UDC 513.88:513.83

MATHEMATICS

V. A. DIKAREV, V. I. MATSAEV

AN EXACT INTERPOLATION THEOREM

(Presented by Academician L. V. Kantorovich on 1 X 1965)

Let \(X\) and \(Y\) be linear topological spaces. We shall agree to write \(A: X \to Y\) if \(A\) is a linear operator acting continuously from \(X\) into \(Y\).

The classical Riesz–Thorin theorem \(\left({}^{1}\right)\) asserts, in particular, that if

\[ A: L^{p_1}\to L^{q_1}, \qquad A: L^{p_2}\to L^{q_2}, \tag{1} \]

then also

\[ A: L^p \to L^q, \]

where \(p_1 < p < p_2\) and
\[ 1/q = 1/q_2 + (1/p - 1/p_2)(1/q_1 - 1/q_2)\times (1/p_1 - 1/p_2)^{-1}. \]
The main question discussed in the present note is the following: does there exist a space \(X\) (depending only on \(p_1, q_1, p_2, q_2\), and \(p\)) narrower than \(L^q\), and such that, if (1) is fulfilled, then \(A: L^p \to X\)? It turns out that the answer to this question is, generally speaking, affirmative. Under the additional restrictions \(p_1 \le q_1\), \(p_2 \le q_2\), we describe the minimal space \(X\) having this property (see item 2).

1. Let \(R\) be a space with a nonnegative countably additive measure \(\mu\). For a \(\mu\)-measurable (complex) function \(f\) on \(R\) we introduce \(f^*(t)\) \((0 \le t < \infty)\)—the nonincreasing right-continuous function equimeasurable with \(|f|\):

\[ \operatorname{mes}\{t: f^*(t)>N\}=\mu\{x: |f|>N\} \quad \text{for every } N>0. \]

In addition, introduce the function \(f^{**}(t)\)

\[ f^{**}(t)=\frac{1}{t}\int_0^t f^*(s)\,ds. \]

Consider the functional \(\|\cdot\|_{\lambda,p}\):

\[ \|f\|_{\lambda,p} = \left(\int_0^\infty t^{p\lambda-1} f^{*p}(t)\,dt\right)^{1/p}, \qquad 0<\lambda<1,\quad 1\le p<\infty, \]

\[ \|f\|_{\lambda,\infty} = \sup_{0<t<\infty} t^\lambda f^*(t), \qquad 0\le \lambda<1, \tag{2} \]

and the functional \(\langle f\rangle_{\lambda,p}\), which is defined like \(\|f\|_{\lambda,p}\), but in the right-hand sides of (2) one must substitute \(f^{**}(t)\) in place of \(f^*(t)\).

The inequality

\[ c_1(\lambda,p)\|f\|_{\lambda,p} \le \langle f\rangle_{\lambda,p} \le c_2\|f\|_{\lambda,p}, \qquad 0<c_1<c_2<\infty \]

is valid.

By \(L_\mu(\lambda,p)\) we denote \(\left({}^{2-4}\right)\) the set of functions \(f\) for which \(\|f\|_{\lambda,p}<\infty\). \(L_\mu(\lambda,p)\) is a Banach space with norm \(\langle\cdot\rangle_{\lambda,p}\).

We also introduce the space \(L_\mu(1,1)\), coinciding with \(L_\mu^1\), setting \(\|f\|_{1,1}=\|f\|_{L^1}\). We note that for \(\lambda=1/p\) the space \(L_\mu(\lambda,p)\) coincides with \(L_\mu^p\).

Theorem 1. If \(p_1<p_2\), then \(L_\mu(\lambda,p_1)\subset L_\mu(\lambda,p_2)\) and
\[ \|f\|_{\lambda,p_2}\leq C\|f\|_{\lambda,p_1}, \]
where \(C=C(\lambda,p_1,p_2)\). In general there are no other inclusion relations between the spaces \(L_\mu(\lambda,p)\).

2. Let \(X,Y,Z\) be linear subsets of a linear space \(V\), and suppose that each element \(z\in Z\) admits a representation (generally not unique) \(z=x+y\), where \(x\in X,\ y\in Y\). If \(T\) is a linear operator defined on a part of \(V\) containing \(X\) and \(Y\), then it is also defined on \(Z\).

In this situation we shall say that \(T\) is naturally extended to \(Z\). In what follows \(V\) will mean the space of all measurable functions.

Theorem 2. Let \(R_1\) and \(R_2\) be spaces with measures \(\nu_1\) and \(\nu_2\), and let
\[ T:\ L_{\nu_1}(\lambda_i,1)\to L_{\nu_2}(\mu_i,\infty)\qquad (i=1,2,\quad 0<\lambda_i\leq 1,\quad 0\leq \mu_i<1,\quad \lambda_1<\lambda_2,\quad \mu_1\ne \mu_2). \]
Then \(T\) is naturally extended to \(L_{\nu_1}(\lambda,p)\), and
\[ T:\ L_{\nu_1}(\lambda,p)\to L_{\nu_2}(\mu,p), \]
where
\[ \lambda_1<\lambda<\lambda_2,\qquad 1\leq p\leq \infty, \]
\[ \mu=\mu_1+(\lambda-\lambda_1)(\mu_2-\mu_1)(\lambda_2-\lambda_1)^{-1}. \tag{3} \]

Let us now return to the question posed at the beginning of the note. Let the operator \(T\) be defined on simple functions on \(R_1\) and take them into \(\nu_2\)-measurable functions on \(R_2\). Suppose, moreover, that \(T:L_{\nu_1}^{p_i}\to L_{\nu_2}^{q_i}\) \((i=1,2)\), where \(1\leq p_i\leq q_i\leq \infty,\ p_i\ne\infty,\ q_i\ne 1\), and \(p_1>p_2,\ q_1\ne q_2\). By Theorem 1,
\[ L_{\nu_1}(1/p_i,1)\subseteq L_{\nu_1}(1/p_i,p_i)=L_{\nu_1}^{p_i}, \]
\[ L_{\nu_2}^{q_i}=L_{\nu_2}(1/q_i,q_i)\subseteq L_{\nu_2}(1/q_i,\infty). \]

Therefore the hypotheses of Theorem 2 are satisfied with \(\lambda_i=1/p_i\) and \(\mu_i=1/q_i\), and hence, for \(p_1>p>p_2\), we have \(T:L_{\nu_1}^{p}\to L_{\nu_2}(\mu,p)\), where \(\mu\) is determined from (3) with \(\lambda=1/p\). Since \(p_i\leq q_i\), we have \(p\leq 1/\mu\), and \(L_{\nu_2}(\mu,p)\subseteq L_{\nu_2}^{1/\mu}\). Thus the range of the operator \(T\) lies in the space \(L_{\nu_2}(\mu,p)\), generally narrower than the space \(L^{1/\mu}\) obtained from the theorem of M. Riesz. As the following theorem shows, the space \(L(\mu,p)\) cannot be narrowed further.

We shall agree to say that a measure \(\nu\) on a space \(R\) is nonatomic if, for any two sets \(E_1\) and \(E_2\) such that \(E_1\subset E_2,\ \nu(E_1)<\nu(E_2)\), and for any number \(s\) such that \(\nu(E_1)<s<\nu(E_2)\), there exists a set \(E,\ E_1\subset E\subset E_2\), such that \(\nu(E)=s\).

Theorem 3. Let \(R_1\) and \(R_2\) be spaces with nonatomic measures \(\nu_1\) and \(\nu_2\), \(\nu_1(R_1)=\infty\), and let arbitrary collections of numbers \(\lambda,\lambda_i,\mu_i,p_i,q_i,p\) \((i=1,2)\) be given for which the spaces \(L(\lambda_i,p_i)\), \(L(\mu_i,q_i)\), \(\lambda_1<\lambda<\lambda_2\), \(\mu_1\ne\mu_2\), \(1\leq p\leq \infty\) make sense, and suppose that for each \(i\) (taken separately) one of the following two conditions is satisfied: a) \(0<\lambda_i<1,\ 0<\mu_i<1,\ 1\leq p_i\leq q_i\leq\infty\); b) \(0\leq \lambda_i\leq 1,\ p_i=1,\ q_i=\infty\). Let \(f\) be an arbitrary function from the space \(L_{\nu_2}(\mu,p)\), where \(\mu\) is determined from (3). Then there exists an operator \(T\) such that
\[ T:\ L_{\nu_1}(\lambda_i,p_i)\to L_{\nu_2}(\mu_i,q_i)\qquad (i=1,2) \]
and, consequently, \(T\) is naturally extended to \(L_{\nu_1}(\lambda,p)\),
\[ T:\ L_{\nu_1}(\lambda,p)\to L_{\nu_2}(\mu,p) \]
and \(Tg=f\), where \(g\in L_{\nu_1}(\lambda,p)\).

Below we give two examples illustrating the application of Theorem 2.

3. Let \(\{\varphi_n\}_1^\infty\) be an orthonormal and uniformly bounded system of functions on a space \(R\) with measure \(\mu\). Consider a space \(S\), whose elements are all positive integers, with measure \(\nu\) equal to one on each one-point set. If \(\{c_n\}_1^\infty\) is a sequence of numbers, then by \(\|c\|_{\lambda,p}\) we shall denote the norm in \(L_\nu(\lambda,p)\) of the function equal to \(c_n\) at the point \(n\).

Theorem 4. 1) Let \(f\in L_\mu(\lambda,p)\) \((1/2<\lambda<1,\ 1\le p\le\infty)\), and let \(\{c_n\}_1^\infty\) be the Fourier coefficients of \(f\) with respect to the system \(\{\varphi_n\}_1^\infty\). Then

\[ \|c\|_{1-\lambda,p}\le A\|f\|_{\lambda,p},\qquad A=A(\lambda,p,\{\varphi\}). \]

2) Let, for the sequence \(\{c_n\}_1^\infty\), the norm \(\|c\|_{\lambda,p}\) be finite \((1/2<\lambda<1,\ 1\le p\le\infty)\). Then there exists a function \(f\) for which the \(c_n\) are the Fourier coefficients with respect to the system \(\{\varphi_n\}_1^\infty\), and

\[ \|f\|_{1-\lambda,p}\le A\|c\|_{\lambda,p},\qquad A=A(\lambda,p,\{\varphi\}). \]

Indeed, let \(T\) be the operator assigning to \(f\) the function \(c(n)\) on the space \(S\), where \(c(n)=c_n\), and \(\{c_n\}\) are the Fourier coefficients of \(f\). It is clear that
\(T:L_\mu(1,1)\to L_\nu(0,\infty)\) and
\(T:L_\mu(1/2,1)\to L_\nu(1/2,\infty)\) (the latter is a consequence of Bessel’s inequality and Theorem 1).

From Theorem 2, 1) now follows; 2) is proved analogously. Theorem 4 contains, as a special case, Paley’s theorem \((^1)\).

Theorem 4 is sharp in a certain sense.

Theorem 5. Let \(c_1\ge c_2\ge c_3\ldots\) be a sequence tending to zero. In order that the function
\(f(x)=\sum c_n\cos nx\) belong to \(L(\lambda,p)\)
\((0<\lambda<1,\ 1\le p\le\infty)\), it is necessary and sufficient that
\(\|c\|_{1-\lambda,p}<\infty\).

This theorem is analogous to the Hardy–Littlewood theorem \((^1)\).

1. Consider the operator \(J\):

\[ (Jf)(x)=\int_{R_n}\frac{f(t)\,dt}{|t-x|^\alpha}\quad \left(0<\frac{\alpha}{n}<1\right), \tag{4} \]

where the integration is over \(n\)-dimensional real space \(R_n\). In what follows, spaces of functions defined on \(R_n\) with Lebesgue measure are considered.

Theorem 6.
\[ J:L(\lambda,p)\to L(\lambda-1+\alpha/n,p) \quad (1-\alpha/n<\lambda<1,\ 1\le p\le\infty). \]

Theorem 6 is obtained with the aid of Theorem 2 from the fact that

\[ J:L(1,1)\to L(\alpha/n,\infty);\qquad J:L(1-\alpha/n,1)\to L(0,\infty). \]

In particular, according to Theorem 6,

\[ J:L^p\to L(1/p-1+\alpha/n,p)\subseteq L^r \]

\[ (1-\alpha/n<1/p<1,\ 1/r=1/p-1+\alpha/n). \]

Thus Theorem 6 yields a strengthening of the well-known Hardy–Littlewood–Sobolev theorem \((^{5,6})\).

Theorem 7. Let \(f(t)\) in (4) range over all \(L(\lambda,p)\)
\((1-\alpha/n<\lambda<1,\ 1\le p<\infty)\). Consider all possible measurable functions \(g(x)\) such that \(|g(x)|\) is majorized by some rearrangement of the function \(|(Jf)(x)|\). The set of all such \(g(x)\) coincides with \(L(\lambda-1+\alpha/n,p)\).

Theorem 7 shows that Theorem 6 cannot be strengthened within the class of spaces whose norm is unchanged under rearrangement of functions.

Kharkov Institute of Mining Engineering,
Automation, and Computer Technology

Physico-Technical Institute of Low Temperatures,
Academy of Sciences of the Ukrainian SSR

Received
25 IX 1965

Cited Literature

\(^1\) A. Zygmund, Trigonometric Series, 2, 1965.
\(^2\) G. G. Lorentz, Ann. Math., 51, 37 (1950).
\(^3\) I. Halperin, Canad. J. Math., 5, 273 (1953).
\(^4\) A. P. Calderon, Sborn. per. Matematika, No. 3, 56 (1965).
\(^5\) G. Hardy, J. Littlewood, G. Polya, Inequalities, IL, 1948.
\(^6\) L. Hörmander, Estimates for Operators Invariant under Translations, IL, 1962.