Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.562983 
Title:  Fourier restriction phenomenon in thin sets  
Author:  Papadimitropoulos, Christos 
ISNI:
0000 0004 2726 8680


Awarding Body:  University of Edinburgh  
Current Institution:  University of Edinburgh  
Date of Award:  2010  
Availability of Full Text: 


Abstract:  
We study the Fourier restriction phenomenon in settings where there is no underlying proper smooth subvariety. We prove an (Lp, L2) restriction theorem in general locally compact abelian groups and apply it in groups such as (Z/pLZ)n, R and locally compact ultrametric fields K. The problem of existence of Salem sets in a locally compact ultrametric field (K,  · ) is also considered. We prove that for every 0 < α < 1 and ǫ > 0 there exist a set E ⊂ K and a measure μ supported on E such that the Hausdorff dimension of E equals α and bμ(x) ≤ Cx−α 2 +ǫ. We also establish the optimal extension of the HausdorffYoung inequality in the compact ring of integers R of a locally compact ultrametric field K. We shall prove the following: For every 1 ≤ p ≤ 2 there is a Banach function space Fp(R) with σorder continuous norm such that (i) Lp(R) ( Fp(R) ( L1(R) for every 1 < p < 2. (ii) The Fourier transform F maps Fp(R) to ℓp′ continuously. (iii) Lp(R) is continuously included in Fp(R) and Fp(R) is continuously included in L1(R). (iv) If Z is a Banach function space with the same properties as Fp(R) above, then Z is continuously included in Fp(R). (v) F1(R) = L1(R) and F2(R) = L2(R).


Supervisor:  Wright, Jim. ; Carbery, Tony.  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.562983  DOI:  Not available  
Keywords:  Fourier restriction phenomenon ; restriction theorem ; abelian groups  
Share: 