Related papers: Modified log-Sobolev inequalities and isoperimetry

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any…

In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $t_{\mu,\nu}(x)$. In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality,…

A Sobolev type embedding for radially symmetric functions on the unit ball $B$ in $\mathbb R^n$, $n\geq 3$, into the variable exponent Lebesgue space $L_{2^\star + |x|^\alpha} (B)$, $2^\star = 2n/(n-2)$, $\alpha>0$, is known due to J.M. do…

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted)…

We establish a Leray- Trudinger Type inequality in the anisotropic setting induced by a strongly convex Finsler norm F. The result generalizes classical exponential integrability inequalities for Sobolev functions to the framework of…

We prove that if the Sobolev embedding $M^{1,p}(X)\hookrightarrow L^q(X)$ holds for some $q>p\geq 1$ in a metric measure space $(X,d,\mu),$ then a constant $C$ exists such that $\mu(B(x,r))\geq Cr^n$ for all $x\in X$ and all $0<r\leq 1,$…

In this paper we deal with an invariant ergodic hyperbolic measure $\mu$ for a diffeomorphism $f,$ assuming that $f$ it is either $C^{1+\alpha}$ or $f$ is $C^1$ and the Oseledec splitting of $\mu$ is dominated. We show that this system…

It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of…

We prove that the canonical sub-Laplacian on $SU(2)$ admits a uniform modified log-Sobolev inequality for all its matrix-valued functions, independent of the matrix dimension. This is the first example of sub-Laplacian that a matrix-valued…

For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $\phi$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle \phi(\lambda_{1}) \mu(\{x \in \Omega : f(x) \geq…

The purpose of this short note is to demonstrate uniform logarithmic Sobolev inequalities for the mean field gradient particle systems associated to an energy functional that is convex in the flat sense. A defective log-Sobolev inequality…

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on $\mathbb{R}^n$ with an explicit expression for the constant. Namely, we show that for all $0<s<\frac{n}{2}$ and $a>0$ we have the…

Let $f,g:\Bbb{R}^{N}\rightarrow (-\infty ,\infty ]$ be Borel measurable, bounded below and such that $\inf f+\inf g\geq 0.$ We prove that with $ m_{f,g}:=(\inf f-\inf g)/2,$ the inequality $||(f-m_{f,g})^{-1}||_{\phi…

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the…

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into $L^q$ spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also…

This paper deals with various questions related to the isoperimetic problem for smooth positive measure $d\mu = \varphi(x)dx$, with $x \in \Omega \subset \mathbb{R}^N$. Firstly we find some necessary conditions on the density of the measure…

Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the…

Simple inequalities for some integrals involving the modified Struve function of the first kind $\mathbf{L}_{\nu}(x)$ are established. In most cases, these inequalities have best possible constant. We also deduce a tight double inequality,…

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional…

We investigate Sobolev and Hardy inequalities, specifically weighted Minerbe's type estimates, in noncompact complete connected Riemannian manifolds whose geometry is described by an isoperimetric profile. In particular, we assume that the…