Very weak solutions to the two dimensional Monge-Amp\'ere equation

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Amp'ere equation with H"older-continuous first derivatives of exponent $\beta <1\mathrm{/}5$. Our approach is based on combining the approach of Lewicka-Pakzad \cite{Lewicka:2015gz} with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by the second author with De Lellis and Inauen in \cite{DeLellis:2015wm} for the isometric immersion problem.