Fourier Multipliers and Littlewood-Paley Theory

Postgraduate Thesis uoadl:2959778 246 Read counter

Κατεύθυνση Θεωρητικά Μαθηματικά
Library of the School of Science
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Balidou Vasiliki
Supervisors info:
Απόστολος Γιαννόπουλος, Καθηγητής, τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Original Title:
Πολλαπλασιαστές Fourier και Θεωρία Littlewood-Paley
Translated title:
Fourier Multipliers and Littlewood-Paley Theory
This dissertation studies the orthogonality properties of the Fourier Transform on Lp spaces. This comes as a generalization of the L2 orthogonality which is given by the Pythagorean Theorem. For the purpose of this generalization, we make use of the Littlewood-Paley operator and prove the Littlewood-Paley theorem. We continue by presenting some basic results on Fourier Multipliers which are applications of this theorem. In chapter 2 we mention some basic concepts that are necessary for the proving of the Littlewood-Paley theorem, like Schwartz functions and tempered distributions. On chapter 3 we introduce Fourier Multipliers and present some basic relevant results. Chapter 4 includes the Hilbert transform and it is proved that it is a bounded linear operator. Chapter 5 is a collection of inequality results for linear operators that are used to prove theorems of the next chapters. On chapter 6 we present 4 different versions of the Littlewood-Paley theorem. Those versions are based on the selection of a Ψ function that determines the Littlewood-Paley operator and the option of defining this function on dyadic rectangles or dyadic annuli. Chapter 7 includes the Marcinkiewicz Theorem and the Hormander-Mikhlin theorem; two theorems that give sufficient conditions for L-infinity functions to be Fourier Multipliers for every p. Chapter 8 includes some additional applications of the Littlewood-Paley theorem.
Main subject category:
Dyadic Annuli Dyadic Rectangles Fourier Multipliers Fourier Transform Hilbert Transform Littlewood-Paley Operator Littlewood-Paley Theorem Marcinkiewicz Theorem Plancherel’s Identity Schwartz Function Tempered Distribution Translation Hormander-Mikhlin Theorem
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