Reports of the Academy of Sciences of the USSR

1. Volume 164, No. 4

MATHEMATICS

E. M. SEMENOV

INTERPOLATION OF LINEAR OPERATORS IN SYMMETRIC SPACES

(Presented by Academician L. V. Kantorovich on 13 III 1965)

Following \((^1)\), a Banach space \(E\) of measurable functions on \([0,1]\) will be called symmetric if: 1) from \(|x(t)| \le |y(t)|\) and \(y(t) \in E\) it follows that \(\|x\|_E \le \|y\|_E\); 2) from \(x(t) \in E\) and the equimeasurability of the functions \(|x(t)|\) and \(|y(t)|\) it follows that \(\|x\|_E = \|y\|_E\). Every symmetric space \(E\), by the formula \(\|\chi_e\|_E = \varphi(me)\), where \(\chi_e(t)\) is the characteristic function of a measurable set \(e \subset [0,1]\), defines a function \(\varphi(t)\), called the fundamental function. In \((^1)\) some properties of the fundamental function of a symmetric space are given. It can be shown that in \(E\) one can always introduce an equivalent norm such that the fundamental function of \(E\) will be concave; therefore throughout what follows the concavity of \(\varphi(t)\) is assumed.

Theorem 1. In order that every linear operator which acts continuously in each space \(L_p\) \((1 < p < \infty)\) act continuously in the symmetric space \(E\), it is necessary, and if \(E\) is separable or conjugate to a separable space, also sufficient, that there exist numbers \(1 < \mu \le \nu < 2\) and \(0 < t_0 \le \tfrac12\) such that

\[ \mu \varphi(t) \le \varphi(2t) \le \nu \varphi(t) \tag{1} \]

for all \(0 \le t \le t_0\).

Obviously, the second condition in the definition of a symmetric space is necessary, up to an equivalent norm of the space \(E\), in order that every linear operator continuous in \(L_p\) \((1 < p < \infty)\) be continuous in \(E\).

B. S. Mityagin \((^2)\) showed that if \(A\) is a linear operator acting continuously in the spaces \(L_1\) and \(L_\infty\), and \(E\) is either a separable symmetric space or the conjugate of a separable symmetric space, then \(A\) acts continuously in \(E\) and

\[ \|A\|_{E \to E} \le \max\bigl(\|A\|_{L_1 \to L_1}, \|A\|_{L_\infty \to L_\infty}\bigr). \]

By virtue of the interpolation theorem of M. Riesz, the continuity of a linear operator \(A\) in \(L_1\) and \(L_\infty\) is equivalent to the continuity of \(A\) in \(L_p\) \((1 \le p \le \infty)\). However, many operators considered in analysis (singular operators, multipliers of Fourier series, etc.) are continuous in every \(L_p\) \((1 < p < \infty)\), but do not act continuously in \(L_1\) and \(L_\infty\). In connection with this, an attempt was undertaken to describe the class of symmetric spaces in which all linear operators bounded in each \(L_p\) \((1 < p < \infty)\) are continuous.

Our principal tool will be the special functional spaces \(\Lambda(\psi)\) and \(M(\psi)\), first introduced for consideration by G. G. Lorentz \((^3)\):

\[ \|x\|_{\Lambda(\psi)} = \int_0^1 x^*(t)\,d\psi(t), \tag{2} \]

\[ \|x\|_{M(\psi)} = \sup_{0<h\le1}\int_0^h x^*(t)\,dt \big/ \psi(h), \tag{3} \]

where \(x^*(t)\) here and everywhere below denotes the rearrangement \((^4)\) of the function \(|x(t)|\), i.e. \(x^*(t)\) is a nonincreasing function equimeasurable

with \(|x(t)|\). In (2) the function \(\psi(t)\) is concave and nondecreasing, and in (3) it is nondecreasing on \([0,1]\).

Scheme of the proof of Theorem 1. Necessity.

Lemma 1. Let the function \(h(t)\) be concave and nondecreasing on \([0,1]\), \(h(0)=0\), and let \(l>1\).

1. If
   \[ \lim_{t\to 0}\frac{h(2t)}{h(t)}=1, \]
   then
   \[ \lim_{t\to 0}\frac{h(lt)}{h(t)}=1. \]

2. If
   \[ \overline{\lim}_{t\to 0}\frac{h(2t)}{h(t)}=2, \]
   then
   \[ \overline{\lim}_{t\to 0}\frac{h(lt)}{h(t)}=l. \]

To prove the necessity of the conditions of Theorem 1, consider the Hilbert operator
\[ \widetilde{x}(t)=\mathrm{v.p.}\int_0^1 \frac{x(\tau)}{t-\tau}\,d\tau . \]

It is well known that the Hilbert operator is continuous in each \(L_p\) \((1<p<\infty)\); therefore, by assumption, it is continuous in \(E\), and a fortiori
\[ \|\widetilde{x}_{[0,h]}\|_E \le c\|x_{[0,h]}\|_E . \]
If \(\varphi(t)\) is the fundamental function of \(E\), then, by virtue of (1), this inequality implies
\[ \|\widetilde{x}_{[0,h]}\|_{M(\psi)} \le c\|x_{[0,h]}\|_{M(\psi)}=c\varphi(h), \]
where \(\psi(t)=t/\varphi(t)\). Computing \(\widetilde{x}_{[0,h]}(t)\), it is easy, using Lemma 1, to obtain from the inequality
\[ \lim_{t\to 0}\frac{\varphi(2t)}{\varphi(t)}>1,\qquad \overline{\lim}_{t\to 0}\frac{\varphi(2t)}{\varphi(t)}<2. \]

Sufficiency.

Lemma 2. If a linear operator \(A\) is continuous in each \(L_p\) \((1<p<\infty)\), and the concave nondecreasing function \(\varphi(t)\) satisfies condition (1), then \(A\) is continuous in \(\Lambda(\varphi)\), and \(\|A\|_{\Lambda(\varphi)\to\Lambda(\varphi)}\) depends only on \(\mu,\nu\) from (1) and \(\|A\|_{L_p\to L_p}\).

In what follows we shall need some definitions from (5). The space \(E'\) associated with \(E\) is defined as the set of functions \(x(t)\) for which
\[ \|x\|_{E'}=\sup_{\|y\|_E=1}\int_0^1 x(t)y(t)\,dt<\infty . \]
The space \(E\) has the Fatou property if
\[ \|x\|_E=\sup_{\substack{|y(t)|\le |x(t)|\\ y(t)\in L_\infty}}\|y\|_E . \]

Using the standard scheme of reasoning (5), one can show: if \(E\) is separable or is conjugate to a separable space, then it has the Fatou property and
\[ \|x\|_E=\sup_{\|y\|_{E'}=1}\int_0^1 x(t)y(t)\,dt . \tag{4} \]

Lemma 3. Let \(x(t)\) and \(y(t)\) be summable on \([0,1]\). In order that \(\|x\|_E\le \|y\|_E\) for every symmetric space \(E\) possessing the Fatou property, it is necessary and sufficient that
\[ \int_0^\tau x^*(t)\,dt \le \int_0^\tau y^*(t)\,dt \]
for all \(0\le \tau\le 1\).

We note that there are various analogues of this assertion (6).

Let \(1<a<b<2\). Denote by \(Q(a,b)\) the set of summable functions \(y(t)\) for which

\[ a\int_0^\tau y^*(t)\,dt \leq \int_0^\tau y^*(t)\,dt \leq b\int_0^\tau y^*(t)\,dt \]

for all \(0\leq \tau \leq 1/2\). On \(E\) define a new symmetric norm

\[ \|x\|_1=\sup_{y\in E'\cap Q(a,b)} \int_0^1 x(t)y(t)\,dt \,/\, \|y\|_{E'} . \]

The central point in the proof of the sufficiency of the conditions of Theorem 1 is

Lemma 4. Let \(E\) satisfy the conditions of Theorem 1. If \(a\) and \(b\) are sufficiently close to 1 and 2 respectively, then \(\|x\|_E\) and \(\|x\|_1\) are equivalent.

We outline the proof of Lemma 4. It is enough to obtain the estimate

\[ \|x\|_E \leq c\|x\|_1 \tag{5} \]

only on functions \(x(t)=x^*(t)\). With the aid of the above-mentioned theorem of B. S. Mityagin it is shown that (5) need only be proved for functions of the form

\[ x(t)=\sum_{k=1}^{\infty} x_k\chi_{[2^{-k},\,2^{-k+1}]}(t), \tag{6} \]

where \(0\leq x_1\leq x_2\leq \cdots\). But for such functions (5) follows from the inequality

\[ \sup_{y\in E'\cap Q(a,b)}\inf \|z-y\|_{E'}<1, \]

where the supremum is taken over all functions \(z(t)\) of the form (6), \(\|z\|_{E'}=1\). With the aid of the theorem on extreme points it is proved that the latter inequality need only be verified for the functions \(z(t)=x_{-l}(t)/\|x_{0,z^{-l}}\|_{E'}\) \((l=1,2,\ldots)\). The validity of this same assertion, and with it the lemma, is proved directly by computation.

We have

\[ \|x\|_E=\|x^*\|_E=\sup_{y\in E'}\int_0^1 x^*(t)y(t)\,dt\,/\,\|y\|_{E'}= \]

\[ =\sup_{y\in E'}\int_0^1 x^*(t)y^*(t)\,dt\,/\,\|y\|_{E'} =\sup_{y\in E'}\|x\|_{\Lambda(\psi)}\,/\,\|y\|_{E'}, \]

where

\[ \psi(t)=\int_0^t y^*(\tau)\,d\tau . \]

The conditions \(y(t)\in Q(a,b)\) and \(a\psi(t)\leq \psi(2t)\leq b\psi(t)\) coincide. Therefore, for certain \(a,b\), by Lemmas 2 and 4 we obtain

\[ \|Ax\|_E \leq c\|Ax\|_1 = c\sup_{y\in E'\cap Q(a,b)} \|Ax\|_{\Lambda(\psi)} /\|y\|_{E'} \leq \]

\[ \leq cc_1\sup_{y\in E'\cap Q(a,b)} \frac{\|x\|_{\Lambda(\psi)}}{\|y\|_{E'}} =cc_1\|x\|_1\leq cc_1\|x\|_E . \]

One can verify that \(\|A\|_{E\to E}\) depends only on the function \(\alpha(p)=\|A\|_{L_p\to L_p}\) and on the constants \(\mu\) and \(\nu\) from condition (1).

Corollary 1. If \(G_\alpha^i\) \((i=1,2)\) are two regular scales (7) of symmetric spaces connecting the spaces \(L_1\) and \(L_\infty\), and a linear operator \(A\) is continuous in each space \(G_\alpha^1\) \((0<\alpha<1)\), then \(A\) is continuous also in each space \(G_\alpha^2\) \((0<\alpha<1)\).

This assertion is a generalization of the theorem on types of operators (8).

The best studied class of symmetric spaces is (apart from \(L_p\)) the class of Orlicz spaces (9).

Corollary 2. In order that every linear operator continuous in each space \(L_p\) \((1<p<\infty)\) be continuous in the Orlicz space \(L_M^*\), it is necessary and sufficient that \(L_M^*\) be reflexive.

This assertion was recently obtained by R. Ryan \((^{10})\). With the aid of Theorem 1 one can obtain some results of A. Zygmund \((^{11})\) and I. B. Simonenko \((^{12})\) on interpolation of linear operators in Orlicz spaces.

In analysis one studies the operator

\[ \widetilde{x}(t)=\frac{1}{\pi}\int_{0}^{\pi} \frac{x(t+\tau)-x(t-\tau)}{2\operatorname{tg}\tau/2}\,d\tau . \]

Corollary 3. In order that the operator \(x(t)\to \widetilde{x}(t)\) be continuous in the symmetric space \(E\), it is necessary, and if \(E\) is separable or conjugate to a separable space, also sufficient, that the fundamental function of the space \(E\) satisfy condition (1).

If \(E\) is an Orlicz space, then we obtain the corresponding results of J. Lamperti \((^{13})\), S. M. Lozinskii \((^{14})\), A. Zygmund \((^{15})\), pp. 116–118, and R. Ryan \((^{10})\). Exactly the same result is also valid for the Hilbert transform.

From Theorem 1 one can obtain a certain assertion of G. G. Lorentz \((^{16})\) on the continuity of the operator

\[ \theta x(t)=\frac{1}{t}\int_{0}^{t}x(\tau)\,d\tau \]

in the spaces \(\Lambda(\psi)\) or the spaces \(\Lambda(p,\psi)\), where

\[ \|x\|_{\Lambda(p,\psi)} =\left(\int_{0}^{1}[x^{*}(t)]^{p}\,d\psi(t)\right)^{1/p}. \]

Corollary 4. In order that every linear operator continuous in each space \(L_p\) \((1<p<\infty)\) be continuous in \(\Lambda(p,\psi)\), it is necessary and sufficient that

\[ \lim_{t\to 0}\frac{\psi(2t)}{\psi(t)}>1. \]

Denote by \(S_n(x)\) the sequence of partial sums of the Fourier series of the function \(x(t)\).

Corollary 5. If, for the space \(E\), the conditions of Theorem 1 are satisfied, then \(\|S_n(x)\|_E\le c\|x\|_E\), where \(c\) does not depend on \(n\).

For some concrete spaces this result is known.

The method used in the proof of Theorem 1 makes it possible to somewhat weaken the assumption in the cited theorem of B. S. Mityagin. It is sufficient to require that (4) hold.

The author expresses sincere gratitude to S. G. Krein for his constant attention to the work.

Voronezh State
University

Received
25 II 1965

REFERENCES

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