Related papers: Maz'ya-type bounds for sharp constants in fraction…

We prove a two-sided estimate on the sharp $L^p$ Poincar\'e constant of a general open set, in terms of a capacitary variant of its inradius. This extends a result by Maz'ya and Shubin, originally devised for the case $p=2$, in the…

We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical…

We prove a lower bound on the sharp Poincar\'e-Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the…

We show that, when $sp>N$, the sharp Hardy constant $\mathfrak{h}_{s,p}$ of the punctured space $\mathbb R^N\setminus\{0\}$ in the Sobolev-Slobodecki\u{\i} space provides an optimal lower bound for the Hardy constant…

We prove fractional Sobolev-Poincar\'e inequalities, capacitary versions of fractional Poincar\'e inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results…

Given a bounded convex open set $\Omega\subseteq \mathbb R^N$, we prove that the Poincar\'e-Sobolev constants $\lambda_{p,q}(\Omega)$ can be bounded from below by the $p$-power of the ratio between the perimeter of $\Omega$ and a suitable…

We study the sharp constant in the Hardy inequality for fractional Sobolev spaces defined on open subsets of the Euclidean space. We first list some properties of such a constant, as well as of the associated variational problem. We then…

We determine the sharp constants for the fractional Sobolev inequalities associated with the conformally invariant fractional powers $\mathcal{L}_{s}(0<s<1)$ of the sublaplacian on H-type groups. From these inequalities we derive a sharp…

We study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodecki\u{\i} spaces on the whole $\mathbb{R}^N$. By generalizing a recent work by Hynd and Seuffert, we prove existence of extremals, together with some…

In this paper, we prove capacitary versions of the fractional Sobolev--Poincar\'e inequalities. We characterize localized variant of the boundary fractional Sobolev--Poincar\'e inequalities through uniform fatness condition of the domain in…

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincar\'e-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is…

We develop an intrinsic, heat-kernel based fractional Sobolev framework on closed Riemannian manifolds and study the critical fractional Sobolev embedding. We determine the optimal coefficient of the lower-order $L^{p}$ term and prove that…

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a non-linear and non-local version of the ground state representation.

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…

On a convex bounded open set, we prove that Poincar\'e-Sobolev constants for functions vanishing at the boundary can be bounded from below in terms of the norm of the distance function in a suitable Lebesgue space. This generalizes a result…

In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce…

We investigate the weighted fractional order Hardy inequality $$ \int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partial\Omega)^{-\alpha}\text{dist}(y,\partial\Omega)^{-\beta}\,dy\,dx\geq…

We prove existence of minimizers for the sharp Poincar\'e-Sobolev constant in general Steiner symmetric sets, in the subcritical and superhomogeneous regime. The sets considered are not necessarily bounded, thus the relevant embeddings may…

In this paper, we study the sharp constants in fractional Sobolev inequalities associated with the regional fractional Laplacian in domains.

Let $W^1L^{p,q}(\mathbb H^n)$, $1\leq q,p < \infty$ denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces $\mathbb H^n$. Our aim in this paper is three-fold. First of all, we establish a sharp Poincar\'e inequality in…