By B. A. Plamenevskii (auth.)

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**Additional resources for Algebras of Pseudodifferential Operators**

**Sample text**

Tx in IRm. 3) + f dx(2) R"'-' t f f ~ R' R' lal =[s] Ily I f3 D;u(y,x(2)-lz If3D~u(z,x(2)12 X "), f :;II; O. We will assume that the norm of a function u(x(l),x(2) = f(x(2)w(x(l) in H~(lRm,lRm-lI) is equivalent to the norm of w in Let E CO' (\R m - 0:HlR n ). 4) 49 We rewrite the expression for 11·;6ip(lR n )11 in terms of the Mellin transform. 5) Ilw(A + inl2, ·);Hs(A,sn -1)11 2 dA + Imh=,8-s [s 1 ~ Ilw(A+inl2,);HJ(A,Sn-l)1I 2 dll.. 1. Let v I E [O,s 1the inequality E CO' (IRn \ 0), s, /3 E R s ~ 0.

The equality Fu = F'u holds if u E

It suffices to convince ourselves that every function in Mp -I is the limit of a sequence of functions from Mp. Let w E M p _ l , wk(r,q,) = k:::':::(n+p)w(k:::':::lr,q,) for /3-s §p +nI2. It M 0 42 Chapter 2. remains to note that W-Wk E Mp and that Ilwk;Hp(Rn)II-,>O . 3 remains valid for /3-s = k +n12 too, we only need modify the definitions of the sequences {Vd,{Wk} (d. [44]). We denote by tions ~ jv(x)x'Ydx The 0 the set of functions from = 0, /y/ = 0,1, Co (Rn \ 0) satisfying the condi- ...