Martin boundary of a fine domain and a Fatou-Naim-Doob theorem for finely superharmonic functions

We construct the Martin compactification ${\bar U}$ of a fine domain $U$ in $R^n$, $n\ge 2$, and the Riesz-Martin kernel $K$ on $U \times{\bar U}$. We obtain the integral representation of finely superharmonic fonctions $\ge 0$ on $U$ in terms of $K$ and establish the Fatou-Naim-Doob theorem in this setting.Comment: Manuscript as accepted by publisher. To appear in Potential Analysi

Sweeping at the Martin boundary of a fine domain

We study sweeping on a subset of the Riesz-Martin space of a fine domain in \RR^n ($n\ge2$), both with respect to the natural topology and the minimal-fine topology, and show that the two notions of sweeping are identical.Comment: Minor correctio

Domains of existence for finely holomorphic functions

We show that fine domains in $\mathbf{C}$ with the property that they are Euclidean $F_\sigma$ and $G_\delta$, are in fact fine domains of existence for finely holomorphic functions. Moreover \emph{regular} fine domains are also fine domains of existence. Next we show that fine domains such as $\mathbf{C}\setminus \mathbf{Q}$ or $\mathbf{C}\setminus (\mathbf{Q}\times i\mathbf{Q})$, more specifically fine domains $V$ with the properties that their complement contains a non-empty polar set $E$ that is of the first Baire category in its Euclidean closure $K$ and that $(K\setminus E)\subset V$, are NOT fine domains of existence.Comment: 13 pages 1 figure. This new version has Bent Fuglede as coauthor. We extended the main result to include that regular fine domains are fine domains of existence and corrected many typo's and inaccuracies. In the third version a mistake at the end of the proof of Proposition 2.6 has been correcte

Plurisubharmonic and holomorphic functions relative to the plurifine topology

A weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions are introduced. Strong will imply weak. The weak concept is studied further. A function f is weakly plurifinely plurisubharmonic if and only if f o h is finely subharmonic for all complex affine-linear maps h. As a consequence, the regularization in the plurifine topology of a pointwise supremum of such functions is weakly plurifinely plurisubharmonic, and it differs from the pointwise supremum at most on a pluripolar set. Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preserved under composition with weakly plurifinely holomorphic maps.Comment: 28 page

Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser

For a finite collection $\mathbf A=(A_i)_{i\in I}$ of locally closed sets in $\mathbb R^n$, $n\geqslant3$, with the sign $\pm1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $\alpha\in(0,2]$, over positive vector Radon measures $\boldsymbol\mu=(\mu^i)_{i\in I}$ such that each $\mu^i$, $i\in I$, is carried by $A_i$ and normalized by $\mu^i(A_i)=a_i\in(0,\infty)$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $\boldsymbol\lambda^{\boldsymbol\xi}_{\mathbf A}=(\lambda^i_{\mathbf A})_{i\in I}$ (also in the presence of an external field) if we restrict ourselves to $\boldsymbol\mu$ with $\mu^i\leqslant\xi^i$, $i\in I$, where the constraint $\boldsymbol\xi=(\xi^i)_{i\in I}$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted vector $\alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $\lambda^i_{\mathbf A}$, $i\in I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $\alpha$-Riesz energy on a set of vector measures associated with $\mathbf A$, as well as on the establishment of an intimate relationship between the constrained minimum $\alpha$-Riesz energy problem and a constrained minimum $\alpha$-Green energy problem, suitably formulated. The results are illustrated by examples.Comment: 35 pages. arXiv admin note: substantial text overlap with arXiv:1711.0548

An alternative concept of Riesz energy of measures with application to generalized condensers

In view of a recent example of a positive Radon measure $\mu$ on a domain $D\subset\mathbb R^n$, $n\geqslant3$, such that $\mu$ is of finite energy $E_g(\mu)$ relative to the $\alpha$-Green kernel $g$ on $D$, though the energy of $\mu-\mu^{D^c}$ relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $0<\alpha\leqslant2$, is not well defined (here $\mu^{D^c}$ is the $\alpha$-Riesz swept measure of $\mu$ onto $D^c=\mathbb R^n\setminus D$), we propose a weaker concept of $\alpha$-Riesz energy for which this defect has been removed. This concept is applied to the study of a minimum weak $\alpha$-Riesz energy problem over (signed) Radon measures on $\mathbb R^n$ associated with a (generalized) condenser ${\mathbf A}=(A_1,D^c)$, where $A_1$ is a relatively closed subset of $D$. A solution to this problem exists if and only if the $g$-capacity of $A_1$ is finite, which in turn holds if and only if there exists a so-called measure of the condenser $\mathbf A$, whose existence was analyzed earlier in different settings by Beurling, Deny, Kishi, Bliedtner, and Berg. Our analysis is based particularly on our recent result on the completeness of the cone of all positive Radon measures $\mu$ on $D$ with finite $E_g(\mu)$ in the metric determined by the norm $\|\mu\|_g:=\sqrt{E_g(\mu)}$. We also show that the pre-Hilbert space of Radon measures on $\mathbb R^n$ with finite weak $\alpha$-Riesz energy is isometrically imbedded into its completion, the Hilbert space of real-valued tempered distributions with finite energy, defined with the aid of Fourier transformation. This gives an answer in the negative to a question raised by Deny in 1950.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1802.07171, arXiv:1711.0548