0
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0answers
24 views

How to get between the equivalent definitions of Newtonian capacity?

Here are the two definitions: (1) $$\mathrm{Cap}(A)=\left[\inf \left\{\int\int |x-y|^{d-2}\mu(dx)\mu(dy):\mu \; \text{a probability measure on} \; A \right\}\right]^{-1}$$ and (2) ...
0
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1answer
18 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
2
votes
1answer
34 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
1
vote
1answer
41 views

Uniform Rectifiability

What is the definition of uniform rectifiability as used in the context of analytic capacity of compact sets in $\mathbb{C}$? The precise context is this paper by Mattila, Mernikov and Verdera.
5
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1answer
287 views

Hausdorff Dimension of Arbitrary Julia Set

I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$. I know this question is known for a number of ...
22
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1answer
614 views

Which sets are removable for holomorphic functions?

[Note: I received a version of this question via email and decided to answer it on MSE, where it might be useful to others.] Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some ...