Abstract
Using a version of Erhling's inequality for L
p
-Sobolev spaces H
s,p
on ℝ
n
, −∞ < s < +∞, 1 ≤ p < ∞ given by Wong in his paper [M.W. Wong, Erhling's inequality and pseudo-differential operators on L
p
(ℝ
n
), Cubo 8 (2006), pp. 97–108]. we establish an analogue of Agmon–Douglis–Nirenberg inequality for M-hypoelliptic pseudo-differential operators perturbed by singular potentials on L
p
(ℝ
n
), 1 < p < ∞. We also state and prove some facts concerning the essential spectra of M-hypoelliptic pseudo-differential operators T
σ on L
p
(ℝ
n
), 1 < p < ∞, perturbed by the operators of the form . A self-adjointness result is also proved for such perturbations of M-hypoelliptic pseudo-differential operators on L
2(ℝ
n
) whose symbols are independent of x in ℝ
n
. Moreover, as in the work [M.W. Wong, Erhling's inequality and pseudo-differential operators on L
p
(ℝ
n
), Cubo 8 (2006), pp. 97–108] by Wong an application to strongly continuous semigroups of contractions generated by M-hypoelliptic pseudo-differential operators on L
p
(ℝ
n
), 1 < p < ∞ is given.