Reports of the Academy of Sciences of the USSR
1963. Volume 151, No. 6

MATHEMATICS

I. B. Simonenko

INTERPOLATION AND EXTRAPOLATION OF LINEAR OPERATORS IN ORLICZ SPACES

(Presented by Academician V. I. Smirnov, 4 III 1963)

A function \(M\) is called an \(N\)-function if

\[ M(u)=\int_0^{|u|} p(t)\,dt, \]

where \(p(t)\) is a nondecreasing positive function; \(N(u)=\int_0^{|u|} q(t)\,dt\), where \(q(t)\) is the right inverse of \(p\), is called complementary to \(M(u)\) \((^{9})\).

Let \(D\) be a measurable set (possibly of infinite measure) in \(n\)-dimensional Euclidean space \(E_n\). Then by the class \(L_M(D)\) we mean the set of functions \(u\) for which

\[ \int_D M[u(P)]\,dP<\infty . \]

To the Orlicz space \(L_M^*(D)\) we assign those functions \(u\) satisfying the condition

\[ \int_D |u|\,|v|\,dP<\infty \]

for all \(v\in L_N(D)\). The norm in the space \(L_M^*(D)\) is introduced by the equality

\[ \|u\|_M=\sup_v \int_D u(P)v(P)\,dP,\qquad \int_D N[v(P)]\,dP\leq 1 . \tag{1} \]

Besides \(N\)-functions, we shall need one more class of functions. An absolutely continuous positive function \(M(u)\) will be called quasipower if

\[ \alpha=\sup_u \frac{uM'(u)}{M(u)}<\infty,\qquad \beta=\inf_u \frac{uM'(u)}{M(u)}>-\infty; \]

these numbers \(\alpha\) and \(\beta\), determined by the function \(M\), will be denoted by \(\alpha(M)\) and \(\beta(M)\). We also define the numbers \(\alpha_\infty(M)\) and \(\beta_\infty(M)\) by the equalities

\[ \alpha_\infty(M)=\overline{\lim}_{u\to\infty}\frac{uM'(u)}{M(u)};\qquad \beta_\infty(M)=\underline{\lim}_{u\to\infty}\frac{uM'(u)}{M(u)}. \]

If \(\beta(M)>1\), then \(M\), although generally speaking not an \(N\)-function, has associated with it an \(N\)-function

\[ \widetilde{M}(u)=\int_0^{|u|}\frac{M(t)}{t}\,dt, \]

satisfying the inequalities

\[ \frac{1}{\alpha(M)}M(u)\leq \widetilde{M}(u)\leq \frac{1}{\beta(M)}M(u). \]

Moreover, \(\widetilde{M}\) is a quasipower function and

\[ \alpha(\widetilde{M})\leq \alpha(M),\qquad \beta(\widetilde{M})\geq \beta(M); \]

analogous inequalities are also valid for \(\alpha_\infty,\beta_\infty\). Let \(\bar D\), generally speaking different from \(D\), be a measurable set in \(\bar n\)-dimensional Euclidean space. For brevity we introduce the notation

\[ \rho(D,M,f)=\int_D M[|f(P)|]\,dP. \]

We have obtained the following interpolation theorem 1.

Theorem 1. Let the linear operator \(K\) act from \(L_{p_i}(D)\) into \(L_{\bar p_i}(\bar D)\) \((1\leq p_1\leq \bar p_1,\ 1\leq p_2\leq \bar p_2)\), and let \(M\) be a quasipower function, with \(p_1<\beta(M)\leq \alpha(M)<p_2\), while \(\bar M\) is related to \(M\) by the relation \(\bar M^{-1}(t)=\)

\[ = t^\delta M^{-1}\left(t^{\Delta/\bar{\Delta}}\right) \left( \Delta=\pi_1-\pi_2,\quad \bar{\Delta}=\bar{\pi}_1-\bar{\pi}_2,\quad \delta=\frac{\pi_1\bar{\pi}_2-\pi_2\bar{\pi}_1}{\pi_1-\pi_2},\quad \pi_i=\frac{1}{p_i},\quad \bar{\pi}_i=\frac{1}{\bar p_i} \right). \]

Then
\[ \rho(\bar D,\bar M,Kf)\leq T[\rho(D,M,f)], \]
where \(T\) is a monotonically increasing function depending on \(p_1,\bar p_1,p_2,\bar p_2,M,\|K\|_{(L_{p_i}(D)\to L_{\bar p_i}(\bar D))}\) \((i=1,2)\), and the dependence on the norms \(\|K\|_{p_i\to\bar p_i}\) is such that, when they tend to zero and the other constants are fixed, \(T(u)\to 0\) (\(u(>0)\) is any number).

From Theorem 1 follows the interpolation theorem 2 for Orlicz spaces.

Theorem 2. Suppose the conditions of Theorem 1 are satisfied and, in addition, the function \(M\) is an \(N\)-function. Then the operator \(K\) also acts from \(L_M^*(D)\) into \(L_{\widetilde M}^*(\bar D)\), where
\[ \widetilde M(u)=\int_0^{|u|}\frac{\bar M(t)}{t}\,dt, \]
and
\[ \|K\|_{(L_M^*(D)\to L_{\widetilde M}^*(\bar D))}\to 0 \]
when
\[ \|K\|_{(p_i\to\bar p_i)}\to 0, \]
while the remaining quantities remain constant.

In the case when at one of the endpoints \(p_1\) or \(p_2\) the operator \(K\) is also completely continuous, there holds a result analogous to that of M. A. Krasnosel’skii \((^3)\) for the spaces \(L_p\).

Theorem 3. Let the linear operator \(K\) act from \(L_{p_1}(D)\) into \(L_{\bar p_1}(\bar D)\) or from \(L_{p_2}(D)\) into \(L_{\bar p_2}(\bar D)\) as a completely continuous operator, and at the other endpoint be simply continuous. Then \(K\) also acts as a completely continuous operator from \(L_M^*(D)\) into \(L_{\widetilde M}^*(\bar D)\).

In the case when \(\operatorname{mes}D<\infty\), Theorems 1, 2, and 3 remain valid under weaker assumptions on \(M\): instead of the inequalities for \(\alpha(M)\), \(\beta(M)\), it is required that the corresponding inequalities hold for \(\alpha_\infty(M)\), \(\beta_\infty(M)\).

The first interpolation theorems of the theory of linear operators in the spaces \(L_p\) are due to M. Riesz \((^1)\). A further development of this theory was given by C. G. Krein \((^6)\). Then A. P. Calderón and A. Zygmund \((^2)\) and S. Koizumi \((^7)\) proved an interpolation theorem in Orlicz spaces for the case \(p_i=\bar p_i\). M. A. Krasnosel’skii \((^3)\) obtained, for the spaces \(L_p\), a theorem according to which the complete continuity of an operator \(K\) at one of the endpoints \(L_{p_1}\) or \(L_{p_2}\) implies the complete continuity of the operator \(K\) in the intermediate spaces \(L_p\) \((p_1<p<p_2)\). Theorems 2 and 3 proved by us generalize these results.

At the basis of the proof of Theorem 1 and of Theorem 4 formulated below lies the idea of A. P. Calderón and A. Zygmund \((^{10})\) of using an estimate of the measure \(E_y(Kf)\) \((P\in E_y(Kf)\), if \(|Kf|>y)\). This idea was realized by them in the case of singular integrals of a special form in order to prove their boundedness in the spaces \(L_p\). A careful analysis of the work \((^{10})\) has allowed us to prove the following result:

Theorem 4 (extrapolation). Let the integral operator
\[ Af=\int_D A(P,Q)f(Q)\,dQ \]
possess the following properties:

1) \(A\) is bounded as an operator acting from \(L_{p_0}(D)\) into \(L_{p_0}(\bar D)\).

2) The kernel \(A(P,Q)\) satisfies the condition: for any concentric spheres \(S\) and \(S_1\) with ratio of radii \(2/1\), for any \(Q\) and \(Q'\) from \(S_1\) the inequality
\[ \int_{S'\cap D}|A(P,Q)-A(P,Q')|\,dP\leq B<\infty, \tag{2} \]
holds,

where \(S'\) is the complement of \(S\), and the constant \(B\) does not depend on \(S,Q\), or \(Q'\).

I. Then the operator \(A\) is bounded as an operator acting from \(L_M^*(D)\) into \(L_M^*(D)\). Here the \(N\)-function \(M\) satisfies the condition

\[ 1<\beta \leq \frac{uM'(u)}{M(u)} \leq \alpha \tag{3} \]

for \(\alpha < p_0\). In this case the norm of the operator \(\|A\|_M\) depends only on \(\|A\|_{p_0}, \alpha, \beta, p_0, D, B\).

II. If instead of condition 2) the condition

\[ 2')\quad \int_{S'\cap D} |A(P,Q)-A(P',Q)|\,dQ \leq B' < \infty, \]

where \(P,P'\in S_1\), is satisfied, and \(B'\) does not depend on \(P, P', S\), then the operator \(A\) is bounded in the spaces \(L_M^*(D)\) for \(\beta>p_0\). In this case the norm of the operator depends only on \(\|A\|_{p_0}, \alpha, \beta, p_0, D, B'\).

III. If conditions 1), 2), 2') are satisfied simultaneously, then the operator \(A\) is bounded in all spaces \(L_M^*\) satisfying condition (3), and \(\|A\|_M\) depends only on \(\alpha, \beta, p_0, \|A\|_{p_0}, D, B, B'\).

Let us note that the main assertion of the theorem is not the boundedness of the operator \(A\), but the fact that its norm depends only on the constants occurring in the hypotheses of the theorem. This is connected with the fact that an extrapolation theorem is usually applied according to the following scheme.

There is a sequence of integral operators

\[ A_m f=\int_D A_m(P,Q)f(Q)\,dQ, \]

converging strongly in some space \(L_{p_0}\) to the operator \(A\) (most often \(p_0=2\)). In order to establish the boundedness of the operator \(A\) in other spaces \(L_p\) and in Orlicz spaces, it is necessary to verify whether the estimates of the extrapolation theorem hold uniformly with respect to \(m\). In this way one can prove boundedness in Orlicz spaces of a certain kind for singular integrals, proved by the author earlier in paper \((^8)\). The application of the extrapolation theorem considerably simplifies the proof.

We give two more results characterizing the operator \(A\) in the limiting cases \(p=1\) and \(p=\infty\).

Theorem 5. If \(\operatorname{mes} D<\infty\) and the operator \(A\) satisfies conditions 1), 2'), then for every measurable bounded function \(f\) there exists a number \(\delta>0\) such that

\[ I=\int_D \exp(\delta |Af|)\,dP<\infty, \tag{4} \]

where \(\delta\) depends only on \(\operatorname{vrai}\sup |f|, B', \|A\|_{p_0}, p_0, n\); \(I\) also depends on \(\operatorname{mes}D\).

This theorem follows from the estimate \(\|A\|_p\leq cp\) for \(p\geq 2p_0\), and from a result of V. I. Yudovich \((^{15})\) concerning operators with such growth in \(p\).

By passing to the conjugate operator, Theorem 3 implies

Theorem 6. If the operator \(A\) satisfies conditions 1) and 2) of the extrapolation theorem and \(\operatorname{mes}D<\infty\), then the estimate

\[ \int_D |A(f)|\,dP \leq \int_D \ln\left(\frac{|f|}{c_1}+1\right) \left(\frac{|f|}{c_1}+1\right)\,dP +c_2, \tag{5} \]

holds, where \(c_1,c_2\) are constants depending only on \(\|A\|_{p_0}, p_0, \operatorname{mes}D, B, n\).

The theorems proved make it possible to obtain a number of new results. We present some of them.

Corollary 1. Potential-type integrals over the whole \(n\)-dimensional space \(E_n\)

\[ Kf\int_D \frac{f(Q)}{r^\lambda(P,Q)}\,dQ \]

were studied by S. L. Sobolev \((^4)\).

They established that the operator \(K\) acts from \(L_p\left(1<p<\dfrac{n}{n-\lambda}\right)\) into \(L_{\bar p}\), where

\[ \frac{1}{p}-\frac{1}{\bar p}=\frac{n-\lambda}{n}. \]

Let \(M(u)\) satisfy the condition

\[ 1<\beta \leq \frac{uM'(u)}{M(u)} \leq \alpha < \frac{n}{n-\lambda}; \]

then, on the basis of the interpolation theorem, the operator \(K\) acts as a bounded operator from \(L_M^{*}\) into \(L_{\widetilde M}^{*}\) (the connection of \(M\) with \(\widetilde M\) is indicated in Theorem 3). If integration is carried out not over the whole space, but over a measurable subset of finite measure, then the operator \(K\), in addition, is completely continuous. It is understood that Theorems 2 and 3 also make it possible to obtain embedding theorems in Orlicz spaces.

Some results on the boundedness of integrals of potential type for domains of finite measure and embedding theorems were obtained by I. V. Gelman \((^{16,17})\).

Corollary 2. Let the operator \(K\) be bounded as an operator acting from \(L_{p_1}(D)\) into \(L_{p_1}(D)\) and from \(L_{p_2}(D)\) into \(L_{p_2}(D)\), and let \(N\) be its right (left) regularizer* in one of the spaces \(L_{p_1}(D)\) or \(L_{p_2}(D)\), with \(N\) also bounded in both spaces \(L_{p_1}\) and \(L_{p_2}\). Then, on the basis of Theorem 3, we obtain that the operator \(N\) regularizes \(K\) on the right (left) also as an operator acting from \(L_M^{*}\) into \(L_M^{*}\), where \(M(u)\) satisfies the condition

\[ p_1<\beta \leq \frac{uM'(u)}{M(u)} \leq \alpha < p_2. \]

Corollary 3. Since regularization is the basis of the investigations of S. G. Mikhlin \((^{11-14})\) on multidimensional singular integral equations, all these investigations, carried out in the spaces \(L_p(D)\), automatically extend to Orlicz spaces**.

The work was carried out in the seminar on nonlinear problems of mechanics. The author expresses his gratitude to its participants, especially V. I. Yudovich, for useful advice.

Rostov-on-Don
State University

Received
2 III 1963

REFERENCES CITED

\(^{1}\) M. Riesz, Acta Math., 49, 465 (1926).
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\(^{4}\) S. L. Sobolev, Matem. sborn., 4 (46), 471 (1938).
\(^{5}\) A. Zygmund, Trigonometric Series, 1939.
\(^{6}\) S. G. Krein, DAN, 130, 491 (1960).
\(^{7}\) Koizumi Simuyuki, Proc. Japan Acad., 134, No. 4 (1958).
\(^{8}\) I. B. Simonenko, DAN, 130, No. 5 (1960).
\(^{9}\) M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, 1958.
\(^{10}\) A. P. Calderon, A. Zygmund, Acta Math., 88, 85 (1952).
\(^{11}\) S. G. Mikhlin, UMN, 3, issue 3, 29 (1948).
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\(^{14}\) S. G. Mikhlin, DAN, 131, No. 6, 1019 (1960).
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\(^{16}\) I. V. Gelman, Izv. vyssh. uchebn. zaved., Matematika, No. 2, 44 (1960).
\(^{17}\) I. V. Gelman, Izv. vyssh. uchebn. zaved., Matematika, No. 4, 55 (1960).

* The operator \(N\) regularizes \(K\) on the right (left) if \(KN=I+B\) (\(NK=I+B\)), where \(B\) is completely continuous.

** When \(\operatorname{mes} D<\infty\), it is sufficient to require that, instead of the inequalities for \(\alpha,\beta\), the corresponding inequalities for \(\alpha_\infty\) and \(\beta_\infty\) be fulfilled. This remark also applies to Theorem 4 (the extrapolation theorem).