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…)

The next application I want to talk about here of Fourier analysis is to (a basic case of) ellipic regularity. Later we will use refinements of these techniques to obtain all kinds of estimates. Anyway, for now, a partial differential operator

\displaystyle P = \sum_{a: |a| \leq k} C_a D^a

is called elliptic if the homogeneous polynomial

\displaystyle \sum_{a: |a| = k} C_a \xi^a, \quad \xi = (\xi_1, \dots, \xi_n)

has no zeros outside the origin. For instance, the Laplace operator is elliptic. Later I will discuss how this generalizes to other PDEs, and how this polynomial becomes the symbol of the operator. For the moment, though let’s define {Q(\xi) = \sum_{a: |a| \leq k} C_a (2 \pi i \xi)^a}. The definition of {Q} such that

\displaystyle \widehat{ Pf } = Q \hat{f},

and we know that {|Q(\xi)| \geq \epsilon |\xi|^k} for {|\xi|} large enough. This is a very important fact, because it shows that the Fourier transform of {Pf} exerts control on that of {f}. However, we cannot quite solve for {\hat{f}} by dividing {\widehat{Pf}} by {Q} because {Q} is going to have zeros. So define a smoothing function {\varphi} which vanishes outside a large disk {D_r(0)}. Outside this disk, an estimate {|Q(\xi)| \geq \epsilon |\xi|^k} will be assumed to hold. (more…)