Related papers: Optimal geometric estimates for fractional Sobolev…

The purpose of this work is the study of solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the…

The normalization condition, average values and reduced distribution functions can be generalized by fractional integrals. The interpretation of the fractional analog of phase space as a space with noninteger dimension is discussed. A…

We prove fractional Sobolev-Poincar\'e inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy…

The paper contains a review of results on linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general inhomogeneous boundary conditions in Sobolev spaces. The character of the…

In this paper, we uncover a novel connection between the Fenchel-Willmore inequality and a new logarithmic Sobolev inequality for mean-convex submanifolds immersed in non-negatively curved manifolds with Euclidean volume growth. Building on…

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional…

Based on the Riemann- and Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group SO(3) to construct a higher dimensional representation of a fractional rotation group with mixed…

In this paper, we develop the theory of Sobolev spaces on locally finite graphs, including completeness, reflexivity, separability, and Sobolev inequalities. Since there is no exact concept of dimension on graphs, classical methods that…

We consider a metric measure space with a local regular Dirichlet form. We establish necessary and sufficient conditions for upper heat kernel bounds with sub-diffusive space-time exponent to hold. This characterization is stable under…

We discuss sharp Sobolev inequalities for vector valued maps.

We discuss recent progress in understanding the dynamical properties of partially hyperbolic diffeomorphisms that preserve volume. The main topics addressed are density of stable ergodicity and stable accessibility, center Lyapunov…

In this paper, we study the stability of fractional Sobolev trace inequality within both the functional and critical point settings. In the functional setting, we establish the following sharp estimate:…

In various analytical contexts, it is proved that a weak Sobolev inequality implies a doubling property for the underlying measure.

It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of…

We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and…

In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansion. This estimate is an analog of a result previously obtained by…

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincar\'{e} inequalities, general Beckner inequalities...). We also discuss the…

We study a non-local eigenvalue problem related to the fractional Sobolev spaces for large values of p and derive the limit equation as p goes to infinity. Its viscosity solutions have many interesting properties and the eigenvalues exhibit…

We prove fractional order Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary. We demonstrate by counterexamples that fatness conditions alone are not sufficient for such Hardy inequalities to…