  The main technical result of the paper is a Bochner type formula for the
sub-laplacian on a quaternionic contact manifold. With the help of this formula
we establish a version of Lichnerowicz' theorem giving a lower bound of the
eigenvalues 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 a
3-Sasakian manifold the lower bound is reached iff the quaternionic contact
manifold is a round 3-Sasakian sphere. Another goal of the paper is to
establish a-priori estimates for square integrals of horizontal derivatives of
smooth compactly supported functions. As an application, we prove a sharp
inequality bounding the horizontal Hessian of a function by its sub-Laplacian
on the quaternionic Heisenberg group.
