Wang, Lifeng
(2023)
A STUDY ON FUNCTION SPACES.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
In this dissertation, we investigate the properties of homogeneous function spaces. We study related basic definitions and prerequisite lemmas. We also study useful properties of the famous Hardy-Littlewood maximal function and the Peetre-Fefferman-Stein maximal function for functions whose distributional Fourier transforms have compact supports. Furthermore, we introduce the classical Plancherel-Polya-Nikol'skij inequality for Schwartz functions and then generalize this inequality to the case of sufficiently smooth tempered distributions. We state and prove a complex interpolation theorem for the homogeneous Triebel-Lizorkin spaces. We prove a Fourier multiplier theorem for sequences of functions and deduce another for homogeneous Triebel-Lizorkin spaces. This dissertation provides improved results of function spaces found in the famous literature by Hans Triebel. The first pair of results provides equivalence characterizations of homogeneous Triebel-Lizorkin and Besov-Lipschitz spaces in terms of maximal functions of the mean values of iterated difference. It also furnishes the reader with inequalities in homogeneous Triebel-Lizorkin spaces in terms of iterated difference and in terms of iterated difference along coordinate axes. The corresponding inequalities in homogeneous Besov-Lipschitz spaces in terms of iterated difference and in terms of iterated difference along coordinate axes are also considered.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
1 September 2023 |
| Date Type: |
Publication |
| Defense Date: |
28 March 2023 |
| Approval Date: |
1 September 2023 |
| Submission Date: |
15 June 2023 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
177 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
Homogeneous Triebel-Lizorkin space, homogeneous Besov-Lipschitz space, iterated difference, Fourier analysis, Hardy-Littlewood maximal function, Peetre-Fefferman-Stein maximal function. |
| Date Deposited: |
01 Sep 2023 19:05 |
| Last Modified: |
01 Sep 2023 19:05 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/44996 |
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