The p -Harmonic Measure of Small Axially Symmetric Sets

Using the first eigenvalue/eigenvector pair of a singular eigenvalue problem (motivated by the Dirichlet eigenvalue problem for the Laplace-Beltrami operator on a spherical cap), we define certain nonnegative p -superharmonic and p -subharmonic functions on a convex cone which are singular at the vertex and vanish on the rest of the boundary. We use these functions to give upper and lower estimates of the p -harmonic measure near the vertex of the cone as well as the p -harmonic measure of a small spherical cap.