Subelliptic Bourgain-Brezis Estimates on Groups

We show that divergence free vector fields which belong to L^1 on stratified, nilpotent groups are in the dual space of functions whose sub-gradient are L^Q integrable where Q is the homogeneous dimension of the group. This was first obtained on Euclidean space by Bourgain and Brezis.Comment: 15 pages, v2 has some typos fixed in lemma 2.

Generalizations of Integral Inequalities Similar to Hardy’s Inequality

In this paper, we established the generalizations of integral inequalities similar to Hardy’s inequality. Keywords:    Hardy’s inequality; Integral inequalities;  similar version;  Hlder’s inequality;  Generalization

Ostrowski type fractional integral operators for generalized (;,,)−preinvex functions

In the present paper, the notion of generalized (;,,)−preinvex function is applied to establish some new generalizations of Ostrowski type inequalities via fractional integral operators. These results not only extend the results appeared in the literature but also provide new estimates on these type

Geometric Harmonic Analysis

This thesis is the compilation of the results obtained during my PhD, which started in January 2018 and is being completed in the end of 2021. The main matter is divided into  ve chapters, Chapters 2 6. Each of these chapters has its own introductory part, some longer some shorter. This chapter is intended to be an introduction to the whole thesis. Without going into technical details, in this Chapter we will not only motivate the results and the content of the dissertation, but we also explain how and why these results came to be studied. We also introduce the main notation and some preliminary concepts that will be used throughout the dissertation

Constrained extremal problems in the Hardy space H2 and Carleman's formulas

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formulation of the problem. We further compute directional derivatives for this functional as a computational means to approach the issue. We then consider a finite-dimensional polynomial version of the bounded extremal problem

On the localization of the magnetic and the velocity fields in the equations of magnetohydrodynamics

We study the behavior at infinity, with respect to the space variable, of solutions to the magnetohydrodynamics equations in ${\bf R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free nonstationnary Navier--Stokes equations when $|x|\to +\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are boundedness criteria for convolution operators in weighted spaces.Comment: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics (to appear) (0000) --xx-

Non-commutative integration for spectral triples associated to quantum groups

This thesis is dedicated to the study of non-commutative integration, in the sense of spectral triples, for some non-commutative spaces associated to quantum groups