In this paper, we consider Poincaré inequalities for non-euclidean metrics on ℝd. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give equivalent functional forms of these Poincaré type inequalities in terms of transportation-cost inequalities and inf-convolution inequalities. Workable sufficient conditions are given...

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions is well known. We give here the correspondence (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for log-concave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial,…)....

Our understanding of the interplay between Poincaré inequalities, Sobolev inequalities and the geometry of the underlying space has changed considerably in recent years. These changes have simultaneously provided new insights into the classical theory and allowed much of that theory to be extended to a wide variety of different settings. This paper reviews some of these new results and techniques and concludes with an example on the preservation of Sobolev spaces by the maximal function.[Proceedings...

Using an inverse system of metric graphs as in [3], we provide a simple example of a metric space X that admits Poincaré inequalities for a continuum of mutually singular measures.

We prove here that the Poincaré-Sobolev pointwise inequalities for the relative rearrangement can be considered as the root of a great number of inequalities in various sets not necessarily vector spaces. In particular, new interpolation inequalities can be derived.

We consider the half-linear differential equation of the form $$\left(p\left(t\right)\right|x^{\text{'}}{|}^{\alpha }\mathrm{sgn}{x}^{\text{'}}{)}^{\text{'}}+q\left(t\right){\left|x\right|}^{\alpha }\mathrm{sgn}x=0,t\ge t_0,$$ under the assumption that $p{\left(t\right)}^{-1/\alpha }$ is integrable on $[t_0,\infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t\to \infty$.

Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $\sigma \left(A\right)={\int }_B{\chi }_A(x,\phi \left(x\right))dx$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^p\left(ℝ³\right)$ to $L^q(\Sigma ,d\sigma )$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell’s theorem using a direct semigroup argument. Our quantitative uniqueness result has various...

Let $n\ge 2$ and $\Omega \subset {ℝ}^n$ be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space $W_0{L}^{\Phi }\left(\Omega \right)$, where the Young function $\Phi$ behaves like $t^n{log}^{\alpha }\left(t\right)$, $\alpha <n-1$, for $t$ large, into the Zygmund space $Z_0^{\frac{n-1-\alpha }{n}}\left(\Omega \right)$. We also study the same problem for the embedding of the generalized Lorentz-Sobolev space $W_0^m{L}^{\frac{n}{m},q}{log}^{\alpha }L\left(\Omega \right)$, $m<n$, $q\in (1,\infty ]$, $\alpha <\frac{1}{q^{\text{'}}}$, embedded into the Zygmund space $Z_0^{\frac{1}{q^{\text{'}}}-\alpha }\left(\Omega \right)$.