The Wiener transform on the Besicovitch spaces
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- by Christopher Heil PDF
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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 $p=2$.References
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Additional Information
- Christopher Heil
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
- Email: heil@math.gatech.edu
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 2065-2071
- MSC (1991): Primary 42A38; Secondary 42A75, 46B03
- DOI: https://doi.org/10.1090/S0002-9939-99-04798-X
- MathSciNet review: 1487371