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Title: Optimisation of Hardy-type inequalities with applications in infinite dimensional Geometry
Author: Guzu, Dorian
ISNI:       0000 0004 9357 2031
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2020
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The following thesis concerns the generalisation of Hardy's integral inequality to multiple dimensions. The structure involves three main papers, two published in journals and one paper in preparation. In the first article, we make use of the classical Hardy inequality away from a point and a multi-dimensional Hardy inequality associated to the Laplacian operator. The inequality's domain is extended to an in finite case and the behaviour of systems is studied when multiple particles are introduced as a concept. The result we obtain resembles the classical Hardy inequality, but, as an extension, we introduce new vector fields and boundary approximations on a many-particle quantum system. The second article presents the interest in the re finement of Hardy inequalities using Fourier expansions applied to a new Hopf system of coordinates. As a starting point, we con figured the coordinates in four dimensions for block - radial functions such that our argument could be easier understood. In comparison to previous research, our result behaves promising in the sense that the inequality discovered is true for functions on the whole space. Moreover, by introducing a new system of coordinates, we are able to apply our result to the extension of the Hopf fi bration to dimension eight. Last but not least, in article three we look at discrete Hardy Inequalities and their link to the continuous Hardy inequalities discovered so far. Considering that Hardy inequalities are more difficult to analyse for discrete operators, we make use of two important results in order to establish whether for the discrete Hardy inequality discovered there is a sharp constant on a closed disk in Z^2. The results involve a specifi c type of Aharonov-Bohm magnetic potential which facilitates the analysis of the constant's behaviour, which is present in the new discrete Hardy inequality.
Supervisor: Laptev, Ari Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral