Parabolic BMO estimates for pseudo-differential operators of arbitrary order

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

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