Low energy resolvent for the Hodge Laplacian: Applications to Riesz transform, Sobolev estimates and analytic torsion

2013/10/17 by Guillarmou, Colin, Sher, David A. · 1 citation
#58J37 #58J40 #58J52 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1310.4694

Abstract

On an asymptotically conic manifold (M,g), we analyze the asymptotics of the integral kernel of the resolvent R_q(k):=(Δ_q+k²)^-1 of the Hodge Laplacian Δ_q on q-forms as the spectral parameter k approaches zero, assuming that 0 is not a resonance. The first application we give is an L^p Sobolev estimate for d+δ and Δ_q. Then we obtain a complete characterization of the range of p>1 for which the Riesz transform for q-forms T_q=(d+δ)Δ_q^-1/2 is bounded on L^p. Finally, we obtain an asymptotic formula for the analytic torsion of a family of smooth compact Riemannian manifolds (Ω_ε,g_ε) degenerating to a compact manifold (Ω₀,g₀) with a conic singularity as ε→ 0.

Cited by

• Analysis on fibred cusp spaces

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