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 $\Omega$ in $\mathbb R^n$ ($n\ge 2$) satisfies, for sufficiently large $k$, the estimate ${\rm meas}\{\vert v\vert>k\}\le k^{-\alpha}\varphi(k)/\psi(k)$, 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 either 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 ${\rm meas}\{\vert v\vert>k\}\le Ck^{-\alpha}(\ln k)^{-\beta}$ with $C,\alpha>0$ and $\beta\ge 0$. 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 interval, $[0,1]$ 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 ${\rm meas}\{\vert v\vert>k\}\le Ck^{-\alpha}(\ln\ln k)^{-\beta}$ with $C,\alpha>0$ and $\beta\ge 0$. 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 second-order nonlinear elliptic equations with right-hand side in some classes close to $L^1(\Omega)$ and defined by the logarithmic function or its double composition.