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 L^α(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.

中文翻译：

---

给定分布函数行为的函数的可积性及其应用

我们建立，如果测函数的分布函数v在ℝ有界域Ω定义^Ñ（Ñ ≥2）满足，对于足够大的ķ，估计MEAS {| v | > ķ }≤ ķ ^-α φ（ķ）/ψ（ķ），其中α > 0，φ：[1，+∞）→ℝ是一个非负非增测函数，使得该函数的积分小号→φ（小号）/ š以上[1，+∞）是有限的，和ψ：[0，+∞）→ℝ是一个正连续函数，具有一些其他性质，然后| v | ^α ψ（| V |）∈大号¹（Ω）。这样，函数ψ可以是有界的或无界的。对于函数ϕ和ψ的某些特定比率，我们给出了相应定理的推论。特别地，我们考虑以下情况：对于足够大的k，可测量函数v的分布函数满足估计值{|。v | > ķ }≤ CK ^-α（LN ķ）^-β与C，α > 0和β ≥0。在这种情况下，我们强化我们以前对结果β > 1，并且对整个中，我们显示该函数的积属性如何v而不同，这取决于间隔，[0，1]或（1，+∞）包含β。我们还考虑以下情况：对于足够大的k，可测量函数v的分布函数满足估计值{|。v | > k } ≤Ck- ^α（ln ln k）^-β，C，α > 0和β≥0我们得到实施例示出了获得的结果的在类对应的尺度的准确度接近大号^α（Ω）。最后，我们将这些结果应用到右手型二阶非线性椭圆方程的Dirichlet问题的熵和弱解中，该类在接近L ¹（Ω）的某些类别中，并由对数函数或其双组合定义。