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…)