MATHEMATICS

M. M. Vainberg and I. V. Shragin

THE NEMYTSKII OPERATOR AND ITS POTENTIAL IN ORLICZ SPACES

(Presented by Academician S. L. Sobolev, February 7, 1958)

1. It is known ((^1)) that, in the study of nonlinear integral equations of Hammerstein type, an important role is played by the properties of the Nemytskii operator (h) and its potential (f). The study of the operator (h) and its potential (f) in the spaces (L^p), begun by one of the authors ((^2)), led to definitive results ((^3)) on the action, boundedness, and continuity of the operator (h), and made it possible to establish the properties of the functional (f) in these spaces that are needed for applications ((^4)). In the present work we study the operator (h) and its potential (f) in Orlicz spaces generalized in the sense of Zaanen ((^{5})), ((^{6-9})). In Theorem 1, necessary and sufficient conditions are established for the action of the operator (h) from some class or Orlicz space into other similar spaces or classes. In the other theorems, conditions are established for the boundedness and continuity of the operator (h), as well as conditions for the continuity and weak lower semicontinuity of the functional (f). The proof of the propositions on the action of the operator (h) uses a functional-theoretic construction generalizing the construction used in proving Theorem 1 of ((^{10})).

We also note that, although the work is joint, part of the results belongs to one of the authors, and part to the other.

1. Let (B) be a set in a finite-dimensional Euclidean space of finite or infinite Lebesgue measure; let (M(u)) and (M_1(u)) be arbitrary Young functions (see ((^{8,5,10}))).

Put

[
d=\sup {u:u\in[0,\infty),\ M(u)<\infty},\qquad
d_1=\sup {u:u\in[0,\infty),\ M_1(u)<\infty}.
]

From the definition of Young functions it follows that (0<d,d_1\leq\infty). Let the real-valued function (g(u,x)) be defined on the topological product ([-\infty,+\infty]\times B), and let (g(u,x)) be allowed to take infinite values as well. We assume that (g(u,x)) is measurable on (B) with respect to (x) for every fixed (u\in(-\infty,+\infty)), and is continuous on ((-\infty,+\infty)) with respect to (u) for almost every fixed (x\in B).

For arbitrary positive (\lambda,\mu,\nu) and (x\in B), put:

[
E_x(\lambda,\mu,\nu)={u:\lambda u\in\langle -d,d\rangle,\ M_1[\mu |g(u,x)|]>\nu M(\lambda |u|)},
]

[
F(x;\lambda,\mu,\nu)=\sup{\mu |g(u,x)|:u\in E_x(\lambda,\mu,\nu)\cup{0}},
]

where (\langle -d,d\rangle=[-d,d]) when (M(d)<\infty), and (\langle -d,d\rangle=(-d,d)) when (M(d)=\infty).

We note the following properties of the function (F(x;\lambda,\mu,\nu)), which are used in the proof of Theorem 1:

(1^\circ.) The function (F(x;\lambda,\mu,\nu)) is measurable in (x) for fixed (\lambda,\mu,\nu).

(2^\circ.) For fixed (x,\lambda,\mu), it is nonincreasing in (\nu).

(3^\circ.) For fixed (x,\mu,\nu), it is nonincreasing in (\lambda).

(4^\circ.) For fixed (x,\lambda,\nu), it is nondecreasing in (\mu).

1. Consider the Nemytskii operator (h): (hu=g(u(x),x)). Let (L^M=L^M(B)) and (L^{M_1}=L^{M_1}(B)) be the Orlicz spaces generated by the Young functions (M(u)) and, respectively, (M_1(u)). As is known (see (8), p. 100, and (5)), the space (L^M) consists of all functions (u(x)), measurable on (B), for which

[
\int_B M[k|u(x)|]\,dx<\infty,
\tag{1}
]

where (k) is a certain positive number, depending on the function (u(x)). The functions (u(x)) satisfying inequality (1) with (k=1) form the class (L_M). The functions (u(x)) satisfying inequality (1) for every (k) form the subspace (L_M^\chi) of the space (L^M) (11) (see also (9)). The space (L_M^\chi) is of interest only in the case (d=\infty), since for (d<\infty) it contains only the zero element. In view of this, everywhere that the action of the operator (h) from (L_M^\chi) into some class or space is discussed, we shall assume that (d=\infty).

We shall say that the operator (h) acts from ((L^M,L_M,L_M^\chi)) into ((L^{M_1},L_{M_1},L_{M_1}^\chi)) if it acts from some class indicated in the first parentheses into some class indicated in the second parentheses.

The following proposition establishes a characteristic property of the function (g(u,x)) that generates the operator (h) from ((L^M,L_M,L_M^\chi)) into ((L^{M_1},L_{M_1},L_{M_1}^\chi)).

Theorem 1. In order that (h) act from ((L^M,L_M,L_M^\chi)) into ((L^{M_1},L_{M_1},L_{M_1}^\chi)), it is necessary and sufficient that the conditions indicated in the following table, respectively, be satisfied:

Let (\mathfrak F) be a family of functions measurable on (B) with the following property: if (u(x)\in\mathfrak F) and a function (v(x)), measurable on (B), satisfies the condition (|v(x)|\le |u(x)|) almost everywhere, then (v(x)\in\mathfrak F). Examples of such families are (L^{M_1}), (L_{M_1}), (L_{M_1}^\chi).

Theorem 2. Let (d<\infty). Then, in order that the operator (h) act from (L^M) into (\mathfrak F), it is sufficient, and if (L^M) consists of the same functions as (L^\infty), then also necessary, that the following condition be satisfied:

[
\sup_{|u|\le \alpha}|g(u,x)|\equiv a_\alpha(x)\in\mathfrak F
]

for every nonnegative (\alpha).

Let us also note the following proposition.

Theorem 3. Let (d=\infty) and (d_1<\infty). Then, in order that the operator (h) act from any one of the classes (L^M,\ L_M,\ L_M^\chi) into (L^{M_1}), it is necessary, and if (L^{M_1}) consists of the same functions as (L^\infty), then also sufficient, that the following condition be satisfied: there exists a constant (C) such that, for almost all (x\in B), (|g(u,x)|\le C) for (u\in(-\infty,+\infty)).

This theorem generalizes the proposition from ((^3)) on the action of the operator (h) from (L^p) into (L^\infty).

1. In ((^3)) a necessary and sufficient condition for the action of the operator (h) from the class (L^p) into the class (L^{p_1}) was published in the form of a certain inequality for the function (g(u,x)). Similar conditions are obtained from Theorem 1 also for generalized Orlicz spaces. We give two such propositions. Put
   [
   M_1^{-1}(v)=\sup{u:u\ge0,\ M_1(u)\le v},\quad v\ge0.
   ]
   (M_1^{-1}(v)) is a nondecreasing and continuous function on ([0,\infty)).

Theorem 4. In order that the operator (h) act from (L^M) into (L^{M_1}), it is necessary and sufficient that the following condition be satisfied: to each positive number (\lambda) there correspond a function (a_\lambda(x)\in L^{M_1}) and positive numbers (\alpha_\lambda) and (\beta_\lambda) such that
[
|g(u,x)|\le a_\lambda(x)+\alpha_\lambda M_1^{-1}[\beta_\lambda M(\lambda |u|)]
]
for all (x\in B,\ \lambda u\in\langle-d,d\rangle).

It is said ((^6)) that the function (M_1(u)) satisfies the (\Delta_2)-condition if (d_1=\infty) and (M_1(2u)\le C M_1(u)) for (u\ge u_0\ge0), where (C=\mathrm{const}). In what follows, when the (\Delta_2)-condition is discussed, we shall assume (u_0=0) if (\operatorname{mes} B=\infty).

Theorem 5. In order that the operator (h) act from (L_M) into (L_{M_1}), it is necessary, and if (M_1(u)) satisfies the (\Delta_2)-condition then also sufficient, that for all (x\in B,\ u\in\langle-d,d\rangle) the inequality
[
|g(u,x)|\le a(x)+M_1^{-1}[bM(|u|)],
\tag{2}
]
hold, where (a(x)\in L_{M_1},\ b>0).

This theorem generalizes the proposition from ((^3)) on the action of the operator (h) from (L^p) into (L^{p_1}), when (p_1<\infty). Propositions analogous to Theorems 4 and 5 hold also in the other cases of Theorem 1.

Remark. If in Theorem 5 inequality (2) is replaced by the inequality
[
|g(u,x)|\le \max{a(x),\ M_1^{-1}[bM(|u|)]},
]
then the reservation about the (\Delta_2)-condition for the function (M_1(u)) is not needed.

1. Let us now consider the question of boundedness and continuity of the operator (h) (cf. ((^{12}))).

Theorem 6. If the operator (h) acts from (Z) into (L^{M_1}), where (Z) coincides with one of the classes: (L^M,\ L_M,\ L_M^\chi), then it is bounded on (S), where (S) coincides with (L^M) for (Z=L^M), with the unit ball (|u|_M\le1) of the space (L^M) for (Z=L_M), and with the ball (D={u(x):|u|_M=\lambda^{-1}}) for (Z=L_M^\chi) ((\lambda) is the number appearing in Theorem 1 in the case of the action of the operator (h) from (L_M^\chi) into (L^{M_1})).

Theorem 7. If the operator (h) acts from (L^M) into (L_{M_1}^\chi), or from (L_M^\chi) into (L_{M_1}^\chi), then it is continuous.

We note that the following proposition also holds: if the operator (h) acts from some ball of the space (L^M) into (L_{M_1}^\chi), then it is continuous at every interior point of this ball.

1. Let the operator (h) act from (L^M) into (L^{M_1}), where (M(u)) and (M_1(u)) are mutually complementary Young functions (see ((^5)) and ((^8)), p. 99). As in ((^4)), consider the functional

[
f(u)=\int_B dy \int_0^{u(y)} g(v,y)\,dv,
\tag{3}
]

for which the following propositions hold.

Theorem 8. If the operator (h) acts from (L^M) into (L^{M_1}), then the functional (f(u)) satisfies the Lipschitz condition in every ball of the space (L^M).

Theorem 9. If the operator (h) acts from (L^M) into (L^{M_1}), then it is the gradient of the functional (3): (\operatorname{grad} f(u)=hu), i.e. (f) is a potential of the operator (h).

Analogous propositions hold for the functional (f(u)) when the operator (h) acts from (L_M) or (L_M^\chi) into (L^{M_1}).

1. The continuity of the functional (3) is used when, by the variational method ((^1)), one proves the existence of eigenfunctions of the Hammerstein operator (\Gamma=Ah), or of solutions of the equations (\Gamma u=u), where (A) is a linear integral operator. In this case one must require that (A) be a completely continuous operator from (L^{M_1}) or (L_{M_1}^\chi) into (L^M). If, however, the functional (f(u)) is weakly semicontinuous, then in proving the existence of eigenfunctions of the operator (\Gamma), or of solutions of the equation (\Gamma u=u), one may dispense with the requirement of complete continuity of the operator (A), replacing it by the requirement that the operator (A) be bounded. In view of this, it is important to establish conditions for the weak semicontinuity of the functional (3). We give one such proposition.

Theorem 10. Let the function (g(u,x)), generating the operator (h) from (L^M) into (L^{M_1}) or into (L_{M_1}^\chi), be nonincreasing (nondecreasing) with respect to (u) for almost every fixed (x\in B). Then the functional (f(u)), defined by equality (3), is weakly lower (upper) semicontinuous in (L^M).

1. In conclusion, we note that the propositions established in the present work, by using topological or variational methods, lead to new criteria for the existence both of eigenfunctions of the operator (\Gamma) and of solutions of the equation (\Gamma u=u).

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
7 II 1958

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