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

**Theorem 1.** Let be differentiable with bounded and continuous derivatives and Then and

In particular if for all then and the chain rule holds provided

Let be open sets in We say is **compactly contained in** denoted if for any such that is compact. We now extend functions in to functions on

**Theorem 2.** Let be bounded, be a -manifold, and Then there exists a linear map such that for all we have

- where depends upon and

*Proof Outline*. Let and be the tangent hyperplane at –which satisfies Since is locally homeomorphic to approximately bisects a small ball around one which intersects with and one that does not. We will respectively call these and Suppose and define

Note that the two parts of the function agree on and that since if we let then

on (where .) Thus derivatives agree on Thus

then allows one to use partitions of unity on to write in local coordinates and establish the bounds in (3). (2) is then established by continuously sending the function to in or a slightly larger set. is then defined as elsewhere.

Above we assumed If we assume more generally that as in the hypotheses, then we can approximate (as seen in the last post) by functions in (a special case of which is on if is bounded and is a -manifold [see 5.3.2 and 5.3.3 in [2]]) which yields the result with consequence (1).

Lastly we begin with a function and attempt to extend it to the boundary.

**Theorem 3.** Let be bounded and be a -manifold. Then there exists a bounded linear operator such that

- if

The function is called the **trace** of on We also have

**Proposition 4.** Again let be bounded and be a -manifold. Let Then

This sort of seems intuitive since it’s like saying the summable extension is on the boundary iff it belongs to the class of functions whose derivatives vanish on the boundary. Recall elements of are functions whose support is a compact subset of or whose support extends to The proof is rather technical in the forward direction (see [2] for details). The backwards direction uses density of in

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 be a an -dimensional, oriented, and compact manifold with boundary such that is an -form on Then

Since its derivatives were integrable (i.e. the left side), and under the trace 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 and be the collection of functions (or if preferred) such that

It turns out are Banach spaces (see [2] for proof), called **Sobolev spaces**, under the above norm. In particular The spaces are Hilbert with inner product

We can also define the **local Sobolev spaces** as the subset of such that for We have

and where under dictionary ordering on provided has a finite Lebesgue measure. Thus in particular

is a -module by a simple boundedness argument following from: if and then is finite in Thus Correspondingly this gives us action on (which denotes the distributions on for us, although the action works on the dual space as well) defined by

**Theorem 1.** Let be bounded in and Then there exists a sequence of functions such that

We can also define (and similarly ) as the set of functions such that the following norm is finite

Thus

I’m wondering if we could consider attempting to define (and similarly ) as the collection of functions such that

is finite. This would require all derivatives to eventually get to in such a way that the sum converges. It would thus include polynomials if is bounded and hence be nonempty. In particular it would obey for all And for all

[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 and denote the set of smooth functions on with compact support.

**Definition 1.** Let and be a sequence in We say **converges to ** denoted if

- for all for a fixed compact
- uniformly on

By we mean

is a vector space, and the above notion of convergence allows us to define closed sets and hence turns into a topological vector space called the **space of test functions** of and we denote it

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

We say a sequence of distributions **converges** to the distribution denoted if for all This gives us a topological vector space structure on as well.

**Definition 3.** Let be a distribution and We define the -th **distributional (weak) derivative** of by

where We may also write

We will omit if The **gradient of** is simply

Let us define the space as the space of functions such that

for all compact We define **strong convergence** on as convergence in and **weak convergence** on as weak (pointwise) convergence in for all compact

Let then clearly

Let and and define

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

where which gives us the continuity of So We also have

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

# Metric Construction of Buildings

Let be a Coxeter group where is generated by elements. Also by for we mean the minimal length is a product decomposition of Let us first use the book definition.

**Definition 1.** A **Weyl distance function** is a map where a set whose elements are called chambers such that

- if and such that then If we also have that then
- if then for any we have a chamber such that and

The pair is called a **– metric space**. The triple is called a **building**.

I’m fairly certain I copied correctly (triple checked), but it looks like or the first sentence in 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 then can be thought of as a product on By condition the element is thus not an identity with respect to this product. Also in this regard one can show that the chambers of a Coxeter complex form a building where

with being the gallery metric we previously defined.

Conversely we can say two elements in are **-adjacen**t if and **-equivalent** if they are either -adjacent or equal. If and are -equivalent, we write This is an equivalence relation since

**Proposition 2.** if one takes a generator value.

*Proof.* Suppose By part of the definition there is a such that and Thus by we have and thus

The equivalence classes under are called **-panels**. A **panel** is an -panel for some Galleries can be defined similarly with this terminology. Thus

[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 with counting measure where and is called the **monetary constant**, is a collection of subsets of such that elements of which are called **owners**, and is called the **worth of ** for an owner

Note we are not requiring to be closed under any operations (i.e. it is not an algebra of sets). Suppose we have two structures on and Let (i.e. a multi-valued map into ). 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** as a map by

Again, need not be in but we can of course still define the counting measure on it.

**Definition 2.** A **composite trade** is a map where and are trades.

Note that since it is defined on the image of simply evaluates on all sets in

**Definition 3.** Let be a continuum of static capital systems. We say is a **capital system** if

- for every and there is a unique trade
- (i.e. );
- if and are trades such that then for all

**Example 4.** A capital system is in a **socialist state** at time if for all We may further say is **socialist** during if is in a socialist state for all A capital system is in a **communist state** at time if Similarly we have the definition for **communist** during a set

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 where is a static capital system for all where and are comparable (in the inclusion sense) for all In particular the function defined by is called the **monetary policy**. If is strictly increasing during an interval, we say is **expansionary** during that interval. Similarly it is **contractionary** if it is strictly decreasing on some interval.

**Definition 6.** A dynamic capital system is **rational** if for all

Of course if is we have and thus the condition is satisfied for this case:

So in a rational dynamic capital system we have the inequality

with If exists and is finite, then the rational dynamic capital system is said to have an **end game**.