The Wiener transform on the Besicovitch spaces

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 $p=2$.