Fourier multipliers and spectral measures in Banach function spaces

Fourier multipliers and spectral measures in Banach function spaces It is classical that amongst all spaces L p (G), 1 ≤ p ≤ ∞, for $$G = {\mathbb{R}}, \mathbb{Z}$$ , or $${\mathbb{T}}$$ say, only L 2 (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L 2 (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L 2 (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L 2 (G); this fails for every p ≠ 2. We show that this special status of L 2 (G) amongst the spaces L p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Fourier multipliers and spectral measures in Banach function spaces

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Publisher
Birkhäuser-Verlag
Copyright
Copyright © 2008 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2210-2
Publisher site
See Article on Publisher Site

Abstract

It is classical that amongst all spaces L p (G), 1 ≤ p ≤ ∞, for $$G = {\mathbb{R}}, \mathbb{Z}$$ , or $${\mathbb{T}}$$ say, only L 2 (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L 2 (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L 2 (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L 2 (G); this fails for every p ≠ 2. We show that this special status of L 2 (G) amongst the spaces L p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).

Journal

PositivitySpringer Journals

Published: Sep 1, 2008

References

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