Related papers: Modified logarithmic Sobolev inequalities in null …

We are interested in the $q$ Logarithmic Sobolev inequality for probability measures on the infinite product of Heisenberg groups. We assume that the one site boundary free measure satisfies either a $q$ Log-Sobolev inequality or a U-Bound…

We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to…

Given a probability measure $\mu$ supported on a convex subset $\Omega$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincar\'e and log-Sobolev type inequalities on $(\Omega,g_0,\mu)$. To this end, we change the…

This article presents near-optimal guarantees for accurate and robust image recovery from under-sampled noisy measurements using total variation minimization. In particular, we show that from O(slog(N)) nonadaptive linear measurements, an…

Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $\Omega=\Omega_1 \times \Omega_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type…

The well known duality between the Sobolev inequality and the Hardy-Littlewood-Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three norms instead of…

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schr\"odinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ d\rho \propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x -…

In this paper we give a geometric condition which ensures that $(q,p)$-Poincar\'e-Sobolev inequalities are implied from generalized $(1,1)$-Poincar\'e inequalities related to $L^1$ norms in the context of product spaces. The concept of…

In a recent work, Klartag gave an improved version of Lichnerowicz' spectral gap bound for uniformly log-concave measures, which improves on the classical estimate by taking into account the covariance matrix. We analyze the equality cases…

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 that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\Omega\subset\mathbb{R}^n$, let $\rho$ be a quasi-metric on $\Omega$,…

We establish the higher differentiability of solutions to a class of obstacle problems for integral functionals where the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher…

We employ a Markov semigroup approach combined with the $\Gamma$-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincar\'e inequality…

For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $\Omega\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_{\Omega}|u|^q…

We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution $\nu = e^{-f}$ on $\mathbb{R}^n$ under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when $\nu$…

We introduce prox-convex for minimizing $F(x)=g(x)+h(C(x))+s(R(x))$, where $g$ and $h$ are convex, $C$ and $s$ are smooth, and each component of $R$ is convex (possibly nonsmooth). Here $g$ captures general convex objectives and indicator…

In this paper, we are concerned with the stability problem for endpoint conformally invariant cases of the Sobolev inequality on the sphere $\mathbb{S}^n$. Namely, we will establish the stability for Beckner's log-Sobolev inequality and…

Extending a result of Caffarelli, we provide global Lipschitz changes of variables between compactly supported perturbations of log-concave measures. The result is based on a combination of ideas from optimal transportation theory and a new…

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…

In this paper, we study the stability of fractional Sobolev trace inequality within both the functional and critical point settings. In the functional setting, we establish the following sharp estimate:…