Related papers: A characterization of BV and Sobolev functions via…
In this article, we study the relation between Sobolev-type embeddings for Sobolev spaces or Besov spaces or Triebel-Lizorkin spaces defined either on a doubling or on a geodesic metric measure space and lower bound for measure of balls…
We introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the maximal weak subslope of an arbitrary causal function, which plays the role of the (Lorentzian) modulus of its differential. It is…
The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincar\'e, Nash, Super Poincar\'e and logarithmic Sobolev…
We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar\'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar\'e and weak Poincar\'e inequalities was studied in…
In this paper, we study the existence of limits at infinity along almost every infinite curve for the upper and lower approximate limits of bounded variation functions on complete unbounded metric measure spaces. We prove that if the…
We give sufficient geometric conditions, not involving capacities, for a compact null set to be removable for the Sobolev functions on weighted $\mathbb R^n$, defined as the closure of smooth functions in the weighted Sobolev norm. Our…
We prove that Sobolev spaces on Cartesian and warped products of metric spaces tensorize, only requiring that one of the factors is a doubling space supporting a Poincar\'e inequality.
In this paper, we introduce and study a new class of fractional modular function spaces, called \emph{Fractional Anisotropic Musielak--Sobolev Spaces}, which generalize both the fractional Anisotropic Orlicz--Sobolev spaces and the…
We study local and global higher integrability properties for quasiminimizers of a class of double-phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure…
Proceeding the study of local properties of analytic functions started in [Br] we prove new dimensionless inequalities for such functions in terms of their Chebyshev degree. As a consequence, we obtain the reverse Holder inequalities for…
This work is devoted to the geometric analysis of metric-measure spaces satisfying a Prekopa-Leindler or a more general Borell-Brascamp-Lieb inequality. Completing the early investigations by Cordero-Erausquin, McCann and Schmuckenschlager,…
Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In a companion article, we introduced the notion of multiscale (first-order) functional inequalities, which extend standard functional inequalities like…
Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean…
We investigate the equivalence of Sobolev inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain their equivalence up to constants by…
We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of…
We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the…
We establish new approximation results in the sense of Lusin for Sobolev functions $f$ with $|\nabla f| \in L\log L$ on infinite-dimensional spaces equipped with Gaussian measures. The proof relies on some new pointwise estimate for the…
In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non-local functionals. One special case shows that rescalings of the…
We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to…
A classical result of Norbert Wiener characterises doubly shift-invariant subspaces for square integrable functions on the unit circle with respect to a finite positive Borel measure $\mu$, as being the ranges of the multiplication maps…