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

In this paper, we study the boundedness of pseudodifferential operators with symbols in the H"ormander class $S_{\rho ,\rho }^0$ 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 $S_{\alpha ,\alpha }^0$ are bounded on all $\alpha$-modulation spaces $M_{p,q}^{s,\alpha }$, for arbitrary $s\in \mathbb{R}$ and for the whole range of exponents $0<p,q\le \mathrm{\infty }$.