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MATHEMATICS
S. G. Krein, Yu. I. Petunin
On the Concept of a Minimal Scale of Spaces
(Presented by Academician I. G. Petrovskii on 17 VII 1963)
The present paper contains a further development of a number of questions in the theory of scales of Banach spaces (see \((^{1-5})\)).
- Let us consider a family of Banach spaces \(E_\alpha\) \((0 \leqslant \alpha \leqslant 1)\) possessing the following property:
I. For \(\alpha < \beta\), the space \(E_\beta\) is densely embedded in \(E_\alpha\) and
\[ \|x\|_{E_\alpha}\leqslant \|x\|_{E_\beta}\qquad (x\in E_\beta). \]
Lemma 1. If for the family \(E_\alpha\) there exists an element \(e\in E_1\) for which \(\|e\|_{E_1}=\|e\|_{E_0}=1\), then for each element \(x_0\in E_1\) one can construct a linear operator \(A\), acting in the space \(E_1\), such that
\[ \|A\|_{E_\alpha}=\|x_0\|_{E_\alpha}. \]
Proof. By the Hahn—Banach theorem there exists a linear functional \(f(x)\) from \(E_0'\) for which \(\|f\|_{E_0'}=f(e)=1\). Since \(\|f\|_{E_1'}\leqslant \|f\|_{E_0'}=1\) and \(\|e\|_{E_1}=1\), it follows that \(\|f\|_{E_1'}=1\), and consequently \(\|f\|_{E_\alpha'}=1\) \((0\leqslant \alpha\leqslant 1)\). For any \(x_0\in E_1\) construct the linear operator \(A\) by the formula
\[ A(x)=x_0 f(x). \]
Then
\[ \|A\|_{E_\alpha} = \sup_{x\in E_1} \frac{\|x_0 f(x)\|_{E_\alpha}}{\|x\|_{E_\alpha}} = \|x_0\|_{E_\alpha} \sup_{x\in E_1} \frac{|f(x)|}{\|x\|_{E_\alpha}} = \|x_0\|_{E_\alpha}. \]
Definition 1. We shall say that a family of Banach spaces \(E_\alpha\) \((0\leqslant \alpha\leqslant 1)\) has the normal (strict) interpolation property with respect to a family of spaces \(F_\alpha\) \((0\leqslant \alpha\leqslant 1)\) if, for every bounded linear operator \(A\) acting from the spaces \(E_0\) and \(E_1\), respectively, into the spaces \(F_0\) and \(F_1\), it follows that it acts from any space \(E_\alpha\) into the space \(F_\alpha\) \((0\leqslant \alpha\leqslant 1)\), and its norm \(\|A\|_{E_\alpha\to F_\alpha}\) satisfies the inequality
\[ \|A\|_{E_\alpha\to F_\alpha} \leqslant \|A\|_{E_0\to F_0}^{\,1-\alpha} \|A\|_{E_1\to F_1}^{\,\alpha} \]
(is a logarithmically convex function of \(\alpha\)).
From the lemma it follows:
Corollary. Let the family \(E_\alpha\) \((0\leqslant \alpha\leqslant 1)\) possess the strict interpolation property with respect to itself; then this family forms a normal scale of spaces (see \((^{2})\)).
Let \(E_\alpha\) and \(F_\alpha\) be two families of Banach spaces possessing property I, and let \(E_\alpha'\) and \(F_\alpha'\) be the families of spaces conjugate to them. If the family \(F_\alpha'\) possesses the strict interpolation property with respect to the family \(E_\alpha'\), then, as is not hard to prove by passing to the adjoint operator, the family \(E_\alpha\) possesses the strict interpolation property with respect to the family \(F_\alpha\).
- Let \(E_\alpha\) \((0\leq \alpha\leq 1)\) be a continuous normal scale. Introduce on the space \(E'_0\), conjugate to \(E_0\), the family of norms \(f_{E'_\alpha}\). The completion of the space \(E'_0\) with respect to the norm \(\|f\|_{E'_\alpha}\) will be denoted by \(\widetilde E_{-\alpha}\).
Definition 2. A continuous normal scale \(E_\alpha\) \((0\leq \alpha\leq1)\) will be called regular if the spaces \(\widetilde E_{-\alpha}\) form a normal scale on the interval \((-1,0)\).
The scale \(\widetilde E_{-\alpha}\) \((0\leq \alpha\leq 1)\) is called conjugate to the scale \(E_\alpha\); \(E_{-\alpha}\) is a continuous normal scale.
A continuous normal scale need not be regular. As an example one may give the following scale: let \(E_\alpha=E_0\) for \(0\leq \alpha\leq \tfrac12\), and \(E_\alpha=F_\alpha\), where \(F_\alpha\) is an arbitrary nontrivial continuous normal scale on the interval \([\tfrac12,1]\), with \(F_{1/2}=E_0\).
Theorem 1. The maximal scale \({}^{(2)}\), constructed from two normally embedded Banach spaces \(E_0\) and \(E_1\), is a regular scale.
- Let \(E_0\) and \(E_1\) be two related spaces. Introduce on the space \(E_1\) the family of norms
\[ \|x\|_{E^0_\alpha}=\sup_{f\in E'_0}\frac{|f(x)|}{\|f\|_0^{\,1-\alpha}\|f\|_1^\alpha}. \tag{1} \]
As shown in \({}^{(3)}\), the completions of the space \(E_1\) with respect to the norms (1) form a continuous normal scale \(E^0_\alpha\) connecting \(E_0\) and \(E_1\). We shall call this scale minimal. This term is connected with the following theorem.
Theorem 2. The minimal scale \(E^0_\alpha\) is majorized by any regular scale \(E_\alpha\) connecting the spaces \(E_0\) and \(E_1\), i.e.
\[ \|x\|_{E^0_\alpha}\leq \|x\|_{E_\alpha}\qquad (x\in E_1;\ 0\leq \alpha\leq 1). \]
Proof. Let \(x\in E_1\). The spaces \(E_\alpha\) and \(E_1\) are related; therefore, from the proof of Theorem 1 in \({}^{(3)}\) it follows that
\[ \|x\|_{E_\alpha}=\sup_{f\in E'_0}\frac{|f(x)|}{\|f\|_{E'_\alpha}}. \]
By virtue of the regularity of the scale \(E_\alpha\),
\[ \|f\|_{E'_\alpha}\leq \|f\|_{E'_0}^{\,1-\alpha}\|f\|_{E'_1}^{\,\alpha},\qquad (f\in E'_0). \]
Then
\[ \|x\|_{E_\alpha}\sup_{f\in E'_0}\frac{|f(x)|}{\|f\|_{E'_0}^{\,1-\alpha}\|f\|_{E'_1}^{\,\alpha}}=\|x\|_{E^0_\alpha}. \]
The theorem is proved.
Theorem 3. The minimal scale has the normal interpolation property with respect to any other minimal scale.
Proof. Let \(E^0_\alpha\) be a minimal scale connecting the spaces \(E_0\) and \(E_1\), and let \(F^0_\alpha\) be a minimal scale constructed from the spaces \(F_0\) and \(F_1\). If a linear operator \(A\), acting in \(E_1\), satisfies the conditions
\[ \|Ax\|_{F_0}\leq C_0\|x\|_{E_0},\qquad \|Ax\|_{F_1}\leq C_1\|x\|_{E_1}, \]
then the conjugate operator \(A^*\) maps the space \(F'_0\) into \(E'_0\) and \(F'_1\) into \(E'_1\), and the inequalities
\[ \|A^*f\|_{E'_0}\leq C_0\|f\|_{F'_0},\qquad (f\in F'), \]
\[ \|A^*f\|_{E'_1}\leq C_1\|f\|_{F'_1}. \]
Let the element \(x \in E_1\); then
\[ \|Ax\|_{F_\alpha} = \sup_{f\in F_0'} \frac{|f(Ax)|}{\|f\|_{F_0'}^{1-\alpha}\|f\|_{F_1'}^\alpha} = \sup_{f\in F_0'} \frac{|A^*f(x)|}{\|f\|_{F_0'}^{1-\alpha}\|f\|_{F_1'}^\alpha} \leq C_0^{1-\alpha}C_1^\alpha \sup_{f\in F_0'} \frac{|A^*f(x)|}{\|A^*f\|_{E_0'}^{1-\alpha}\|A^*f\|_{E_1'}^\alpha} \leq \]
\[ \leq C_0^{1-\alpha}C_1^\alpha \sup_{g\in E_0'} \frac{|g(x)|}{\|g\|_{E_0'}^{1-\alpha}\|g\|_{E_1'}^\alpha} = C_0^{1-\alpha}C_1^\alpha\|x\|_{E_\alpha}; \]
the theorem is proved.
Corollary. Every regular scale has the normal interpolation property with respect to any minimal scale.
- Let \(E_\alpha\) \((0 \leq \alpha \leq 1)\) be a continuous normal scale. Consider the family of Banach spaces \(E_{-\alpha}\). Each element \(x\in E_1\) gives rise in the natural way to a continuous linear functional \(x(f)=f(x)\) \((f\in E_{-1})\) on the space \(\widetilde E_{-1}\). From the proof of Theorem 1 it follows that the norm of this functional \(x(f)\) coincides with the norm \(\|x\|_{E_1}\). If all continuous linear functionals on the space \(\widetilde E_{-1}\) are exhausted by functionals of the form \(x(f)\) \((x\in E_1,\ f\in \widetilde E_{-1})\), then we shall call the scale \(E_\alpha\) reflexive. For a reflexive scale, in a certain sense, the second conjugate scale coincides with the original one.
Lemma 2. If a minimal scale \(E_\alpha\) is regular and reflexive, then the scale conjugate to it forms a maximal scale joining the spaces \(\widetilde E_0\) and \(\widetilde E_{-1}\).
Theorem 4. Every regular scale has the strict interpolation property with respect to a minimal scale whose conjugate is maximal.
- For the spaces \(L_1(0,1)\) and \(L_\infty(0,1)\), a maximal scale was constructed in [4]. It consists of Lorentz spaces \(S_\alpha\), in which the norm is defined by the formula
\[ \|x\|_{S_\alpha} = \alpha\int_0^1 t^{\alpha-1}x^*(t)\,dt, \]
where \(x^*(t)\) is the rearrangement of the function \(|x(t)|\) in nonincreasing order.
The scale \(S_\alpha^*\) is conjugate to the scale \(M_\alpha^0\), where the space \(M_\alpha^0\) consists of all measurable functions for which
\[ \|x\|_{M_\alpha^0} = \sup \frac{\displaystyle\int_E |x(t)|\,dt}{(\operatorname{mes} E)^\alpha} <\infty, \]
\[ \lim_{\operatorname{mes} E\to 0} \frac{\displaystyle\int_E |x(t)|\,dt}{(\operatorname{mes} E)^\alpha} = 0 \tag{5} \]
The scale \(M_\alpha^0\) is a minimal scale joining the spaces \(L_1(0,1)\) and \(L_\infty(0,1)\).
It follows from Theorem 4 that every regular scale has the strict interpolation property with respect to the scale \(M_\alpha^0\).
We note that regular scales include the scales of the spaces \(L_p\), \(M_\alpha^0\), \(S_\alpha\), Hilbert scales, etc. Every minimal scale has the normal interpolation property with respect to the scale \(M_\alpha^0\).
Consider now the space \(C_\alpha^0\) \((0 \leq \alpha \leq 1)\), consisting of all functions \(x(t)\), \(0 \leq t \leq 1\), for which
\[ \lim_{t\to \tau} \frac{|x(t)-x(\tau)|}{|t-\tau|^\alpha} = 0 \qquad (t,\tau\in[0,1]). \]
Identifying all functions that differ by a constant, introduce in the space \(C_\alpha^0\) the norm
\[ \|x\|_{C_\alpha^0}=\sup_{0\le t,\tau\le 1}\frac{|x(t)-x(\tau)|}{|t-\tau|^\alpha} \qquad (0\le \alpha \le 1). \]
The spaces \(C_\alpha^0\) \((0\le \alpha \le 1)\) form a continuous normal scale \(({}^{2,6})\), joining the spaces \(C(0,1)\) and \(C_1(0,1)\). If, from these spaces, one constructs the minimal scale \(E_\alpha^0\), then the norm
\[ \|x\|_{C_\alpha^0}\le \|x\|_{E_\alpha^0} \qquad (x\in C_1(0,1),\ 0\le \alpha \le 1). \tag{2} \]
The scale \(\Phi_\alpha=\widetilde E_{-\alpha}\), conjugate to the scale \(C_\alpha^0\) \((0\le \alpha \le 1)\), consists of spaces of absolutely additive functions on the interval \([0,1]\), considered in \(({}^{7})\).
It is shown there that the space conjugate to \(\Phi_\alpha^0\) \((0<\alpha\le 1)\) is the Hölder space \(C_\alpha\).
The family of Hölder spaces, as shown in \(({}^{8})\), has the strict interpolation property with respect to itself. As was indicated, it follows from this that the family of spaces \(\Phi_\alpha\) also has the strict interpolation property with respect to itself and therefore (Lemma 1) forms a normal scale. Thus, the scale \(C_\alpha^0\) is proper. By Theorem 2, the inequality inverse to (2) is valid, i.e. the scale \(C_\alpha^0\) coincides with the minimal scale constructed from the spaces \(C[0,1]\) and \(C_1(0,1)\).
The scale \(C_\alpha^0\) is reflexive; therefore Lemma 2 on the maximality of the conjugate scale is valid for it.
Corollary. Every proper scale (minimal scale) has the strict (normal) interpolation property with respect to the scale \(C_\alpha^0\).
Voronezh State
University
Received
6 VII 1963
CITED LITERATURE
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\({}^{3}\) S. G. Kreĭn, Yu. I. Petunin, DAN, 139, No. 6 (1961).
\({}^{4}\) S. G. Kreĭn, E. M. Semenov, DAN, 138, No. 4 (1961).
\({}^{5}\) E. M. Semenov, DAN, 148, No. 5 (1963).
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\({}^{8}\) A. P. Calderon, Studia Math., Special ser. No. 1, Conference on Func. An., Warsaw, 1960.