Almost-periodic Functions in Abstract Spaces (Research Notes by S. Zaidman

By S. Zaidman

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D) L is an elliptic operator, with Coo-coefficients. That is, we have O. 3) implies cplx=0 assuming T real-valued. Real characteristics do not exist. oxn . , e) L = P(D) is a constant coefficient hyperbolic operator (cf. ), with respect to some given real vector h# 0. 9. 5. ,n . The characteristic equation is of the form PN((plx)=0. ,0,1), as always may be achieved by a linear transformation of independent variables. 15) will give its own family of characteristic surfaces and bicharacteristics.

3). 3), with PIx'O. 3), defined in B={Ix-x°l0 . 3) on the surface t only. It is clear from the above that a characteristic surface r is fibered by characteristic integral strips: There is precisely of 42 0. Introductory discussions one strip through each point x E r . 1) if r is given by cp(x)=0 with g(x)=0 . , , aN(x,cplx(x))=0 on r , as above, and in addition aN1,(x,T,x(x))#0 on r . 3) near r . Indeed, we may assume that (x°,c)lx(x°))#0, and then solaNI ve aN(x,T;)=0 for ' .

7). 9) is replaced by elx . 1 follows. 4. 13) Ka (supp v x supp w)fl{jx+yIsa} is compact for every a>0 . Here supp v and supp w are regarded as subsets of in of the variables x and y, respectively (cf. [Schwl] or [C1], 1,8). 13) holds under these assumptions. Moreover, P(D)v=g follows, leading to a solution of the free space problem, and the related initial-boundary problems. , , Now let us attempt a detailed construction of the proper e. , the potential equation Au=f. For n2:3 the function -IX is Lpol, hence a distribution in S'.

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