Some Integrability Properties of $m$-Subharmonic Functions

Let $1\leq m<n$ and let $u$ be an $m$-subharmonic function on a domain in $\mathbb{C}^n$. 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 $L^p$ 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 $L^1_{loc}$, even on $\text{SH}_m$. Their polynomial openness conjecture also fails through an explicit power-logarithmic endpoint example.
Finally, we introduce a scale of local Hessian-capacity conditions ($C_{m,\delta}$). The volume-capacity inequality and the layer-cake formula give
$$u\in L^s_{loc}\quad\text{for every}\quad s<\frac{(m+\delta)n}{n-m}.$$
The critical member $\mathrm C_{m,0}=\mathrm C_m$ contains the finite-mass and radial cases. More generally, the energy class ${E}_{p,m}$ satisfies $\mathrm C_{m,p}$, 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.