# Questions tagged [potential-theory]

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### Demailly regularisation on singular complex spaces

Let $X$ be a compact (Hausdorff reduced) complex space. It is asserted (and used in an essential way) in a famous paper by Demailly and Păun ("Numerical characterization of the Kähler cone of a ...
227 views

### Newtonian potentials of balls and spheres

This is a simple question whose answer was probably known to Poisson, but I was not able to find it by searching. I need explicit formulas for the Newtonian potential of the unit ball $\mathbb{B}^n$ ...
85 views

### Subharmonic distributions on the plane

A subharmonic (Schwartz) distribution on $\mathbf R^n$ is a distribution $u$ satisfying $\Delta u\ge0$. This implies $\Delta u$ is a positive Radon measure $\mu$, thus for any ball $B$ the convolution ...
163 views

### Value of $\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}$

I wonder if any of you knows how to find the value of the series $$\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}.$$ This function shows up while solving a magnetostatic problem with complex-valued ...
1 vote
135 views

### A characterization of plurisubharmonic functions

Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function. Recall that $u$ is called plurisubharmonic (psh) if its restriction ...
135 views

### A possible characterization of subharmonic functions

Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function. If $u$ is subharmonic then for any point $x\in \Omega$ and any $C^2$-...
115 views

### Proof of a theorem in degenerate Monge Ampère equation by Vincent Guedj and Ahmed Zeriahi

$\DeclareMathOperator\PSH{PSH}$This question is about Proposition 9.25 page 252 from the book "Degenerate Complex Monge-Ampère Equations" by Vincent Guedj and Ahmed Zeriahi (see picture ...
29 views

### Functional inequality for fractional Laplacian

Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...
90 views

### L¹ norm of Riesz potentials on flat tori

Let $g$ be the distribution whose Fourier coefficients are given by $$\hat{g}(k) = \begin{cases} 0, & {k=0} \\ |k|^{s-d}, & {k\in \mathbb{Z}^d\setminus\{0\}},\end{cases} \qquad 0\leq s<d,$$ ...
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### What does a Lipschitz barrier imply about boundary regularity of a domain?

Consider the Dirichlet problem for Laplace's equation in a bounded domain $\Omega \subset \mathbb R^n$: $$-\Delta u = 0, \quad x \in \Omega,$$ with $u = \phi$ on $\partial\Omega$, and $\phi$ is ...
80 views

### Convergence of Riesz measure of SH function

Let $u$ be a subharmonic function in a domain $\Omega$ pf $\mathbb{C}$. The functions $u_{j} := \max(u, -j)$ still subharmonic. Let $\mu := \Delta u$ and $\mu_{j} := \Delta u_{j}$ be the associated ...
65 views

### When a strictly positive log pluriharmonic function $g$ is equal to the norm of holomorphic function?

Suppose $V$ is a local analytic variety (singular). Suppose $g$ a strictly positive log pluriharmonic function on $V$, i.e. $\log g$ is pluriharmonic. I wonder when $g=|f|^2$, where $f$ is a ...
199 views

### Show those PSH functions belongs to Sobolev space

Let u be a plurisubharmonic function defined on the unit ball $\mathbb{B}$ of $\mathbb{C}^{k}$ such that $u \ge 1$. Question: why the partial derivates $\frac{\partial u}{\partial x_{i}}$ (which are ...
102 views

### On the definition of Cauchy transform [closed]

I have seen two different definitions of the Cauchy transform of a smooth function one is with respect to the line integral (for eg. in. the book "The Cauchy transform and potential theory")...
171 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-...
1 vote
77 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 , i.e., with a finite atlas $\mathcal{A}$ so that for ...
149 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 ...
153 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 ...
1 vote
122 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?
1 vote
44 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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### 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 -...
237 views

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### Core for a Sobolev space

Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected 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
49 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 ...