arXiv subjects

Andrei Velicu

Publications and source records attributed to Andrei Velicu.

6 recordsLinked to original sources

arXiv2020

Logarithmic Sobolev inequalities for Dunkl operators with applications to functional inequalities for singular Boltzmann-Gibbs measures

In this paper we study several inequalities of log-Sobolev type for Dunkl operators. After proving an equivalent of the classical inequality for the usual Dunkl measure $μ_k$, we also study a number of inequalities for probability measures of Boltzmann type of the form $e^{-|x|^p} dμ_k$. These are obtained using the method of $U$-bounds. Poincaré inequalities are obtained as consequences of the log-Sobolev inequality. The connection between Poincaré and log-Sobolev inequalities is further examined, obtaining in particular tight log-Sobolev inequalities. Finally, we study application to exponential integrability and to functional inequalities for a class of singular Boltzmann-Gibbs measures.

math.AP↗

arXiv2020

Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities

In this paper we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, $L^p$ inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli-Kohn-Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalised root systems are proved, which in the particular case of root systems $A_{N-1}$ improve some well-known results.

math.FA↗

arXiv2019

Infinite dimensional systems of particles with interactions given by Dunkl operators

Firstly we consider a finite dimensional Markov semigroup generated by Dunkl laplacian with drift terms. Using gradient bounds we show that for small coefficients this semigroup has an invariant measure. We then extend this analysis to an infinite dimensional semigroup on $(\mathbb{R}^N)^{\mathbb{Z}^d}$ which we construct using gradient bounds, and finally we study the existence of invariant measures and ergodicity properties.

math-ph↗

arXiv2019

Rellich, Gagliardo-Nirenberg, Trudinger and Caffarelli-Kohn-Nirenberg inequalities for Dunkl operators and applications

In this paper we obtain weighted higher order Rellich, weighted Gagliardo-Nirenberg, Trudinger, Caffarelli-Kohn-Nirenberg inequalities and the uncertainty principle for Dunkl operators. Moreover, we introduce an extension of the classical Caffarelli-Kohn-Nirenberg inequalities. Furthermore, we give an application of Gagliardo-Nirenberg inequality to the Cauchy problem for the nonlinear damped wave equations for the Dunkl Laplacian.

math.AP↗

arXiv2019

Spectral inequalities for a class of integral operators

We obtain inequalities for the Riesz means for the discrete spectrum of a class of self-adjoint compact integral operators. Such bounds imply some inequalities for the counting function of the Dirichlet boundary problem for the Laplace operator. The paper is an extension of the results previously obtained in [5].

math.AP↗

arXiv2019

Sobolev-Type Inequalities for Dunkl Operators

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. We also find estimates of the sharp constants for the Sobolev inequality for Dunkl gradient. Related inequalities and some improvements are also considered (Nash inequality, Besov space embeddings).

math.FA↗