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The Wiener transform
on the Besicovitch spaces

Author: Christopher Heil
Journal: Proc. Amer. Math. Soc. 127 (1999), 2065-2071
MSC (1991): Primary 42A38; Secondary 42A75, 46B03
Published electronically: February 26, 1999
MathSciNet review: 1487371
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Abstract: In his fundamental research on generalized harmonic analysis, Wiener proved that the integrated Fourier transform defined by $Wf(\gamma ) = \int f(t) \, (e^{-2\pi i \gamma t} - \chi _{[-1,1]}(t))/(-2\pi i t) \, dt$ is an isometry from a nonlinear space of functions of bounded average quadratic power into a nonlinear space of functions of bounded quadratic variation. We consider this Wiener transform on the larger, linear, Besicovitch spaces ${\mathcal{B}}_{p,q}({\mathbf{R}})$ defined by the norm $\|f \|_{{\mathcal{B}}_{p,q}} = \bigl (\int _{0}^{\infty }\bigl (\frac{1}{2T} \int _{-T}^{T} |f(t)|^{p} \, dt\bigr )^{q/p} \frac{dT}{T}\bigr )^{1/q}$. We prove that $W$ maps ${\mathcal{B}}_{p,q}({\mathbf{R}})$ continuously into the homogeneous Besov space ${\dot {B}}^{1/p'}_{p',q}({\mathbf{R}})$ for $1 < p \le 2$ and $1 < q \le \infty $, and is a topological isomorphism when $p=2$.

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Additional Information

Christopher Heil
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Keywords: Besicovitch spaces, Besov spaces, Marcinkiewicz spaces, Wiener--Plancherel formula, Wiener transform
Received by editor(s): August 20, 1996
Received by editor(s) in revised form: October 8, 1997
Published electronically: February 26, 1999
Additional Notes: This research was supported by National Science Foundation Grant DMS-9401340.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society

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