January 20, 2010

Towards Malgrange-Ehrenpreis: Bad-tempered distributions and polynomial inequalities

Posted by Akhil Mathew under analysis, complex analysis | Tags: , , , , |
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Apologies for the embarrassingly bad pun in the title.

Distributions in general

First, it’s necessary to talk about distributions on an arbitrary open set {\Omega \subset \mathbb{R}^n}, which are not necessarily tempered. In particular, they may “grow arbitrarily” as one approaches the boundary. So, instead of requiring a functional on a Schwarz space, we consider functionals on {C_0^{\infty}(\Omega),} the space of smooth functions compactly supported in {\Omega}. However, we need some notion of continuity, which would require a topology on {C_0^{\infty}(\Omega)}. There is now the tricky question of how we would require completeness of the topological vector space {C_0^{\infty}(\Omega)}, which we of course desire. We can get such a topology by talking about “strict inductive limits” and whatnot, but since I don’t really find that particularly fun, I’ll sidestep it (but not really—most of the ideas will still remain).

Anyway, the idea here will be to consider auxiliary spaces {C^{\infty}(K)} for {K \subset \Omega} compact. This is the space of smooth functions {f: \Omega \rightarrow \mathbb{R}} which are supported in {K}. We give the space a Frechet topology by the family of seminorms

\displaystyle ||f||_a := \sup_K |D^a f|. (more…)

 

January 17, 2010

Towards constant-coefficient elliptic regularity: parametrices and generalities on the singular support

Posted by Akhil Mathew under analysis | Tags: , , , , , , |
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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…)

 

January 5, 2010

Tempered distributions

Posted by Akhil Mathew under analysis, harmonic analysis | Tags: , , , , |
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The next basic tool we’re going to need is the theory of distributions.

Distributions

Distributions are extremely useful because they are both fairly general (including both all integrable functions and things like the Dirac delta function) but also allow for operations such as differentiation. So oftentimes we can obtain distribution solutions to differential equations we are interested in.

Actually, we’ll only discuss here tempered distributions. A tempered distribution is a linear functional {\phi: \mathcal{S} \rightarrow \mathbb{C}}. Clearly the tempered distributions form a vector space {\mathcal{S}'}; it is a locally convex space if we endow it with the weak* topology. It must now be seen how distributions generalize functions. So, if {f \in L^p(\mathbb{R}^n)} for any {p, 1 \leq p \leq \infty}, then {f} can be made into a distribution

\displaystyle w \rightarrow \int_{\mathbb{R}^n} wf dx, \ w \in \mathcal{S}.

In fact, we could just assume that {f} grows polynomially. In particular, we have an imbedding {\mathcal{S} \rightarrow S'}.

An example of a distribution that is not a function is the Dirac distribution {\delta} mapping {f \rightarrow f(0)}. (more…)