A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin

2023/02/18 by Batu Güneysu, Güneysu, Batu, Stefano Pigola +5
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2302.09423

openalex publication_date 2023/02/18 · openalex created_date 2023/02/22 · openalex updated_date 2026/07/28

Abstract

Given a strongly local Dirichlet space and λ≥ 0, we introduce a new notion of λ--subharmonicity for L¹_\loc--functions, which we call local λ--shift defectivity, and which turns out to be equivalent to distributional λ--subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional L^q-solutions of Δf≤ f for complete Riemannian manifolds.

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