Related papers: An extension operator on bounded domains and appli…

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $\lambda=n-\alpha$ (that…

Let $\Omega$ be either $\mathbb{R}^n$ or an unbounded strongly Lipschitz domain of $\mathbb{R}^n$, and $\Phi$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical…

Let $\Omega$ be a strongly Lipschitz domain of $\reel^n$. Consider an elliptic second order divergence operator $L$ (including a boundary condition on $\partial\Omega$) and define a Hardy space by imposing the non-tangential maximal…

Given a bounded Lipschitz domain $\Omega\subset\mathbb R^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. In this paper we introduce some new…

In this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators. This family includes among others the classical Poisson extension operator and the extension operator with Riesz kernel.…

We introduce a general $L_p$-solvability result for the Poisson equation in non-smooth domains $\Omega\subset \mathbb{R}^d$, with the zero Dirichlet boundary condition. Our sole assumption on the domain $\Omega$ is the Hardy inequality:…

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: \mu=n and p=q, by limiting the inequality…

We consider a type of Hardy-Sobolev inequality, whose weight function is singular on the whole domain boundary. We are concerned with the attainability of the best constant of such inequality. In dimension two, we link the inequality to a…

We prove the unique solvability for the Poisson and heat equations in non-smooth domains $\Omega\subset \mathbb{R}^d$ in weighted Sobolev spaces. The zero Dirichlet boundary condition is considered, and domains are merely assumed to admit…

Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1<p< (2n)/(n-2). We prove a reverse-Holder inequality for functions realizing equality in the Sobolev inequality, which finds a lower bound for their (p-1)-norm…

In this paper we prove the existence of extremal functions for the Adams-Moser-Trudinger inequality on the Sobolev space $H^{m}(\Omega)$, where $\Omega$ is any bounded, smooth, open subset of $\mathbb{R}^{2m}$, $m\ge 1$. Moreover, we extend…

In this paper, we obtain the reversed Hardy-Littlewood-Sobolev inequality with vertical weights on the upper half space and discuss the extremal functions. We show that the sharp constants in this inequality are attained by introducing a…

We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also…

Let $\Omega\subset\dR^d$ be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space $H^1(\Omega)$ up to functions in the Sobolev space $H^1(\dR^d)$ via a linear…

For a strongly elliptic second-order operator $A$ on a bounded domain $\Omega\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(\Omega)$-realizations of $A$ as representing boundary conditions…

Let $\Omega\subset{\mathbb R}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/\delta^2\in L^1(\Omega)$ if $u\in H^1_0(\Omega)$, where $\delta$ denotes the distance to $\partial\Omega$. The…

In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and…

Let $\Omega$ be a strongly Lipschitz domain of $\mathbb{R}^n$, whose complement in $\mathbb{R}^n$ is unbounded. Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet boundary condition, and the…

This paper is mainly devoted to the study of the reversed Hardy-Littlewood-Sobolev (HLS) inequality on Heisenberg group $\mathbb{H}^n$ and CR sphere $\mathbb{S}^{2n+1}$. First, we establish the roughly reversed HLS inequality and give a…

Some special Hilbert spaces are introduced to present the class of infinitesimal operators with complete minimal non-basis family of eigenvectors. The discrete Hardy inequality plays an important role in the proposed approach. The…