Liouville-Riemann-Roch Theorems on Abelian Coverings by Minh Kha & Peter Kuchment

Author:Minh Kha & Peter Kuchment
Language: eng
Format: epub
ISBN: 9783030674281
Publisher: Springer International Publishing

(b)

Under the assumption of Theorem 2.​8, the Liouville-Riemann-Roch inequality (2.​2) holds for any N ≥ 0 if and only if p ≥ 2. Indeed, suppose that d ≥ 5 and p < 2, then (2 − d)p ≥−d and therefore,

(3.25)

This implies that L p(μ, −Δ, 0) = {0}, where μ is the same point divisor mentioned in the previous remark. So (2.​2) fails, since,

(3.26)

3.4.1 Proof of Proposition 2.​9

We evoke the operators and and their corresponding domains from the proof of Theorem 2.​2. Now we recall from our discussion in Sect. 1.​8 the notations of the operators

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