Parabolic BMO estimates for pseudo-differential operators of arbitrary order (1408.2343v1)

Abstract: In this article we prove the BMO-$L_{\infty}$ estimate $$ |(-\Delta)^{\gamma/2} u|{BMO(\mathbf{R}^{d+1})}\leq N |\frac{\partial}{\partial t}u-A(t)u|{L_{\infty}(\mathbf{R}^{d+1})}, \quad \forall\, u\in C^{\infty}_c(\mathbf{R}^{d+1}) $$ for a wide class of pseudo-differential operators $A(t)$ of order $\gamma\in (0,\infty)$. 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 $u\in C^{\infty}_c(\mathbf{R}^{d+1})$ $$ |u_t|{L_p(\mathbf{R}^{d+1})}+|(-\Delta)^{\gamma/2}u|{L_p(\mathbf{R}^{d+1})}\leq N|u_t-A(t)u|_{L_p(\mathbf{R}^{d+1})}, $$ where $p\ in (1,\infty)$ and the constant $N$ is independent of $u$.