Related papers: Fractional Korn's inequality on subsets of the Euc…

We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry…

We study the fractional Hardy inequality on the integers. We prove the optimality of the Hardy weight and hence affirmatively answer the question of sharpness of the constant.

We prove several interesting equalities for the integrals of higher order derivatives on the homogeneous groups. As consequences, we obtain the sharp Hardy--Rellich type inequalities for higher order derivatives including both the…

We prove a critical Hardy inequality on the half-space by using the harmonic transplantation. Also we give an improvement of the subcritical Hardy inequality on the half-space, which converges to the critical Hardy inequality. Sobolev type…

We give new proofs of Hardy space estimates for fractional and singular integral operators on weighted and variable exponent Hardy spaces. Our proofs consist of several interlocking ideas: finite atomic decompositions in terms of $L^\infty$…

In this short article we obtain some necessary conditions for a so-called fractional Hardy-Sobolev's inequalities in multidimensional case. We also give some examples to show the sharpness of these inequalities.

We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $\Sigma\subset\mathbb R^d$, $d \ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy…

We consider shells of non-constant thickness in three dimensional Euclidean space around surfaces which have bounded principal curvatures. We derive Korn's interpolation (or the so called first and a half (The inequality first introduced in…

We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The…

Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.

We establish a new improvement of the classical $L^p$-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one dimensional Hardy inequality.…

We prove fractional boundary Hardy's inequality in dimension one for the critical case $sp =1$. Optimality of the inequality is obtained for any $p$. The extra logarithmic correction term appears in usual fashion. We also provide a concrete…

We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions…

We review recent progress in the fractional Calder\'on problem, where one tries to determine an unknown coefficient in a fractional Schr\"odinger equation from exterior measurements of solutions. This equation enjoys remarkable uniqueness…

We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.

We establish fractional Hardy-type inequalities in a bounded domain with plump complement. In particular our results apply in bounded C^\infty domains and Lipschitz domains.

New Hardy type inequality with double singular kernel and with additional logarithmic term in a ball $B\subset \mathbb{R}^n$ is proved. As an application an estimate from below of the first eigenvalue for Dirichlet problem of p-Laplacian in…

The goal of this work is to show that the generalized Korn inequality that replaces the symmetric part of the differential matrix in the classical Korn inequality by its trace-free part is valid over John domains and weighted Sobolev…

In this paper we prove an invariant Harnack inequality on Carnot-Carath\'eodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an "abstract" formulation of a technique recently introduced by Caffarelli…

In this paper, we present the geometric Hardy inequalities on the starshaped sets in the Carnot groups. Also, we obtain the geometric Hardy inequalities on half-spaces for general vector fields.