And Push – Forward Theorem

The Singular Asymptotics Lemma by Brüning and Seeley and the Push-Forward Theorem by Melrose lie at the very heart of their respective approaches to singular analysis. We review both and show that they deal with the same basic problem, giving solutions that emphasize different aspects of it. This also points to a possible common extension. 1. An Example As a simple example, consider f : R+ → R+, (x, y) 7→ xy. (1.1) The level lines {f = c} of f are shown in Figure 1. As c tends to 0, the corresponding level line approaches the boundary {0}×R+∪R+×{0} (where we write R+ = [0,∞)). We wish to integrate a compactly supported smooth density over the level lines of f , i.e. to compute the push-forward of u dxdy under f , for some u ∈ C c (R 2 +) (that is, u is the restriction to R+ of a compactly supported function on R ): we pair with v ∈ Cc (R+) to get 〈f∗(u dxdy), v〉 = 〈u dxdy, f v〉 = ∫∫ R 2 + u(x, y)v(xy) dxdy