Pseudodifferential Operators of Type (ρ,δ)
In order to generalize the parametrix construction of Section 1, Chapter I to certain classes of nonelliptic (yet hypoelliptic) equations, one must enlarge the pool of amplitudes that one is willing to use. Such a construction is carried out in Section 1 of this chapter. In an embryonic form it was first, I believe, attempted in Treves . The classes of symbols best suited for such an extension were introduced and described by Hörmander in , and given the name S ρ,δ classes. Roughly speaking the properties of pseudodifferential operators of type (1, 0), our “standard” pseudodifferential operators (Chapter I), generalize well to type (ρ, δ) as long as we keep 0 ≤ δ < ½ < ρ, as shown in Section 2.
KeywordsCompact Subset Bounded Linear Operator Pseudodifferential Operator Principal Symbol Positive Continuous Function
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