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I. Idempotent Multipliers of H¹ on the Circle. II. A Mean Oscillation Inequality for Rearrangements

Citation

Klemes, Ivo (1985) I. Idempotent Multipliers of H¹ on the Circle. II. A Mean Oscillation Inequality for Rearrangements. Dissertation (Ph.D.), California Institute of Technology. https://resolver.caltech.edu/CaltechTHESIS:01222019-095854306

Abstract

H1(T) is the space of integrable functions f on the circle T such that the Fourier coefficients f̂(n) vanish for negative integers n. A multiplier is by definition a map m of H1 to itself such that the Fourier transform diagonal izes m. Let m̂(n) denote the diagonal coefficients of m for nonnegative n. Then m is called idempotent if each coefficient is zero or one.

Theorem: If m is idempotent, then the set of n for which m̂(n) = 1 is a finite Boolean combination of sets of nonnegative integers of the following three types: finite sets, arithmetic sequences, and lacunary sequences.

By definition, a sequence is lacunary if there is a real number q > 1 such that each term of the sequence is at least as large as q times the preceding term. The theorem implies a classification of the projections in H1 which commute with translations, or, what is equivalent on the circle (but not on the line), of the closed, translation invariant subspaces which are complemented in H1. In the course of the proof, a 1 ower bound is obtai ned on the operator norm of a multiplier whose coefficients are 0 or greater than 1 in magnitude. This bound implies that the number of nonzero coefficients in disjoint intervals of the same length is the same, up to some factor depending on the norm of m, provided that both intervals are shorter than their distance from 0.

Part II is unrelated to Part I. There it is proved that a general expression measuring the oscillation of a function on an interval is minimized by the decreasing rearrangement of the function. A special case of this expression is the BMO norm for functions of bounded mean oscillation.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Wolff, Thomas H.
Thesis Committee:
  • Wolff, Thomas H. (chair)
  • Katok, Anatole
  • Kechris, Alexander S.
  • DePrima, Charles R.
  • Smith, Brent
Defense Date:10 May 1985
Record Number:CaltechTHESIS:01222019-095854306
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:01222019-095854306
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11349
Collection:CaltechTHESIS
Deposited By: Melissa Ray
Deposited On:29 Jan 2019 18:55
Last Modified:02 Dec 2020 01:34

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