Parabolic BMO estimates for pseudo-differential operators of arbitrary order

TL;DR

This paper establishes BMO-$L_{ ext{infinity}}$ estimates for a broad class of pseudo-differential operators of arbitrary order with measurable coefficients, and applies these results to derive $L_p$ estimates for related evolution equations.

Contribution

It introduces novel BMO-$L_{ ext{infinity}}$ estimates for pseudo-differential operators of any order with minimal regularity assumptions on coefficients.

Findings

Proves BMO-$L_{ ext{infinity}}$ estimate for pseudo-differential operators of arbitrary order.

Derives $L_p$ estimates for solutions to evolution equations involving these operators.

Results hold with coefficients measurable in time, broadening applicability.

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 $u$.