The Nelson conjecture and chain rule property

Abstract: Let $p\ge 1$ and let $\mathbf v \colon \mathbb R^d \to \mathbb R^d$ be a compactly supported vector field with $\mathbf v \in L^p(\mathbb R^d)$ and $\operatorname{div} \mathbf v = 0$ (in the sense of distributions). It was conjectured by Nelson that it $p=2$ then the operator $\mathsf{A}(\rho) := \mathbf v \cdot \nabla \rho$ with the domain $D(\mathsf A)=C_0^\infty(\mathbb R^d)$ is essentially skew-adjoint on $L^2(\mathbb R^d)$. A counterexample to this conjecture for $d\ge 3$ was constructed by Aizenmann. From recent results of Alberti, Bianchini, Crippa and Panov it follows that this conjecture is false even for $d=2$. Nevertheless, we prove that for $d=2$ the condition $p\ge 2$ is necessary and sufficient for the following chain rule property of $\mathbf v$: for any $\rho \in L^\infty(\mathbb R^2)$ and any $\beta\in C^1(\mathbb R)$ the equality $\operatorname{div}(\rho \mathbf v) = 0$ implies that $\operatorname{div}(\beta(\rho) \mathbf v) = 0$. Furthermore, for $d=2$ we prove that $\mathbf v$ has the renormalization property if and only if the stream function (Hamiltonian) of $\mathbf v$ has the weak Sard property, and that both of the properties are equivalent to uniqueness of bounded weak solutions to the Cauchy problem for the corresponding continuity equation. These results generalize the criteria established for $d=2$ and $p=\infty$ by Alberti, Bianchini and Crippa.