Related papers: On Choquet integrals and pointwise estimates

We consider integrals in the sense of Choquet with respect to the $\delta$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<\delta\le n$. In…

We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content $\mathcal{H}_\infty^{\delta}$. In particular, if $\Omega$ is a bounded John domain in $\mathbb{R}^n$, $n\geq 2$, and $0 <\delta \le n$, we prove…

We introduce Riesz potentials for non-Lebesgue measurable functions by taking the integrals in the sense of Choquet with respect to Hausdorff content and prove boundedness results for these operators. Some earlier results are recovered or…

Let $\Omega$ be a bounded John domain in $\mathbb R^n$ with $n\ge 2$, and let $\mathcal{H}_{\infty }^{\delta}$ denote the Hausdorff content of dimension $\delta\in (0,n]$. In this article, the authors prove the Poincar\'e and the…

We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density…

We study Poincare-Sobolev type inequalities for compactly supported smooth functions which are defined in the Euclidean $n$-space and whose absolute value of gradient are Choquet $\delta /n$-integrable with respect to the…

In this note the Choquet type operators are introduced, in connection to Choquet's theory of integrability with respect to a not necessarily additive set function. Based on their properties, a quantitative estimate for the nonlinear…

In this article we study some new pointwise inequalities between rough singular integral operators, weighted maximal functions of the gradient and weighted Morrey spaces. These pointwise estimates will naturally lead us to a new class of…

We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means,…

This article discusses several matters related to Sobolev, Poincare, and isoperimetric inequalities in various settings.

We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. The contents: 1. Estimate for a quadratic form of…

We establish sharp pointwise inequalities for the Riesz potential and its gradient in $\mathbb{R}^{n}$ and indicate their usefulness for potential analysis, moment theory and other applications.

The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…

We study weak-type estimates and exponential integrability for the variable order Riesz potential. As an application we prove an exponential integrability result with respect to the Hausdorff content for functions from variable exponent…

The Riesz-Sobolev inequality provides an upper bound, in integral form, for the convolution of indicator functions of subsets of Euclidean space. We formulate and prove a sharper form of the inequality. This can be equivalently phrased as a…

We study in this article a new pointwise estimate for ''rough'' singular integral operators. From this pointwise estimate we will derive Sobolev type inequalities in a variety of functional spaces.

The integral representation of Choquet operators defined on a space C(X) is established by using the Choquet-Bochner integral of a real-valued function with respect to a vector capacity.

For distributions, we build a theory of higher order pointwise differentiability comprising, for order zero, {\L}ojasiewicz's notion of point value. Results include Borel regularity of differentials, higher order rectifiability of the…

We study the extension of Hill's method of infinite determinants to the case of integro-differential equations with periodic coefficients and kernels. We develop the analytical theory of such methods, and we obtain certain qualitative…

We consider nonlocal equations of order larger than one with measure data and prove gradient regularity in Sobolev and H\"older spaces as well as pointwise bounds of the gradient in terms of Riesz potentials, leading to fine regularity…