By Marchenko V. M.

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**Example text**

H the argument is repeated k times we find that a can be replaced by a symbol of order m+k(fJ-e)- tribution u-I~(au) is a O~ function. 00. k_ oo • so the dis- Proof of the lemma. pply a. partition of unity on the sphere bundle in rcxxRN. extended to a system of homo- 36 FOURIER INTEGRAL OPERATORS. I 93 geneous functions of degree 0, to obtain a global solution. If (xo' ( 0 ) Er, 00 =1=0, the existence of a solution in a conic neighborhood is obvious unless (xo, Oo)EG as we shall now assume. The functions 1' ...

This is clear for the first term since L 1 ... LkaE Sg"-kQ• Since Lj+1 ... £m-\ the inductive hypothesis shows that the terms in the sum are in sm-l+m'-(k-})0-(j-1)0 C S:;'+ m'-kO, which proves the assertion. 4) and [Lj, b] E sm-1, so the same proof by induction ean be applied. It follows in particular that if F' is the SO module generated by F and S-t, then F'S;' = FS;', We can therefore always assume without restriction that F is a module containing S-1. If LjE F it follows then that the principal part LYEF, hence [Lj,b]=LYbeS;'-o if bEFS;'.

This is a module over the ring SO(XXRN). 6. By iteration it follows that Ll ... LkaE~-ke(x x RN) if Ll , ... l'. Conversely, if we just know that L l ... L~ E s;:'-ke (X x RN) for all L l , ... l', taking these operatol'H to be differentiations with respect to x, and 0, variables we conclude that aE~(X x RN). This corinects our earlier definitions with the following one. 1. l' we denote by FB';(X x RN) the set of all aE S~(X x RN) such that for arbitrary Lr, ... , LkEF, k= I, 2, ... we have 38 95 FOURIER INTEGRAL OPERATORS.

### A Brief Review of the Development of Qualitative Control Theory in Belarus by Marchenko V. M.

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