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Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2016-09-21

2016

Σκαρμόγιαννης Νικόλαος

Απόστολος Γιαννόπουλος Καθηγητής

Σημειαική σύγκλιση της σειράς Fourier-Το θεώρημα του Carleson

Greek

Pointwise convergence of Fourier series – Carleson’s theorem

The aim of this thesis is to provide a complete a self-contained proof of the

Carleson-Hunt theorem: if f is a p-integrable 2π-periodic function, where p>1,

then the Fourier series S[f](x) converges to f(x) almost everywhere.

In the first chapter we present some interpolation theorems for sublinear

operators that will be needed later on: among them are the theorem of

Marcinkiewicz and the interpolation theorem of Stein-Weiss. Then we formulate a

basic lemma-theorem and show that accepting this we can easily deduce the

Carleson-Hunt theorem.

In the second chapter we introduce two linear operators that help us to prove

that the Hilbert transform and the modified Hilbert transform are of type p for

every p >1, as well as some exponential estimates for these two

transforms.καθώς

In the third chapter we introduce the notion of dyadic intervals that are

needed in the sequel, and some modified Hilbert transforms which correspond to

them. Then, we introduce the modified Fourier coefficients and reduce the

question to the problem to show that a suitable operator M* is of type p on the

indicator functions of measurable sets.

In the last more technical chapter we define some exceptional sets whose

measure is of "type p" and using them we show that M* has the desired property.

This completes the proof.

Carleson-Hunt theorem: if f is a p-integrable 2π-periodic function, where p>1,

then the Fourier series S[f](x) converges to f(x) almost everywhere.

In the first chapter we present some interpolation theorems for sublinear

operators that will be needed later on: among them are the theorem of

Marcinkiewicz and the interpolation theorem of Stein-Weiss. Then we formulate a

basic lemma-theorem and show that accepting this we can easily deduce the

Carleson-Hunt theorem.

In the second chapter we introduce two linear operators that help us to prove

that the Hilbert transform and the modified Hilbert transform are of type p for

every p >1, as well as some exponential estimates for these two

transforms.καθώς

In the third chapter we introduce the notion of dyadic intervals that are

needed in the sequel, and some modified Hilbert transforms which correspond to

them. Then, we introduce the modified Fourier coefficients and reduce the

question to the problem to show that a suitable operator M* is of type p on the

indicator functions of measurable sets.

In the last more technical chapter we define some exceptional sets whose

measure is of "type p" and using them we show that M* has the desired property.

This completes the proof.

Series, Fourier, Pointwise, Convegence, p-integrable

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