# A Canonical Form of Vector Control Systems by Korovin S. K., Il’in A. V., Fomichev V. V. By Korovin S. K., Il’in A. V., Fomichev V. V.

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A tempered distribution u belongs to Hs(Rn) if its Fourier transform u is a measurable function such that \H2H- = fwi)\20- + \Z\2)'<% is finite. Pseudodifferential operators have simple mapping properties on L2-Sobolev spaces. The following theorem summarizes the basic properties of the KohnNirenberg class of pseudodifferential operators. 1. Suppose that A and B are Kohn-Nirenberg pseudodifferential operators. (a) [Symbol filtered algebra] If A e *™(Rn) and Be ^(W1), then AoB e tf£J+m'(Rn). (b) [Composition formula for principal symbols] Let o-m(A),o~mi(B) be the principal symbols of A and B then the principal symbol of the composite is given by the pointwise product:

If CM is everywhere non-degenerate, then M is strictly pseudoconvex. 39 Exercise 7. Let B\ denote the unit ball in C n and let U be a neighborhood of a point on the boundary. Suppose that :U C\B\ —> C" is an injective, holomorphic map, smooth up to U n B~[. If (j)(U n S2n~l) C 5 2 " - 1 then

The five-periodicity of, say, Ai follows from the calculation xi+1 = Yi- 4>h{Xi) = -Xi-! - 4>h{xt) Therefore, \-Xi) = -Xi+1 - X^ Taking into account (23), we have 4> (Xi) = Xi — Xi+i — xt_i 18 Now we can use these identities to transform A^: Ai+5 = A+4 = Bi+3 - H(-Xi+3) = Ei+2 - 4>h(-Xi+3) = = Ci+1 + 4>h(Xi+1) - h(-Xi+3) = Ai + ^(Xi) + cj>h(Xi+l) - cj>h{-Xi+3) = = Ai + (Xi — Xi-i — Xi+\) + (Xi+i - Xi — Xi+2) + (Xi+4 + Xi+2) = = Ai + Xj + 4 — Xi-i = Ai. We have shown that the five-periodicity of Ai (and therefore the one of Bi, Ci, Di, Ei, and Yi) follows form the five-periodicity of Xi.