Pseudodifferential operators with symbols in the Hörmander class \(S^0_{\alpha ,\alpha }\) on \(\alpha \)-modulation spaces (Q1736098)

The authors study the boundedness of pseudodifferential operators with symbols in the Hörmander class $S^0_{\rho,\rho}$ on $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}$. In particular, it is show that if $0 \le \alpha <1$, $0 < p, q \le \infty$ and $s \in \mathbb{R}$ then, all pseudo-differential operators with symbols in $S^0_{\alpha,\alpha}$ are bounded on $M^{s,\alpha}_{p,q}(\mathbb{R}^n)$. In consequence, more generally, all pseudo-differential operators with symbols in $S^b_{\delta,\rho}$, $0\le \delta\le \alpha\le \rho \le 1$ are bounded from $M^{s,\alpha}_{p,q}(\mathbb{R}^n)$ to $M^{s-b,\alpha}_{p,q}(\mathbb{R}^n)$.