Consider the scalar elliptic equation of divergence form $$div((1+a)\nabla\pi)=div F\ \ in\ \ R^3,$$ where $a$ is a Schwartz function with $1+a\geq c=const>0$, $F=(F_1,F_2,F_3)$ is a vector-valued Schwartz function.

Now can we give a counterexample to show that for $1<p<\infty, p\neq2$, the solution map $F\mapsto\nabla\pi$ is not bounded on $L^p$, namely, there does not hold $$\|\nabla\pi\|_{L^p}\leq C\|F\|_{L^p}.$$

Let's give a remark. Denote by $Q=\nabla(-\Delta)^{-1}div$, a $C-Z$ singular integral operator on $L^p$, then $$(1+a)\nabla\pi=[a,Q]\nabla\pi+QF,$$ where $[a,Q]\nabla\pi=aQ\nabla\pi-Q(a\nabla\pi)$ is a commutator.

By *R. R. Coifman, R. Rochberg, and Guido Weiss*, **Factorization theorems for Hardy spaces in several variables**, *Ann. of Math. (2)* **103** (1976), no. 3, 611--635, the commutator operator $[a,Q]$ is bounded on $L^p$ if and only if $a\in BMO$
$$\|[a,Q]\nabla\pi\|_{L^p}\leq C\|a\|_{BMO}\|\nabla\pi\|_{L^p}.$$ Thus if $\|a\|_{BMO}$ is sufficiently small, then the gradient estimate is valid. But what happens if we remove the smallness condition for $a$?