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March 3, 2026Interdisciplinary Information Sciences0 citationsOpen Access

Boundary Growth Rates of Superharmonic Functions Satisfying a Nonlinear Inequality in the Unit Ball of ℂN

KHKentaro HirataHiroshima University

Key Points

  • The aim is to estimate the Hausdorff dimension of a specific set on the boundary of the unit ball where superharmonic functions behave under a nonlinear inequality.
  • Assessment of growth rates of Δ𝔹-superharmonic functions
  • Use of the Laplace–Beltrami operator associated with Bergman metrics
  • Analysis of conditions on parameters c, τ, and p
  • Determined the nonisotropic Hausdorff dimension for the specified set
  • Established relationships between parameters influencing growth rates
  • Showed that certain superharmonic functions grow faster than a prescribed order

Abstract

Let Δ𝔹 be the Laplace–Beltrami operator associated with Bergman metric on the unit ball 𝔹 of ℂN. We estimate the nonisotropic Hausdorff dimension of a set in ∂𝔹 where a positive Δ𝔹-superharmonic function u satisfying -Δ𝔹u (z) ≤c (1-|z|) τu (z) p in 𝔹 for some p∈ (0, N/ (N-1) ), τ>maxα (p−1), N (p−1) and c>0 grows faster than a prescribed order α.

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Cite This Study

Kentaro Hirata (2026) studied this question.

synapsesocial.com/papers/69a67e0ef353c071a6f0a034https://doi.org/10.4036/iis.2026.a.10
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