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The main technical result of the paper is a Bochner type formula for thesub-laplacian on a quaternionic contact manifold. With the help of this formulawe establish a version of Lichnerowicz' theorem giving a lower bound of theeigenvalues of the sub-Laplacian under a lower bound on the $Sp(n)Sp(1)$components of the qc-Ricci curvature. It is shown that in the case of a3-Sasakian manifold the lower bound is reached iff the quaternionic contactmanifold is a round 3-Sasakian sphere. Another goal of the paper is toestablish a-priori estimates for square integrals of horizontal derivatives ofsmooth compactly supported functions. As an application, we prove a sharpinequality bounding the horizontal Hessian of a function by its sub-Laplacianon the quaternionic Heisenberg group.

The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold