Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X({\mathbb{R}}^n)$ and on its associate space $X^{\mathrm{\prime }}({\mathbb{R}}^n)$, then a pseudodifferential operator $\mathrm{Op}(a)$ is bounded on $X({\mathbb{R}}^n)$ 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 $S_{\rho ,\delta }^{n(\rho -1)}(\mathrm{\varkappa },n)$ with $0\le \delta \le \rho \le 1$, $0\le \delta <1$, and $\mathrm{\varkappa }>0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot )}({\mathbb{R}}^n)$.