Boundedness of pseudodifferential operators on Banach function spaces

We show that if the Hardy-Littlewood maximal operator is bounded on aseparable Banach function space $X(\R^n)$ and on its associate space$X'(\R^n)$, then a pseudodifferential operator $\Op(a)$ is boundedon $X(\R^n)$ whenever the symbol $a$ belongs to the H\"ormander class$S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$with $0\le\delta\le\rho\le 1$, $0\le\delta<1$, and $\varkappa>0$.This result is applied to the case of variable Lebesgue spaces$L^{p(\cdot)}(\mathbb{R}^n)$.