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By B. A. Plamenevskii (auth.)

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IT(A)V )(w,s) can be analytically extended to the halfplane Ims :§; 0 for almost all w. IT(A)V)OIC-1(W) = (E(A)V)(W), W = g/ Igl, can be analytically extended as a homogeneous function of degree - iA - n / 2 to the halfplane 1m gn :§; O. Thus we have a monomorphism E(A):H;lmA(A,Sn-l)~X+(-iA-nl2). 4. e. «P(t~ = ta«p(~ for ~ E IR m \ 0, t > 0. We study the problem of continuity of the maps A:Hs(lRm) ~Hp -Rea (IRm) and A:Hp(lRm,lRm -n) ~ Hp - Rea (IRm ,lR m -n). 2, =0 x(l) = (xj, ... ,xn), x(2) m n = m the space Hp(lRm,lR -n) coincides with Hp(lRn).

5. 6) realizes a continuous map Hp (JR n) -? Hp - Rea (JR n) if and only if the function Sn-I 3 ()t-+«II(fJ) is a multiplier in the Sobolev-Slobodetskii space HP(sn-l) (cf. 1). 6) to be continuous. 6. 4) cancel each other. 47 §4. •. ,xn ) ERn, Rm. Put X(2) CQ(Rm,R m - n) = (Xn+l, •.. ,Xm ) E Rm - n, = CQ (R m \{x=(x(1),x(2»: =O}). 1) = m the space Hp(Rm,R m -n) coincides with Hp(Rm). We denote by it = §"u the Fourier transform of a function u with respect to the X (2) = (Xn+l, ···,xm ) variable, and we put Z = Ix(l) 111, For n V(Z,l1) = 111ls-p-nl2it(zlll1l,l1).

3» and the line Imi\ = 0 by Imi\ = T. p. 3), the inner integral in 1+, containing "1 +, is an analytic function on the whole i\-plane and decreases faster than any power of Ii\ I in every strip I1m i\ I < N. p(w,e-l())Jo(t, w)dt. 4) IS analytic in the halfplane Imi\q+nI2). 6) (where, again, we put 55 §5. 8). 4) fall within the strip 0< ImA < T+ Rea. 5) imply that the poles disappear after an application of E(A) -1 to this integral. 12) remains valid also under the single restriction - n 12 < T + Re a.

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