On a question of Louis Nirenberg (Q1429377)

This interesting paper answers a question on the local solvability of a particular second-order linear operator with smooth coefficients. More precisely it considers smooth real vector fields \(A,B\) on an open subset \(\Omega\) of \(\mathbb{R}^3\) so that \(A,B\), and the commutator \([A,B]\) are linearly independent. Given any smooth real vector field \(C\) and any smooth function \(\varphi\) defined in \(\Omega\), let \(L=AB+ C+\varphi\). The main result in this paper asserts that for each \(x_0\in\Omega\) and each \(\varepsilon>0\) there exists a neighborhood of zero \(U=U_{x_0, \varepsilon}\) and a linear operator \(G=G_{x_0, \varepsilon}\) mapping the Sobolev space \(H^{-1}(U)\) into itself continuously with norm \(\leq \varepsilon\), so that \(LGf=f\) for every distribution \(f\in H^{-1}(U)\). The starting point in the proof of this result is to write the vector field \(C\) as the linear combination with smooth coefficients \[ C=\alpha (x)A+\beta(x) B+\gamma (x)[A,B]. \] The author shows that when \(\gamma(0)\neq-\frac 12\), that is when the operator \(C+\frac 12[A,B]\) is elliptic on the double characteristics of \(L\), the right local inverse \(G\) is indeed a linear and continuous operator from \(H^{-1}(U)\) to \(L^2(U)\). The proof is based on the existence of constants \(K\), \(\delta>0\) such that \[ \| ABu\| +\| BAu \|+\|[A,B] u\|\leq K\| L^*u\| \] for every \(u\in C^\infty_0({\mathcal B}_\delta)\), where \({\mathcal B}_\delta\) is the open ball in \(\mathbb{R}^3\) centered at zero with radius \(\delta\), \(\|\cdot\|\) is the \(L^2\) norm and \(L^*\) is the adjoint of the operator \(L\). When \(\gamma (0)=-\frac 12\) the author shows the existence of a local right inverse on \(H^{-1}(U)\). The paper also discusses extensions to local solvability in the Sobolev space \(H^s\) for any \(s\in\mathbb{R}\). This is done by extending the main result to particular first-order pseudo-differential operators. The paper ends by posing several open questions.