| Abstract: |
| In this talk we discuss several notions of subharmonic functions in the context of regular Dirichlet forms. A particular emphasis is put on their regularity. We extend recent results by G\uneysu, Pigola, Stollmann and Veronelli `24 to a class of possibly non-local Dirichlet forms and put their attempt at subharmonic functions via so-called locally shift defective functions into perspective. The Bravermann, Milatovic, Shubin conjecture (BMS conjecture, recently confirmed by Pigola, Veronelli `21) asks, whether on a complete Riemannian manifold, for $f \in L^2(M)$ the distributional inequality $\Delta f \leq f$ implies $f \geq 0$. We use our regularity theory for subharmonic functions to verify the validity of this conjecture for a large class of `distributional operators` derived from regular Dirichlet forms. |
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