Related papers: Modified logarithmic Sobolev inequalities on R

We give a simple proof of the existence of a minimizer for the Sobolev inequality. Our proof is based on a representation formula via a cut-off fundamental solution.

In this note, we characterize the equality case of the sharp $L^2$-Euclidean logarithmic Sobolev inequality with monomial weights, exploiting the idea by Bobkov and Ledoux \cite{Bob}. Our approach is new even in the unweighted case. Also,…

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…

We prove that for a probability measure on $\mathbb{R}^n$, the Poincar\'e inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan et al. and…

We give a new proof of the compactness of minimizing sequences of the Sobolev inequalities in the critical case. Our approach relies on a simplified version of the concentration-compactness principle, which does not require any refinement…

We show that the convolution of a compactly supported measure on $\mathbb{R}$ with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). We use this result to give a new proof of a classical result in random matrix theory…

This survey-type paper provides a common framework for a larger number of higher order concentration results (i.\,e., concentration results for non-Lipschitz functions which have bounded derivatives of higher order) in the spirit of…

We give necessary and sufficient conditions for the Chebyshev inequality to be an equality.

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 [12] it has been shown that $(p,q)$ Sobolev inequality with $p>q$ implies the doubling condition on the underlying measure. We show that even weaker Orlicz-Sobolev inequalities, where the gain on the left-hand side is smaller than any…

In this work we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. For this we use optimal transport methods and the Borell-Brascamp-Lieb inequality. These refinements can be written as a…

We present a general subadditivity inequality for log-Sobolev constants of convolution measures. As a corollary, we show that the log-Sobolev constant is monotone along the sequence of standardized convolutions in the central limit theorem.

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

We give a new proof of Aubin's improvement of the Sobolev inequality on $\mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem…

In this short article we obtain some necessary conditions for a so-called fractional Hardy-Sobolev's inequalities in multidimensional case. We also give some examples to show the sharpness of these inequalities.

In the paper we pursue the analysis from the section 5 of the Talagrand's paper "Sample boundedness of stochastic processes under increment conditions." Ann. Probab. 18, No. 1, 1-49. In particular we give the proof of some Sobolev…

We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices.…

If one thinks of a Riemannian metric, $g_1$, analogously as the gradient of the corresponding distance function, $d_1$, with respect to a background Riemannian metric, $g_0$, then a natural question arises as to whether a corresponding…

We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is…

We prove that for a symmetric Markov semigroup, Ricci curvature bounded from below by a non-positive constant combined with a finite $L_\infty$-mixing time implies the modified log-Sobolev inequality. Such $L_\infty$-mixing time estimates…