One-Parameter Semigroups Generated by Strongly M-Elliptic Pseudo-Differential Operators on Euclidean Spaces

We begin with a recall of the definitions and basic properties of the standard Hörmander classes of pseudo-differential operators on Rn. Then we introduce a new class of pseudo-differential operators that can be traced back to Taylor, generalized by Garello and Morando and further developed by M. W. Wong. A related class of pseudo-differential operators depending on a complex parameter on an open subset of the complex plane is constructed. We tease out from this related class the strongly M – elliptic pseudo-differential operators and prove that they are infinitesimal generators of holomorphic and hence strongly continuous one-parameter semigroups of bounded linear operators on Lp(Rn), 1<p<ꚙ. The proof is based on careful refinements of the Agmon – Douglis – Nirenberg estimates for the pseudo-differential operators in the book by M. W. Wong. In the case when p=2, we give another proof that strongly (ρ,Λ) – elliptic pseudo-differential operators, which include strongly M – elliptic ones, are infinitesimal generators of strongly continuous one-parameter semigroups of bounded linear operators on L2(Rn) by first proving Gårding’s Inequality for strongly (ρ,Λ) – elliptic pseudo-differential operators on Rn.