Related papers: A geometric characterization of a sharp Hardy ineq…

We give sharp limiting case Hardy inequalities on the sphere $\mathbb{S}^{2}$ and show that their optimal constants are unattainable by any $f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$. The singularity of the problem is related to…

We show that for bounded domains in $\mathbb C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial \Omega \setminus F$ is $\mathcal C^{2}$-smooth and locally…

It is shown that the Carath\'eodory distance and the Lempert function are almost the same on any strongly pseudoconvex domain in $\C^n;$ in addition, if the boundary is $C^{2+\eps}$-smooth, then $\sqrt{n+1}$ times one of them almost…

We prove that if an infinite, discrete semigroup has the property that every right syndetic set is left syndetic, then the semigroup has a left invariant mean. We prove that the weak*-closed convex hull of the two-sided translates of every…

We prove Rellich and improved Rellich inequalities that involve the distance function from a hypersurface of codimension $k$, under a certain geometric assumption. In case the distance is taken from the boundary, that assumption is the…

In this paper, we unify and improve existing results on characterizing strict and almost stricty convex functions via subdifferential mapping, Moreau envelope, and proximal mappings. In particular, it is shown that if a convex function is…

The celebrated Hardy inequality can be written in the form $$\int_0^\infty \mathcal{P}_p \big(f|_{[0,x]}\big)dx \le (1-p)^{-1/p} \int_0^\infty f(x)\:dx \qquad \text{ for }p\in(0,1)\text{ and }f \in L^1\text{ with }f\ge0,$$ where…

We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.

In this paper, we study a part of approximation theory that presents the conditions under which a \Ceby\sev set in a Banach space is convex. To do so, we use Gateaux differentiability of the distance function.

In this paper we present a unified simple approach to anisotropic Hardy inequalities in various settings. We consider Hardy inequalities which involve a Finsler distance from a point or from the boundary of a domain. The sharpness and the…

We show that, on an oriented compact surface, two sufficiently $C^2$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows, and same marked boundary distance, are…

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.

We prove a Hardy inequality for ultraspherical expansions by using a proper ground state representation. From this result we deduce some uncertainty principles for this kind of expansions. Our result also implies a Hardy inequality on…

It is shown that, given a point $x\in\mathbbm{R}^d$, $d\ge 2$, and open sets $U_1,...,U_k$ containing $x$, any convex combination of the harmonic measures for $x$ with respect to $U_n$, $1\le n\le k$, is the limit of a sequence of harmonic…

We prove that the gradient of any bounded subharmonic function is upper semi-continuous, provided that its super-level sets can be touched from the exterior by uniform $C^{1,\text{Dini}}$ domains at every point. This idea extends to a class…

For any \alpha in (0, 2), a truncated symmetric \alpha-stable process is a symmetric Levy process with no diffusion part and with a Levy density given by c|x|^{-d-\alpha} 1_{|x|< 1} for some constant c. In previous paper we have studied the…

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive…

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…

Let $E$ be a Banach space such that $E'$ has the Radon-Nikod\'ym property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(\mu,E)$ to a convex compactness criterion in a weaker…

An important result in the theory of harmonic maps is due to Benoist--Hulin: given a quasi-isometry $f:X\to Y$ between pinched Hadamard manifolds, there exists a unique harmonic map at a finite distance from $f$. Here we show existence of…