Related papers: Logarithmic Sobolev Trace Inequalities
We investigate Sobolev inequalities for several rough operators. We prove that several operators satisfy a pointwise bound by the Riesz potential applied to the gradient. From this inequality, we derive several new Sobolev-type inequalities…
Consider a Lipschitz domain $\Omega$ and a measurable function $\mu$ supported in $\overline\Omega$ with $\left\|{\mu}\right\|_{L^\infty}<1$. Then the derivatives of a quasiconformal solution of the Beltrami equation $\overline{\partial} f…
In this paper we provide a proof of the Sobolev-Poincar\'e inequality for variable exponent spaces by means of mass transportation methods. The importance of this approach is that the method is exible enough to deal with different…
We answer an open problem posed by Mossel--Oleszkiewicz--Sen regarding relations between $p$-log-Sobolev inequalities for $p\in(0,1]$. We show that for any interval $I\subset(0,1]$, there exist $q,p\in I$, $q<p$, and a measure $\mu$ for…
We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $L^{p}(d\gamma)$ distance for $p>1$. To this end, we construct a sequence of centered probability measures…
In this short article we show a particular version of the Hedberg inequality which can be used to derive, in a very simple manner, functional inequalities involving Sobolev and Besov spaces in the general setting of Lebesgue spaces of…
We prove the continuity of Sobolev functions $\varphi \in W^{1,n}_{\mathrm{loc}}(\Omega)$, $\Omega \subset \mathbb{R}^n$, that satisfy \[ \lvert\nabla \varphi(x)\rvert^n \le K(x)\bigl(\langle \nabla \varphi(x), \xi(x)\rangle + A(x)\bigr),…
We study the boundary traces of Newton-Sobolev, Hajlasz-Sobolev, and BV (bounded variation) functions. Assuming less regularity of the domain than is usually done in the literature, we show that all of these function classes achieve the…
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…
A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit…
The goal of this short note is to prove qualitative stability for a family of trace Sobolev inequalities first proven by Carlen \& Loss for $p=2$ and by Maggi and the author for $p\in (1,n)$. This answers an open problem raised in a recent…
We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaces $W^{1,p}(\Omega,w) \to L^p(\Omega,w),$ where $w$ is some weight on a domain $\Omega \subset \Real^n$.
We prove several Sobolev-type inequalities related to the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, which can be viewed as a $\bar\partial$-version of the classical Sobolev inequality and its various generalizations, and…
We study traces of elements of fractional Sobolev spaces $H_p^\alpha(\mathbb{R}^n)$ on closed subsets $\Gamma$ of $\mathbb{R}^n$, given as the supports of suitable measures $\mu$. We prove that if these measures satisfy localized upper…
In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brezis-Gallouet-Wainger logarithmic type inequalities revealing Sobolev embeddings in…
In this paper, we investigate the approximation problem for functions in Gaussian Sobolev spaces $W^s_p(\mathbb{R}^d, \gamma)$ of smoothness $s > 0$, where the approximation error is measured in the Gaussian Lebesgue space…
We study the possibility of a gradual improvement as time progresses of the regularity of solutions to evolution problems of parabolic type driven by L\'evy-type operators, not necessarily translation invariant. In the course of our…
We study the weak regularity of mappings inverse to weighted Sobolev homeomorphisms $\varphi:\Omega\to\widetilde{\Omega}$, where $\Omega$ and $\widetilde{\Omega}$ are domains in $\mathbb R^n$. Using the weak regularity of inverse mappings…
This paper is concerned with developing a theory of traces for functions that are integrable but need not possess any differentiability within their domain. Moreover, the domain can have an irregular boundary with cusp-like features and…
By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN) inequalities on the half space $\mathbf{R}^{n-1}\times\mathbf{R}_+$, $n\geq 1$…