Fourth Problem set for 18.155
Due November 7 in class or 2-174.

Especially for undergraduates I will accept just one of these problems as enough work for the week. Notice that the first problem is considered the deepest result in `elementary' distribution theory.

Problem 1 (Schwartz kernel theorem)  

1. Show (mostly recalling things from class) that the topology of $\mathcal{S}(\mathbb{R}^n)$ can be expressed as the projective limit of the Hilbert topologies on the weighted Sobolev spaces $\langle x\rangle^{-k}H^k(\mathbb{R}^n)$ as $k\to\infty$
2. Conclude that the dual space $\mathcal{S}'(\mathbb{R}^n)$ can be topologized as the inductive limit of the weighted Sobolev spaces $\langle x\rangle^{k}H^{-k}(\mathbb{R}^n)$ as $k\to\infty.$
3. Show that the Fourier transform, multiplication by $\langle x\rangle ^m$ for any $m$ and the map $\langle D\rangle ^m$ (defined as the preceeding map conjugated by the Fourier transform) are all isomorphisms of both $\mathcal{S}(\mathbb{R}^n)$ and $\mathcal{S}(\mathbb{R}^n).$
4. Show that for a linear map (an operator)

   $$A:\mathcal{S}(\mathbb{R}^n)\longrightarrow \mathcal{S}'(\mathbb{R}^n)$$ (1)

   
   
   continuity with respect to these topologies is equivalent to the existence of some $k$ such that $A$ extends by continuity (in the Hilbert norms) to $A:\langle x\rangle^{-k}H^k(\mathbb{R}^n)\longrightarrow \langle x\rangle^{k}H^{-k}(\mathbb{R}^n).$
5. Show how to compose $A$ on the left and right with maps from (3) and arrive at a continuous linear map $A'$ from $H^{-n}(\mathbb{R}^n)$ to $H^n(\mathbb{R}^n).$
6. Recall the Sobolev embedding theorem and that the delta `function' at any point is in $H^{-n}(\mathbb{R}^n)$ and use this to conclude that the formula $a(x,y)=(A'(\delta _y))(x)$ defines a continuous bounded function on $\mathbb{R}^{2n}).$
7. Show that the map

   $$\mathcal{S}(\mathbb{R}^n)\times \mathcal{S}(\mathbb{R}^n)\longrig... ...bb{R}^{2n}), (\phi,\psi )\longmapsto \phi \boxtimes\psi (x,y)=\phi (x)\psi (y)$$

   
   
   is jointly continuous (i.e. is continuous in the product metric topology).
8. Show that if $\beta \in\mathcal{S}'(\mathbb{R}^{2n})$ then the formula

   $$(B\psi)\phi) =\beta(\phi \boxtimes\psi ) \forall \phi,\psi \in\mathcal{S}(\mathbb{R}^n)$$ (2)

   
   
   defines a continuous linear map $B:\mathcal{S}(\mathbb{R}^n)\longrightarrow\mathcal{S}'(\mathbb{R}^n).$
9. Going back to (6) show that $a$ thought of as a distribution defines $A'$ in this way.
10. Conclude that every continuous linear operator (0.1) arises from the construction (0.2).
11. If you get this far and still have the energy, show that conversely $\beta$ is determined by $B$ through (0.2) and finally arrive at Schwartz kernel theorem:

12. Theorem 0.1   There is a 1-1 correspondence between continuous linear maps

    $$A:\mathcal{S}(\mathbb{R}^n)\longrightarrow \mathcal{S}(\mathbb{R}^{n'})$$ (3)

    
    

    and elements of $\mathcal{S}'(\mathbb{R}^{n+n'}).$

Problem 2   Work out the elementary behavior of the heat equation.

i)

Show that the function on $\mathbb{R}\times\mathbb{R}^n,$ for $n\ge1,$

$$F(t,x)=\begin{cases}t^{-\frac n2}\exp\left(-\frac{\vert x\vert^2}{4t}\right) &t>0\ 0& t\le0 \end{cases}$$

is measurable, bounded on the any set $\{\vert(t,x)\vert\ge R\}$ and is integrable on $\{\vert(t,x)\vert\le R\}$ for any $R>0.$

ii)

Conclude that $F$ defines a tempered distibution on $\mathbb{R}^{n+1}.$

iii)

Show that $F$ is $\mathcal{C}^{\infty}$ outside the origin.

iv)

Show that $F$ satisfies the heat equation

$$(\partial_t-\sum\limits_{j=1}^n\partial^2_{x_j})F(t,x)=0 (t,x)\not=0.$$

v)

Show that $F$ satisfies

$$F(s^2t,sx)=s^{-n}F(t,x) \mathcal{S}'(\mathbb{R}^{n+1})$$ (4)

where the left hand side is defined by duality `` $F(s^2t,sx)=F_s$'' where

$$F_s(\phi)=s^{-n-2}F(\phi_{1/s}), \phi_{1/s}(t,x)=\phi(\frac t{s^2},\frac xs).$$

vi)

Conclude that

$$(\partial_t-\sum\limits_{j=1}^n\partial^2_{x_j})F(t,x)=G(t,x)$$

where $G(t,x)$ satisfies

$$G(s^2t,sx)=s^{-n-2}G(t,x) \mathcal{S}'(\mathbb{R}^{n+1})$$ (5)

in the same sense as above and has support at most $\{0\}.$

vii)

Hence deduce that

$$(\partial_t-\sum\limits_{j=1}^n\partial^2_{x_j})F(t,x)=c\delta (t)\delta (x)$$ (6)

for some real constant $c.$

Hint: Check which distributions with support at $(0,0)$ satisfy (0.5).

viii)

If $\psi \in\mathcal{C}^{\infty}_c(\mathbb{R}^{n+1})$ show that $u=F\star\psi$ satisfies

$$\begin{multline} u\in\mathcal{C}^{\infty}(\mathbb{R}^{n+1})\text{ and }\\ \sup\... ...u(t,x)\vert<\infty\ \forall S>0,\alpha \in\mathbb{N}^{n+1}, N. \end{multline}$$

ix)

Supposing that $u$ satisfies (0.7) and is a real-valued solution of

$$(\partial_t-\sum\limits_{j=1}^n\partial^2_{x_j})u(t,x)=0$$

in $\mathbb{R}^{n+1},$ show that

$$v(t)=\int_{\mathbb{R}^n} u^2(t,x)$$

is a non-increasing function of $t.$

Hint: Multiply the equation by $u$ and integrate over a slab $[t_1,t_2]\times\mathbb{R}^n.$

x)

Show that $c$ in (0.6) is non-zero by arriving at a contradiction from the assumption that it is zero. Namely, show that if $c=0$ then $u$ in viii) satisfies the conditions of ix) and also vanishes in $t<T$ for some $T$ (depending on $\psi ).$ Conclude that $u=0$ for all $\psi.$ Using properties of convolution show that this in turn implies that $F=0$ which is a contradiction.

xi)

So, finally, we know that $E=\frac1cF$ is a fundamental solution of the heat operator which vanishes in $t<0.$ Explain why this allows us to show that for any $\psi\in\mathcal{C}^{\infty}_c(\mathbb{R}\times\mathbb{R}^n)$ there is a solution of

$$(\partial_t-\sum\limits_{j=1}^n\partial^2_{x_j})u=\psi, u=0 t<T T.$$ (7)

What is the largest value of $T$ for which this holds?

xii)

Can you give a heuristic, or indeed a rigorous, explanation of why

$$c=\int_{\mathbb{R}^n}\exp(-\frac{\vert x\vert^2}4)dx?$$