The behavior of harmonic functions at singular points of $$\mathsf {RCD}$$ spaces

Philippis, Guido De; Núñez-Zimbrón, Jesús

doi:10.1007/s00229-021-01365-9

Cited by 10 publications

(3 citation statements)

References 29 publications

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“…Here a relevant technical point is that there is no topology on the 'tangent bundle' or, to put it differently, it is totally unclear what it means for a tangent vector field to be continuous or continuous at a point (in fact, not even the value of a vector field at a point is defined in our setting!). In this direction we also remark that the recent result in [7] suggests that it might be pointless to look for 'many' continuous vector fields already on finite dimensional Alexandrov spaces, thus a fortiori on RCD ones.…”

Section: Introductionmentioning

confidence: 79%

Quasi-Continuous Vector Fields on RCD Spaces

2020

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Section: Introductionmentioning

confidence: 79%

Quasi-Continuous Vector Fields on RCD Spaces

2020

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“…Example 6.2). Note that such a result is rather delicate, as on Alexandrov spaces it may happen that |∇ f | 2 * need not be continuous [49], which follows essentially from the results obtained in [22,47].…”

Section: Introductionmentioning

confidence: 88%

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

Güneysu,

Pigola,

Stollmann

et al. 2024

Math. Ann.

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Given a strongly local Dirichlet space and $$\lambda \geqslant 0$$ λ ⩾ 0 , we introduce a new notion of $$\lambda $$ λ -subharmonicity for $$L^1_\textrm{loc}$$ L loc 1 -functions, which we call local$$\lambda $$ λ -shift defectivity, and which turns out to be equivalent to distributional $$\lambda $$ 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 $$L^q$$ L q -solutions of $$\Delta f\leqslant f$$ Δ f ⩽ f for complete Riemannian manifolds.

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“…Example 6.2). Note that such a result is rather delicate, as on Alexandrov spaces it may happen that |∇f | 2 ˚need not be continuous [47], which follows essentially from the results obtained in [21,48]. The regularity of harmonic and subharmonic functions is a central theme in analysis and beyond, ever since Weyl's seminal work [61].…”

Section: Introductionmentioning

confidence: 95%

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

Güneysu¹,

Pigola²,

Stollmann³

et al. 2023

Preprint

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Given a strongly local Dirichlet space and λ ě 0, we introduce a new notion of λ-subharmonicity for L 1 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 ∆f ď f for complete Riemannian manifolds.

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The behavior of harmonic functions at singular points of $$\mathsf {RCD}$$ spaces

Cited by 10 publications

References 29 publications

Quasi-Continuous Vector Fields on RCD Spaces

Quasi-Continuous Vector Fields on RCD Spaces

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

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

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