# Conservative, solenoidal vector fields and Hardy space

Theorem 1 Let ${E \in L^p ({\mathbb R}^n; {\mathbb R}^n), B \in L^q ({\mathbb R}^n ; {\mathbb R}^n)}$ with ${1 < p < \infty, \frac{1}{p} + \frac{1}{q} = 1}$, such that ${B = \nabla \phi}$, where ${\phi \in L^{\frac{nq}{n-q}}}$, if ${q < n}$ ; ${\phi \in L_{loc}^s}$ for all ${s < \infty}$, if ${q \ge n}$. If ${E}$ satisfies

$\displaystyle div E = 0 \quad \text{ in } D'({\mathbb R}^n),$

then ${E \cdot B \in \mathcal{H}^1}$.

# Two fixed point theorems in Banach space

Here is a famous fixed point theorem in finite dimension by Brouwer:

Theorem 1 (Brouwer fixed point theorem) Let ${M \subset {\mathbb R}^n}$ be a convex compact set, for any continuous function ${f : M \rightarrow M}$, there exists a point ${x_0 \in M}$ such that ${f (x_0) = x_0}$.

There are couple of ways to extend this theorem to Banach spaces. First recall that a mapping between two Banach spaces is called compact, if the mapping is continuous (not necessarily linear) and the images of bounded sets are pre-compact.

# 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.

# Harnack inequality implies Holder continuity

We will show a basic argument why Harnack inequality implies Holder continuity. Assume ${u \in H^1(B_2)}$ is a weak solution to a uniform elliptic equation

$\displaystyle -\partial_i(a^{ij} \partial_j u) = 0,$

in ${B_2}$. Assume we have Harnack inequality (Indeed we do have due to Moser), i.e. for any positive weak solution ${u}$ we will have

$\displaystyle \sup_{B_1} u \le C \inf_{B_1} u,$

where ${C}$ is a constant independent of ${u}$.

First one will know ${u \in L^\infty(B_1)}$, see this note.

If ${u}$ is a weak solution, so are ${\sup_{B_2} u - u}$ and ${u - \inf_{B_2} u}$. And they are strictly positive in ${B_1}$ due to the strong maximum principle, otherwise ${u}$ will be a constant. By Harnack inequality, we will have

$\displaystyle \sup_{B_2} u - \inf_{B_1} u \le C(\sup_{B_2} u - \sup_{B_1} u),$

$\displaystyle \sup_{B_1} u - \inf_{B_2} u \le C(\inf_{B_1} u - \inf_{B_2} u).$

Adding these two inequalities up we will have

$\displaystyle \text{osc}_{B_2} u + osc_{B_1} u \le C(osc_{B_2} u - osc_{B_1}),$

hence

$\displaystyle osc_{B_1} u \le \frac{C-1}{C+1} osc_{B_2} u.$

Denote ${\theta = \frac{C-1}{C+1} < 1}$, and choose ${\alpha > 0}$ such that ${\theta = 2^{- \alpha}}$, by interation we know

$\displaystyle osc_{B_{2^{-k}}} u \le \theta^k osc_{B_1} u \le 2 \theta^k \|u\|_{L^\infty(B_1)}.$

Then for any ${0 < r < 1}$, choose ${k \in {\mathbb N}}$ such that ${2^{-k-1} < r \le 2^{-k}}$, we will have

$\displaystyle osc_{B_r} u \le osc_{B_{2^{-k}}} u \le 2^{-\alpha k + 1} \|u\|_{L^\infty(B_1)} = 2^{1+\alpha} \cdot 2^{- \alpha ( k +1)} \|u\|_{L^\infty(B_1)}< 2^{1+\alpha} \|u\|_{L^\infty(B_1)} r^\alpha.$

So Holder continuity follows.

# Newtonian Potential

— 1. Definition of Newtonian Potential —

Definition 1 Let ${\Omega}$ be a bounded open set, we define the Newtonian potential of ${f}$ is the function ${N_f}$ on ${{\mathbb R}^n}$ by

$\displaystyle N_f(x) = \int_\Omega \Gamma(x-y) f(y)\, dy,$

where ${\Gamma}$ is the fundamental solution of Laplace’s equation.

# Fredholm Alternative and Riesz Schauder Theorem

Theorem (Analytic Fredholm Alternative) Let ${D}$ be a connected open subset of ${{\mathbb C}}$ and ${\mathcal{H}}$ be a separable Hilbert space. Suppose that ${f : D \longrightarrow \mathcal{L} (\mathcal{H})}$ is an analytic operator-valued function such that ${f(z)}$ is compact ${\forall z \in D}$. Then either

1. ${(I - f(z)) ^{-1}}$ does not exist ${\forall z \in D}$.
2. ${(I - f(z)) ^{-1}}$ exists ${\forall z \in D\setminus S}$, where ${S}$ is discrete.

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 ${p>0}$, ${0<\theta <1}$,
$\displaystyle \sup_{B_{\theta R}} v \le C(n, \lambda, \Lambda, p) (1- \theta)^{- \frac{n}{p}} \left( \frac{1}{|B_R|} \int_{B_R} (v^+)^p \right)^{\frac{1}{p}}.$