  Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class
of elliptic equations on compact hyperhermitian manifolds. By adapting the
approach of Sz\'ekelyhidi to the hypercomplex setting, we prove some a priori
estimates for solutions to such equations. In the estimate of the Laplacian we
assume the existence of a flat hyperk\"ahler metric. As an application of our
results we prove that the quaternionic analogue of the Hessian equation can
always be solved on compact flat hyperk\"ahler manifolds.
