By Seiichiro Wakabayashi

The booklet develops "Classical Microlocal research" within the areas of hyperfunctions and microfunctions, which makes it attainable to use the tools within the distribution class to the experiences on partial differential equations within the hyperfunction type. the following "Classical Microlocal research" signifies that it doesn't use "Algebraic research. the most device within the textual content is, in a few feel, integration by way of elements. The reports on microlocal area of expertise, analytic hypoellipticity and native solvability are diminished to the issues to derive power estimates (or a priori estimates). the writer assumes easy figuring out of idea of pseudodifferential operators within the distribution classification.

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1 with F0 = K~ x R and F1 = K1 x {0}, we can choose ¢1 E C ° ° ( R ~+1 \ ( g l M/(2) x {0}) so t h a t ¢1(X, Xn-}-l) = 0 0 1 if I(x,x=+l)l > 1, near K2 x R \ K1 x {0), near (/~'1 \/4"2) X {0}. Then ¢1U ( resp. (1-(~I)U) can be regarded as a function in C°° (~'~oU~~l) ( resp. C°°(12o U f~2)), where 121 = R TM \ ( ( g l r3 K2) x {0) u supp ¢1) and ~2 = R TM \ ( ( g l M K2) x {0) U supp (1 - ¢1)). Moreover, (1 Ax,x,+~)(¢lU) can be regarded as a function in C°°(120 U ~1 U ~2) and satisfies (1-Ax,xn+l)(~)lV) :-0 in f~l U f~2.

We shall prove this fact ( analytic pseudolocality) for general analytic pseudodifferential operators in Theorem 2 . 6 . 5 . Now assume t h a t v is analytic in X. 15), we can see t h a t V(x,x,~+l) can be extended to a real analytic function in a neighborhood of X × [0, oc). This proves the assertion (ii). (iii) By assumption U(x, Xn+l) is real analytic in X~ x R and satisfies ( 1 - A,,,,,+l)U(x,x,,+l ) = 0. 4 it follows that U(x,x,~+l) can be continued analytically to {(z,z~+l) E C~+l; IRe z - x I + I(Im z, im Zn+i)l < (f for some x E X}.

16) is Cl~l+l,lal,l~,l,l=f I ~ Cl~l,lsl,l~,l-i-l,l~ I - ~'<~ • iZ~l,lal+l,lZ,l,151 x(2Bo)h-h~+~, 3~<3 [] t h e n I z'z'~'~< 1. This proves the l e m m a . (~) 1-1~1 1, i/I~l + 13t _> 1 in ~, where ~ is an open subset of R n × R "~', ¢(x,~), Co(x,~), Ao(x,~) (~) and Bo(x,~) are function defined in ~ and A(z)(x,~ ) = O~D~h(x,~). Define ~,~A(x,~) ~(~,) ~o~(A; x , ~ ) : = ,~-A(~,O ,. t'J(Z)" CHAPTER 2. (~'>- H - M ~ P(X,~)k<~)k/k! 17) k=0 in n if p(x,~) > O, X(x,~) >_ 2 and Al(x,~) >_X(x,~)Ao(x,~), Bt(x,~) >_X(x,~)Bo(x,~), A2(x,~) >_ 3d(x,~)Ao(x,~), Sz(x,~) >_ 3d(x,~)Bo(x,~), d(x, ~) = max{X (x, ~), ~(x, ~)/p(x, ~), 16Co(x, ~)/(p(x, ~)X(x, c))}.

### Classical Microlocal Analysis in the Space of Hyperfunctions by Seiichiro Wakabayashi

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