Chain Rule, Extensions, and Trace in W^{1,p}

Theorem 1.  Let $f:\mathbb{R}^n\to\mathbb{C}$ be differentiable with bounded and continuous derivatives and $g\in W_{loc}^{1,p}(\Omega).$  Then $h=f\circ g\in W_{loc}^{1,p}(\Omega)$ and

$\displaystyle\partial_i h=\sum_j\partial_j f(g)\,\partial_i g_j.$

In particular if $g_j\in W^{1,p}(\Omega)$ for all $j,$ then $h\in W^{1,p}(\Omega)$ and the chain rule holds provided $f(0)=0.$

Let $U,V$ be open sets in $\mathbb{R}^n.$  We say $U$ is compactly contained in $V,$ denoted $U\sqsubset V$ if for any $U'$ such that $U\subset U'\subset V,$ $U'$ is compact.  We now extend functions in $W^{1,p}(\Omega)$ to functions on $W^{1,p}(\mathbb{R}^n).$

Theorem 2.  Let $\Omega$ be bounded, $\partial\Omega$ be a $C^1$-manifold, and $\Omega\sqsubset V.$  Then there exists a linear map $E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n)$ such that for all $f\in W^{1,p}(\Omega)$ we have

1. $Ef=f\mbox{~a.e. in~}U,$
2. $\mbox{supp~}E\subseteq V,$
3. $\|Ef\|_{W^{1,p}(\mathbb{R^n})}\leq C\|f\|_{W^{1,p}(\Omega)}$ where $C$ depends upon $p,\Omega,$ and $V.$

Proof Outline.  Let $x_0\in\partial\Omega$ and $\Pi$ be the tangent hyperplane at $x_0$–which satisfies $x_n=0.$  Since $\partial\Omega$ is locally homeomorphic to $\mathbb{R}^{n-1},$ $\Pi$ approximately bisects a small ball around $x_0:$ one which intersects with $\Omega$ and one that does not.  We will respectively call these $B^{+}$ and $B^{-}.$  Suppose $f\in C^1(\overline{\Omega})$ and define

$\displaystyle\overline{f}(x)=\left\{\begin{array}{lcl}\displaystyle f(x)&\mbox{if}&x\in B^{+}\\\displaystyle -3f(x_1,...,x_{n-1},-x_n)+4f(x_1,...,x_{n-1},\frac{-x_n}{2})&\mbox{if}&x\in B^{-}\end{array}\right.$

Note that the two parts of the function agree on $\partial\Omega$ and that $\overline{f}\in C^1(B)$ since if we let $f^{-}=\overline{f}|_{B^{-}},$ then

$\displaystyle\partial_nf^{-}(x)=3\frac{\partial f}{\partial x_n}(x_1,...,-x_n)-2\frac{\partial f}{\partial x_n}(x_1,...,\frac{-x_n}{2})=\partial_n f$

on $\partial\Omega$ (where $x_n=0$.)  Thus derivatives agree on $\Pi.$  Thus $\overline{f}\in C^1(B).$

$\Omega\sqsubset V$ then allows one to use partitions of unity on $\partial\Omega$ to write $f$ in local coordinates and establish the bounds in (3).  (2) is then established by continuously sending the function to $0$ in $B^{-},$ or a slightly larger set.  $\overline{f}$ is then defined as $0$ elsewhere.

Above we assumed $f\in C^1(\overline{\Omega}).$  If we assume more generally that $f\in W^{1,p}(\Omega)$ as in the hypotheses, then we can approximate (as seen in the last post) by functions in $C^1(\Omega)$ (a special case of which is on $C^1(\overline{\Omega})$ if $\Omega$ is bounded and $\partial\Omega$ is a $C^1$-manifold [see 5.3.2 and 5.3.3 in [2]]) which yields the result with consequence (1).

Lastly we begin with a function $f\in W^{1,p}(\Omega)$ and attempt to extend it to the boundary.

Theorem 3.  Let $\Omega$ be bounded and $\partial\Omega$ be a $C^1$-manifold.  Then there exists a bounded linear operator $T:W^{1,p}(\Omega)\to L^p(\partial\Omega)$ such that

1. $Tf=f|_{\partial\Omega}$ if $f\in W^{1,p}(\Omega)\cap C(\overline{\Omega}),$
2. $\|Tf\|_{L^p(\partial\Omega)}\leq C\|f\|_{W^{1,p}(\Omega)}.$

The function $T$ is called the trace of $f$ on $\partial\Omega.$  We also have

Proposition 4.  Again let $\Omega$ be bounded and $\partial\Omega$ be a $C^1$-manifold.  Let $f\in W^{1,p}(\Omega).$  Then

$\displaystyle Tf=0\mbox{~on~}\partial\Omega\Leftrightarrow f\in W_0^{1,p}(\Omega).$

This sort of seems intuitive since it’s like saying the summable extension is $0$ on the boundary iff it belongs to the class of functions whose derivatives vanish on the boundary.  Recall elements of $W_0^{1,p}(\Omega)$ are functions whose support is a compact subset of $\Omega$ or whose support extends to $\partial\Omega.$  The proof is rather technical in the forward direction (see [2] for details).  The backwards direction uses density of $C_c^\infty(\Omega)$ in $W^{1,p}(\Omega).$

Regarding Theorem 3, at first glance this reminded me of Stokes’ theorem on manifolds, although Evans doesn’t mention anything about it (nor uses it in either of the two proofs).  Recall the statement.

Stokes’ Theorem.  Let $M$ be a an $n$-dimensional, $C^k,$ oriented, and compact manifold with boundary such that $\omega$ is an $(n-1)$-form on $M.$  Then

$\displaystyle\int_M\,d\omega=\int_{\partial M}\omega.$

Since $f\in W^{1,p}(\Omega),$ its derivatives were integrable (i.e. the left side), and under the trace $f$ is sent to a function which is integrable on the boundary.

[1]  Lieb, Elliot and Michael Loss.  Analysis.  2nd Edition.  Graduate Studies in Mathematics.  Vol. 14.  American Mathematical Society.  2001.

[2]  Evans, Lawrence.  Partial Differential Equations.  Graduate Studies in Mathematics.  Vol. 19.  American Mathematical Society.  1998.

Sobolev Spaces

Let $\Omega\subseteq\mathbb{R}^n$ and $W^{k,p}(\Omega)$ be the collection of functions $f:\Omega\to\mathbb{R}$ (or $\mathbb{C}$ if preferred) such that

$\displaystyle\|f\|_{k,p}=\left(\sum_{|\alpha|\leq k}\left\|D^\alpha f\right\|_p^p\right)^{1/p}<\infty.$

It turns out $W^{k,p}(\Omega)$ are Banach spaces (see [2] for proof), called Sobolev spaces, under the above norm.  In particular $W^{0,p}(\Omega)=L^p(\Omega).$  The spaces $H^k(\Omega)=W^{k,2}(\Omega)$ are Hilbert with inner product

$\displaystyle\langle f,g\rangle_{H^k}=\sum_{|\alpha|\leq k}\int_\Omega\overline{D^\alpha f}D^\alpha g\,dx$

We can also define the local Sobolev spaces $W_{loc}^{k,p}(\Omega)$ as the subset of $L_{loc}^p(\Omega)$ such that $D^\alpha f\in L_{loc}^p(\Omega)$ for $|\alpha|\leq k.$  We have

$W_{loc}^{r,s}(\Omega)\subseteq W_{loc}^{k,p}(\Omega)\subseteq L_{loc}^p(\Omega),$

$W^{r,s}(\Omega)\subseteq W^{k,p}(\Omega)\subseteq W^{0,p}(\Omega)=L^p(\Omega)\subseteq L_{loc}^p(\Omega),$

and $W^{r,s}(\Omega)\subseteq W_{loc}^{r,s}(\Omega)$  where $(k,p)\leq (r,s)$ under dictionary ordering on $\mathbb{N}^2$ provided $\Omega$ has a finite Lebesgue measure.  Thus in particular

$W^{r,s}(\Omega)\subseteq W_{loc}^{r,s}(\Omega)\cap W^{k,p}(\Omega).$

$C_c^\infty(\Omega)$ is a $C^\infty(\Omega)$-module by a simple boundedness argument following from:  if $f\in C^\infty(\Omega)$ and $\phi\in C_c^\infty(\Omega),$ then $\sup f$ is finite in $\mbox{supp~}\phi.$  Thus $f\phi\in C_c^\infty(\Omega).$  Correspondingly this gives us $C^\infty(\Omega)$ action on $D(\Omega)^*$ (which denotes the distributions on $\Omega$ for us, although the action works on the dual space as well) defined by

$(fT)(\phi):=T(f\phi).$

Theorem 1.  Let $\Omega$ be bounded in $\mathbb{R}^n$ and $f\in W^{k,p}(\Omega).$  Then there exists a sequence of functions $f_m\in C^\infty(\Omega)\cap W^{k,p}(\Omega)$ such that

$\|f-f_m\|_{k,p}\to 0.$

We can also define $W^{k,\infty}(\Omega)$ (and similarly $W_{loc}^{k,\infty}(\Omega)$) as the set of functions $f:\Omega\to\mathbb{R}$ such that the following norm is finite

$\displaystyle\|f\|_{k,\infty}=\sum_{|\alpha|\leq k}\mbox{ess}\,\sup|D^\alpha f|.$

Thus $W^{0,\infty}(\Omega)=L^\infty(\Omega).$

I’m wondering if we could consider attempting to define $W^{\infty,p}(\Omega)$ (and similarly $W^{\infty,\infty}(\Omega)$) as the collection of functions $f:\Omega\to\mathbb{R}$ such that

$\displaystyle\|f\|_{\infty,p}=\left(\sum_{\alpha\in\mathbb{N}^n}\|D^\alpha f\|_p^p\right)^{1/p}$

is finite.  This would require all derivatives to eventually get to $0$ in such a way that the sum converges.  It would thus include polynomials if $\Omega$ is bounded and hence be nonempty.  In particular it would obey $W^{\infty,p}(\Omega)\subseteq W^{k,p}(\Omega)$ for all $k.$  And $W^{\infty,\infty}(\Omega)\subseteq W^{k,p}(\Omega)$ for all $k,p.$

[1]  Lieb, Elliot and Michael Loss.  Analysis.  2nd Edition.  Graduate Studies in Mathematics.  Vol. 14.  American Mathematical Society.  2001.

[2]  Evans, Lawrence.  Partial Differential Equations.  Graduate Studies in Mathematics.  Vol. 19.  American Mathematical Society.  1998.

Distributions

Let $\Omega\subseteq\mathbb{R}^n$ and $C_c^\infty(\Omega)$ denote the set of smooth functions on $\Omega$ with compact support.

Definition 1.  Let $\alpha\in\mathbb{N}^n$ and $\{\phi_k\}$ be a sequence in $C_c^\infty(\Omega).$  We say $\{\phi_k\}$ converges to $\phi\in C_c^\infty(\Omega),$ denoted $\phi_k\to\phi,$ if

1. $\mbox{supp~}\phi_k-\phi\subseteq K$ for all $k$ for a fixed compact $K\subseteq\Omega,$
2. $D^\alpha\phi_m\to D^\alpha\phi$ uniformly on $K.$

By $D^\alpha\phi$ we mean

$\displaystyle\frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}}\cdots\frac{\partial^{\alpha_n}}{\partial x_n^{\alpha_n}}\,\phi.$

$C_c^\infty(\Omega)$ is a vector space, and the above notion of convergence allows us to define closed sets and hence turns $C_c^\infty(\Omega)$ into a topological vector space called the space of test functions of $\Omega,$ and we denote it $D(\Omega).$

Definition 2.  A distribution is a continuous linear functional on $D(\Omega).$  We will denote the set of distributions on $\Omega$ by $D(\Omega)^*$–noting that we are only considering continuous linear functionals.

We say a sequence of distributions $\{T_k\}$ converges to the distribution $T,$ denoted $T_k\to T,$ if $T_k(\phi)\to T(\phi)$ for all $\phi\in D(\Omega).$  This gives us a topological vector space structure on $D(\Omega)^*$ as well.

Definition 3.  Let $T$ be a distribution and $\alpha\in\mathbb{N}^n.$  We define the $\alpha$-th distributional (weak) derivative of $T$ by

$D^\alpha(T)(\phi)=(-1)^{|\alpha|}T(D^\alpha\phi)$

where $|\alpha|=\sum\alpha_i.$  We may also write

$\displaystyle\partial_i^kT=(-1)^kT\left(\frac{\partial^k}{\partial x_i^k}\phi\right).$

We will omit $k$ if $k=1.$  The gradient of $T$ is simply

$\nabla T=(\partial_1T,...,\partial_nT).$

Let us define the space $L_{loc}^p(\Omega)$ as the space of functions $f:\Omega\to\mathbb{R}$ such that

$\displaystyle\|f\|_{L^p(K)}=\left(\int_K|f|^p\,dx\right)^{1/p}<\infty$

for all compact $K\subseteq\Omega.$  We define strong convergence on $L_{loc}^p(\Omega)$ as convergence in $L^p(\Omega)$ and weak convergence on $L_{loc}^p(\Omega)$ as weak (pointwise) convergence in $L^p(K)$ for all compact $K\subseteq\Omega.$

Let $1\leq p\leq q,$ then clearly

$L^q(\Omega)\subseteq L_{loc}^q(\Omega)\subseteq L_{loc}^p(\Omega).$

Let $f\in L_{loc}^1(\Omega)$ and $\phi\in D(\Omega)$ and define

$\displaystyle T_f(\phi)=\int_\Omega f\phi\,dx.$

This integral is finite since both functions are integrable (use integration by parts and establish bounds).  Also if $\phi_k\to\phi$ uniformly, then

$\begin{array}{lcl}|T_f(\phi)-T_f(\phi_k)|&=&\displaystyle\left|\int_\Omega(\phi(x)-\phi_k(x))f(x)\,dx\right|\\&\leq&\displaystyle\sup_{x\in K}|\phi(x)-\phi_k(x)|\int_K|f(x)|\,dx\\&<&\infty\end{array}$

where $K=\mbox{supp~}(\phi-\phi_k),$ which gives us the continuity of $T_f.$  So $T_f\in D(\Omega)^*.$  We also have

$\displaystyle (D^\alpha T_f)(\phi):=(-1)^{|\alpha|}\int_\Omega(D^\alpha\phi)f\,dx=\int_\Omega(D^\alpha f)\phi\,dx=T_{D^\alpha f}(\phi).$

[1]  Lieb, Elliott and Michael Loss.  Analysis.  2nd Edition.  Graduate Studies in Mathematics.  Vol. 14.  American Mathematical Society.  2001.

Metric Construction of Buildings

Let $G$ be a Coxeter group where $G$ is generated by $n$ elements.  Also by $\ell(g)$ for $g\in G$ we mean the minimal length is a product decomposition of $g.$  Let us first use the book definition.

Definition 1.  A Weyl distance function is a map $\delta:C\times C\to G$ where $C$ a set whose elements are called chambers such that

1. $\delta(x,y)=1\Leftrightarrow x=y;$
2. if $\delta(x,y)=g$ and $w\in C$ such that $\delta(w,x)=r_i,$ then $\delta(w,y)\in\{r_ig,g\}.$  If we also have that $\ell(r_ig)=\ell(g)+1,$ then $\delta(w,y)=r_ig;$
3. if $\delta(x,y)=g,$ then for any $i$ we have a chamber $w\in C$ such that $\delta(w,x)=r_i$ and $\delta(w,y)=r_ig.$

The pair $(C,\delta)$ is called a $W$– metric space.  The triple $(C,G,\delta)$ is called a building.

I’m fairly certain I copied correctly (triple checked), but it looks like $3\Rightarrow 2,$ or the first sentence in $2$ at least.  Of importance is the fact we previously mentioned:  that chambers of a Coxeter complex coincide with elements of the Coxeter group (since they are (standard) cosets of the trivial standard subgroup).  If we thus let $C=C(G_\Delta),$ then $\delta:G\times G\to G$ can be thought of as a product on $G.$  By condition $1,$ the element $1\in G$ is thus not an identity with respect to this product.  Also in this regard one can show that the chambers $C(G_\Delta)$ of a Coxeter complex form a building where

$d(x,y)=\ell(\delta(x,y))$

with $d$ being the gallery metric we previously defined.

Conversely we can say two elements in $C$ are $r_i$-adjacent if $\delta(x,y)=r_i$ and $r_i$-equivalent if they are either $r_i$-adjacent or equal.  If $x$ and $y$ are $r_i$-equivalent, we write $x\sim_{r_i}y.$  This is an equivalence relation since

Proposition 2.  $\delta(x,y)=\delta(y,x)$ if one takes a generator value.

Proof.  Suppose $\delta(x,y)=r_i.$  By part $3$ of the definition there is a $w\in C$ such that $\delta(w,x)=r_i$ and $\delta(w,y)=r_i^2=1.$  Thus by $1$ we have $w=y$ and thus

$\delta(y,x)=\delta(w,x)=r_i=\delta(x,y).$

The equivalence classes under $\sim_{r_i}$ are called $r_i$-panels.  A panel is an $r_i$-panel for some $i.$  Galleries can be defined similarly with this terminology.  Thus $(C,G,\delta)=\left(C(G_\Delta),d\right).$

[1]  Abramenko, Peter and Kenneth Brown.  Buildings.  Graduate Texts in Mathematics.  Vol. 248.  Springer Science and Business Media.  2008.

Capitalism

Definition 1.  We define a static capital system as a triple $(X,\Sigma,\mu)$ with counting measure $\mu$ where $\mu(X)=m$ and  is called the monetary constant, $\Sigma$ is a collection of subsets of $X$ such that $\mu(\cup_{A\in\Sigma}A)=m,$ elements of which are called owners, and $\mu(A)$ is called the worth of $A$ for an owner $A.$

Note we are not requiring $\Sigma$ to be closed under any operations (i.e. it is not an algebra of sets).  Suppose we have two structures on $X,$ $(X,\Sigma_1,\mu)$ and $(X,\Sigma_2,\mu).$   Let $f:\Sigma_1\to P(\Sigma_2)$ (i.e. a multi-valued map into $\Sigma_2$).  Such a function is called a trade (and may correspondingly be thought of as a change of ownership).  We define the trade utility of a trade $f$ as a map $u_f:\Sigma_1\to\mathbb{Z}$ by

$\displaystyle u_f(A)=\mu\left(\bigcup f(A)\right)-\mu(A).$

Again, $\cup f(A)$ need not be in $\Sigma_2,$ but we can of course still define the counting measure on it.

Definition 2.  A composite trade is a map $g\circ f:\Sigma_1\to P(\Sigma_3)$ where $f:\Sigma_1\to P(\Sigma_2)$ and $g:\Sigma_2\to P(\Sigma_3)$ are trades.

Note that $g\,\circ:P(\Sigma_2)\to P(\Sigma_3)$ since it is defined on the image of $f.$  $g\,\circ$ simply evaluates $g$ on all sets in $f(A).$

Definition 3.  Let $(X,\Sigma_t,\mu)_{t\geq 0}$ be a continuum of static capital systems.  We say $(X,\Sigma_t,\mu)_{t\geq 0}$ is a capital system if

1. for every $t\geq 0$ and $\varepsilon\geq t$ there is a unique trade $f_{t,\varepsilon}:\Sigma_t\to P(\Sigma_{t+\varepsilon});$
2. $f_{t,0}=1$ (i.e. $f_{t,0}(A)=\{A\}$);
3. if $f_{t,\varepsilon_1}$ and $f_{\varepsilon_1,\varepsilon_2}$ are trades such that $\varepsilon_1+\varepsilon_2=\varepsilon,$ then $f_{t,\varepsilon}=f_{\varepsilon_1,\varepsilon_2}\circ f_{t,\varepsilon_1}$ for all $t,\varepsilon_1,\varepsilon_2\geq 0.$

Example 4.  A capital system is in a socialist state at time $t$ if $\mu(A)=\mu(B)$ for all $A,B\in\Sigma_t.$  We may further say $(X,\Sigma_t,\mu)_{t\geq 0}$ is socialist during $T\subseteq[0,\infty)$ if $(X,\Sigma_t,\mu)$ is in a socialist state for all $t\in T.$  A capital system is in a communist state at time $t$ if $\Sigma_t=\{X\}.$ Similarly we have the definition for communist during a set $T\subseteq[0,\infty).$

Note that by this definition a communist state implies a socialist state.  In the above regards, a communist state can be thought of as having a single owner (say, “the people”), and socialist state has owners with equal worth.

Definition 5.  A dynamic capital system is a capital system $(X_t,\Sigma_t,\mu)_{t\geq 0}$ where $(X_t,\Sigma_t,\mu)$ is a static capital system for all $t$ where $\mu(X_t)=m_t$ and $X_t,X_s$ are comparable (in the inclusion sense) for all $s,t\geq 0.$  In particular the function $m:[0,\infty)\to\mathbb{N}$ defined by $m(t)=m_t$ is called the monetary policy.  If $m_t$ is strictly increasing during an interval, we say $(X_t,\Sigma_t,\mu)$ is expansionary during that interval.  Similarly it is  contractionary if it is strictly decreasing on some interval.

Definition 6.  A dynamic capital system $(X_t,\Sigma_t,\mu)_{t\geq 0}$ is rational if $u_{f_{t,\varepsilon}}\geq 0$ for all $t,\varepsilon\geq 0.$

Of course if $\varepsilon$ is $0$ we have $f_{t,0}=1$ and thus the condition is satisfied for this case:

$\displaystyle u_{f_{t,0}}(A)=\mu\left(\bigcup \{A\}\right)-\mu(A)=0.$

So in a rational dynamic capital system we have the inequality

$\displaystyle\mu(A)\leq\mu\left(\bigcup f_{t,\varepsilon}(A)\right)\leq m_{t+\varepsilon}$

with $A\in\Sigma_t.$  If $\lim_{t\to\infty}m_t$ exists and is finite, then the rational dynamic capital system $(X_t,\Sigma_t,\mu)$ is said to have an end game.