Higher-dimensional Monge–Ampère decomposition conjecture

Let $comega\textsuperscript{}\subset\mathbb{R}^d$ be an open, bounded, sufficiently regular set. Let $m\geq 0$ and $\gamma\in(0,1]$. The claim concerns a linear proper subspace $E_d\varsubsetneqq\mathbb{R}^{d\times d}_{\mathrm{sym}}$ and linear maps

$$\bar\Psi:\mathcal{C}^{m,\gamma}(\bar\omega,\mathbb{R}^{d\times d}_{\mathrm{sym}})\to\mathcal{C}^{m+1,\gamma}(\bar\omega,\mathbb{R}^{d}),\qquad \bar A:\mathcal{C}^{m,\gamma}(\bar\omega,\mathbb{R}^{d\times d}_{\mathrm{sym}})\to\mathcal{C}^{m,\gamma}(\bar\omega,E_d),$$

continuous for all $m\geq0$ and $\gamma\in(0,1]$.

Higher-dimensional decomposition conjecture. For every $D\in\mathcal{C}^{m,\gamma}(\bar\omega,\mathbb{R}^{d\times d}_{\mathrm{sym}})$,

$$D+\operatorname{sym}\nabla\big(\bar\Psi(D)\big)=\bar A(D),$$

and the maps satisfy

$$\bar\Psi(\operatorname{Id}_d)\equiv0,\qquad \bar A(\operatorname{Id}_d)\equiv\operatorname{Id}_d$$

in $\omega$.

The paper establishes this decomposition in dimension two and notes that validating the analogous construction in general dimension would be necessary for the same approach to work there. The higher-dimensional assertion is presented as something to be validated and is not resolved in the supplied text.