By B. A. Plamenevskii (auth.)

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**Extra resources for Algebras of Pseudodifferential Operators**

**Sample text**

Let «P be positively homogeneous function of degree a on IR m \ 0 that is infinitely differentiable on Sm -I. We denote the operator of multiplication by «P by the same letter «P. 1) where F is the Fourier transform on IR m and u u= E Co (Rm,lR m - n). We denote, as §it the partial Fourier transform with respect to the variables We put 0 = r 2)/lr2)1, X = x(l)lr 2)1, Y = y(l)lr 2)1. By using the fact that «P is homogeneous we obtain that the quantity Ir2) l- a(A0(X Ir2) I-I, r2) is equal to before, by Xn+l, ···,Xm .

Then = (0'"j)(w,s+iT) is,jorT";:;;O, a continuous junction with values in L 2(sn-2,0'"H"+(IR». ytic in the halfplane T < 0, and +00 f 1(0'"f)(w,s + iT) 12(1 +s2 +~)'ds ~ c(W) < 00, -00 where c (w) is independent of T ~ 0. 12) 35 §6. 12), then f < 00 is also independent of 7' ~ o. Converse~y, if for the function ('fI/)( w,·) is anarytic in the halfplane indicated and E L 2 (sn -2,H+ (IR)). The proposition remains valid replaced by H _ and 7' < 0 by 7' > if H + is 0, everywhere. Let X+(,u) (resp.

H;ImA (A,sn -I) --7 HO(sn -I) is continuous. 2(w,t). ' (A,Sn -I )11. ::U = T(A)U. 11) can be extended to all complex A = i (k + n 12), k = 0,1, .... 9), which clearly is a monomorphism. 9) is epimorphic. ,sn-2) is continuous. Put U = T(A)-Ij for a given j E L 2(sn-2,H"±lmA(IR». :: to u. e. ::j = T(A)-Ij = u, we E obtain that H"±ImA(A,Sn-I). Thus it has been proved that T(A) is epimorphic if ImA";:;; 0. If, however ImA > 0, we choose a sequence {fk} converging to j in L 2(sn -2 ,H"± ImA (IR» such that T(A)-IA E HO(sn-I).