Fourier integral operators with fold singularities.

are locally diffeomorphisms. In particular dx = dY-=d. Then &εΙ(Χ,Υ,<€') maps ^a.compOO into L> tloc(X) if β ^ a — μ. This was shown by H rmander s a consequence of the calculus in [7]. By composing 3F with a fractional integral operator it is easy to see that ^G/*(jr,r,<jf') maps La,comp into I£Ioc, 2 ^ < 7 < o o , if β £ αμd/2 + d/q. More general if dx ^ dy and dnL has maximal rank 2dx then the same mapping properties hold for Fourier integral operators in the class ̂ 6/ + ~(T, 7,«")·

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