UDC 513.88 : 513.83

MATHEMATICS

S. G. KREIN, Yu. I. PETUNIN, E. M. SEMENOV

HYPERSCALES OF BANACH STRUCTURES

(Presented by Academician L. V. Kantorovich, 4 XII 1965)

In this paper we shall use the terminology and notation of article (^1).

1. Let \(\Omega\) be some set with a finite normalized measure \(m\). Denote by \(\mathcal A\) the topological vector space consisting of all functions \(x(t)\) measurable with respect to the measure \(m\) \((t \in \Omega)\), with the topology of convergence in measure. In the space \(\mathcal A\) the notion of semi-order is introduced naturally: \(x \leq y\) when \(x(t) \leq y(t)\) for almost all \(t \in \Omega\).

A Banach space \(E \subset \mathcal A\), which together with every function \(x(t)\) contains the function \(|x(t)|\), and for which from the inequality \(|x(t)| \leq |y(t)|\) it follows that \(\|x\|_E \leq \|y\|_E\), is called a functional Banach structure (see (^2)). By an ideal structure we shall mean a functional Banach structure for which the following condition is satisfied: if \(y(t) \in E\), \(x(t) \in \mathcal A\), and \(|x(t)| \leq |y(t)|\), then \(x(t) \in E\).

Theorem 1. Let \(x(t), z(t)\) be elements of an ideal structure \(E\). Then the function \(|x(t)|^\theta |z(t)|^{1-\theta}\) \((0 \leq \theta \leq 1)\) belongs to \(E\), and

\[ \bigl\|\, |x(t)|^\theta |z(t)|^{1-\theta} \,\bigr\|_E \leq \|x\|_E^\theta \|z\|_E^{1-\theta}. \]

The operator \(A(x,z)=|x|^\theta |z|^{1-\theta}\) is continuous from \(E \times E\) into \(E\).

In what follows we shall everywhere use the following notation:

\[ \mu = (\gamma-\beta)/(\gamma-\alpha),\quad \nu = (\beta-\alpha)/(\gamma-\alpha). \]

Definition 1. A continuous normal scale (see (^1)) of ideal structures \(H_\alpha\) \((0 \leq \alpha \leq 1)\), possessing the property

\[ \bigl\|\, |x(t)|^\mu |z(t)|^\nu \,\bigr\|_{H_\beta} \leq \|x\|_{H_\alpha}^{\mu}\, \|z\|_{H_\gamma}^{\nu}, \qquad (0 \leq \alpha \leq \beta \leq \gamma \leq 1) \tag{1} \]

for all \(x,z \in H_1\), is called a hyperscale.

We note that it is sufficient to verify inequality (1) only on a dense set of the space \(H_1\).

If the ideal structures \(H_0\) and \(H_1\) are related Banach spaces, then the minimal scale constructed from the spaces \(H_0\) and \(H_1\) (see (^1)) is a hyperscale.

It is not difficult to prove that among all hyperscales joining the ideal structures \(H_0\) and \(H_1\) there exists a hyperscale \(H_\alpha^{\max}\) \((0 \leq \alpha \leq 1)\) possessing the following property: if \(H_\alpha\) \((0 \leq \alpha \leq 1)\) is an arbitrary hyperscale joining the spaces \(H_0\) and \(H_1\), then

\[ \|x\|_{H_\alpha} \leq \|x\|_{H_\alpha^{\max}} \qquad (x \in H_1). \]

The hyperscale \(H_\alpha^{\max}\) \((0 \leq \alpha \leq 1)\) will be called maximal. It turns out that the maximal hyperscale joining the spaces \(\mathcal L_{p_0}(\Omega)\) and \(\mathcal L_{p_1}(\Omega)\) \((1 \leq p_0 \leq p_1 \leq \infty)\) is the scale of spaces \(\mathcal L_p(\Omega)\), where

\[ p = p_0p_1/[\,p_1-\alpha(p_1-p_0)\,],\quad 0 \leq \alpha \leq 1. \]

1. An operator \(T\) acting from a Banach structure \(E\) into the topological space \(\mathcal A\) is called convex on the cone of nonnegative functions if for all \(x_1,x_2 \in K\) and \(\lambda \in [0,1]\)

\[ T(\lambda x_1+(1-\lambda)x_2) \leq \lambda T(x_1)+(1-\lambda)T(x_2). \]

Theorem 2. Let \(H_\alpha^{\max}\) \((0 \leq \alpha \leq 1)\) be a maximal hyperscale, \(F_\alpha\) \((0 \leq \alpha \leq 1)\) an arbitrary hyperscale, and let \(T\) be a homogeneous, monotone convex operator on the cone \(K\), acting from the space \(H_i^{\max}\) into \(F_i\) \((i=0,1)\).

If for the operator \(T\)

\[ \|T\|_{H_i^{\max}\to F_i}=M_i<\infty \qquad (i=0,1), \]

then it maps the space \(H_\alpha^{\max}\) into \(F_\alpha\) and

\[ \|Tx\|_{F_\alpha}\leq M_0^{1-\alpha}M_1^\alpha\|x\|_{H_\alpha^{\max}} \qquad (x\in H_\alpha^{\max}). \]

An analogous theorem can be proved for a family of linear integral operators

\[ T_\alpha(x)=\int_\Omega K_\alpha(t,s)x(s)\,dm, \]

where \(K_\alpha(t,s)\geq 0\) (almost everywhere) and, for almost all \(t,s\in\Omega\), \(K_\alpha(t,s)\) is logarithmically convex in \(\alpha\).

1. Denote by \(E(\alpha)\) \((0<\alpha\leq 1)\) the set of all \(x\) from the ideal structure \(E\) with unit \(e(t)\equiv 1\), for which the functional

\[ \Phi_\alpha(x)=\bigl\||x|^{1/\alpha}\bigr\|_E^\alpha<\infty . \]

The functional \(\Phi_\alpha(x)\) is a norm defined on \(E(\alpha)\), and \(E(\alpha)\) is complete with respect to the norm \(\Phi_\alpha(x)\).

Using the results of A. P. Calderón \((^3)\), one can show that the space \(E(\alpha)\) is reflexive if the space \(E\) is reflexive.

We note that in the case of \(KB\)-spaces the norm \(\Phi_\alpha(x)\) was considered earlier in \((^4)\).

It is not difficult to see that for any bounded function

\[ \lim_{\alpha\to 0}\bigl\||x|^{1/\alpha}\bigr\|_E^\alpha = \|x\|_{\mathscr L_\infty(\Omega)}, \]

therefore it is natural to take \(E(0)=\mathscr L_\infty(\Omega)\).

The generalized Hölder inequality holds:

\[ \|xy\|_E\leq \|x\|_{E(\alpha)}\|y\|_{E(1-\alpha)} \qquad (x\in E(\alpha),\ y\in E(1-\alpha)). \]

If the space \(\mathscr L_\infty(\Omega)\) is normally embedded (see (1)) in \(E\), then the spaces \(E(1-\alpha)\) form a maximal hyperscale joining \(E\) and \(\mathscr L_\infty(\Omega)\).

1. A. P. Calderón (see \((^3)\)) proposed the following construction of intermediate spaces \(K_\alpha\) for ideal structures \(H_0\) and \(H_1\):

\[ \|x\|_{K_\alpha} = \inf_{\|x_0\|_{H_0}\leq 1,\ \|x_1\|_{H_1}\leq 1} \operatorname*{vrai\,max}_{t\in\Omega} \frac{|x(t)|}{|x_0(t)|^{1-\alpha}|x_1(t)|^\alpha}. \]

If \(H_1=\mathscr L_\infty(\Omega)\) and \(H_0=E\), then \(K_\alpha=E(1-\alpha)\).

Theorem 3. Let \(K_\theta\) \((0\leq\theta\leq 1)\) be Calderón’s family of intermediate spaces joining the ideal structures \(H_0\) and \(H_1\). For every interval \([\alpha_0,\beta_0]\subset[0,1]\) the space \(\overline K_\alpha\) of Calderón’s family \(\overline K_\alpha\) \((0\leq\alpha\leq 1)\), joining the spaces \(K_{\alpha_0}\) and \(K_{\beta_0}\), coincides with the space \(K_\theta\), \(\theta=\alpha_0(1-\alpha)+\beta_0\alpha\), and has the same norm.

It can be shown that for any two related ideal structures \(H_0\) and \(H_1\), Calderón’s spaces \(K_\alpha\) form a hyperscale on the interval \([0,1]\). This hyperscale is proper (see \((^1)\)).

Theorem 4. Let \(H_0\) and \(H_1\) be related ideal structures, and suppose that the set \(M\) of all bounded measurable functions is dense in the space \(H_1\). Then Calderón’s scale \(K_\alpha\) \((0\leq\alpha\leq 1)\) coincides with the maximal hyperscale constructed from the spaces \(H_0\) and \(H_1\).

Corollary. Let \(H_\alpha\) \((0 \leq \alpha \leq 1)\) be an arbitrary hyperscale connecting two ideal structures \(H_0\) and \(H_1\), for which the conditions of Theorem 4 are satisfied. If \(x_0 \in H_0\), \(x_1 \in H_1\), and \(0 \leq \alpha \leq 1\), then the function
\[ x(t)=|x_0(t)|^{1-\alpha}|x_1(t)|^\alpha \]
belongs to the space \(H_\alpha\).

1. Let \(E_\alpha\) \((0 \leq \alpha \leq 1)\) be a continuous normal scale of ideal structures connecting the ideal structures \(H_0\) and \(H_1\). Suppose that the set of all bounded measurable functions is dense in \(H_1\).

The family of Banach spaces \(E_\alpha(1/p)\) \((p \geq 1)\) will be called the power transform of the scale \(E_\alpha\) \((0 \leq \alpha \leq 1)\). It is easy to see that \(E_\alpha(1/p)\) is a continuous normal scale.

Theorem 5. Let \(E_0\) and \(E_1\) be related ideal structures, and let the set \(M\) be dense in \(E_1\). Then the minimal scale constructed from the spaces \(E_0(1/p)\) and \(E_1(1/p)\) majorizes the scale \(E_\alpha^{\min}(1/p)\) (see \((1)\)).

We give an example of an interpolation theorem that is a consequence of Theorem 5.

Consider the minimal scale \(E_\alpha^{\min}\), connecting the spaces \(\mathscr L_1\) and \(\mathscr L_\infty\), which is the Marcinkiewicz scale \(M_\alpha^0\), consisting of all functions \(x(t) \in \mathcal A\) for which
\[ \lim_{\operatorname{mes} e\to 0}\left[\int_e |x(t)|\,dt/(\operatorname{mes} e)^\alpha\right]=0 \]
with norm
\[ \|x\|_{M_\alpha^0} = \sup_{\operatorname{mes} e\ne 0} \left[\int_e |x(t)|\,dt/(\operatorname{mes} e)^\alpha\right]<\infty \]
(see \((1)\)). Every continuous normal scale \(F_\alpha\) \((0 \leq \alpha \leq 1)\), majorizing the minimal scale \(F_\alpha^{\min}\) \((0 \leq \alpha \leq 1)\), possesses the normal interpolation property with respect to the power transform of the Marcinkiewicz scale \(M_\alpha^0(1/p)\).

Let us note that the maximal scale connecting two related ideal structures consists of ideal structures.

Theorem 6. The power transform of the maximal scale connecting two related ideal structures \(H_0\) and \(H_1\) such that the set \(M\) is dense in \(H_1\) is a regular scale and, consequently, possesses the strict interpolation property with respect to every minimal scale.

As an example one may give the power transform of the Lorentz scale \(\Lambda_\alpha\) (see \((5)\)), which is the maximal scale connecting the spaces \(\mathscr L_1\) and \(\mathscr L_\infty\) (see \((1)\)).

The space conjugate to \(\Lambda_\alpha(1/p)\) consists of measurable functions \(f(t)\) for which
\[ \|f\|_{\Lambda_\alpha(1/p)'} = \inf\left\{(1-\alpha)\int_0^1 t^{-\alpha}D(t)^q\,dt\right\}^{1/q}<\infty, \]
where the infimum is taken over all functions \(D(t)\) such that
\[ (1-\alpha)\int_0^s t^{-\alpha}D(t)\,dt \geq \int_0^s f^*(t)\,dt \]
for all \(s \in [0,1]\), \(1/p+1/q=1\), \(p>1\).

Voronezh State University

Received
18 XI 1965

CITED LITERATURE

1. S. G. Kreĭn, Yu. I. Petunin, Scales of Banach spaces, UMN 21, no. 2 (128) (1966).
2. L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semiordered Spaces, Moscow–Leningrad, 1950.
3. A. P. Calderon, Studia Math., 24, No. 2 (1964); A. P. Calderon, Sbornik, Transl., Mathematics, 9 (3) (1965).
4. G. Ya. Lozanovskii, DAN, 158, No. 3 (1964).
5. G. G. Lorentz, Ann. Math., (2), 51, 37 (1950).