相关论文: The $L_{p}$-Brunn-Minkowski inequalities for varia…

We show that certain functional inequalities, e.g.\ Nash-type and Poincar\'e-type inequalities, for infinitesimal generators of $C_0$ semigroups are preserved under subordination in the sense of Bochner. Our result improves \cite[Theorem…

The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schr\"odinger operator in terms of an $L^p$ norm of the potential. This is dual to a bound on the $H^1$-norms of a system of orthonormal functions. Here we extend…

We completely characterize $L^p-L^q$ boundedness of two classes of Forelli-Rudin type operators on the unit ball of $\mathbb{C}^n$ for all $(p, q)\in [1, \infty]\times [1, \infty]$. The results are not only a complement to some previous…

The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent…

We discuss the behavior of $(\lambda_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $\lambda_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$.…

We define fractional power of the Dunkl Laplacian, fractional modulus of smoothness and fractional $K$-functional in $L^p$-space with the Dunkl weight. As application, we prove direct and inverse theorems of approximation theory, and some…

Poincar\'{e}-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$…

In this paper we study spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains $\Omega\subset\mathbb R^2$. This study is based on (weighted) Poincar\'e-Sobolev inequalities. The main technical tool is the…

We prove Lp Poincare inequalities for functions on the discrete cube and their discrete gradient. We thus recover an exponential inequality and the concentration phenomenon for the uniform probability on the cube first obtained by Bobkov…

We present a Korn-Poincar\'e-type inequality in a planar setting which is in the spirit of the Poincar\'e inequality in SBV due to De Giorgi, Carriero, Leaci. We show that for each function in SBD$^2$ one can find a modification which…

In this paper, we discuss various basic properties of moduli of smoothness of functions from $L_p(\mathbb{R}^d)$, $0<p\le \infty$. In particular, complete versions of Jackson-, Marchaud-, and Ulyanov-type inequalities are given for the…

We give necessary and sufficient conditions for interpolation inequalities of the type considered by Marcinkiewicz and Zygmund to be true in the case of Banach space-valued polynomials and Jacobi weights and nodes. We also study the…

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ (with…

The current state of art concerning the $L_p$ Minkowski problem as a Monge-Ampere equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the $p=0$ case are surveyed and connections to…

We prove the local logarithmic Brunn-Minkowski inequality for bodies of revolution. Furthermore, we give a generalization for one origin symmetric body of revolution and one body of revolution that does not need to be symmetric and restrict…

For $1<p\le 2$, we establish sharp inequalities for the Fourier transform of the characteristic function of the $l^p$-unit ball $B_p\subset\mathbb{R}^2$. We show that $$ \sup_{\boldsymbol{\omega} \in \mathbb{R}^2} \|\boldsymbol{\omega}…

We study weighted Poincar\'e and Poincar\'e-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form $$ \left…

Sharp affine fractional $L^p$ Sobolev inequalities for functions on $\mathbb R^n$ are established. The new inequalities are stronger than (and directly imply) the sharp fractional $L^p$ Sobolev inequalities. They are fractional versions of…

We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<\infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among…

We investigate the interactions of functional rearrangements with Prekopa-Leindler type inequalities. It is shown that that a general class of integral inequalities tighten on rearrangement to "isoperimetric" sets with respect to a relevant…