E. M. SEMENOV

EMBEDDING THEOREMS FOR BANACH SPACES OF MEASURABLE FUNCTIONS

(Presented by Academician A. N. Kolmogorov on 31 I 1964)

Recently, various authors \((^{1,2})\) have studied Banach spaces of measurable functions possessing the following property: if \(x(t)\in E\), then \(|x(t)|\in E\), and the norms of these functions in \(E\) coincide.

In the present note we consider spaces of measurable functions \(E\) satisfying the requirements:

1) If \(x(t)\in E\) and the functions \(|x(t)|\) and \(|y(t)|\) are equimeasurable, then \(y(t)\in E\) and
\[ \|x\|_E=\|y\|_E . \]

2) If \(|x(t)|\leq |y(t)|\) and \(y(t)\in E\), then \(x(t)\in E\) and
\[ \|x\|_E\leq \|y\|_E . \]

We shall call such spaces symmetric. We shall assume that the functions from \(E\) are defined on the interval \([0,1]\).

It is obvious that \(\chi_e(t)\), the characteristic function of any measurable subset \(e\subset[0,1]\), belongs to \(E\), whence it follows that every finitely valued function belongs to \(E\). It can be shown (see \((^3)\) or \((^4)\)) that the second condition in the definition of a symmetric space follows from the first if it is known that the set of finitely valued functions is dense in \(E\).

Among symmetric spaces, the spaces \(\Lambda(\varphi)\) and \(M(\varphi)\), first considered by Lorentz \((^5)\) and Halperin \((^6)\), possess important extremal properties. By \(\Lambda(\varphi)\) is denoted the Banach space of functions measurable on \([0,1]\) for which
\[ \|x\|_{\Lambda(\varphi)}=\int_0^1 x^*(t)\,d\varphi(t), \]
where \(x^*(t)\) is a nonincreasing function equimeasurable with \(|x(t)|\), and \(\varphi(t)\) is a nondecreasing concave function on \([0,1]\). In \(M(\varphi)\) the norm is introduced by the formula
\[ \|x\|_{M(\varphi)}=\sup_{0<h\leq 1}\frac{\int_0^h x^*(t)\,dt}{\varphi(h)} . \]

Here \(\varphi(t)\) does not decrease on \([0,1]\), and \(\varphi(t)>0\) for \(t>0\).

By definition, \(\|\chi_e\|_E\) is determined by specifying the measure of the set \(e\subset[0,1]\). In other words, for every symmetric space \(E\) there exists a function \(\varphi(t)\) \((0\leq t\leq 1)\) such that
\[ \|\chi_e\|_E=\varphi(me). \]
From the definition of a symmetric space it follows that \(\varphi(0)=0\). In this case we shall call the function \(\varphi(t)\) fundamental for the space \(E\). For example, \(\varphi(t)=\sqrt[p]{t}\) is the fundamental function of the space \(\mathcal L_p\) \((1\leq p<\infty)\).

Theorem 1. In order that a function \(\varphi(t)\) be fundamental for some symmetric space, it is necessary and sufficient that the following conditions be satisfied: a) \(\varphi(t)\) does not decrease on \([0,1]\); b) \(\varphi(t)/t\) does not increase on \((0,1]\).

Proof. Necessity. Let \(0<t_1<t_2\leq 1\), \(e_1\subset e_2\subset[0,1]\), \(me_1=t_1\), \(me_2=t_2\), and \(\|\chi_e\|_E=\varphi(me)\). Then

\[ \varphi(t_1)=\|\chi_{e_1}\|_E=\frac12\|(\chi_{e_1}-\chi_{e_2/e_1})+\chi_{e_2}\|_E \leq \frac12\|\chi_{e_1}-\chi_{e_2/e_1}\|_E+\frac12\|\chi_{e_2}\|_E =\|\chi_{e_2}\|_E=\varphi(t_2). \]

Put
\[ e_{k,n,m}(t)=\chi_{\left[0,\frac{k}{n}\right]}(t)-\chi_{\left[\frac{m-1}{n},\frac{m}{n}\right]}(t), \]
where \(1\leq m\leq k\leq n\). From the obvious identity
\[ (k-1)\chi_{\left[0,\frac{k}{n}\right]}(t)=\sum_{m=1}^{k} e_{k,n,m}(t) \]
we have, by definition,
\[ (k-1)\left\|\chi_{\left[0,\frac{k}{n}\right]}\right\|_E = \left\|\sum_{m=1}^{k} e_{k,n,m}\right\|_E \leq \sum_{m=1}^{k}\|e_{k,n,m}\|_E = k\left\|\chi_{\left[0,\frac{k-1}{n}\right]}\right\|_E \]
or
\[ (k-1)\varphi\left(\frac{k}{n}\right)\leq k\varphi\left(\frac{k-1}{n}\right), \]
whence we obtain
\[ \frac{\varphi\left(\frac{k}{n}\right)}{\frac{k}{n}} \leq \frac{\varphi\left(\frac{k-1}{n}\right)}{\frac{k-1}{n}}. \]

Thus, if \(r<R\) are rational, then
\[ \frac{\varphi(R)}{R}\leq\frac{\varphi(r)}{r}. \tag{1} \]

Now let \(0<t_1<t_2\leq 1\). Construct two sequences of rational numbers \(r_k\) and \(R_k\) such that \(r_k\uparrow t_1,\ R_k\downarrow t_2\). Then, by virtue of a) and (1):
\[ \frac{\varphi(t_2)}{t_2} \leq \frac{R_k}{t_2}\frac{\varphi(R_k)}{R_k} \leq \frac{R_k}{t_2}\frac{\varphi(r_k)}{r_k} \leq \frac{R_k t_1}{r_k t_2}\frac{\varphi(t_1)}{t_1}. \tag{2} \]

Passing in (2) to the limit as \(k\to\infty\), we obtain the required inequality.

Sufficiency. Let \(\varphi(t)\) satisfy conditions a) and b). We shall prove that as \(E\) one may take the space \(M(\psi)\), \(\psi(t)=t/\varphi(t)\). Indeed,
\[ \|\chi_e\|_{M(\psi)} = \sup_{0<h\leq 1} \frac{\int_0^h \chi_e^*(t)\,dt}{\psi(h)}. \]

Since \(\psi(h)\) is nondecreasing in \(h\), it follows that
\[ \|\chi_e\|_{M(\psi)} = \sup_{0<h\leq me} \frac{\int_0^h dt}{\psi(h)} = \sup_{0<h\leq me}\varphi(h). \]

But \(\varphi(h)\) also is nondecreasing in \(h\), therefore
\[ \|\chi_e\|_{M(\psi)}=\varphi(me). \]
The theorem is proved.

Theorem 2. If \(E\) is symmetric, then \(\mathscr L_\infty\subset E\subset \mathscr L_1\), and both embeddings are continuous.

Proof. If \(x(t)\in \mathscr L_\infty\), then, by the definition of a symmetric space, \(x(t)\in E\) and
\[ \|x\|_E\leq \|\operatorname{vrai\ sup}|x(t)|\|_E=\varphi(1)\|x\|_{\mathscr L_\infty}. \]

Let

\[ y_0(t)=\sum_{k=0}^{N-1} x_k \chi_{e_k}(t), \quad \text{where } x_k \geq 0,\quad m e_k=\frac1N . \]

Put

\[ y_j(t)=\sum_{k=0}^{N-1} x_{k+j(\bmod N)}\chi_{e_k}(t) \]

\((j=1,2,\ldots,N-1)\). Obviously, the functions \(y_0(t)\) and \(y_j(t)\) are equimeasurable, and therefore \(\|y_0\|_E=\|y_j\|_E\). Since

\[ \sum_{k=0}^{N-1} x_k=\sum_{j=0}^{N-1} y_j(t), \]

we have

\[ \sum_{k=0}^{N-1} x_k\varphi(1)= \left\|\sum_{j=0}^{N-1}y_j\right\|_E \leq \sum_{j=0}^{N-1}\|y_j\|_E = N\|y_0\|_E, \]

whence

\[ \|y_0\|_E \geq \frac1N \sum_{k=0}^{N-1} x_k\varphi(1) = \varphi(1)\|y_0\|_{\mathcal L_1}. \tag{3} \]

Now let \(z(t)\) be a nonnegative function from \(E\). There exists a nondecreasing sequence of nonnegative simple functions \(z_k(t)\) converging to \(z(t)\) almost everywhere on \([0,1]\). Moreover, one may assume that the function \(z_k(t)\) is constant on sets of equal measure. Since \(z_k(t)\leq z(t)\), we have \(\lim_{k\to\infty}\|z_k\|_E\leq \|z\|_E\). By Levi’s theorem it follows that \(\lim_{k\to\infty}\|z_k\|_{\mathcal L_1}=\|z\|_{\mathcal L_1}\) (if \(z(t)\notin \mathcal L_1\), then, obviously, \(\lim_{k\to\infty}\|z_k\|_{\mathcal L_1}=\infty\)). From (3) it follows that

\[ \|z\|_{\mathcal L_1} = \lim_{k\to\infty}\|z_k\|_{\mathcal L_1} \leq \varphi(1)\lim_{k\to\infty}\|z_k\|_E \leq \varphi(1)\|z\|_E . \tag{4} \]

Replacing an arbitrary function from \(E\) by its modulus and using (4), we obtain the required inequality. The theorem is proved.

It follows from Theorem 1 that the fundamental function \(\varphi(t)\) is continuous on \((0,1]\). It can be shown (see \((^3)\) or \((^4)\)) that for the continuity of \(\varphi(t)\) at \(t=0\) it is necessary and sufficient that the norm of the space \(E\) be weaker than the norm of \(\mathcal L_\infty\); similarly, the boundedness on \((0,1]\) of the function \(\varphi(t)/t\) is equivalent to the fact that the norm of \(E\) is stronger than the norm of \(\mathcal L_1\).

Theorem 3. Let \(\varphi(t)\) be the fundamental function of the symmetric space \(E\). Then there exists a concave function \(\psi(t)\) satisfying the conditions \(\psi(0)=\varphi(0)\), \(\psi(1)=\varphi(1)\), \(\varphi(t)\leq \psi(t)\leq 2\varphi(t)\), and such that \(E\supset \Lambda(\psi)\) and

\[ \|x\|_E \leq \|x\|_{\Lambda(\psi)}. \]

If \(\varphi(t)\) is concave, then \(\psi(t)=\varphi(t)\).

In the proof of the theorem the following simple fact is used: the functions

\[ \frac{1}{\psi(m e\cup g)}[\chi_e(t)-\chi_g(t)], \]

where \(e,g\subset [0,1]\), \(e\cap g=\varnothing\), and only they, are the extreme points of the unit ball of the space \(\Lambda(\psi)\).

In what follows (in the corollaries of Theorems 3 and 7) we shall use the definition from \((^7,^8)\).

Corollary 1. If \(\varphi_0(t)\leq \varphi_1(t)\), then the spaces \(\Lambda(\varphi_1^{-\alpha}\varphi_0^\alpha)\) \((0\leq \alpha\leq 1)\) form a maximal continuous normal scale of spaces connecting \(\Lambda(\varphi_0)\) and \(\Lambda(\varphi_1)\).

Corollary 2 (interpolation theorem in the spaces \(\Lambda(\varphi)\)). If \(\varphi_0(t)\leq \varphi_1(t)\), \(\psi_0(t)\leq \psi_1(t)\), \(A\) is a linear operator and

\[ \|Ax\|_{\Lambda(\varphi_i)}\leq C_i\|x\|_{\Lambda(\psi_i)} \quad (i=0,1), \]

then for all \(\alpha \in (0,1)\) and all \(x \in \Lambda(\psi_0^{1-\alpha}\psi_1^\alpha)\) the inequality

\[ \|Ax\|_{\Lambda(\varphi_0^{1-\alpha}\varphi_1^\alpha)} \leq C_0^{1-\alpha}C_1^\alpha \|x\|_{\Lambda(\psi_0^{1-\alpha}\psi_1^\alpha)} \]

is satisfied.

The proof follows directly from Corollary 1 and the interpolation theorem of S. G. Kreĭn \((^7)\).

Theorem 4. If \(\varphi(t)\) is the fundamental function of a symmetric space \(E\), then \(E \subset M(\psi)\) and

\[ \|x\|_{M(\psi)} \leq \|x\|_{E}, \]

where \(\psi(t)=t/\varphi(t)\).

Theorem 4 cannot be improved in the sense that \(\|\chi_e\|_{M(\psi)}=\varphi(me)\); the same can be said also of Theorem 3, if \(\varphi(t)\) is concave.

From Theorems 3 and 4 one can obtain new embedding theorems for the basic classes of symmetric spaces: \(\mathscr L_M(^9)\), \(\Lambda(p,\varphi)\), and \(M(p,\varphi)\) \((^5)\).

Theorem 5. Let \(E_i\) \((i=0,1)\) be symmetric spaces and let \(\varphi_i(t)\) be their corresponding fundamental functions. If \(\varphi_0(t)\) and \(t/\varphi_1(t)\) are concave on \([0,1]\) and

\[ C=\int_0^1 \varphi_0'(t)\,d\,\frac{t}{\varphi_1(t)}<\infty, \]

then \(E_0 \supset E_1\) and

\[ \|x\|_{E_0} \leq C\|x\|_{E_1}. \]

Theorem 6. If the function \(\psi(t)=t x_0^*(t)\) is nondecreasing in some neighborhood of \(0\), \(x^*(1)>0\), and

\[ \varlimsup_{h\to 0} \frac{\displaystyle \int_0^h x_0^*(t)\,dt} {h x_0^*(h)} <\infty, \]

then from the inclusion \(x_0(t)\in E\), where \(E\) is symmetric, it follows that \(x_0(t)\in M(\psi)\subset E\), and the embedding is continuous.

Let \(E\) be a symmetric space. A set of functions \(M\subset E\) will be called symmetric if \(x(t)\in E\) whenever \(y(t)\in E\) and \(x^*(t)\leq y^*(t)\). It is obvious that the unit ball of the space \(E\) is a symmetric set. We give one property of embeddings of symmetric spaces.

Theorem 7. Let the set of finitely valued functions be dense in the symmetric spaces \(E_0\) and \(E_1\), let \(\|x\|_{E_0}\leq \|x\|_{E_1}\), and let \(M\) be a symmetric set from \(E_1\). If \(M\) is closed in the norm of \(E_1\), then it is also closed in the norm of \(E_0\).

Applying the theorem to the unit ball of the space \(E_1\), we obtain, by virtue of \((^8)\),

Corollary. If for the spaces \(E_0\) and \(E_1\) the conditions of Theorem 7 are fulfilled, then \(E_0\) and \(E_1\) are conjugate.

The author expresses sincere gratitude to S. G. Kreĭn for guidance.

Voronezh State University

Received
31 I 1964

CITED LITERATURE

\(^1\) W. A. J. Luxemburg, A. C. Zaanen, Math. Ann., 14, 150 (1963).
\(^2\) Yu. I. Gribanov, P. K. Belobrov, Izv. vyssh. uchebn. zaved., Matematika, No. 4, 44 (1963).
\(^3\) E. M. Semenov, Scales of Banach spaces connecting the spaces \(\mathscr L_1\) and \(\mathscr L_\infty\), Dissertation, Voronezh, 1964.
\(^4\) B. S. Mityagin, A. S. Shvarts, UMN, 19 (2), No. 2 (1964).
\(^5\) G. G. Lorentz, Pacific J. Math., 1, 411 (1950).
\(^6\) J. Halperin, Canad. J. Math., 5, 273 (1953).
\(^7\) S. G. Kreĭn, DAN, 132, No. 3 (1960).
\(^8\) S. G. Kreĭn, Yu. I. Petunin, DAN, 139, No. 6 (1961).
\(^9\) M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, 1958.