Integrability Properties of Functions with a Given Behavior of Distribution Functions and Some Applications

We establish that if the distribution function of a measurable function v defined on a bounded domain Ω in ℝn (n ≥ 2) satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ k−αϕ(k)/ψ(k), where α > 0, ϕ: [1,+∞) → ℝ is a nonnegative nonincreasing measurable function such that the integral of the function s → ϕ(s)/s over [1,+∞) is finite, and ψ: [0,+∞) → ℝ is a positive continuous function with some additional properties, then |v|αψ(|v|) ∈ L1(Ω). In so doing, the function ψ can be either bounded or unbounded. We give corollaries of the corresponding theorems for some specific ratios of the functions ϕ and ψ. In particular, we consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ Ck−α(ln k)−β with C, α > 0 and β ≥ 0. In this case, we strengthen our previous result for β > 1 and, on the whole, we show how the integrability properties of the function v differ depending on which interval, [0, 1] or (1,+∞), contains β. We also consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ Ck−α(ln ln k)−β with C, α > 0 and β ≥ 0. We give examples showing the accuracy of the obtained results in the corresponding scales of classes close to Lα(Ω). Finally, we give applications of these results to entropy and weak solutions of the Dirichlet problem for second-order nonlinear elliptic equations with right-hand side in some classes close to L1(Ω) and defined by the logarithmic function or its double composition.


INTRODUCTION
By definition (see, for instance, [1,2]), the distribution function of a measurable function v defined on a bounded domain Ω ⊂ R n (n ≥ 2) is the correspondence s → meas{|v| > s}, s ≥ 0. An estimate of the values of the distribution function makes it possible to establish a certain integrability on Ω of the original function or a function depending on it. The study of this question is of interest, in particular, to clarify the integrability properties of solutions of elliptic equations and variational inequalities with right-hand side in the space L 1 (Ω) or classes close to L 1 (Ω). For instance, it was shown in [2] that, for k > 0, the distribution function of the entropy solution u of the Dirichlet problem for a second-order nonlinear elliptic equation with right-hand side f ∈ L 1 (Ω) satisfies an estimate of the form meas{|u| > k} ≤ Ck −α (0. 1) with positive constants C and α, the first of which depends on the space dimension n, an exponent p characterizing the growth of the coefficients of the equation, and the norm of the function f in L 1 (Ω) and the second depends only on n and p. A similar estimate was also established for the gradient of the entropy solution. The obtained estimates imply the integrability of certain powers of the moduli of the entropy solution and its gradient. In particular, it follows from the estimate (0.1) that u ∈ L λ (Ω) for any λ ∈ (0, α). In addition, in the case p > 2 − 1/n, the established estimates provide the belonging of the entropy solution to the Sobolev spaces with any exponent less than a limit one. If, for sufficiently large k, the distribution function of a measurable function v : Ω → R satisfies the estimate meas{|v| > k} ≤ Ck −α (ln k) −β , (0. 2) where C, α > 0 and β > 1, or the more general estimate where α > 0 and ϕ : [1, +∞) → R is a nonnegative nonincreasing measurable function such that the integral of the function s → ϕ(s)/s over [1, +∞) is finite, then, as shown, for instance, in [3,4], the function v belongs to L α (Ω). Using these results, in the mentioned papers, we established conditions on the right-hand side of a second-order nonlinear elliptic equation under which the entropy solution of the corresponding Dirichlet problem and its gradient belong to some limit Lebesgue spaces. Based on the result proved in [4] that a measurable function v belongs to the space L α (Ω) if its distribution function satisfies the estimate (0.3) with the above function ϕ, in the present paper, we study the case where, for sufficiently large k, the distribution function of a measurable function v satisfies the estimate meas{|v| > k} ≤ ϕ(k) k α ψ(k) (0. 4) with α > 0, a function ϕ as above, and a positive continuous function ψ on [0, +∞) having some additional properties. It turns out that, in this case, the inclusion |v| α ψ(|v|) ∈ L 1 (Ω) holds, and the function ψ can be either bounded or unbounded. The corresponding results are proved in Section 1 (see Theorems 1 and 2). Consequences of these general results for some specific ratios of the functions ϕ and ψ in the estimate (0.4) are given in Section 2. In particular, we consider the case where the distribution function of a measurable function v satisfies the estimate (0.2) with C, α > 0 and β ≥ 0. In so doing, we strengthen the result obtained in [3] for β > 1 and, on the whole, show how the integrability properties of the function v differ depending on which interval, [0, 1] or (1, +∞), contains β (see Corollaries 1 and 2). We also consider the case where the distribution function of a measurable function v satisfies the estimate meas{|v| > k} ≤ Ck −α (ln ln k) −β with C, α > 0 and β ≥ 0 (see Corollaries 3 and 4). In Section 3, we give examples showing the accuracy of the results of the previous section in the corresponding scales of classes close to L α (Ω). The closeness of a class of functions K to the space L α (Ω) is understood as the validity of one of the following conditions: (i) K ⊂ L α (Ω) and K ⊂ L α+ε (Ω) for any ε > 0; (ii) K ⊂ L α (Ω) and K ⊂ L α−ε (Ω) for any ε ∈ (0, α). Finally, in Section 4, we give applications of the results of Section 2 S114 KOVALEVSKY to entropy and weak solutions of the Dirichlet problem for second-order nonlinear elliptic equations with right-hand side in some classes close to L 1 (Ω) and defined by the logarithmic function or its double composition. As a result, we strengthen known and obtain new results on the integrability properties of the moduli of the specified solutions.

GENERAL THEOREMS
Let n ∈ N, n ≥ 2, and let Ω be a bounded open set in R n . We give a proposition on which the subsequent theorems are based.
Essentially, the stated proposition coincides with Lemma 2.1 in [4] and does not require a separate proof.
We pass to the statement and proof of theorems.
The difference between the statement of the following theorem and the statement of Theorem 1 is that condition (d) of the first theorem is replaced by the requirement of boundedness of the function ψ.
Proof. By the boundedness of the function ψ, there exists c > 0 such that We define It is clear that the function ϕ 1 is nonnegative, nonincreasing, and measurable and the function ψ 1 is positive and continuous. By condition (a), we have S116 KOVALEVSKY +∞ 1 ϕ 1 (s) s ds < +∞, and, in view of conditions (b) and (c), the following assertions hold: This and the fact that the function ϕ 1 is nonincreasing imply the inequality ϕ 1 (s) ≤ ϕ 1 (t). Finally, by condition (d), for any k ≥ k 0 , we have .

COROLLARIES
We give corollaries of the general Theorems 1 and 2 for some specific ratios of the functions ϕ and ψ.
Proof. We fix γ > 1 − β, and let ϕ : [1, +∞) → R be the function such that It is clear that the function ϕ is nonnegative, nonincreasing, and measurable. Let ψ : [0, +∞) → R be the function such that It is clear that the function ψ is positive, bounded, and continuous. We define k 0 = max{e, e γ/α } and show that conditions (a)-(d) of Theorem 2 are satisfied. Since β + γ > 1, for an arbitrary N > e, we obtain Hence, We further note that, for any λ, s > 0, the inequality λ ln s < s λ holds. Using this, for an Finally, by (2.1) and the definition of the functions ϕ and ψ, for any k ≥ k 0 , the following inequality holds: Thus, conditions (a)-(d) of Theorem 2 are satisfied. Hence, by this theorem, Proof. We fix γ ∈ (0, β − 1), and let ϕ : [1, +∞) → R be the function such that It is clear that the function ϕ is nonnegative, nonincreasing, and measurable. Let ψ : [0, +∞) → R be the function such that It is clear that the function ψ is positive and continuous. Define k 0 = e and σ = (1 + γ/α) β−γ . Now, we note that conditions (a)-(e) of Theorem 1 are satisfied. Indeed, condition (a) of Theorem 1 is satisfied because β − γ > 1. The fulfillment of conditions (b) and (c) of Theorem 1 is obvious. Next, let s > k 0 and Thus, condition (d) of Theorem 1 is satisfied. Finally, by (2.2) and the definition of the functions ϕ and ψ, condition (e) of Theorem 1 is satisfied. We now deduce from this theorem We note that if the conditions of Corollary 2 are satisfied, then, according to Lemma 2 in [3] with equivalent conditions, we have only the inclusion v ∈ L α (Ω). Thus, the conclusion of Corollary 2 is stronger than the conclusion of the mentioned lemma in [3].

EXAMPLES
We consider examples showing the accuracy of the results of the previous section in the corresponding scales of the classes introduced above.

APPLICATIONS
We can specify a number of applications of the results of Sections 1 and 2 to the study of the integrability properties of solutions of elliptic equations and variational inequalities with right-hand side in classes close to L 1 (Ω). However, in this section, we restrict ourselves to several applications of the results of Section 2 to entropy and weak solutions of the Dirichlet problem for second-order nonlinear elliptic equations with right-hand side in some classes close to L 1 (Ω) and defined by the logarithmic function or its double composition. Other applications of the results of this paper will be given in our forthcoming publications.

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
Vol. 308 Suppl. 1 S122 KOVALEVSKY In addition, we assume that, for almost all x ∈ Ω and any ξ, ξ ∈ R n , ξ = ξ , Let f ∈ L 1 (Ω). We consider the following Dirichlet problem: We note that if p > 2 − 1/n, then, according to Theorem 1 in [5], there exists a weak solution of problem (4.1) belonging to • W 1,λ (Ω) for any λ, 1 ≤ λ < n(p − 1)/(n − 1). Further, for any k > 0, let the function T k : R → R be defined as follows: We denote by , then δv is the mapping from Ω to R n such that, for any x ∈ Ω and any i ∈ {1, . . . , n}, we have (δv(x)) i = δ i v(x).
Definition 4. An entropy solution of problem (4.1) is a function u ∈ • T 1,p (Ω) such that, for any v ∈ C ∞ 0 (Ω) and any k > 0, By Theorem 6.1 in [2], there exists a unique entropy solution of problem (4.1). We also note that if p > 2 − 1/n and u is the entropy solution of problem (4.1), then u is a weak solution of this problem (see, for instance, [2,4]).
As a result, we come to the required conclusion.
Using Corollary 3 and Proposition 11, we obtain the following results.