Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · lemma · f4bd2b11

Lemma: Wavefront of the boundary square root

The diffracted coefficient and its nonvanishing criterion · lines 1076–1101 · proof included or linked

I-B-VP-2@1#lem:sqrtWF

At $s = 0$ ,

WF ((s + i 0)^{- 1/2}) = {(0, λ) : λ > 0},

and the lower boundary value has $λ < 0$ . Multiplication by a smooth coefficient nonzero on $s = 0$ preserves that conormal wavefront.

Proof. The representation

(s + i 0)^{- 1/2} = \frac{e ^{- i π /4}}{π} \int_{0}^{\infty} e^{i λ s} λ^{- 1/2} d λ

is understood as a damped oscillatory integral. It gives the one-sided wavefront and an elliptic conormal amplitude at every positive $λ$ . The lower boundary value is obtained by conjugating the scalar model. Smooth elliptic multiplication preserves the wavefront HormanderI.

Displayed formulas in this passage

unlabeled · line 1079
$WF ((s + i 0)^{- 1/2}) = {(0, λ) : λ > 0},$
unlabeled · line 1090
$(s + i 0)^{- 1/2} = \frac{e ^{- i π /4}}{π} \int_{0}^{\infty} e^{i λ s} λ^{- 1/2} d λ$

Lemma: Wavefront of the boundary square root

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