Definition of $C^{m,k/2}$-capacity of a point.

Question

I hve come across the following notation and a new term $C^{m,k/2}$-capacity of a point. I'd appreciate some reference, where I can find the definition and relevant theory.

Answer 1

The $C^{m,k/2}$ capacity of a point $x \in \mathbb R^d$ might be the number: $$ \inf\{ \|\varphi\|_{W^{m,k/2}(\mathbb R^d)} \mid \varphi \ge 1 \text{ in an open superset of } \{x\}\}. $$ (Often, this is raised to the power $k/2$). Note that this number does not depend on $x$, since everything is translation invariant.

For further reading, I would suggest the books

• Nonlinear Potential Theory of Degenerate Elliptic Equations by Juha Heinonen, Tero Kipelainen, Olli Martio
• Function Spaces and Potential Theory by Adams, David R., Hedberg, Lars I.