leqslant \gamma \leqslant \beta_0) and (x \in E_\gamma), then

[
|x|\beta \leqslant |x|\alpha^{\frac{\gamma-\beta}{\gamma-\alpha}}
|x|_\gamma^{\frac{\beta-\alpha}{\gamma-\alpha}} .
\tag{2}
]

As was indicated in (¹), property (2) is satisfied for spaces of an analytic scale. In order that (1) also be satisfied, it is necessary to impose an inessential additional restriction on the operators (T(z)). The classical scales of spaces considered in analysis, after a certain renormalization, become normal. We give examples.

In an (n)-dimensional domain (G), consider the spaces (L_p) of functions summable with the (p)-th power ((p \geqslant 1)). Denote by (L^\alpha) the space (L_p) for

[
p=\frac{2}{1-\alpha}
]

with norm

[
|f|\alpha = (\operatorname{mes} G)^{\frac{\alpha-1}{2}} |f|
\quad
\left(p=\frac{2}{1-\alpha},\ f \in L_p\right).
\tag{3}
]

The spaces (L^\alpha) form a normal scale for (-1 \leqslant \alpha \leqslant 1). A normal scale for (0 \leqslant \alpha \leqslant 1) is formed by the spaces (C_\alpha) of functions continuous in (\overline G) and satisfying in (G) a Hölder condition, with norm

[
|f|{C\alpha}
=
\max\left{
\max_{x \in \overline G} |f(x)|,\,
D^\alpha \sup_{x,y \in G}
\frac{|f(x)-f(y)|}{|x-y|^\alpha}
\right},
\tag{4}
]

where (D) is the diameter of the domain (G).

Finally, a normal scale for (-\infty<\alpha<\infty) is formed by the spaces (L_{p,\alpha}) (see (1)) with norm

[
|f|{L,}}=D^\alpha\left{\int \frac{|f|^p}{r^{p\alpha}}\,dx\right}^{1/p
\tag{5}
]

where (r) is the distance from the point (x) to some fixed point of (\overline G) or to some manifold lying in (\overline G).

Let us indicate the simplest properties of normal scales. If, instead of the index (\alpha), one introduces an index (\alpha') linearly related to (\alpha): (\alpha=k\alpha'+\theta) ((k>0)), and denotes (\widetilde E_{\alpha'}=E_{k\alpha'+\theta}=E_\alpha), then the spaces (\widetilde E_{\alpha'}) form an (\alpha_0)-normal scale on the interval

[
\left[\frac{\alpha_0-\theta}{k},\,\frac{\beta_0-\theta}{k}\right].
]

Thus, if (\alpha_0) and (\beta_0) are finite, then without loss of generality they may be assumed equal to 0 and 1.

If in the spaces (E_\alpha) of a normal scale one makes an equivalent renorming, then, generally speaking, inequalities (1) and (2) are violated. However, there exists a class of renormings that does not violate (1) and (2). Let (q(\alpha)) be a nondecreasing logarithmically convex function on the interval ([\alpha_0,\beta_0]); then the spaces (E_\alpha) form a normal scale with respect to the norms (|x|{\alpha,1}=q(\alpha)|x|\alpha).

Let (E_\alpha) ((\alpha_0\leqslant\alpha\leqslant\beta_0)) be a family of Banach spaces forming a normal scale on the intervals ([\alpha_0,\delta]) and ([\delta,\beta_0]), where (\alpha_0<\delta<\beta_0). In order that these spaces form a normal scale on the whole interval ([\alpha_0,\beta_0]), it is necessary and sufficient that inequality (2) hold for (\beta=\delta) and arbitrary (\alpha<\delta) and (\gamma>\delta) ((\alpha,\gamma\in[\alpha_0,\beta_0])).

If the interval ([\alpha_0,\beta_0]) is covered by intervals ([\alpha_0,\beta_1)), ((\beta_1,\beta_2),\ldots,(\beta_{k-1},\beta_0)), and the family of Banach spaces (E_\alpha) ((\alpha_0\leqslant\alpha\leqslant\beta_0)) forms a normal scale on each of the intervals ([\alpha_0,\beta_1)), ((\beta_1,\beta_2),\ldots,(\beta_{k-1},\beta_0)), then the family (E_\alpha) will be a normal scale of spaces on the entire interval ([\alpha_0,\beta_0]).

If (E_\alpha) is a normal scale on ([\alpha_0,\beta_0]) and (F_\alpha) is a normal scale on ([\delta,\beta_0]) ((\alpha_0<\delta<\beta_0)) such that (F_\delta=E_\delta), (F_\alpha\subset E_\alpha), and (|x|{E\alpha}\leqslant|x|{F\alpha}) ((x\in F_\alpha)) for (\delta<\alpha\leqslant\beta_0), then the spaces (\widetilde E_\alpha), coinciding with (E_\alpha) for (\alpha\in[\alpha_0,\delta]) and with (F_\alpha) for (\alpha\in[\delta,\beta_0]), form a normal scale on ([\alpha_0,\beta_0]).

A widespread construction of a normal scale is the following: one considers a certain linear set (M), and on it specifies a family of norms (|x|_\alpha) ((x\in M,\ \alpha_0\leqslant\alpha\leqslant\beta_0)) satisfying conditions (1) and (2).

If (E_\alpha) denotes the completion of (M) in the norm (|x|\alpha) for (\alpha<\beta_0), and (E) denotes the completion of (M) in the norm

[
|x|{\beta_0}=\lim|x|_\alpha,
]

then the spaces (E_\alpha) will form a normal scale on the interval ([\alpha_0,\beta_0]). If, however, (\widetilde E_{\beta_0}) denotes the completion of (M) in the norm (|x|{\beta_0}), which, generally speaking, is larger than (|x|) by the quotient space (E'}), then (\widetilde E_{\beta_0}) may fail to be contained in the spaces (E_\alpha) for (\alpha<\beta_0). However, there exists a homomorphic mapping of (\widetilde E_{\beta_0}) into (E_\alpha), and if one replaces (\widetilde E_{\beta_0{\beta_0}) by the kernel of this homomorphism, then the resulting family of spaces ((E\alpha) for (\alpha<\beta_0) and (E'{\beta_0})) will form a normal scale on ([\alpha_0,\beta_0]). If one constructs the scale (L^\alpha) by completing a certain set of bounded functions, defining (|f|\alpha) for (\alpha<1) by formula (3) and putting (|f|1=\sup|f(x)|), then the passage from the norm (|f||f(x)|). This passage to the quotient space corresponds to identifying functions that differ on a set of measure zero.}) to the norm (|f|'_{\beta_0}) corresponds to the passage from the norm (|f|_1) to the norm (|f|'_1=\operatorname{vrai\,sup

2°. Consider two Banach spaces (F_0) and (F_1) such that

[
F_1 \subset F_0,\qquad F_1\ \text{is dense in }F_0,\qquad |x|{F_0}\leq |x|\quad (x\in F_1).
\tag{6}
]

Suppose that on the interval ([0,1]) a normal scale of spaces (E_\alpha) is given such that

[
F_0 \subset E_0,\qquad F_1 \subset E_1,\qquad
|x|{E_0}\leq |x|\quad (x\in F_0),\qquad
|x|{E_1}\leq |x|\quad (x\in F_1);
\tag{7}
]

[
\lim_{\beta\to 1}|x|{E\beta}=|x|_{E_1}\quad (x\in E_1).
\tag{8}
]

We shall say that the scale (E_\alpha) is based on the spaces (F_0) and (F_1). One scale based on (F_0) and (F_1) exists—this is the trivial scale (E_\alpha \equiv F_0).

Theorem 1. Among the normal scales based on the spaces (F_0) and (F_1), there exists a scale (G_\alpha) ((0\leq \alpha\leq 1)) having the following properties: a) (G_0=F_0); b) (F_1) is dense in (G_1); c) for any normal scale (E^\alpha) based on (F_0) and (F_1), the inequality

[
|x|{E\alpha}\leq |x|{G\alpha}\qquad (0\leq \alpha\leq 1,\ x\in F_1)
]

holds.

The scale whose existence is asserted in the theorem will be called the maximal scale constructed from the spaces (F_0) and (F_1). If there exists some scale based on (F_0) and (F_1) such that (E_1=F_1), then for the maximal scale we shall also have (G_1=F_1). In this case the spaces (F_0) and (F_1) will be called related.

Remark. If (G_\alpha) ((0\leq \alpha\leq 1)) is the maximal scale constructed from the spaces (F_0) and (F_1), then on any interval ([\alpha_0,\beta_0]\subset[0,1]) it will be the maximal scale constructed from the spaces (G_{\alpha_0}) and (G_{\beta_0}).

Theorem 2 (interpolation). Let (F_0) and (F_1) be two spaces satisfying conditions (6); let (G_\alpha) ((0\leq \alpha\leq 1)) be the maximal scale constructed from them; and let (E_\alpha) be an arbitrary normal scale on ([0,1]) satisfying condition (8).

If a linear operator (A) is defined on (F_1) and satisfies the conditions

[
|Ax|{E_0}\leq C_0|x|,\qquad
|Ax|{E_1}\leq C_1|x|\qquad (x\in F_1),
]

then for (0\leq \alpha\leq 1) the inequality

[
|Ax|{E\alpha}\leq C_0^{1-\alpha}C_1^\alpha|x|{G\alpha}\qquad (x\in F_1)
\tag{9}
]

holds.

Theorem 2 is naturally reformulated for the case when the scale (G_\alpha) is defined on the interval ([\alpha_0,\beta_0]), and the scale (E_\alpha) on the interval ([\alpha_1,\beta_1]).

We note that in the proof of the theorem one uses not the fact that the operator (A) is linear, but only the fact that the quantity (|Ax|_\alpha) has the properties of a norm.

3°. Consider, as an example, the spaces (L_1) and (L_\infty) of functions defined on the interval ([0,1]). These spaces satisfy conditions (6) and are related, since the scale of spaces (L^\alpha) is based on them (see (3)), and (L^1=L_\infty). It would have been natural to suppose that the scale (L^\alpha) is maximal. However, this supposition is false. As E. M. Semenov showed jointly with the author, the maximal scale constructed from the spaces (L_1) and (L_\infty) is the scale of spaces (S_\alpha), consisting of summable functions (f(t)) on ([0,1]) for which

[
|f|{S\alpha}=(1-\alpha)\int_0^1 t^{-\alpha} f^*(t)\,dt<\infty,
\tag{10}
]

where by (f^*(t)) is denoted the rearrangement of the function (|f(t)|) in decreasing order (see (2), p. 332).

Similarly, for the space (V_a) of absolutely continuous functions on ([0,1]) with norm (|f|{V_a}=\max{\max |f(t)|,\operatorname{Var} f(t)}) and the space (C_1) of functions satisfying the Lipschitz condition, with norm (4), the maximal scale based on them is the scale of spaces (S^1\alpha), consisting of absolutely continuous functions for which

[
|f|{S^1\alpha}
=
\max\left{
\max |f(t)|,\,
(1-\alpha)\int_0^1 t^{-\alpha} f^*(t)\,dt
\right}<\infty.
\tag{11}
]

From Theorem 2 one can obtain a number of theorems on operators defined in the spaces (S_\alpha) and (S^1_\alpha).

Let us formulate some of them:

Theorem 3. If the operator (A) is a bounded operator acting from the space (L_1) into the space (R_\alpha) and from the space (L_\infty) into the space (R_\beta), where (R_\alpha) and (R_\beta) are two spaces either from the scale (L^\alpha) (see (3)), or from the scale (C_\alpha) (see (4)), or (L_{p,\beta}) (see (5)), or (S_\alpha) (see (10)), or (S^1_\alpha) (see (11)), with the corresponding indices, then the operator (A) is a bounded operator acting from the space (S_\mu) ((0\le \mu\le 1)) into the space (R_{\alpha(\mu)}), where (\alpha(\mu)=\alpha(1-\mu)+\beta\mu).

Using the remark to Theorem 2, one can prove a more general theorem:

Theorem 4. If the operator (A) is a bounded operator acting from the space (S_{\alpha'}) into the space (R_\alpha) and from (S_{\beta'}) into (R_\beta), where (R_\alpha) and (R_\beta) are the same as in Theorem 3, then the operator (A) is a bounded operator acting from the space (S_{\alpha'(\mu)}) into the space (R_{\alpha(\mu)}), where (\alpha'(\mu)=\alpha'(1-\mu)+\beta'\mu), (\alpha(\mu)=\alpha(1-\mu)\beta\mu) ((0\le \mu\le 1)).

Using inequality (9), in Theorems 3 and 4 one can estimate the norms of the operator (A) in the corresponding spaces.

Received
27 I 1960

References

(^{1}) S. G. Krein, DAN, 130, No. 3 (1960).
(^{2}) G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, IL, 1948.