Download e-book for iPad: Algebras of Pseudodifferential Operators by B. A. Plamenevskii (auth.)

By B. A. Plamenevskii (auth.)

ISBN-10: 9400923643

ISBN-13: 9789400923645

ISBN-10: 9401075646

ISBN-13: 9789401075640

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15) O(/x/-n-Rea-Iyl) for Rea>-n-/y/, hold. Everything said remains valid for the functions Z y(, 0) also. Proof. 5 we find that this term decreases faster than any power of / x / as x ~ 00. 9). The inner integral has a meromorphic extension onto the whole A-plane. The points A = i ( / y / + /K / +n 12), / K / = 0,1, ... ) -I aKcp(O, O)w K • As A~ 00 such that / 1m A/ < N for some N, then this extension decreases faster than any power of / A /. 3). 9) the line of integration ImA = by the line ImA = h.

Unless stated otherwise, A is considered for arbitrary functions «II. 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).

10) j where Xj E coo(lRn) and the sum is finite. 10), II(I +8)2'Y; HO(lR n )11 2 :s;; :s;; 1(1+p2)2spn-1dP '" c 1(1 + [7JIFXj'Y12d~r [JIF'Y12d~r-s:s;; p')"p" -Idp f [~:I Fx/VI' + F~l'l dol 1 26 Chapter 1. (we have used Young's inequality aSb 1-s ~ sa +(1-s)b). Hence 11(1 HY'v: HO(R")II' "" C Since Xj E Cco(lRn) and "\~r,cp) [lie" H"(R")II' + 117X/V: H"(R')II'l. = 0 for r > 312, we have 11(1 +8Y'Y; HO(lRn)11 ~ cll'\; H 2s (lR n)ll. 7) has been proved. Let us verify the second. We have J = II'Y; H 2s (lR n)11 2 ~ co 1(1 + p2)2s pn ° -I dp 1 I(I + 8)-SF(1 + 8)S(YI2d~.

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Algebras of Pseudodifferential Operators by B. A. Plamenevskii (auth.)

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