Sharp L p Bounds on Spectral Clusters for Lipschitz Metrics

Abstract

Abstract. We establish L p bounds on L 2 normalized spectral clusters for selfadjoint elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we obtain best possible bounds for all 2 ≤ p ≤ ∞, up to logarithmic losses for 6 < p ≤ 8. In higher dimensions we obtain best possible bounds for a limited range of p.

From spectral cluster to uniform resolvent estimates on compact manifolds

“…The relevant issue is whether one has perfect estimates for the Sobolev exponent q = 2n/(n−2). The results in [27] only yield perfect estimates for larger exponents q > (6n − 2)/(n − 1). It is conjectured that perfect estimates should hold for q ≥ 2(n + 2)/(n − 1).…”

“…This accounts for the resolvent estimate in (ii). The higher dimensional bounds of [27] do not yield uniform resolvent bounds. The relevant issue is whether one has perfect estimates for the Sobolev exponent q = 2n/(n−2).…”

“…Remark 2.14. For Lipschitz metrics, Koch, Smith and Tataru [27] proved sharp spectral cluster estimates in two dimensions. Perfect estimates hold for q > 8 (with a logarithmic loss at the endpoint q = 8).…”

“…Most modern proofs of spectral cluster and resolvent estimates rely on wave equation techniques. One popular approach is via square function estimates for the wave equation [34,39,2,9,40], another one is via Strichartz estimates for a half wave equation [28,7,31,27]. The latter is purely stationary, i.e.…”

Lqbounds on restrictions of spectral clusters to submanifolds for low regularity metrics

“…He also proved that the bound (1.1) holds when q = ∞. By interpolation, this shows (1.1) with a loss of σ/q derivatives when 2(n+1) n−1 ≤ q ≤ ∞, though a subsequent work of Koch-Smith-Tataru [12] improves upon this.…”