AI 中文总结
研究\(1\leq m < n\)时\(\mathbb{C}^n\)中\(m\)-次调和函数的可积性，通过显式径向例子指出相关准则不成立，分类奇点并计算\(L^p\)区间，回答相关问题，引入局部黑塞容量条件尺度，对布洛基猜想取得进展并部分证实。
AI 中文摘要
设\(1\leq m < n\)，\(u\)是\(\mathbb{C}^n\)中一个区域上的\(m\)-次调和函数。研究了三种局部可积性形式：指数可积性、多项式可积性以及布洛基猜想预测的尖锐指数。通过显式径向例子表明，当\((m < n)\)时，直接的关 - 周强开放性陈述和基于\(m\)-勒隆数的直接斯科达准则都不成立。对一类径向幂 - 对数奇点进行了分类并计算了其精确的\(L^p\)区间，包括端点行为。回答了贝纳利 - 吉卢菲提出的两个问题。归一化球最大值极限总是等于\(m\)-勒隆数；这是通过将其球平均公式与切线的强唯一性相结合得出的。逐点可积性指数在基点处是下半连续的，但作为\(L^1_{loc}\)上的泛函不是下半连续的，即使在\(\text{SH}_m\)上也是如此。他们的多项式开放性猜想也通过一个显式的幂 - 对数端点例子不成立。最后，引入了局部黑塞容量条件\((C_{m,\delta})\)的尺度。体积 - 容量不等式和层蛋糕公式给出\(u\in L^s_{loc}\)对于每个\(s < \frac{(m + \delta)n}{n - m}\)。关键成员\(\mathrm C_{m,0}=\mathrm C_m\)包含有限质量和径向情况。更一般地，能量类\({E}_{p,m}\)满足\(\mathrm C_{m,p}\)，恢复了完整的阿哈格 - 齐兹索伯列夫指数。我们的结果朝着已经开放二十年的布洛基猜想取得了进展并部分得到了证实。
英文摘要
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 Błocki'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 $C_{m,δ}$. The volume-capacity inequality and the layer-cake formula yield $$u\in L^s_{loc}\quad\text{for every}\quad s<\frac{(m+δ)n}{n-m}.$$ 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 Åhag--Czy{ż} Sobolev exponent. These results provide partial progress toward Błocki's conjecture, which has remained open for more than two decades.
