Spectral asymptotics for nonsmooth singular Green operators

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Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω⊂ Rⁿ, and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order -t on a bounded domain, its eigenvalues or s-numbers have the behavior (*) sⱼ(G) c j⁻ᵗ/⁽ⁿ⁻¹⁾ for j→ ∞, governed by the boundary dimension n-1. In some nonsmooth cases, upper estimates (**) sⱼ(G) ≤ Cj⁻ᵗ/⁽ⁿ⁻¹⁾ are known. We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely Hölder continuous in x. We also show (*) with t=2 for the boundary term in the Krein resolvent formula comparing the Dirichlet and a Neumann-type problem for a strongly elliptic second-order differential operator (not necessarily selfadjoint) with coefficients in W¹ₚ(Ω) for some p>n.