Boundary Toeplitz type operators in weighted harmonic Sobolev spaces

Journal article, 2025

We consider the space Dnα(Ω) consisting of functions u(x), harmonic in a bounded domain Ω⊂Rd+1 with smooth boundary Γ, satisfying |∇nu(x)|2ρ(x)α∈L1(Ω), α>-1, where ρ(x) is the distance from x to the boundary. For a Borel measure μ on Γ and a weight function V, μ-measurable, we study the operator defined by means of the quadratic form μV[u]=∫V(x)|γu(x)|2μ(dx), where γ is, properly defined, operator of restriction of functions u∈Dnα(Ω) to Γ. Main interest is directed to the case of a singular measure μ possessing some Ahlfors regularity properties. For such operators, we establish two-sided estimates for singular values, and, under some geometrical conditions, Weyl type eigenvalue asymptotics.