Uncertainty principles as embeddings of modulation spaces (Q1856848)

The classical Heisenberg uncertainty principle states that \(\frac{1}{2\pi} \|f\|_2^2 \leq \|xf \|_2^2 +\|\xi \hat f \|_2^2\), i.e., it provides the lower bound for the time and frequency concentration of a function \(f \in L^2(\mathbb R)\). In the paper under review, the authors generalize the above uncertainty principle by studying inequalities where the lower bound is replaced by a term that measures the joint time-frequency concentration of \(f\in L^2(\mathbb R)\). As such a measure they use the modulation space norm: \[ \|f\|_{M_{\alpha, \beta}^{r,s}} = \left( \int_{{\mathbb R}^d} \left( \int_{{\mathbb R}^d} |V_gf(x,\xi)|^r(1+|x|)^{\alpha r}dx \right)^{s/r} (1+|\xi|)^{\beta s}\right)^{1/s}, \] where \(V_g\) is the short-time Fourier transform (also known as the continuous Gabor transform) \(V_g(f)(x,\xi) = \int_{{\mathbb R}^d} f(t)\overline{g(t-x)}e^{-2\pi i\xi t}dx\), and \(g\) is a Schwartz function on \({\mathbb R}^d\). The main result of this paper provides conditions for which \[ \|f\|_{M_{\alpha, \beta}^{r,s}} \leq C(\|(1+|x|)^af\|_p + \|(1+|\xi|)^b\hat f\|_q), \] which can be seen as an embedding result of an appropriate space into the modulation space \(M_{\alpha, \beta}^{r,s}\). Counterexamples show that such embedding results are optimal.