By R. Rüster
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Shilov was the first who began to study the spaces of ultradistribution and who applied it to the Cauchy problem for partial differential equations with constant coefficients. ) E rb and , sufficiently large. 47) holds when is sufficiently large. '&;/12 =] o 19(xo's") 12 exp 2xocllf" 11/ 3 dxO d s" . 48) Since g( x O' t;") decreases exponentially in sf! for Xo > 0 its inverse Fourier transform g( xo,x") belongs to a Gevrey class for ~O > O. Denote by G the inverse Fourier transform of the space G.
27) is given in [7 ]. §3. 1) is the pseudodifferential operator with the symbol To obtaAi~ (211) we use that Re(l- i)(N+ AI) = Nl + Al + N2 ~ Nl + N2 ? cis IK , where Nl = Re N, N2 =ImN • f Let ~(x' ,S') be a C~-function for S' 0 which is homogeneous in S' of order O. Since the symbol of the commutator of N + A and ~ has order K(s' ,'q we obtain that WF(V)CWF(h). when Therefore it remains to describe the wave front set of Al(x' ,s") is not a nonnegative function. v We shall describe the wave front set of the solution u(x' ,x ) assuming that the wave front set of WF(h) is given.
Where We assume that h(x') is given distribution on ofl and hex') = 0 for Xo < O. 3) is well-posed. The initial-boundary problem is called well19 H. G. ), Singularities in Boundary Value Problems, 19-54. Copyright @ 1981 by D. Reidel Publishing Company. ESKIN 20 posed if for any h E 8'(on) there exists a unique solution u E 8' ( n). We note that the boundary ost is noncharacteristic with respect to the operator A(x,D); therefore the restriction B(x,D)u1an exists for any distribution such that A(x,D)u = 0, since A(x,D) is partially hypoelliptic with respect to the normal to on.