# Questions tagged [potential-theory]

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184
questions

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### 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-...

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 ...

5
votes

0
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130
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### 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 ...

3
votes

0
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127
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### 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 ...

1
vote

1
answer

115
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### 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?

1
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0
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77
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### 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, ...

1
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0
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33
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### 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; ...

11
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0
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163
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### 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)...

1
vote

0
answers

48
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### 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 ...

0
votes

0
answers

138
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### 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\...

0
votes

1
answer

94
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### 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 ...

2
votes

0
answers

85
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### 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 | ...

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 -...

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\...

4
votes

2
answers

391
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### $\log |f|$ 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 ...

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 \...

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 ...

0
votes

0
answers

113
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### 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.

0
votes

0
answers

71
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### 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 ...

2
votes

0
answers

61
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### 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 $...

3
votes

1
answer

230
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### 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 ...

1
vote

0
answers

58
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### 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 ...

3
votes

0
answers

164
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### 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 $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...

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})...

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 ...

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 $D$ ...

1
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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 $...

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(...

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 ...

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 $...

5
votes

1
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128
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### 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 ...

0
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0
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### 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(...

1
vote

1
answer

157
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### 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 $=|x|^{2-m}$ ...

4
votes

0
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134
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### 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
...

0
votes

0
answers

57
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### 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 $f$ ...

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 $\...

4
votes

0
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### 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) = \...

4
votes

1
answer

58
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### 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 ...

10
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0
answers

271
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### 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 $...

3
votes

0
answers

141
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### 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 ...

3
votes

1
answer

107
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### 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) ...

0
votes

1
answer

133
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### 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 $D$...

0
votes

1
answer

106
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### 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&...

4
votes

1
answer

102
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### 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 $u\in \mathcal{H}^1(\mathbb{R})$,...

1
vote

1
answer

120
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### Subharmonic function in unbounded regions

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

1
vote

0
answers

32
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### 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 ...

4
votes

1
answer

346
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### 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 ...

2
votes

0
answers

86
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### 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+\...

1
vote

1
answer

85
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### 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*}
...

0
votes

1
answer

53
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### 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}...