Kernel and capacity estimates in Dirichlet spaces

2019 
Abstract We study the capacity in the sense of Beurling–Deny associated with the Dirichlet space D ( μ ) where μ is a finite positive Borel measure on the unit circle. First, we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of D ( μ ) . It allows us to give an estimate of the capacity of points and arcs of the unit circle. We also provide new conditions on closed sets to be polar. Our method is based on sharp estimates of norms of some outer functions, which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted Sobolev space.
    • Correction
    • Source
    • Cite
    • Save
    • Machine Reading By IdeaReader
    18
    References
    3
    Citations
    NaN
    KQI
    []