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…)