Parabolic BMO estimates for pseudo-differential operators of arbitrary order

Abstract

In this article we prove the BMO-$L_{\infty }$ estimate $$\Vert (-\Delta {)}^{\gamma /2}u{\Vert }_{BMO({\mathbf{R}}^{d+1})}\le N\Vert \frac{\partial }{\partial t}u-A(t)u{\Vert }_{L_{\infty }({\mathbf{R}}^{d+1})},\forall u\in C_c^{\infty }({\mathbf{R}}^{d+1})$$ 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 $$\frac{\partial }{\partial t}u=A(t)u+f,t\in \mathbf{R}$$ we prove that for any $u\in C_c^{\infty }({\mathbf{R}}^{d+1})$ $$\Vert u_t{\Vert }_{L_p({\mathbf{R}}^{d+1})}+\Vert (-\Delta {)}^{\gamma /2}u{\Vert }_{L_p({\mathbf{R}}^{d+1})}\le N\Vert u_t-A(t)u{\Vert }_{L_p({\mathbf{R}}^{d+1})},$$ where $pin(1,\infty )$ and the constant $N$ is independent of $u$.

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