In this paper we introduce the hybrid class of the so-called SG−M-hypoelliptic symbols, and consider the corresponding pseudo-differential operators.With any SG−M-elliptic pseudo-differential operator with positive order, we associate the minimal and maximal operators on L p(Rn), 1 < p < ∞. Further on, we prove that the minimal and maximal operators are equal and we compute their domains in terms of a family of suitable Sobolev spaces. In the last section, we show that an SG−M-elliptic pseudo-differential operator is Fredholm. Moreover, we discuss the essential spectra of SG−M-elliptic pseudo-differential operators with suitable orders.
