The global kernel bound omits a bulk contribution
Where the issue occurs
Section 3.6, proof of Lemma 3.9, paragraph estimating the second circle kernel · p. 40
Its integral in one variable is at most C/d(s).
The issue is the unqualified global integral bound; the endpoint estimate alone does not include the bounded bulk contribution.
Issue & evidence
The unqualified C/d(s) integral estimate in the quoted sentence is too strong for the full second kernel R(s,v)=θ′(s)θ′(v)/sin²((θ(s)+θ(v))/2). Its integral over v∈[−A,A] equals 4θ′(s)/sinθ(s). At s=0 this tends to 4πvarrho(0)>0, whereas C/d(0) tends to zero. The endpoint estimate is valid locally, but a global estimate needs a bounded bulk contribution.
Proposed repair by OSVerify
The exact integral 4θ′(s)/sinθ(s), together with θ′=πλ_L and sinθ(s) comparable to λ_L(s)min(1,d(s)), gives the corrected global bound C(1+1/d(s)). By symmetry of R, this yields ∫∫R(s,v)|Q(s)+Q(v)|²dsdv≤C(‖Q‖₂²+∫_{d(s)≤2}|Q(s)|²/d(s)ds). Thus the existing L² term absorbs the bulk contribution, preserving the upper comparison displayed earlier in the proof of Lemma 3.9 and its use in the subsequent gap and counting argument.
Related context
Background Section 3.6, same proof, definition of d(s) and the two circle kernels
Identifies d(s) = A − |s| and the full second kernel to which the sentence refers.
Background Section 3.5, definitions of λ_L and θ; Section 3.1, density varrho
Supplies the change of variables and positive bulk density used to check the integral.
Downstream use Section 3.6, upper comparison and subsequent gap/counting argument
The existing L² term can absorb the missing bulk contribution, preserving the comparison and its applications.