# Half Solved: A problem on the heat operator not being elliptic with a weakened version of elliptic regularity

I should first mention this: in my studies of Sobolev spaces I have completed all the questions of chapter 9 from Folland's real analysis with the help of this site and this is my last one, which is also very interesting, but seemingly difficult so I am posting here in the hopes of getting some help, but I should first mention some notation conventions used here.

$\mathcal{S}$ is the Schwartz class of functions and $\mathcal{S}'$ is the class of distributions ("generalized functions") on $\mathcal{S}$

$\partial ^ {\alpha}$ is the multi-index distribution derivative on $L^2$

$\Lambda _ s f = { [(1+|\xi|^2)^{\frac{s}{2}} \widetilde{f} ]^{\vee} }$ is a continuous linear operator on the Schwartz distributions.

The Sobolev space $H_s = \{ f \in S' | \Lambda_s f \in L^2 \}$

Now define the following differential operator $D^{\alpha} = (2i\pi)^{-\alpha}\partial^{\alpha}$ along with its symbol $P(D) = \sum_{|\alpha|<m} (2\pi i)^{|\alpha|}a_{\alpha} D^{\alpha}$, this of course has the property that the Fourier transform of $P(D)[f]$ is equal to $P\widehat{(f)}$

And now the elliptic regularity theorem as I know it:

Suppose that L is a constant-coefficient elliptic differential operator of order m and $\Omega$ is open in $R^n$ and $u \in D'(\Omega)$. If $Lu \in H_{s}^{loc}(\Omega)$ for some $s \in R$, then $u \in H_{s+m}^{loc}(\Omega)$. Also, if $Lu \in C^{\infty}(\Omega)$ then $u \in C^{\infty}(\Omega)$.

I can honestly say this theorem makes sense to me. Now for the exercise at hand:

The heat operator $\partial_t - \Delta$ is not elliptic but a weakened version of the elliptic regularity theorem does hold for it. Here we are workign on $R^{n+1}$ with coordinates $(x,t)$ and dual coordinates $(\xi,\tau)$, and $\partial_t - \Delta = P(D)$ where $P(\xi,\tau) = 2i\pi\tau + 4 {\pi}^2 |\xi|^2$.

a. There exist $C,R > 0$ such that $|\xi| |(\xi,\tau)|^{\frac{1}{2}} \leq C |P(\xi,\tau)|$ for $|(\xi,\tau)| > R$ (We are to try to consider the region $|\tau| \leq |\xi|^2$ and its complement separately).

b. If $f \in H_s$ and $(\partial_t - \Delta)f \in H_s$ then we are to show $f \in H_{s+1}$ and $\partial _ {x_i} f \in H_{s+(1/2)}$ for $1 \leq i \leq n$

c. If $\zeta \in C_C^{\infty}(R^{n+1})$, then we are to show: $[\partial_t - \Delta,\zeta]f = (\partial_t \zeta - \Delta \zeta)f - 2 \sum{(\partial_{x_i}\zeta)(\partial_{x_i}f)}$

d. We are to show that if $\Omega \in R^{n+1}$ is open and $u \in \mathcal{D}'(\Omega)$ and if $(\partial_t-\Delta)u \in H_{s}^{loc}(\Omega)$ then $u \in H_{s+1}^{loc}(\Omega)$. We are given this hint for part d:

Here is the proof of theorem 9.26 the elliptic regularity theorem:

My problem is this is fairly advanced stuff and I cannot seem to find it in me to write this down rigorously despite being able to see the geometric intuition behind it, but still this is hard analysis and I am a novice... I am positng this here following the successes I had and knowing this is the last question I have before I can truly say I completed this chapter's exercises. Thanks in advance to all helpers.

********** Progressed I think I am only truly stuck on parts b and d where for part d I cannot seem to imitate the proof referenced.

• As a humble physicist, wish I could only decipher the above .. – Han de Bruijn Nov 1 '15 at 19:24
• @HandeBruijn what do you mean exactly? – kroner Nov 1 '15 at 19:32
• If I'm understanding correctly $b$ is false in general: If $u$ is a constant function, then $(\partial_t-\Delta)u=0\in L^2(\mathbb{R}^{n+1})$, but $u\notin L^2(\mathbb{R}^{n+1})$. – Jose27 Nov 15 '15 at 2:46
• @Jose27 No but the constant function does not satisfy the second assumption despite it being zero under heat operator – kroner Nov 15 '15 at 3:32
• Right, thanks. For some reason I missed the hypothesis $f\in H_s$ completely. – Jose27 Nov 15 '15 at 4:22