This note verifies a conjecture of Král, that a continuously differentiable function, which is subharmonic outside its critical set, is subharmonic everywhere.

2000 Mathematics Subject Classification: Primary 26A33; Secondary 47G20, 31B05We study a singular value problem and the boundary Harnack principle for the fractional Laplacian on the exterior of the unit ball.

Let ũ denote the conjugate Poisson integral of a function $f\in L^p\left(ℝ\right)$. We give conditions on a region Ω so that $li{m}_{(v,\epsilon )\to (0,0)}(v,\epsilon )\in \Omega ũ(x+v,\epsilon )=Hf\left(x\right)$, the Hilbert transform of f at x, for a.e. x. We also consider more general Calderón-Zygmund singular integrals and give conditions on a set Ω so that $su{p}_{(v,r)\in \Omega }\left|{ʃ}_{\left|t\right|>r}k(x+v-t)f\left(t\right)dt\right|$ is a bounded operator on $L^p$, 1 < p < ∞, and is weak (1,1).

Let $A$ be an elliptic differential operator of second order with variable coefficients. In this paper it is proved that any $A$-superharmonic function in the Riesz-Brelot sense is locally summable and satisfies the $A$-superharmonicity in the sense of Schwartz distribution.

The classical Mittag-Leffler theorem on meromorphic functions is extended to the case of functions and hyperfunctions belonging to the kernels of linear partial differential operators with constant coefficients.

This paper shows that some characterizations of the harmonic majorization of the Martin function for domains having smooth boundaries also hold for cones.

The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator $W_k:=\Delta +\frac{k}{x_n}\frac{\partial }{\partial x_n}$ on ${ℝ}^n$ is proved. In this note there is shown that in the cases $k\ne 0$, $k\ne 2$ no other transforms of this kind exist and for case $k=2$, all such transforms are described.

We construct bounded domains D not equal to a ball in n ≥ 3 dimensional Euclidean space, Rn, for which ∂D is homeomorphic to a sphere under a quasiconformal mapping of Rn and such that n - 1 dimensional Hausdorff measure equals harmonic measure on ∂D.

The main result of the present paper is : every separately-subharmonic function $u(x,y)$, which is harmonic in $y$, can be represented locally as a sum two functions, $u=u^{*}+U$, where $U$ is subharmonic and $u^{*}$ is harmonic in $y$ , subharmonic in $x$ and harmonic in $(x,y)$ outside of some nowhere dense set $S$.

We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition$${\int }_1^{\infty }\frac{P\left(t\right)}{t^2}\mathrm{d}t=\infty .$$∫ 1 ∞ P ( t ) t 2   d t = ∞ .

We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition ${\mathrm{\int }}_{\mathrm{1}}^{\mathrm{\infty }}\frac{\mathit{P}\mathrm{\left(}\mathit{t}\mathrm{\right)}}{{\mathit{t}}^{\mathrm{2}}}\hspace{0.17em}\mathrm{d}\mathit{t}\mathrm{=}\mathrm{\infty }\mathit{.}$