Related papers: An Approximate Inverse Riesz-Sobolev Inequality

Young's convolution inequality provides an upper bound for the convolution of functions in terms of $L^p$ norms. It is known that for certain groups, including Heisenberg groups, the optimal constant in this inequality is equal to that for…

Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. The estimates are uniform with respect to all graphs of a given finite length, and the constant factors in the…

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev…

We consider the Sobolev norms of the pointwise product of two functions, and estimate from above and below the constants appearing in two related inequalities.

Alexandrov's inequalities imply that for any convex body $A$, the sequence of intrinsic volumes $V_1(A),\ldots,V_n(A)$ is non-increasing (when suitably normalized). Milman's random version of Dvoretzky's theorem shows that a large initial…

We study the generalized existence of extremizers for the sharp $p$-Sobolev inequality on noncompact Riemannian manifolds in connection with nonnegative curvature and Euclidean volume growth assumptions. Assuming a nonnegative Ricci…

In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case $p=2$, as well as an extension to the coefficient $p=1$ that was previously unknown.…

Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals…

We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic functions. We introduce a new large class of measures, Euclidean regular and…

We show that, for $0<s<1$, $0<p<\infty$, $0<q<\infty$, Haj\l asz-Besov and Haj\l asz-Triebel-Lizorkin functions can be approximated in the norm by discrete median convolutions. This allows us to show that, for these functions, the limit of…

This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\mathbb R^n$…

We show how an operation of inf-convolution can be used to approximate convex functions with $C^{1}$ smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some…

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems:…

We prove that if a triplet of functions satisfies almost equality in the Pr\'ekopa-Leindler inequality, then these functions are close to a common log-concave function, up to multiplication and rescaling. Our result holds for general…

In this paper we will study the equivalence between super-Poincar\'e inequality and some log-Sobolev type inequalities, including weak log-Sobolev inequality and super log-Sobolev inequality. The explicit relations between associated rate…

We consider integrals in the sense of Choquet with respect to the $\delta$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<\delta\le n$. In…

We consider Hardy inequalities in $I R^n$, $n \geq 3$, with best constant that involve either distance to the boundary or distance to a surface of co-dimension $k<n$, and we show that they can still be improved by adding a multiple of a…

We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant…

This is a substantially generalized version of the preprint arXiv:1105.4214 by Lifshits and Tyurin. We prove that for any pair of i.i.d. random vectors $X, Y$ in $R^n$ and any real-valued continuous negative definite function $g: R^n\to R$…

We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for…