Some Integrability Properties of $m$-Subharmonic Functions

Abstract

Let $1\le m<n$ and let $u$ be an $m$-subharmonic function on a domain in $\mathbb{C}^n$. We study local exponential and polynomial integrability, with particular attention to the sharp polynomial exponent predicted by B{\l}ocki's conjecture. Explicit radial examples show that direct analogues of the Guan--Zhou strong openness theorem and Skoda's integrability criterion formulated in terms of the $m$-Lelong number fail when $m<n$. We classify a family of radial power-logarithmic singularities and determine the exact $L^p$-integrability range for each member, including endpoint behavior. We resolve two problems posed by Benali--Ghiloufi. The normalized limit of the ball maximum always equals the $m$-Lelong number; this follows by combining their spherical-mean formula with the strong uniqueness theorem for tangents. The pointwise integrability exponent is lower semicontinuous in the base point. However, even when restricted to $SH_m$, it is not lower semicontinuous with respect to the $L^1_{\loc}$ topology. We also disprove their polynomial openness conjecture using an explicit power-logarithmic endpoint example. Finally, we introduce a scale of local Hessian-capacity conditions, denoted by $C_{m,\delta}$. The volume-capacity inequality and the layer-cake formula yield $$u\in L^s_{loc}\quad\text{for every}\quad s<\frac{(m+\delta)n}{n-m}.$$ The critical condition $\mathrm C_{m,0}=\mathrm C_m$ holds for negative functions of finite total Hessian mass with relatively compact deep sublevel sets, and for radial functions. More generally, functions in the energy class $\mathcal E_{p,m}$ satisfy $\mathrm C_{m,p}$, recovering the full {\AA}hag--Czy{\.z} Sobolev exponent. These results provide partial progress toward B{\l}ocki's conjecture, which has remained open for more than two decades.