Back to elliptic regularity. We have a constant-coefficient partial differential operator ${P = \sum_{a: |a| \leq k} C_a D^a}$ which is elliptic, i.e. the polynomial

$\displaystyle Q(\xi) = \sum_{a: |a| \leq k} C_a \xi^a$

satisfies ${|Q(\xi)| \geq \epsilon |\xi|^k}$ for ${|\xi|}$ large. We used this last property to find a near-fundamental solution to ${P}$. That is, we chose ${E}$ such that ${\hat{E} = (1-\varphi) Q^{-1}}$, where ${\varphi}$ was our arbitrary cut-off function equal to one in some neighborhood of the origin. The point of all this was that

$\displaystyle P(E) = \delta - \hat{\varphi}.$

In other words, ${E}$ is near the fundamental solution. So given that ${Pf = g}$, we can use ${E}$ to “almost” obtain ${f}$ from ${g}$ by convolution ${E \ast g}$—if this were exact, we’d have the fundamental solution itself.

We now want to show that ${E}$ isn’t all that badly behaved.

The singular locus of the parametrix

We are going to show that ${\mathrm{sing} E = \{0\}}$. The basic lemma we need is the following. Fix ${m}$. Consider a smooth function ${\phi}$ such that, for each ${a}$, there is a constant ${M_a}$ with

$\displaystyle |D^a \phi(x)| \leq M_a (1+|x|)^{m-|a|};$

then this is a distribution, but it is not necessarily a Schwarz function. And ${\hat{\phi}}$ cannot be expected to be one, thus. Nevertheless:

Lemma 1 ${\hat{\phi}}$ is regular outside the origin. (more…)

Yesterday I defined the Hilbert space of square-integrable 1-forms ${L^2(X)}$ on a Riemann surface ${X}$. Today I will discuss the decomposition of it. Here are the three components:

1) ${E}$ is the closure of 1-forms ${df}$ where ${f}$ is a smooth function with compact support.

2) ${E^*}$ is the closure of 1-forms ${{}^* df}$ where ${f}$ is a smooth function with compact support.

3) ${H}$ is the space of square-integrable harmonic forms.

Today’s goal is:

Theorem 1 As Hilbert spaces,

$\displaystyle L^2(X) = E \oplus E^* \oplus H.$

The proof will be divided into several steps. (more…)

Theorem 1 (Weyl) Let ${f \in L^2(U)}$, where ${U}$ is the unit disk with Lebesgue measure. If  $\displaystyle \int_U f \Delta \phi = 0$  for all ${\phi \in C^{\infty}(U)}$ with compact support, then ${f}$ is harmonic (in particular smooth).

I dropped out of the groove for a couple of days due to other activities; I’m back today to talk about Weyl’s lemma (for the Laplacian—it generalizes to elliptic operators), a tool we will need for the special case of the Hodge decomposition theorem on Riemann surfaces.   The result states that a “weak” solution to the Laplace equation is actually a strong one. (more…)