A.A. Kovalevsky. Integrability properties of functions with a given behavior of distribution functions and some applications ... P. 78-92

Vol. 25, no. 1, 2019

We establish that if the distribution function of a measurable function $v$ given on a bounded domain $\Omega$ of $\mathbb R^n$ ($n\geqslant 2$) satisfies, for sufficiently large $k$, the estimate ${\rm meas}\{\vert v\vert>k\}\leqslant k^{-\alpha}\varphi(k)/\psi(k)$, where $\alpha>0$, $\varphi\colon[1,+\infty)\to\mathbb R$ is a nonnegative nonincreasing measurable function such that the integral of the function $s\to\varphi(s)/s$ over $[1,+\infty)$ is finite, and $\psi\colon[0,+\infty)\to\mathbb R$ is a positive continuous function with some additional properties, then $\vert v\vert^\alpha\psi(\vert v\vert)\in L^1(\Omega)$. In so doing, the function~$\psi$ can be bounded or unbounded. We give corollaries of the corresponding theorems for some specific ratios of the functions $\varphi$ and $\psi$. In particular, we consider the case where the distribution function of a measurable function $v$ satisfies, for sufficiently large $k$, the estimate ${\rm meas}\{\vert v\vert>k\}\leqslant Ck^{-\alpha}(\ln k)^{-\beta}$ with $C,\alpha>0$ and $\beta\geqslant 0$. In this case, we strengthen our previous result for $\beta>1$ and, on the whole, we show how the integrability properties of the function $v$ differ depending on which of the intervals $[0,1]$ or $(1,+\infty)$ contains $\beta$. We also consider the case where the distribution function of a measurable function $v$ satisfies, for sufficiently large $k$, the estimate ${\rm meas}\{\vert v\vert>k\}\leqslant Ck^{-\alpha}(\ln\ln k)^{-\beta}$ with $C,\alpha>0$ and $\beta\geqslant 0$. We give examples showing the accuracy of the obtained results in the corresponding scales of classes close to~$L^\alpha(\Omega)$. Finally, we give applications of these results to entropy and weak solutions of the Dirichlet problem for nonlinear elliptic second-order equations with right-hand side in some classes close to~$L^1(\Omega)$ and defined by the logarithmic function or its double composition.

Keywords: integrability, distribution function, nonlinear elliptic equations, right-hand side in classes close to $L^1$, Dirichlet problem, weak solution, entropy solution

Received October 16, 2018

Revised November 1, 2018

Accepted November 5, 2018

Funding Agency: This work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Aleksandr Al’bertovich Kovalevsky, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: alexkvl71@mail.ru

Cite this article as:  A.A. Kovalevsky. Integrability properties of functions with a given behavior of distribution functions and some applications, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp.  78–92. 

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