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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 39184 (02)
- University: Sharif University of Technology
- Department: Mathematical Science
- Advisor(s): Ranjbar Motlagh, Alireza
- Abstract:
- The Kakeya needle problem was stated by a Japanese mathematician, Kakeya, in 1917 as follows: \What is the least area of a planar shape in which a needle of unit length can be turned 180 degrees?" Later, this problem was closely related to some problems in Fourier analysis and harmonic analysis, theory of partial dierential equations, theory of arithmetic combinatorics, and analytic number theory. In this thesis, after stating some elementary considerations in chapter one, we describe the Kakeya needle problem and Besicovitch's solution in chapter two. In chapter three there will be some discussions about the Fourier multiplier operator and in particular, disk multiplier operator, and we will show how Charles Feerman used the Kakeya needle problem to prove that in dimensions higher than one, the disk multiplier operator is unbounded. In chapter four, we will discuss about the restriction theorems, which are among the most important subjects in harmonic analysis (and also has some applications in partial dierential equations), and state the Stein restriction conjecture. In chapter ve, using Besicovitch's solution for the Kakeya needle problem, we will dene the Besicovitch sets in dimensions higher than one, and state the Kakeya conjecture. Next we will explain the relationship between the restriction conjecture and the Kakyea conjecture. Finally, there will be a short summary of the eorts done to prove the Kakeya conjecture
- Keywords:
- Besicovitch Set ; Fourier Multiplier Operator ; Restriction Conjecture ; Kakeya Conjecture ; Kakeya Maximal Function
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