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2 LOUIS NIRENBER G "reflection", o f th e wav e fron t set , assuming certai n "boundar y conditions" . Thi s resul t is part o f joint work , in progress , of P . D. La x an d th e author . We shall use rathe r standar d notation . Fo r function s u(x) define d fo r x = (x2,"\ x") in a domain i n R n , w e us e th e notatio n d j = B/dxJ, D xf = 9 X /M D = (Dj, *• • , D n ). Occasionally w e will us e subscript s i n writin g derivative s u x = du/dx, etc . Fo r a multi-inde x of nonnegative integer s a = (a v •• • , a„), Da = D®1 •• • D% n i s a differentiation o f orde r | a | = £"*,- • Th e invers e o f a ! = a1! • " a n ! wil l enter a s a coefficient i n Taylo r serie s expansions. £ = (Sj , •• • . ?„) e R n wil l play a role i n th e Fourie r transform , a s dual vari- able, and w e set { = £^ ! •• • Z* n . A linear partia l differentia l operato r i s a polynomial i n D wit h coefficient s dependin g (alway s C°° here ) on x. P(x,D) = Z a a {x)D°. \a\m The correspondin g polynomia l P(x, {• ) = 2fl a (x)fa i s called th e (full ) symbo l o f P\ th e leading homogeneous par t £| a |=m2a(*)£a denote d b y p , i s called th e leadin g symbo l of P The reade r ma y find muc h mor e materia l o n th e topic s discusse d her e i n th e bibliog - raphy. W e call attention , especially , to th e lecture s b y Hormande r [17 ] whic h describ e many o f th e curren t developments , an d als o th e Proceeding s (t o appear ) o f th e America n Mathematical Societ y Summe r Institut e o n Partial Differentia l Equations , Berkeley , Augus t 1971.