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The theory of energy for sub-Laplacians with an application to quasi-continuity. (English) Zbl 1131.35006
Summary: In this paper, we provide a suitable theory for the energy \[ \int\int \Gamma(x,y)\,d\mu(x)\,d\mu(y), \] where \(\mu\) is a Radon measure and \(\Gamma\) is the fundamental solution of a sub-Laplacian \(\Delta_{\mathbb Q}\) on a stratified group \(\mathbb G\). As a significant application, we prove the quasi-continuity of superharmonic functions related to \(\Delta_{\mathbb G}\). The proofs are elementary and mostly rely on the use of appropriate mean-value formulas and mean-integral operators relevant to the Potential Theory for \(\Delta_{\mathbb G}\).

MSC:
35H20 Subelliptic equations
35A08 Fundamental solutions to PDEs
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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