Parabolic BMO estimates for pseudo-differential operators of arbitrary order

In this article we prove the BMO-$L_{\mathrm{\infty }}$ estimate $$\mathrm{\parallel }(-\mathrm{\Delta }{)}^{\gamma \mathrm{/}2}u{\mathrm{\parallel }}_{BMO({\mathbf{R}}^{d+1})}\le N\mathrm{\parallel }\frac{\mathrm{\partial }}{\mathrm{\partial }t}u-A(t)u{\mathrm{\parallel }}_{L_{\mathrm{\infty }}({\mathbf{R}}^{d+1})},\mathrm{\forall }u\in C_c^{\mathrm{\infty }}({\mathbf{R}}^{d+1})$$ for a wide class of pseudo-differential operators $A(t)$ of order $\gamma \in (0,\mathrm{\infty })$. The coefficients of $A(t)$ are assumed to be merely measurable in time variable. As an application to the equation $$\frac{\mathrm{\partial }}{\mathrm{\partial }t}u=A(t)u+f,t\in \mathbf{R}$$ we prove that for any $u\in C_c^{\mathrm{\infty }}({\mathbf{R}}^{d+1})$ $$\mathrm{\parallel }{u}_t{\mathrm{\parallel }}_{L_p({\mathbf{R}}^{d+1})}+\mathrm{\parallel }(-\mathrm{\Delta }{)}^{\gamma \mathrm{/}2}u{\mathrm{\parallel }}_{L_p({\mathbf{R}}^{d+1})}\le N\mathrm{\parallel }{u}_t-A(t)u{\mathrm{\parallel }}_{L_p({\mathbf{R}}^{d+1})},$$ where $pin(1,\mathrm{\infty })$ and the constant $N$ is independent of $u$.