By B. A. Plamenevskii (auth.)

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Additional resources for Algebras of Pseudodifferential Operators

Example text

From this we obtain that the second term also decreases rapidly as x ~ 00. 15). Using the fact that the poles of E(A)-I are located at the points A = -i(k +nI2), k = 0,1, ... 9) we find that this term is 0(1). Consider the first term. The inner integral extends as a meromorphic func- tion onto the whole A-plane, and only the points A = i (/ y / + a - / a/ + n 12) (are first order) poles (in order to see this we must expand CP(w, e -/0) by Taylor's formula at (w,O». By moving the line of integration downwards, we find that the first 59 §5.

48 Chapter 2. Sobolev-Slobodetskii space HS(lRm). 2) 12 1y - z I-n + 2([sl-s)l{vdz + We assume that also for arbitrary s the norm in H~(lRm,lRm-n) is homogeneous of degree f3-s+mI2. This means that the norm is multiplied by tf3-s+mI2 as the result of a similitude transformation x .... 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 ).

For sgno+sgnt = 0 the expression between brackets (on the right) vanishes, sgno = sgnt it equals (e 7T(A-/l)+e- 7T(A-I'))sgno. 6) . • We introduce the space H~(A,sn-l) as the closure of the set CO(S~-I) with respect to the norm of HS(A,Sn -1). Denote by H~ (lR) the completion of the set (IR:±:) with respect to the norm of HS(IR). 3. For A =¥= i (k + n 12), k = 0,1, ... , the operator T(A): H"±ImA(A,sn -1) ~L2(sn -2,H"±ImA(IR)) realizes an isomorphism Proof. 9) and topologicalry). 5 the map 34 Chapter 1.