Consider on . Notify has a corner at the origin and is not smooth at the origin. when and when . We want to resolve by blowing up the origin through a map . After blowing up, looks like the following picture.

Denote and is the blow down map. On (near A), takes the form

where is the boundary defining function for ff and is boundary defining function for rb. Similarly on (near B), takes the form

where is a bdf for lb and is a bdf for ff. One can verify that is a diffeomorphism . Let . Then is a polyhomogeneous conormal function on . Its index can be denoted correspond to lb, ff and rb.

Suppose is the projection to coordinate. Consider , is actually a fibration. Then the push forward map maps to a polyhomogeneous function on .

In order to make is integrable, let us assume support and . What is the index for on ?

For the first integral, letting

For the second integral, letting

Since the Taylor series

Consequently

Combining all the above analysis, the index for is

From another point of view, the vanishing order of on each boundary hypersurface of are , and . maps lb and ff to the boundary of . Therefore the index of is contained in

**Remark:** Daniel Grieser, Basics of Calculus.

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