Fourier Analysis and Partial Differential Equations

✍ Scribed by Jose Garcia-Cuerva

• Publisher: CRC Press
• Year: 2017
• Tongue: English
• Leaves: 336
• Edition: 1
• Category: Library

✦ Synopsis

Fourier Analysis and Partial Differential Equations presents the proceedings of the conference held at Miraflores de la Sierra in June 1992. These conferences are held periodically to assess new developments and results in the field. The proceedings are divided into two parts. Four mini-courses present a rich and actual piece of mathematics assuming minimal background from the audience and reaching the frontiers of present-day research. Twenty lectures cover a wide range of data in the fields of Fourier analysis and PDE. This book, representing the fourth conference in the series, is dedicated to the late mathematician Antoni Zygmund, who founded the Chicago School of Fourier Analysis, which had a notable influence in the development of the field and significantly contributed to the flourishing of Fourier analysis in Spain.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
PART I-MEMORIAL ARTICLES
1 Antoni Zygmund 1900-1992
2 Stylianos Pichorides 1940-1992
2.1 Life and works
2.2 Style et personnalit
PART II-MAIN LECTURES
3 Band-Limited Wavelets
3.1 Introduction
3.2 Necessary and sufficient conditions for the completeness of a band-limited wavelet system
3.3 The systems of Lemari and Meyer
3.4 The local bases of Coifman and Meyer and their connection with the Lemarie-Meyer basis and Wilson bases
3.5 The non-existence of smooth wavelets bases of Hsup(2) and other observations about the support of band-limited wavelets
3.6 Concluding remarks
4 A Family of Degenerate Differential Operators
4.1 Introduction
4.2 Reduction to a first-order system
4.3 Solutions of L.
4.4 A variant
4.5 Wronskian estimate
4.6 Proof of Proposition 3
4.7 Proof of Proposition 1
4.8 Proof of Proposition 2
5 Solvability of Second-Order PDO's on Nilpotent GroupsA Survey of Recent Results
5.1 Introduction
5.2 Some historical background
5.3 Second-order PDO's on two-step nilpotent groups
5.4 On the proof of Theorem 1
5.5 Remarks on more general doubly characteristic operators and re-interpretation of the results in Section 5.3
6 Recent Work on Sharp Estimates in Second Order Elliptic Unique Continuation Problems
6.1 Introduction and counterexamples
6.2 Carleman inequalities
6.3 A modification of the Carleman method
PART IIICONTRIBUTION ARTICLES
7 Weighted Lipschitz Spaces Defined by a Banach Space
7.1 Introduction
7.2 The theorems and their proofs
8 A Note on Monotone Functions
8.1 Introduction
8.2 Distribution formula
8.3 Increasing functions
9 Hilbert Transforms in Weighted Distribution Spaces
9.1 Introduction
9.2 Notation and preliminary lemmas
9.3 Continuity
9.4 Inversion
9.5 Inversion in subspace
10 Failure of an Endpoint Estimate for Integrals Along Curves
10.1 Introduction
10.2 The positive results
10.3 Proof of the negative result
11 Spline Wavelet Bases of Weighted Spaces
11.1 Introduction
11.2 Basic definitions
11.3 Sufficient conditions for unconditional basisness in Hsup(p)
11.4 Necessary conditions for basisness in Lsup(p)
11.5 On Meyer's reniarkable identity for splines
12 A Note on Hardy 'S Inequality in Orlicz Spaces
12.1 Introduction
12.2 Hardy inequalities
13 A Characterization of Commutators of Parabolic Singular Integrals
13.1 Introductio
13.2 Preliminaries and statement of results
13.3 Proof of Theorem 1
13.4 Proof of Theorem 2
14 Inequalities for Classical Operators in Orlicz Spaces
14.1 Introduction
14.2 Maximal functions and Riesz transforms
15 On the Herz Spaces with Power Weights
15.1 Introduction
15.2 Applications to boundedness of operators
15.3 Hardy spaces associated with K[sub(q)]((omitted)[sup(n)], (|x|[sup(.)]) and its characterization
16 On the One-Sided Hardy-Littlewood Maximal Function in the Real Line and in Dimensions Greater than One
16.1 Introduction
16.2 Theorem 1 and the one-sided fractional maximal functions
16.3 The case of equal weights
16. A[sub(.)][sub(+)] weights and one-sided sharp functions
16.5 Weights for one-sided maximal functions in (omitted)[sup(n)], n > 1
16.6 The one-sided maximal function associated to Bore1 measures in (omitted)[sup(n)], n > 1
17 Characterization of the Besov Spaces
18 Oscillatory Singular Integrals on Hardy Spaces
18.1 Introduction
18.2 Polynomial phase
18.3 Generic phase
18.4 Degenerate phase
18.5 Non-convolution operators
19 Three Types of Weighted Inequalities for Integral Operators
19.1 Three types of weighted inequalities
19.2 Maximal operator-the basic result
19.3 The classes A. and E.
19.4 Hilbert transform and related operators
19.5 Strong type inequalities
19.6 Hardy type operators
20 Boundary Value Problems for Higher Order Operators in Lipschitz and C[sup(1)] Domains
20.1 Introduction
20.2 The Dirichlet Problem on nonsmooth domains
21 A[sub(p)] and Approach Regions
21.1 Introduction
21.2 Definitions and previous results
21.3 Translations of a region
22 Maximal Operators Associated to Hypersurfaces with One Nonvanishing Principal Curvature