Semiclassical spectral asymptotics on foliated manifolds

We consider a (hypo)elliptic pseudodifferential operator A(h) on a closed foliated manifold (M,F), depending on a parameter h > 0, of the form A(h) = A + h(m) B, where A is a formally self-adjoint tangentially elliptic operator of order mu > 0 with the nonnegative principal symbol and B is a formally self-adjoint classical pseudodifferential operator of order m > 0 on M with the holonomy invariant transversal principal symbol such that its principal symbol is positive, if mu < m, and its transversal principal symbol is positive, if p > m. We prove an asymptotic formula for the eigenvalue distribution function N-h(lambda) of the operator A(h) when h tends to 0 and lambda is constant.