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### Contents

The Wave Front Set of DA p0 | 8 |

The Solution Operator | 17 |

The Upper and Lower Bounds for DAp0p | 26 |

1 other sections not shown

### Common terms and phrases

2n+l less assume bicharacteristic through y,s bijection closed conic subset COEFFICIENT DETERMINATION PROBLEM coloumn compact set compact subset construct the Wave continuous map Cornell University Corollary cross the plane DA(pQ)p defined Department of Mathematics df G diffeomorphism dimensional manifold distribution kernel Duhamel's Principle eikonal equation evaluated exists a tf Fourier Integral Operators G_(p Hamilton-Jacobi theory Hence the map hence the rank homogeneous canonical relation homogeneous of degree implies impulse response inverse Lemma lies lower bounds map DA(pQ map Pr non-degenerate null bicharacteristic operator of order parametrised phase function pQ)p pQ{x principal symbol Rakesh rank 2n+l rank n+1 rays starting real number reflected bicharacteristic Rice University Rn+1 smooth functions smooth map smoothing operator solution operator Stephen's College sum of Fourier supp Suppose upper and lower upto a smoothing vector wave front set wave speed WF(f WF(uQ WF{f xERn