The Wiener transform on the Besicovitch spaces

Heil, Christopher

doi:10.1090/S0002-9939-99-04798-X

The Wiener transform on the Besicovitch spaces
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Proc. Amer. Math. Soc. 127 (1999), 2065-2071 Request permission

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

J. Benedetto, G. Benke, and W. Evans, An $n$-dimensional Wiener-Plancherel formula, Adv. in Appl. Math. 10 (1989), no. 4, 457–487. MR 1023944, DOI 10.1016/0196-8858(89)90025-0
John J. Benedetto, The spherical Wiener-Plancherel formula and spectral estimation, SIAM J. Math. Anal. 22 (1991), no. 4, 1110–1130. MR 1112069, DOI 10.1137/0522072
George Benke, A spherical Wiener-Plancherel formula, J. Math. Anal. Appl. 171 (1992), no. 2, 418–435. MR 1194091, DOI 10.1016/0022-247X(92)90355-H
Arne Beurling, Construction and analysis of some convolution algebras, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 2, 1–32. MR 182839, DOI 10.5802/aif.172
Yong Zhuo Chen and Ka-Sing Lau, Wiener transformation on functions with bounded averages, Proc. Amer. Math. Soc. 108 (1990), no. 2, 411–421. MR 990416, DOI 10.1090/S0002-9939-1990-0990416-0
Hans G. Feichtinger, An elementary approach to Wiener’s third Tauberian theorem for the Euclidean $n$-space, Symposia Mathematica, Vol. XXIX (Cortona, 1984) Sympos. Math., XXIX, Academic Press, New York, 1987, pp. 267–301. MR 951188
P. Hartman and A. Wintner, The $(L^{2})$-space of relative measure, Proc. Nat. Acad. Sci. 33 (1947), 128–132.
J. Marcinkiewicz, Une remarque sur les espaces de M. Besikowitch, C. R. Acad. Sci. Paris 208 (1939), 157–159.
Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR 1163193, DOI 10.1007/978-3-0346-0419-2
N. Wiener, Generalized harmonic analysis, Acta Math. 55 (1930), 117–258.