This paper studies a family of Hilbert spaces of real harmonic functions on bounded regions in Rn and will show that, for a range of values of s, they are reproducing kernel Hilbert spaces. The spaces are characterized by their boundary traces and the inner products are defined via their expansions in the harmonic Steklov eigenfunctions of the region. The reproducing kernels will then be descri...

In this paper we explore the metric-measure spaces introduced by Laakso in 2000. Building upon the work of Barlow and Evans we are able to show the existence of a large supply of Dirichlet forms, or alternatively Markov Processes, on these spaces. The construction of Barlow and Evans allows us to justify the use of a quantum graph perspective to identify and describe a Laplacian operator genera...

For a finite positive Borel measure μ on the unit circle, let D(μ) be associated harmonically weighted Dirichlet space. A shift invariant subspace M recognizes strong approximate spectral cosynthesis if there exists sequence of subspaces M_k, with codimension, such that orthogonal projections onto M_k converge in operator topology to projection M. If is sum atoms, then we show admits cosynthesis.

Abstract This paper aims at proving the local boundedness and continuity of solutions heat equation in context Dirichlet spaces under some rather weak additional assumptions. We consider symmetric regular forms, which satisfy mild assumptions concerning (1) existence cut‐off functions, (2) a ultracontractivity hypothesis, (3) off‐diagonal upper bound. In this setting, equation, their time deriv...