By B. A. Plamenevskii (auth.)

ISBN-10: 9400923643

ISBN-13: 9789400923645

ISBN-10: 9401075646

ISBN-13: 9789401075640

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**Sample text**

Under F= F fi-s' F- 1, which are defined on the additional restriction /3 - s =1= - (k + n 12) the operators F and F- 1 are inverse to each other. 9) for I y I = 0, ... 9) can be omitted). The closure of this set of functions coincides with the subspace Hp,k(R n), and the map F:Hp,k(R n) -,> Hf,k(R n) is an iso- 2) Let /3-s morphism. The operators F and F fi-s coincide on Hfi,k(R n). All assertions remain valid if /3 and s are interchanged and F- 1 is written instead ofF. Proof. 3). On 6j)(F) (resp.

F wY+"d W X sn-I +00 X As x ~ e(n+I"I+lyl)t

Another partition of unity and another, equivalent, atlas lead to an equivalent norm in HS(''Jrl). For s 1 < s 2 the space H S2 (~) compactly belongs to H S1 (~). The spaces H S(~) and H-S(~) are dual with respect to the scalar product in HO(~). 2. Expansion in spherical hannonics. In the sequel we conveniently denote the spherical harmonics of order m (m =0,1 ... ) by Ymk . The index k enumerates the harmonics of the same order, k = 1, ... e. k m = O(m n - 2 ) as m --i>00. The functions Y mk form a complete orthonormal system in HO(sn-l) = L 2 (sn-I).

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

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