2022/01/09 by Sze-Man Ngai, Yangyang Xu, Ngai, Sze-Man +1 · 1 citation
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2201.02952
openalex publication_date 2022/01/09 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
Peres and Solomyak proved that on \mathbb Rn, the limits defining the Lq-dimension for any q∈(0,∞)∖\1\, and the entropy dimension of a self-conformal measure exist, without assuming any separation condition. By introducing the notions of heavy maximal packings and partitions, we prove that on a doubling metric space the Lq-dimension, q∈(0,∞)∖\1\, is equivalent to the generalized dimension. We also generalize the result on the existence of the Lq-dimension to self-conformal measures on complete Riemannian manifolds with the doubling property. In particular, these results hold for complete Riemannian manifolds with nonnegative Ricci curvature. Moreover, by assuming that the measure is doubling, we extend the result on the existence of the entropy dimension to self-conformal measures on complete Riemannian manifolds.