By B. A. Plamenevskii (auth.)

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**Sample text**

U} is a finite open cover of ~ and X: U --i> IR n are coordinate maps. Let {f} be a partition of unity subordinate to this cover. 2) ° where ~xu = ~UOX-I on X(U), and ~xu = outside X(U). 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 ...

M ,k Umk 12 holds. ' n-j aona-j. 2)), ql = 1, qj = (sin On -I ... sin On _ j + 1)2. The Laplace operator t:. 3) On HO(sn -I), the operator 8 is nonnegative and has a selfadjoint Friedrichs extension, which we will also denote by 8. The spectrum of 8 consists of the eigenvalues Am = m (m + n - 2), m = 0, 1, .... To each Am correspond the k m orthonormal eigenfunctions Ymk. The domain of definition of a power (I + 8Y, s > 0, 24 Chapter 1. 4) m,k converges. 4) converges belong to the domain of definition of (I + oy also for s < O.

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. From this we obtain that the second term also decreases rapidly as x ~ 00. 15). Using the fact that the poles of E(A)-I are located at the points A = -i(k +nI2), k = 0,1, ... 9) we find that this term is 0(1). Consider the first term. The inner integral extends as a meromorphic func- tion onto the whole A-plane, and only the points A = i (/ y / + a - / a/ + n 12) (are first order) poles (in order to see this we must expand CP(w, e -/0) by Taylor's formula at (w,O».