Questions tagged [potential-theory]

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6 votes
0 answers
132 views

Sard's theorem for superharmonic functions: less regularity required?

A function $f:\mathbb{R}^d \to \mathbb{R}$ must be at least $C^d$ in order to guarantee in general that $$\{\phi\in \mathbb{R}|\,\exists x\in \mathbb{R}^d:\,f(x)=\phi,\,(\nabla f)(x)=0\}$$ is a zero-...
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1 vote
0 answers
70 views

Target space of Green's operator on $L^p$-differential forms on closed manifolds

Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott [1], i.e., with a finite atlas $\mathcal{A}$ so that for ...
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5 votes
0 answers
130 views

Potential theory as a tool in extrinsic flows

Let $M \subseteq \mathbb{R}^n$ be a submanifold. For a point $x$ disjoint from $M$, we can define the electric potential $\Phi(x) = \int_M \frac{dM}{|x-m|^{n-2}}$, which is smooth and harmonic where ...
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3 votes
0 answers
127 views

A question on the proof of Bedford-Taylor theorem in Demailly's book

I am trying to understand a proof of the Bedford-Taylor theorem on the weak convergence of Monge-Ampere operators of decreasing sequences of plurisubharmonic functions. I am reading a proof in the ...
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  • 19k
1 vote
1 answer
115 views

Equality of two subharmonic functions

Let $u\leq v$ be two locally bounded subharmonic functions in a domain in $\mathbb{R}^n$. Assume that $u=v$ on a dense subset. Is it true that $u=v$ everywhere?
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  • 19k
1 vote
0 answers
77 views

Minimizer for a certain variational problem

In the process of estimating certain hitting probabilities for Brownian motion I've run into the following variational problem. Let $\mathbb{D}\subseteq{\mathbb{C}}$ be the unit disk in the plane, ...
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1 vote
0 answers
33 views

Stability of Hajłasz-Sobolev class under post-composition

Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev? Assumptions/Setup Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
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11 votes
0 answers
163 views

Factorization of metric space-valued maps through vector-valued Sobolev spaces

Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that $$ \int_{x\in X}\,d(y_0,f(x)...
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1 vote
0 answers
48 views

Are sharper lower bounds known for these potentials on the sphere?

Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that $$ \sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq\frac{(2\ell-1)!!(d-2)!!}{(d+2\ell-2)!!}\cdot n^2, $$ with ...
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0 votes
0 answers
138 views

On a harmonic coordinate for a differential operator

I am looking for a harmonic coordinate for a degenerate differential operator on the unit ball. We write $\langle \cdot,\cdot \rangle$ for the standard inner product on $\mathbb{R}^d$. Set $\lvert\...
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0 votes
1 answer
94 views

Definition of Martin kernels

Let $\Omega \subset \mathbb{R}^n$ $(n \ge 3)$ be a bounded $C^{1,1}$ domain and let $X$ be a Markov process in $\Omega$. My question is regarding the existence of the Green function and Martin kernel ...
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2 votes
0 answers
85 views

Riesz potential on the boundary of a smooth domain

Let $\Omega \subseteq \mathbb{R}^n$ be a measurable set of finite measure. It is well-known that there holds $$ \sup_{x \in \mathbb{R}^n} \int_{\Omega} \frac{d z}{| x - z |^{n - 1}} \leqslant c_n | ...
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  • 306
2 votes
0 answers
52 views

Second order estimates for Dirichlet problem for complex Monge-Ampere equation

Let $\Omega\subset \mathbb{C}^n$ be a bounded pseudo-convex domain. Let $0<f\in C^{\infty}(\bar\Omega)$, $\phi\in C^\infty(\partial \Omega)$. Consider the Dirichlet problem for the complex Monge -...
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  • 19k
0 votes
1 answer
165 views

Harmonic functions in infinite domain in Euclidean space

EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...
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  • 19k
4 votes
2 answers
391 views

$\log |f|$ is subharmonic

It is known that the logarithm of the modulus of an analytic function $f: D \subset \mathbb C \rightarrow \mathbb C$ ($D$ is a domain) is subharmonic. I have two questions: (1) Are there some weaker ...
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  • 165
3 votes
0 answers
141 views

$p$-capacity of the closure

The $p$-capacity of a condenser $(K,\Omega)$ with $K$ compact and $\Omega$ open bounded is defined as $$ \mathrm{Cap}_p(K,\Omega)=\inf \left\lbrace \int_{\Omega} |\nabla u|^p \mathrm{d} x : u \in \...
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2 votes
1 answer
110 views

Comparing integrals of bounded subharmonic functions

Let $\Omega \subset \mathbb{R}^n$ be an open open subset. Let $u,v\colon \Omega\to \mathbb{R}$ be two functions such that at least one of them is compactly supported. Assume each of $u$ and $v$ can be ...
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  • 19k
0 votes
0 answers
113 views

Harmonic measure of a punctured disc

Let $D$ be a disc in $\mathbb{C}\cong\mathbb{R}^2 $ and $z_0$ a fixed point of $D$. Is the harmonic measure for $V=D\setminus\{z_0\}$ known? Any reference would also be welcome.
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  • 381
0 votes
0 answers
71 views

Extension of super harmonic functions

The following is stated as an exercise in the "Classical Potential Theory" of Armitage and Gardiner (pg 195). Let $K$ be a compact of $\mathbb{R}^m$ ($m\geq2$) and $\Omega$ an open set ...
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  • 381
2 votes
0 answers
61 views

A polar open set in a topological subspace?

Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar? A set $...
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  • 381
3 votes
1 answer
230 views

Harmonic interpolation with analytic initial condition

Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold. Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function. Is there a Harmonic ...
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1 vote
0 answers
58 views

Extension of the Kelvin transform

Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect ...
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  • 381
3 votes
0 answers
164 views

Regularity of harmonic functions for a degenerate elliptic operator

This is a question on a degenerate elliptic operator. Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
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  • 703
2 votes
0 answers
97 views

Divergence-free constraint for a boundary integral equation

Consider the system $$ \begin{cases} \operatorname{curl} \operatorname{curl} \mathbf{u} = 0 \qquad B \cup (\mathbb{R}^3 \setminus \overline{B}) \\ c_1 (\operatorname{curl} \mathbf{u} \times \mathbf{n})...
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  • 163
1 vote
0 answers
55 views

Superharmonicity of the distance function

Suppose $V$ is a convex open proper subset of $\mathbb{R}^m$ ($m\geq2$). It is known that the function $u(x)=$dist$(x,\partial V)$ is superharmonic on $V$. Is there a similar result without $V$ being ...
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  • 381
0 votes
2 answers
135 views

A question on minimum principle

Suppose $D$ be an unbounded domain of $\mathbb{R}^m$ for $m\geq3$, and $u$ is superharmonic on $D$. We know that if $\liminf_{x\to y}u(x)\geq0$ for all $y$ in $\partial^\infty D$ (the boundary of $D$ ...
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  • 381
1 vote
0 answers
55 views

An open set whose complement is non-thin at infinity

Let $x^*$ designate the inverse of a point $x\in\mathbb{R}^m$ under the Kelvin transformation with respect to the circle of center 0 and radius 1. Recall that $$x^*=|x|^{-2}x.$$ For a set $E$, we set $...
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  • 381
6 votes
1 answer
190 views

Core for a Sobolev space

Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connceted open subset of $\mathbb{R}^d$. The first-order Sobolev space $W^{1,2}(D)$ on $D$ is defined by \begin{align*} W^{1,2}(D)=\{f \in L^2(...
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  • 703
1 vote
1 answer
45 views

Modifying a superharmonic function on a neighborhood of infinity

Let $u(x)=\alpha+\beta U(x)$, where $U(x)=|x|^{2-m}$ ($N\geq3$) is the fondamental harmonic function, $\alpha<0$ and $\beta>0$. We know that $u$ is superharmonic on $\mathbb{R}^m$ and harmonic ...
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  • 381
0 votes
0 answers
81 views

Diffusion process, hitting times, and harmonic functions

Let $E$ be a locally compact metric space. We consider a diffusion process $X=(\{X_t\}_{t \ge0 },\{P_x\}_{x \in E})$ on $E$ whose lifetime $\zeta$ may be finite: $P_x(\zeta<\infty)>0$ for some $...
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  • 703
5 votes
1 answer
128 views

Superharmonicity at infinity

Some authors define superharmonicity at infinity in the following way. A function $u$ is superharmonic on an open set $V\subset\mathbb{R}^m\cup\{\infty\}$ (one point compactification), containing ...
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  • 381
0 votes
0 answers
37 views

Superharmonic extension 3

This question is related to the MO post Superharmonic extension 2. Let $u$ be a superharmonic function on $\mathbb{R}^m$ ($m>2$) such that for some $\alpha\in\mathbb{R}$ and $\beta$, $R>0$, $$u(...
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  • 381
1 vote
1 answer
157 views

Superharmonic extension 2

This question is a simplified version of the one in the MO post Superharmonic extension. Suppose $K$ is a compact of $\mathbb{R}^m$ ($m\geq2$), and $U(x)=\log\frac{1}{|x|}$ if $m=2$, and $=|x|^{2-m}$ ...
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  • 381
4 votes
0 answers
134 views

Okada-Schur functions and the Martin boundary of the Young-Fibonacci lattice

This question is related to three earlier posts addressing properties of the Young-Fibonacci lattice $\Bbb{YF}$, namely: Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality ...
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0 votes
0 answers
57 views

Inverse of the Riesz potential of a measure

Let $0<\alpha<d$ and let $I_{\alpha}(f)$ be the Riesz potential of a function $f$ on $\mathbb{R}^{d}$, $$ I_{\alpha}(f)(x)=\int_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}dy. $$ Assuming $f$ ...
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  • 3,408
0 votes
0 answers
144 views

Superharmonic extension

We know the following classic result. If $K\subset\mathbb{R}^m$ ($m>1$) is a compact set and $u$ is superharmonic on a neighborhood of $K$, then we can extend $u$ to a superharmonic function $\...
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  • 381
4 votes
0 answers
100 views

Restricted Perron-Bremermann envelopes

Consider an upper semicontinuous function $\phi: \Omega \to (-\infty, \infty]$, in the sense that $\phi = \phi^*$, where $\phi^*$ denotes the upper semicontinuous regularization $$ \phi^*(z) = \...
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  • 141
4 votes
1 answer
58 views

Singularities on null capacity sets are removable — Wiener or Bouligand?

A classical theorem on harmonic functions states that singularities of bounded harmonic functions are removable if the singular set is of null capacity. This theorem is sometimes attributed to ...
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  • 3,212
10 votes
0 answers
271 views

Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality

$\newcommand\rank[1]{\lvert#1\rvert}$Let $\Bbb{P}$ be a 1-differential poset with a unique bottom element $\emptyset \in \Bbb{P}$. With some minor abuse in terminology, The Plancherel measure state $...
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3 votes
0 answers
141 views

Does the maximum principle hold in this pluriharmonic setting?

Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and ...
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  • 3,594
3 votes
1 answer
107 views

A function $V : \mathbb{R}^2 \to \mathbb{R}$ is a (logarithmic) potential

I'm looking for references given some sort of inverse problem in logarithmic potential theory. That is, given a function $V : \mathbb{R}^2 \to \mathbb{R}$, what is a sufficient (and perhaps necessary) ...
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  • 153
0 votes
1 answer
133 views

Does convergence of a sequence of subharmonic functions imply the vague convergence of their Riesz measures?

Suppose $D$ is a bounded domain of $\mathbb{R}^m$ for $m>1$ and $\{u_n\}_{n\geq1}$ is a sequence of subharmonic functions on $D$. Assume $u_n\to u_0$ pointwise on $D$ and $u_0$ is subharmonic on $D$...
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  • 381
0 votes
1 answer
106 views

A simple clarification on Riesz decomposition theorem

Let $D$ be a domain of $\mathbb{R}^{m}$ and let $K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions&...
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  • 381
4 votes
1 answer
102 views

Boundedness of Riesz potential on Hardy space

I encounter the following claim in one paper: If $(-\Delta)^{\frac14}u\in L^{2,\infty}(\mathbb{R})$, then $u\in BMO(\mathbb{R})$. Equivalently in its dual version, if $u\in \mathcal{H}^1(\mathbb{R})$,...
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  • 553
1 vote
1 answer
120 views

Subharmonic function in unbounded regions

The harmonic majorization for a subharmonic function $h$ is well-known for bounded regions $\Omega \subset \mathbb{C}$: $$h \le 0 \text{ in }\partial \Omega \Longrightarrow h \le 0 \text{ in }\Omega.$$...
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  • 165
1 vote
0 answers
32 views

When is inverse geodesic distance positive definite (in a compact manifold)?

We work on a closed smooth Riemannian manifold $(M,g)$ and let $K:M\times M\to \mathbb R\cup\{+\infty\}$ be a kernel, which we assume to be integrable and lower semicontinuous. We say it is positive ...
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4 votes
1 answer
346 views

Elliptic, parabolic and hyperbolic Riemann surfaces: classification?

In the book of Kra and Farkas on Riemann surfaces the following (slightly unusual) definition is given: Definition IV.3.2 (Section IV.3). Let $M$ be a Riemann surface. We will call $M$ elliptic if and ...
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  • 13.2k
2 votes
0 answers
86 views

Hitting measure/overshoot for random walk in $\mathbb{Z}$ with heavy-tail

Let $\alpha \in (0,2)$, (or for simplicity just $\alpha \in (1,2)$) and let $X_1,X_2,\dots$ be an i.i.d collection of random variables with common distribution $$ p(x,y)= \frac{c_\alpha}{|x-y|^{1+\...
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  • 406
1 vote
1 answer
85 views

Coupling times of subordinate Brownian motions

This is a question about coupling times of subordinate Brownian motions. We fix $y \in \mathbb{R}^d$ with $y \neq x$ and define a map $R_{x,y} \colon \mathbb{R}^d \to \mathbb{R}^d$ by \begin{align*} ...
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  • 703
0 votes
1 answer
53 views

Harmonic functions for subordinate Brownian motions and the Hölder continuity

This question is about harmonic functions of subordinate Brownian motions. We write $B=(\{B_t\}_{t \ge 0}, \{P_x\}_{x \in \mathbb{R}^d})$ for the $d$-dimensional Brownian motion. Let $\{S_t\}_{t \ge 0}...
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