The range of localization operators and lifting theorems for modulation and Bargmann-Fock spaces (Q2847043)

The short-time Fourier transform of a function or distribution \(f\) with respect to a window function \(g\in {\mathcal S}({\mathbb R}^d)\) is defined by \(V_gf(z) = \left<f, \pi(z)g\right>\), where \(z = (x,\xi)\in {\mathbb R}^{2d}\) and \(\pi(z)g(t) = e^{2\pi i \xi t}g(t-x)\) denotes a time-frequency shift. The modulation spaces are defined imposing a norm on the short-time Fourier transform, for instance, NEWLINE\[NEWLINE \| f\|_{M^p_m} = \| m\cdot V_g f\|_{L^p}, NEWLINE\]NEWLINE where \(1\leq p\leq \infty\) and \(m\) is a nonnegative weight function on \({\mathbb R}^{2d}\). The localization operator \(A^g_m\) with window \(g\) and symbol \(m\) is defined formally by the integral NEWLINE\[NEWLINE A^g_mf = \int_{{\mathbb R}^{2d}}m(z)V_gf(z)\pi(z)g\, dz. NEWLINE\]NEWLINE A special case of the main result of the paper is the following.NEWLINENEWLINETheorem. Let \(m\) be a nonnegative continuous symbol on \({\mathbb R}^{2d}\) satisfying \(m(z_1 + z_2) \leq e^{a |z_1|^{b}}m(z_2)\) for \(z_1, z_2\in {\mathbb R}^{2d}\), \(0\leq b < 1\), and assume that \(m\) is radial in each time-frequency coordinate. Then, for suitable test functions \(g\), the localization operator \(A^g_m\) is an isomorphism from the modulation space \(M^p_{\mu}({\mathbb R}^{d})\) onto \(M^p_{\frac{\mu}{m}}({\mathbb R}^{d})\) for all \(1\leq p\leq \infty\) and moderate weights \(\mu\).NEWLINENEWLINE In the case that the symbol \(m\) has polynomial growth, a proof was obtained by the authors in [J. Anal. Math. 114, 255--283 (2011; Zbl 1243.42037)] based on a deep result of \textit{J.-M. Bony} and \textit{J.-Y. Chemin} [Bull. Soc. Math. Fr. 122, No.~1, 77--118 (1994; Zbl 0798.35172)] which strongly depends on polynomial growth. The situation studied in this article has necessitated the development of new techniques, the main tool being the construction of a canonical isomorphism between modulation spaces of Hilbert type in terms of localization operators with a Gaussian window. As an application, the authors obtain a new isomorphism theorem for Gabor multipliers as well as an isomorphism theorem for Toeplitz operators between weighted Bargmann-Fock spaces of entire functions.