Essential spectra and semigroups of perturbations of M-hypoelliptic pseudo-differential operators on L^p(ℝⁿ)

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.

Keywords:

• M-hypoelliptic pseudo-differential operators
• Agmon–Douglis–Nirenberg inequality
• Erhling's inequality
• essential spectra
• strongly continuous semigroups
• dissipative operators

AMS Subject Classifications: :

• 35S05
• 47G05