By B. A. Plamenevskii (auth.)

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Additional info for Algebras of Pseudodifferential Operators

Example text

2), we are led to the inequality IIF,B-su ;Hf(Rn)11 ~ cllu ;Hfi(Rn)ll. 7) has been proved; continuity of the second is verified similarly. 2) The restriction of E(A) (resp. E(A)-I) to the lineal does not have poles on the line ImA = k +n12 (resp. 3). Therefore we can reason similarly to the proof of assertion et et 1). • Let Mp be the set of functions v from Co (Rn \ 0) subject to the condition v(i(n +q),q,) Pro p 0 0, q = 0, ... ,p. 3. The set Mp is dense in Hfi (Rn) for any nonnegative integer p, and for /3-s *- k +ni2, k = 0,1, ....

An analog of the Paley-Wiener theorem for the operator E(Ar) Put S"t-- I = {x=(x',xn) E IR n : Ix I = 1, Xn >O}, S~-I = {x=(x',xn) E II" n: Ix I = 1, Xn < O}. ) establishes an isomorphism between the set of functions with support in the halfsphere S"t--I (or in S~ - I) and the set of homogeneous functions of degree - iA - n 12 having an analytic extension in the last coordinate to the lower (resp. upper) complex halfplane. First we will prove several helpful propositions. Iwl = 1, w'¥=O. ) This paragraph is used in Chapter 6 only.

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. 6), in which T = 0, T' = T . 3. Let T must be replaced by . Proof. 1 implies that the integral f +00 eiN(etw,O)Jo(t,w)dt -00 is an analytic function in the halfplane ImA