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.
We prove the second commutation lemma for the Lebesgue spaces $L^p({\bf R}^d)$, $1<p<\infty$, on the whole unbounded domain, extending the $L^2$ theory of Tartar (1990) to the Banach-space framework of H-distributions. Unlike the $L^2$ setting, where the Plancherel isometry and Hilbert-space compactness are available, the global $L^p$ setting has neither. We control the non-local tail through Calder\'on--Zygmund kernel estimates, an explicit Taylor-remainder identity, and spatial truncation, obtaining order-$(-\epsilon)$ smoothing of the remainder into the Besov scale. Pairing weakly convergent sequences with their canonical Nemyckij duals, we then use the lemma to derive $L^p$ transport equations. For first-order scalar equations we establish the phase-space bicharacteristic (Vlasov) flow of the associated H-distribution when $p \ge 2$; for the quasilinear $p$-wave system we lift the local energy identity to the microlocal level, obtaining a Poynting-flux transport of microlocal energy in the linear core $p = 2$ and isolating the structural obstruction to its closure when $p \neq 2$. These results provide functional-analytic tools for tracking the propagation of singularities in degenerate nonlinear and non-local partial differential equations.
We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $\pi$ in its essential range, then $\exp(\widetilde f)$ is not integrable, where $\widetilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $$ \int_0^\infty Y(x) e^{-x}\, dx = \infty. $$ We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, $\widetilde f$ fails to be $Y$-integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Finally, we obtain corresponding results for complex-valued functions.
For every $n\geq 3$, we construct a nonconstant real-valued function $U\in C^\infty(\overline{\mathbb R^n_+})$, harmonic in the upper half-space, whose complete boundary jet vanishes on a compact nowhere dense subset of $\partial\mathbb R^n_+$ of positive $(n-1)$-dimensional measure. The essential construction takes place in two dimensions and yields the case $n=3$; higher-dimensional examples follow by cylindrical lifting. In every dimension, the exceptional set may occupy an arbitrarily large proportion of a fixed boundary cube. This resolves, in the negative, the smooth case of the boundary unique-continuation problem left open by Bourgain and Wolff in 1990 \cite[p.~260]{BourgainWolff1990}. In dimension three, a M\"obius--Kelvin transfer gives the corresponding counterexample in the unit ball. It disproves Nadirashvili's smooth unit-ball conjecture on boundary singular sets \cite[Conjecture~4, p.~232]{Nadirashvili1997} and, a fortiori, disproves the gradient-only formulation subsequently recorded by Logunov and Malinnikova \cite[Section~7.4]{LogunovMalinnikova2020} and by Lin \cite[Conjecture~3, pp.~15--16]{Lin2020Current}. The proof uncovers a hidden flexibility principle for nonlocal elliptic equations: microscopic modifications can exert macroscopic control over exterior data. A quantitative correction mechanism for the half-Laplacian, iterated across scales, produces flat nonlocal Cauchy data on a set of positive measure. Thus nonlocality has a striking dual character: the same long-range interaction that drives unique-continuation rigidity can also furnish the flexibility through which that rigidity fails in the smooth category.
We study the Chebyshev extremal problem on a compact Riemann surface $X$ of genus $g>0$. As an analog to monic polynomials, we consider admissible meromorphic functions having no poles away from a marked point $P_{\infty}$ (with adequate normalisation). For a nonpolar compact set $E\subset X\setminus\{P_{\infty}\}$, we show that the $n$-th root of the Chebyshev constant $t_n(E)$ converges to the capacity of $E$, and we establish the corresponding Bernstein-Walsh inequality. In the second part of the paper, using a Cauchy kernel adapted to Riemann surfaces, we study the refined Szeg\H{o}--Widom asymptotics for the extremals as well as the Widom factors $$ W_n(E)=\frac{t_n(E)}{\operatorname{cap}(E)^n},$$ assuming that $E$ is a finite union of $p$ closed discs with analytic boundaries. The geometry of the Schottky double, which has genus $2g+p-1$, enters explicitly into these asymptotics. We find that there is no analogue of Faber-type asymptotics even for one single boundary curve. We conclude by constructing explicit examples in genus $1$ using the Weierstrass-$\wp$ function.
For every $1\leq p<\infty$ and even integer $m\geq4$, we determine the optimal order of the $p$-moment torus inequality for $L_1$: it is $m^p n^{(1-p/2)_+}+n$, with comparison constants independent of $p,m,n$ after taking $p$-th roots. Consequently, for every $2\leq q<\infty$ and $1\leq p\leq q$, the corresponding metric cotype inequality holds at the sharp scale $m=O(n^{1/q})$. In particular, the quadratic inequality holds with $m=O(\sqrt{n})$, answering the $L_1$ sharp metric cotype question of Mendel and Naor. The proof combines finite cut representations with a nonlinear smoothing estimate and an exact ternary rounding identity.
Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step, and the underlying group structure of $\mathbb{G}$. Our result provides a noncommutative generalization of the dimension-free Euclidean theorem of Ouyang, Spector, and Stockdale arXiv:2608.18068, with the same universal constant. Our proof relies upon a fractional obstacle problem adapted to stratified Lie groups by using the functional calculus of $\mathcal L$ instead of the Fourier transform. As a consequence, by interpolation we obtain uniform $L^p$ bounds for the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ for $p \in (1,2]$. By contrast, for every $p>2$, we construct a sequence of stratified Lie groups with fixed horizontal dimension $5$ and steps tending to infinity for which the $L^p$ norms of the horizontal Riesz transforms diverge.
Sheng-Chen Mao, Yaojun Wang, Ye Zhang· 0 citations
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