Existence and nonexistence of homotopy formulas for the Mizohata complex (Q1817923)

The canonical Mizohata structure \({\mathcal L}\) in \({\mathbb R}^{n+1}={\mathbb R}^n_t\times {\mathbb R}_x,\) \(n=1,2,\ldots,\) is generated by the vector fields \(L_j=\partial/ \partial t_j-it_j\partial/\partial x,\) \(j=1,2,\ldots,n.\) The structure \({\mathcal L}\) is locally integrable, that is its orthogonal bundle \({\mathcal L}^\perp\) is everywhere generated by \(dZ,\) where \(Z=Z(t,x)=x+i(|t|^2/2).\) It is the basic model for the class of Mizohata structures with positive definite Levi form [see \textit{F. Treves}, Hypo-analytic structures: local theory. Princeton Mathematical Series. Princeton, NJ: Princeton University Press (1992; Zbl 0787.35003)]. These structures are fundamental instances to study in connection to problems related to local integrability and local solvability. The exterior derivative in \({\mathbb R}^{n+1}\) defines, when passing to the quotient by \({\mathcal L},\) a complex of differential operators \[ \begin{multlined}\delta_p: C^\infty \bigl({\mathbb R}^{n+1},\Lambda^p({\mathbb C}\otimes T^*{\mathbb R}^{n+1}/{\mathcal L}^\perp)\bigr)\longrightarrow C^\infty\bigl({\mathbb R}^{n+1}, \Lambda^{p+1}({\mathbb C}\otimes T^*{\mathbb R}^{n+1}/{\mathcal L}^\perp)\bigr), \\ \text{ for \(p=0,1,\ldots,n-1.\)}\end{multlined}\tag{1} \] In the special cases \(p=0,1,\) one has \[ \delta_0(u)=\sum_{j=1}^n(L_ju) dt_j,\quad\delta_1\Biggl(\sum_{j=1}^nf_j dt_j \Biggr)=\sum_{k<j}\bigl(L_kf_j-L_jf_k\bigr) dt_k\wedge dt_j. \] A homotopy formula for the complex \((1)\) at the level \(p=1\) is a pair of operators \(K_p:C^\infty({\mathbb R}^{n+1},\Lambda^p({\mathbb C}\otimes T^*{\mathbb R}^{n+1}/{\mathcal L}^\perp))\rightarrow C^\infty({\mathbb R}^{n+1}, \Lambda^{p-1}({\mathbb C}\otimes T^*{\mathbb R}^{n+1}/{\mathcal L}^\perp))\), \(p=1,2,\) such that \[ \omega=\delta_0K_1\omega+K_2\delta_1\omega,\quad \text{for any given 1-form }\omega=\sum_{j=1}^n\omega_j dt_j. \] Homotopy formulas are a key ingredient in Kuranishi's approach to local integrability. In this paper, the authors construct homotopy formulas for the Mizohata structure for \(n>2,\) and discuss some of its functional properties. They then consider an elliptic structure related to Mizohata's one and indicate how the use of the homotopy formulas yields its semiglobal integrability. Finally, they show that when \(n=2\) there are no homotopy formulas, and give another proof of \textit{A. Meziani}'s result [Commun. Partial Differ. Equations 20, No. 3-4, 499-539 (1995; Zbl 0824.58004)], about the local integrability of Mizohata structures when \(n\geq 2.\) In particular, the canonical Mizohata structure of rank \(n=2\) seems to be the only known example of a locally integrable structure for which no homotopy formulas exist and yet its flat formally integrable perturbations remain locally integrable. The final picture they get (in parallel with the integrability of \(2n+1\)-dimensional CR-structures) when the Levi form is positive definite is the following: when \(n=1\) there is no local integrability (in general) for both Mizohata and CR-structures, when \(n>2\) there is local integrability in both cases, when \(n=2\) there are no homotopy formulas in both cases, but Mizohata structures are locally integrable whereas the CR case remains open.