Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. The approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian.Comment: 35 pages. Revised versio

An extension problem and trace Hardy inequality for the sublaplacian on HH-type groups

In this paper we study the extension problem for the sublaplacian on a $H$-type group and use the solutions to prove trace Hardy and Hardy inequalities for fractional powers of the sublaplacian.Comment: 39 page

Mixed norm estimates for the Ces\`aro means associated with Dunkl--Hermite expansions

Our main goal in this article is to study mixed norm estimates for the Ces\`{a}ro means associated with Dunkl--Hermite expansions on $\mathbb{R}^d$. These expansions arise when one consider the Dunkl--Hermite operator (or Dunkl harmonic oscillator) $H_{\kappa}:=-\Delta_{\kappa}+|x|^2$, where $\Delta_{\kappa}$ stands for the Dunkl--Laplacian. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Ces\`{a}ro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of three lines lemma.Comment: 24 pages. Revised version following referee's comments. To appear in Transactions of the American Mathematical Societ

Hardy-type inequalities for fractional powers of the Dunkl-Hermite operator

We prove Hardy-type inequalities for a fractional Dunkl–Hermite operator, which incidentally gives Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use h-harmonic expansions to reduce the problem in the Dunkl–Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by Frank et al. [‘Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc. 21 (2008), 925–950’] in the Euclidean setting, to obtain a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a ‘good’ spectral theorem and an integral representation for the fractional operators involved.MTM2015-65888-C04-4-

On extension problem, trace hardy and Hardy’s inequalities for some fractional Laplacians

We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.2017 Leonardo grant for Researchers and Cultural Creators, BBVA Foundatio

Mixed norm estimates for the Cesàro means associated with Dunkl-Hermite expansions

Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions on $\mathbb{R}^d$. These expansions arise when one considers the Dunkl--Hermite operator (or Dunkl harmonic oscillator) $H_{\kappa}:=-\Delta_{\kappa}+|x|^2$, where $\Delta_{\kappa}$ stands for the Dunkl--Laplacian. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Ces\`{a}ro means for Laguerre expansions with shifted parameter. In order to obtain such vector-valued inequalities, we develop an argument to extend these Laguerre operators for complex values of the parameters involved and apply a version of three lines lemma

Mussel culture experiments in Ennore estuary, Chennai

The present paper gives a detailed account on experiments of mussel culture carried out by the Institute in 1996 in association with fishermen of Ennore in an estuarine environment by adopting the long-line and rack culture methods