# Hausdorff Measure and Capacity

Definition 1 Let ${E \subset {\mathbb R}^n, 0 \le d < \infty, 0 < \delta \le \infty}$. Define

$\displaystyle \mathcal{H}^d_\delta(E) : = \inf \bigg\lbrace (1/2)^d \alpha(d) \sum_{i=1}^\infty diam(A_i)^d | E\subset \bigcup_{i=1}^\infty A_i, diam(A_i)\le \delta \bigg\rbrace,$

where ${\alpha(d) := \frac{\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2} + 1)}}$.

$\displaystyle \mathcal{H}^d (E) := \lim_{\delta \rightarrow 0} \mathcal{H}^d_\delta(E) = \sup_{\delta >0} \mathcal{H}^d_\delta(E)$

Then ${\mathcal{H}^d}$ is called d-dimensional Hausdorff measure in ${{\mathbb R}^n}$.

Remark 1 Hausdorff measure is a Borel outer measure, it is not Radon for ${0 \le d < n}$ because ${{\mathbb R}^n}$ is not ${\sigma-}$finite with respect to ${\mathcal{H}^d}$. And it is a measure if restricted on Lebesgue measurable sets (by Caratheodory condition: We say a set ${E}$ satisfies Caratheodory condition, if for any set ${A \subset {\mathbb R}^n}$, ${\mathcal{L}^n(A) = \mathcal{L}^n(A \cap E) + \mathcal{L}^n(A \setminus E)}$). Moreover, note that ${\alpha(d)}$ gives the volume of unit ball in ${d}$ dimension if ${d}$ is an integer, we naturally have (not trivially) ${\mathcal{H}^n = \mathcal{L}^n }$, where ${\mathcal{L}^n}$ is the n-dimensional Lebesgue measure.

# Regularity of scalar elliptic equation by Moser iteration

Theorem Suppose ${v \in W^{1,2}(B_R)}$ is a subsolution of ${-\partial_i (a^{ij}(x) \partial_j u) =0}$, ${\lambda I \le (a^{ij}(x)) \le \Lambda I}$. Then for all ${p>0}$, ${0<\theta <1}$,

$\displaystyle \sup_{B_{\theta R}} v \le C(n, \lambda, \Lambda, p) (1- \theta)^{- \frac{n}{p}} \left( \frac{1}{|B_R|} \int_{B_R} (v^+)^p \right)^{\frac{1}{p}}.$

# High order BV function has continuous representative

As known, bounded variation (BV) functions are not continuous, but evidently higher order BV functions on corresponding dimension have continuous representative. As a consequence, ${W^{n,1}({\mathbb R}^n) \hookrightarrow C^0({\mathbb R}^n)}$.

Definition A function ${f \in BV_n({\mathbb R}^n)}$, if ${f \in W^{n-1,1}({\mathbb R}^n)}$, and the nth order distributional derivative ${D^n f}$ is a finite Radon measure.

Theorem If ${f \in BV_n({\mathbb R}^n)}$, then ${f}$ has a continuous representative.

Proof: Note that ${C_c^\infty({\mathbb R}^n)}$ is dense in ${BV_n({\mathbb R}^n)}$, i.e. for any ${f \in BV_n({\mathbb R}^n)}$, there exists a sequence ${f_k \in C_c^\infty({\mathbb R}^n)}$ such that ${\lim_{k \rightarrow \infty} f_k = f}$ in ${W^{n-1,1}}$, and ${\lim_{k\rightarrow \infty} \| \nabla^n f_k\|_{L^1({\mathbb R}^n)} = \|D^n f \| ({\mathbb R}^n)}$. (See for example, book by Evans and Gariepy). So it suffice to show for all ${f \in C_c^\infty({\mathbb R}^n)}$,

$\displaystyle \|f\|_{L^\infty({\mathbb R}^n)} \le \| \nabla^n f\|_{L^1({\mathbb R}^n)}$

Indeed,

$\displaystyle f(x_1, \cdots , x_n) = \int_{-\infty}^{x_1} \partial_1 f(s_1,x_2, \cdots , x_n)\,ds_1 = \int_{-\infty}^{x_1} \cdots \int_{-\infty}^{x_n} \partial_1 \cdots \partial_n f$

This gives us the desire inequality. Then by density result we know that ${BV_n({\mathbb R}^n)}$ admits continuous representative for every element. $\Box$

Corollary ${W^{n,1}({\mathbb R}^n) \hookrightarrow C^0({\mathbb R}^n)}$.

Remark If the dimension is greater than the order of BV function, then the above theorem will fail. For example, let ${f(x) = |x|^{-\frac{1}{2}}}$ be defined in a neighborhood ${\Omega}$ around 0, with smooth boundary in ${{\mathbb R}^3}$. ${|\nabla f| \sim |x|^{-\frac{3}{2}}}$ and ${|\nabla^2 f| \sim |x|^{-\frac{5}{2}}}$. So ${f \in W^{2,1}(\Omega)}$. Then we can extend ${f}$ on ${{\mathbb R}^3}$, since ${\partial \Omega}$ is smooth (See for example, PDE book by Evans). But then ${f \rightarrow \infty}$ as ${|x| \rightarrow 0}$, which doesn’t admit a continuous representative.

# Some embedding results

Definition ${X,Y}$ are Banach spaces, we say ${X}$ is embedded in ${Y}$, denoted by ${X \hookrightarrow Y}$, if ${X \subset Y}$ and ${\| \cdot \|_Y \le C \| \cdot \|_X}$.

Notation: ${\langle \xi \rangle := \sqrt{1+\xi^2}}$.

Here are some embedding results.