2026/07/23 by Genglong Lin
Mathematics · #math.CV
Let 1≤ m<n and let u be an m-subharmonic function on a domain in ℂn. We study three related forms of local integrability: exponential integrability, polynomial integrability, and the sharp exponent predicted by Błocki's conjecture. Explicit radial examples show that the direct Guan-Zhou strong-openness statement and the direct Skoda criterion in terms of the m-Lelong number both fail when (m<n). We classify a family of radial power-logarithmic singularities and compute its exact Lp intervals, including endpoint behavior. We answer both problems posed by Benali-Ghiloufi. The normalized ball-maximum limit always equals the m-Lelong number; this follows by combining their spherical-mean formula with strong uniqueness of tangents. The pointwise integrability exponent is lower semicontinuous in the base point, but it is not lower semicontinuous as a functional on L1loc, even on SHm. Their polynomial openness conjecture also fails through an explicit power-logarithmic endpoint example. Finally, we introduce a scale of local Hessian-capacity conditions (Cm,δ). The volume-capacity inequality and the layer-cake formula give u∈ Lsloc\quadfor every s<((m+δ)n)/(n-m). The critical member \mathrm Cm,0=\mathrm Cm contains the finite-mass and radial cases. More generally, the energy class Ep,m satisfies \mathrm Cm,p, recovering the full Åhag-Czyż Sobolev exponent. Our result gives an advance and a partial comfirmation towards Błocki's conjecture, which has been open for twenty years.