Restriction Lemmas, Spherical Summation, Maximal Functions, Square Functions and All That

This chapter begins with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discusses their implications for radial multipliers and associated maximal functions. It focuses on the Littlewood–Paley bounds for two square functions introduced by Stein, who had stressed their importance in harmonic analysis and many important variants and generalizations in various monographs. The chapter proves new endpoint estimates for these square functions, for the maximal Bochner–Riesz operator, and for more general classes of radial Fourier multipliers. The majority of the chapter is devoted to these proofs, such as for convolutions with spherical measures.

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On plate decompositions of cone multipliers

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150700048x ◽

2009 ◽

Vol 52 (3) ◽

pp. 631-651 ◽

Cited By ~ 17

Author(s):

Gustavo Garrigós ◽

Andreas Seeger

Keyword(s):

Wave Equation ◽

Finite Type ◽

Maximal Functions ◽

Square Functions ◽

Local Smoothing ◽

Multiplier Operator ◽

Type Curves ◽

Low Dimensions ◽

Bergman Projections

AbstractAn important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for sharp Lp results on cone multipliers, local smoothing for the wave equation, convolutions with radial kernels, Bergman projections in tubes over cones, averages over finite-type curves in ℝ3 and associated maximal functions. We observe that the range of p in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L4 bound for the cone multiplier operator in ℝ3.

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Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

Bulletin of Symbolic Logic ◽

10.2178/bsl/1130335207 ◽

2005 ◽

Vol 11 (4) ◽

pp. 517-525

Author(s):

Juris Steprāns

Keyword(s):

Harmonic Analysis ◽

Set Theory ◽

Harmonic Functions ◽

Maximal Functions ◽

Pigeonhole Principle ◽

Cardinal Invariants ◽

Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

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