The Algebra of Pseudo-Differential Operator on the Functional Space Wλ Sp,σm

For popularizing the functional space Swhich is common in use, a Frechet functional space WSis defined in this paper and an exploration is attempted on the algebraic characteristics of the pseudo-differential operators stipulated by the functional space WS. Definition 1 We say P（x,）∈C(R×R) is a symbol of the functional set WS（-∞<m<+∞, 0≤δ≤p≤1, δ<1）, if p（x,） has the property that for any multi-indices α,β∈Nthere exists a constant Csuch that