Pseudodifferential operators with symbols in the H\"ormander class $S^0_{\alpha,\alpha}$ on $\alpha$-modulation spaces

In this paper, we study the boundedness of pseudodifferential operators with symbols in the H\"ormander class $S^0_{\rho,\rho}$ on $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$, and consider the relation between $\alpha$ and $\rho$. In particular, we show that pseudodifferential operators with symbols in $S^0_{\alpha,\alpha}$ are bounded on all $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}$, for arbitrary $s\in\mathbb{R}$ and for the whole range of exponents $0 < p,q \leq \infty$.