Get Algebras of Pseudodifferential Operators PDF

By B. A. Plamenevskii (auth.)

ISBN-10: 9400923643

ISBN-13: 9789400923645

ISBN-10: 9401075646

ISBN-13: 9789401075640

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Extra resources for Algebras of Pseudodifferential Operators

Example text

R 2u) = - 2 (r 2u)+---a (r 2u)-ou, ar r r it remains to convince ourselves that For this we must integrate by parts the two last terms . • Let 1) E C""(IR), 1) = 1 on the (closed) interval [3/4, 514] and 1) = 0 outside the interval [112, 3/2]. For v E c",,(sn-l), put C'\{rcp) = 1)(r)v(cp). It is easy to check that §5. The spaces H S CA, Sn -1). 6) with constants c1,c2 not depending on v. 3. For 0 < s < 1 the estimates hold. Proof. 2, 00 11(1 +8r'"'V; HO(lR n)11 2 = J pn -1dp J I(I +8rF'Y12d~. F'Y12d~:S;; [J I(I +8)F'Y12d~r [J IF'Y12d~r-s.

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). ::jk = T(A)-Ijk. we have U = T(A)-Ij = limT(A)-ljk E H"±lmA(A,sn-I) .

Put s = [s]+t. 3, c l ll(I+o)[sl'Y;H 21 (lR n )11 ,;;;; ,,;;:; 11(1+~Y,\;HO(Rn)11 ';;;;c211(I+~)[slcV;H2t(Rn)ll. 19). The case s < 0 is obtained by transition to the dual spaces with respect to duality in HO(sn -I). • 3. The operator E(A) on the spaces HS(A,sn-I). 20) is equivalent to the usual norm in the Sobolev-Slobodetskii space Hs(sn-I). 5. 2) (or A =1= -i(k +nI2)), where k=O,l,···, then the map E(A):Hs(A,sn-I)~Hs+ImA(A,sn-l) (resp. E(A)-I:Hs(A,sn-I)~Hs-ImA (A,sn-I)) is continuous. 2)), k = 0, I, ...

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


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