Can I get a little math help in here?

Xkdjdskjsdkjsfkjsfkjsf

In class recently, my professor gave a proof that any constant coefficient partial differential equation has a weak solution on a bounded subset of $R^n$. That is, if
$|\alpha| = \alpha_1 + \alpha_2 + ... + \alpha_n$
and
$L = \sum_{|\alpha|<k}\frac{\partial^{\alpha_1}}{\parti al x_1}\frac{\partial^{\alpha_2}}{\partial x_2}...\frac{\partial^{\alpha_n}}{\partial x_n}$
then for any bounded $\Omega \subset R^d$ and $f \in L^2(\Omega)$, there exists some $u \in L^2(\Omega)$ that is a solution to the PDE $Lu = f$ in the weak sense. In fact, there is a bounded operator $K$ such that $LKf = f$ in the weak sense for any $f$.

Now, I'm going to outline the proof and point out what I don't understand and why, and hopefully someone can help me.

$\textbf{Lemma: }$ $\exists c \in R$ such that $\varphi \in C^\infty_0(\Omega) \Rightarrow \|\varphi\|_{L^2(\Omega)} \le c\|L^*\varphi\|_{L^2(\Omega)}$.

That is, for any infinitely differentiable function of compact support, the norm of the image of the function under the adjoint of $L$ is bounded below by a constant multiple of the norm of the original function, and this constant depends only on the space $\Omega$.

Thing I Don't Understand #1: Why is this the most difficult part of the theorem? (We'll get into more detail with this later when I return to prove the lemma.)

Now, let's assume the lemma for now. It seems that part of the purpose of this lemma is to demonstrate that $L^*$ is one-to-one, and therefore $L$ is onto. However, I have no idea why it is important that we know this. That's Thing I Don't Understand #2.

So we consider the pre-Hilbert space $H_0$ which is the space of all functions that are $C^\infty_0(\Omega)$, and let $H$ be the closure of that space. Take the norm to be $\|f\|_0 = \|L^*f\|_{L^2(\Omega)}$ (notice that $L^*: H \rightarrow L^2(\Omega)$.

So we consider the linear functional $l: \varphi \rightarrow (\varphi, f)$ for some fixed $f$ and $\varphi \in H_0$ (the usual $L^2$ inner product, I believe). Then

$|(\varphi, f)| \le \|\phi\|_{L^2(\Omega)}\|f\|_{L^2(\Omega)} \le c\|L^*\varphi\|_{L^2(\Omega)}\|f\|_{L^2(\Omega)} = c\|\varphi\|_0\|f\|_{L^2(\Omega)}$.

I don't get the point of this, though I can see why it is true (Cauchy-Schwartz and the Lemma). I also don't get what follows, though I do know the Riesz representation theorem. The claim is that there exists some $U \in H$ such that $l(\varphi) = (L^*\varphi, L^*U)$.

Then we pick $u = L^*U$, and thus $(\varphi, f) = (L^*\varphi, u)$, so $u$ is the weak solution we wanted.



Now, we also need to prove that lemma. Observe that if $(u, L^*\varphi) = (f, \varphi)$ for all $\varphi \in C^\infty_0(\Omega)$, then we can use the Plancherel identity to see that $(\widehat{u}, \widehat{L^*\varphi}) = (\widehat{f}, \widehat{\varphi})$. But derivatives get turned into polynomials, so what we'd like to see is that $\|\widehat{\varphi}\| \le c\|P\widehat{\varphi}\|$ where $P$ is the polynomial corresponding to $L^*$. (Stein calls it the characteristic polynomial; is this really the characteristic polynomial of that linear operator, by the standard lin alg definition?)

Anyway, we now have two sub-lemmas.

Lemma 1a: suppose $\Omega = (-M, M)$ (a subset of the real line) and $g \in L^2(R)$ is supported on $[-M, M]$. Then
$\widehat{g}(\xi) = \int_{-M}^M g(x) e^{-2\pi i\xi x}dx$
can be extended to an analytic function on the complex plane, and furthermore
$\int |\widehat{g}(\xi + i\eta)|^2d\xi \le e^{4\pi M|\eta|}\|g\|^2_{L^2}$.

Lemma 1b: If $F$ is holomorphic on the complex plane, then for any polynomial $Q(z) = z^m + a_{m-1}z^{m-1} + ... + a_1z + a_0$, we have

$|F(0)|^2 \le \frac{1}{2\pi}\int_0^{2\pi}|Q(e^{i\theta})F(e^{i\t heta})|^2d\theta$.

If we take $Q = P$, $F(z) = \widehat{g}(\xi + i\eta)$, $\eta = \sin(\phi)$, shift $\xi$ by $\cos(\phi)$, and finally apply Lemma 1b to the result of Lemma 1a, then we have the result of the original Lemma 1.

Now, there was a lot of information and very little proof near the end there. I'm not even certain everything is right as I don't have any intuition for functions on the complex numbers, so I can't check this very well. Lemma 1a seems reasonable to me, but I have no idea about the reason for 1b, except that the professor was saying some vague thing about the polynomial not causing any harm when $|z| = 1$ and this is just like the much more basic averaging result about holomorphic functions.

So, I'm not looking for a full explanation of this theorem, because it is pretty long and hardcore, but if there's any particular part of my confusion that you think you might be able to address, that would be great. Thanks!