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D. In section I we a l r e a d y m e n t i o n e d the space CO(M) is a B a n a c h space and a Banach algebra tion so that . n Here w e use CO fR ), w h i c h (without unit), under p o i n t w i s e m u l t i p l i c a - (uv) (x) = u(x)v(x). The space CO@9 n) consists of all functions in CB~9 n) n having the limit zero, as Ixl ÷ ~. 14). 5, above. 6. " in the lower row. (i) The space S w i t h "~" is a subalgebra of L 1 (ii) the space S w i t h ". , the isomorphism the relative u ÷ u A is a contraction.

Notice that this implies c o n v e r g e n c e of x ~I~% the r e s t r i c t e d functions ~k = ~ILxX~'J to x = ~iLxX~'j. D.. 5. 23) supp u @ v = supp u × supp v The p r o o f is straight forward, and its d i s c u s s i o n is omitted. Problems: I) Show that all d i s t r i b u t i o n s of the form @(J) @ TE~' ~ 2 ) , Dirac m e a s u r e in ~, 2 = ~ w i t h 6 the and any T£~' 0R) have their support o n the line {(x,y):y=O} in x ~, w i t h the first and second c o o r d i n a t e d e n o t e d b y x and y, respectively.

F o r m u l a is not e n t i r e l y satisfactory, since it used the F o u r i e r transform. 22) In p a r t i c u l a r w e shall require the c o n v o l u t i o n p r o d u c t for a d i s t r i b u t i o n s not in S', thus not a l l o w i n g a F o u r i e r transform. 1) = < u ~ v , ~ > , ~ ( x , y ) = ( 2 ~ ) - n / 2 ~ ( x + y ) , ~EDGR n) Note that ~ as d e f i n e d is in E ~ 2n) b u t not in D~R2n). 1) is b u t not n e c e s s a r i l y for u,vsD'. O n the other h a n d the support of ~ always is c o n t a i n e d in a 'strip' Ix+y[ ~ a < ~.

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Aubry-Mather theory

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