6 LOUIS NIRENBER G condition i s that a t every point , [P % , P 2 ] i s a linear combinatio n o f P { an d P 2 se e [12, §6.1] . The followin g operato r i n R 2 i s perhaps th e simplest cas e in which th e above condi- tion doe s not hold : (1.7) P=^-+ixf. ox oy Here P t = 3/dx, P2 = xd/dy, an d [P { , P2] = b/by i s a linear combinatio n o f P { an d P2 excep t o n x = 0. A number o f simple examples o f nonsolvable equation s hav e bee n published. We will now describe one , due to Grushin [11] , which i s a modification o f an example of Garabedian [10] . I t is for th e equation dw dw 0-8) + ix = f(x, y). ox oy We remark firs t tha t i f f(x, y) i s real analyti c (i.e. R e/ an d I m / ar e analytic) the n (1.8) possesses , in fact, analyti c solution s in a neighbourhood o f th e origin, as follows fro m the Cauchy-Kowalewsk i theorem . Furthermore , if / i s a ° function , the n i t follow s from ou r earlier discussion s tha t (1.8 ) is solvable o n a neighborhood o f any point no t on the j-axis. Example of nomolvability. Le t D n , n = 1 , 2, ••• , b e an arbitrary sequenc e o f closed nonoverlapping disc s in the right hal f o f the (x, y) plane , JC 0 , wit h centre s (x f1 , 0 ) xn 0 an d x n - 0. Le t f(x, y) b e an arbitrarily chose n C°° functio n wit h compac t support whic h is an even functio n o f x vanishin g inJ C 0 outsid e o f the discs D n , an d which is such tha t r \fdxdy*0 fo r n = 1 , 2, / / ' Such a function / i s easily constructed . THEOREM 1 . / / / satisfies the conditions above there is no C l solution o f (1.8) in any neighbourhood of the origin. The proof o f the theorem i s easily extende d (se e [11] ) to show tha t ther e i s no dis- tribution solutio n i n any neighbourhood o f the origin. PROOF. Suppos e w i s a solution o f (1.8) in a neighbourhood1 2 o f the origin. De - compose w = u 4 - v a s a sum of its odd and even part s in x respectively . Sinc e / i s even in x w e see that th e even part o f the equation (1.8 ) is du du (19) T x +iX Yy=f- In particula r (1.9 ) holds i n x 0, while u(0 , y) = 0. I f in the region x 0 w e introduce as new variables 5 = x2/2, s o that 9/3 s = x_1d/Bx, w e find on dividing (1.9) by x du du 1 , + / - = f{sfS.y\ fo r s 0 , u = 0 fo r s = 0.
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