Optimal regularity of solutions to no-sign obstacle-type problems for the sub-Laplacian

We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ \Delta_{H}u=f\chi_{\{u\ne0\}}, \] where $\Delta_{H}$ denotes sub-Laplacian operator in a stratified group. assume weakest condition on $f$, namely $f*\Gamma$ is $C_{H}^{1,1}$, $\Gamma$ fundamental solution $\Delta_{H}$. The understood sense Folland and Stein. In classical Euclidean setting, first seeds above problem are already present 1991 paper Sakai also related quadrature domains. As special instance our results, when $u$ nonnegative satisfies equation we recover obstacle groups, that was previously established by Danielli, Garofalo Salsa. Our result sharp: it can be seen as subelliptic counterpart $C^{1,1}$ due Andersson, Lindgren Shahgholian.