Classical Microlocal Analysis in the Space of Hyperfunctions by Seiichiro Wakabayashi

By Seiichiro Wakabayashi

The e-book develops "Classical Microlocal research" within the areas of hyperfunctions and microfunctions, which makes it attainable to use the equipment within the distribution classification to the reviews on partial differential equations within the hyperfunction type. right here "Classical Microlocal research" signifies that it doesn't use "Algebraic research. the most instrument within the textual content is, in a few experience, integration through elements. The experiences on microlocal area of expertise, analytic hypoellipticity and native solvability are lowered to the issues to derive strength estimates (or a priori estimates). the writer assumes uncomplicated realizing of concept of pseudodifferential operators within the distribution type.

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Define W(x,,Xn+X) : - W ( x , - X n + l ) for Xn+ 1 < O. 5. A P P L I C A T I O N S O F T H E R U N G E T H E O R E M 37 Let K be a compact subset of R n, and choose ¢K E C °O( R n+l \ O K × {0)) such that OK(X, xn+l) = OK(X,-x,~+l) and ¢K(X,x,~+l) = 0 I 0 if 1(~,~+1)1 >> 1, near ( K \ O K ) x {0}, near ( R n \ K ) × {0}. Let us apply the same argument as in the proof of Theorem 1 . 3 . 3 . +I)(CKW) in R n+l \ O K x {0}, and W K -- ¢ K W - - f K E COO(R'~+I\K×{O}) satisfies (1-Ax,~,+I)WK = 0 in R n+i \ K x {0), where ¢ K W and (1 - Ax,z,+~)(CKW) are regarded as functions in Coo ( R n +1 \ K x {0 )) and Coo ( R n +1 \ 0 K x { 0 }), respectively.

EA be a family of open sets in X such that X = U;~eAX;~. Assume that ~ > 0 and u~ E B~(X~) ( A E A) satisfy u~]xxnx, = u~,lxxnx, for every A,# E A. 16) Then there is a unique u E Be(X) such that uIx x = u~ for every A E A. 4. H Y P E R F U N C T I O N S 31 P r o o f Uniqueness of u E Be(X) is obvious. In order to prove the proposition it suffices to show the following: If v~ E ~-e, supp v~ C X~ and supp (v~ - v~) M (X~ n X , ) = q} for any A,# E A, then there is v • }'e satisfying supp v C X and supp (v - v~) O X~ -- 0 for any A E A.

If we put gR(~) = ~R'(~) with R' --- ~ - 1/2, gR(~) has the properties of the lemma. (~) ~-Ic'l in for I~1 + IZl > 1, and IV~,(e,z,~)t(~) -' + IV~,(e,z,~)l ¢: 0 in ~, where nj E Z+ ( j = 1,2,3), ~ is an open subset of R TM × R =z × R =s, Co(O, x, ~), Oo(e, x, ~), Ao(O, x, ~) and Bo(0, x, ~) are functions defined in ft. Put A(e, x,~) = [V=~o(O,x,~)[ ~ + (~}2[V¢~o(o, x,~)[ a. 9) wh~r~ #(O, x, () = 3Co (O, x, ~)~ (n~Bo (O, x, ~)2 + n3(Ao(O, x, ~) + 1/2) 2) (~>2. 1. P R E L I M I N A R Y LEMMAS Proof 47 Note that J ~[~(k + 1 ) ( j + 1 - k) = (j + 1)(j + 2)(j + 3)/6 < 5 .

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