Manuscript XI-F@2 · XI-F-v2.tex · lemma · f4899be0

Lemma: Inverse of A on compact plateau sources

Copied sheet › Conventions · lines 438–450 · proof included or linked

XI-F@2#lem:GA, XI-F@2#eq:GA

For $s$ smooth, compactly supported in $U$ ,

G_{gr}^{R} A s = s - 2 R E_{0}^{R} D s .

Proof. $G_{gr}^{R} A s = (- P E_{0}^{R}) (- □ P s) + (- P E_{0}^{R}) (- 2 D^{*} D s) = PP s + 2 P D^{*} E_{0}^{R} D s = s - 2 R E_{0}^{R} D s$ , using (eq:algebra), $E_{0}^{R} □ s = s$ , and $E_{0}^{R} D^{*} = D^{*} E_{0}^{R}$ on $D s$ , which has zero data.

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XI-F@2#eq:GA · line 441
$G_{gr}^{R} A s = s - 2 R E_{0}^{R} D s .$

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section: Conventions