Abstract Generated abstract
This paper develops the theory of scales of Banach spaces, focusing on minimal scales and their interpolation properties. It defines regular and reflexive continuous normal scales, proves that maximal scales constructed from normally embedded endpoint spaces are regular, and shows that a minimal scale is majorized by any regular scale joining the same endpoints. The authors establish normal interpolation for minimal scales with respect to other minimal scales, derive strict interpolation results involving scales whose conjugates are maximal, and apply the theory to Lorentz type spaces, spaces between L1 and Linfinity, and Hölder spaces C_alpha^0, identifying the latter as a minimal scale under the stated endpoint spaces.
Full Text
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
\({}^{1}\) S. G. Kreĭn, DAN, 130, No. 3 (1960).
\({}^{2}\) S. G. Kreĭn, DAN, 132, No. 3 (1960).
\({}^{3}\) 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).
\({}^{6}\) J. Musielak, Z. Semadeni, Studia math., 20, No. 3 (1961).
\({}^{7}\) L. V. Kantorovich, G. M. Rubinshtein, Vestn. LGU, No. 7 (1958).
\({}^{8}\) A. P. Calderon, Studia Math., Special ser. No. 1, Conference on Func. An., Warsaw, 1960.