Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 2 Laws of large numbers.
 3 Characteristic function.
 4 Central limit theorem (CLT) II.
 5 Random walk.
 6 Conditional probability II.
 7 Martingales and stopping times.
 8 Markov process.
 9 Levy process.
 10 Weak derivative. Fundamental solution. Calculus of distributions.
 11 Functional Analysis.
 12 Fourier analysis.
 13 Sobolev spaces.
 A. Convolution and smoothing.
 B. Approximation by smooth functions.
 C. Extensions of Sobolev spaces.
 D. Traces of Sobolev spaces.
 E. Sobolev inequalities.
 F. Compact embedding of Sobolev spaces.
 G. Dual Sobolev spaces.
 H. Sobolev spaces involving time.
 I. Poincare inequality and Friedrich lemma.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

Compact embedding of Sobolev spaces.

efinition

Let and be Banach spaces and . The is "compactly embedded" in (notation ) if any bounded sequence has a -convergent subsequence.

Proposition

(Uniformly smooth approximation lemma) Suppose is an open bounded subset of and admits a locally continuously differentiable parametrization. For any -bounded sequence there is a family of functions such that for each . Furthermore, the sequence is uniformly bounded and equicontinuous for every .

Proof

We seek such family using the standard mollifiers, see the definition ( Standard mollifier definition ). We introduce : we substitute , : Note that the argument lies outside of for close to the boundary . We solve this problem via the proposition ( Extension theorem ): we extend the to a slightly larger set and perform the convolution using the extended functions.

We initially assume that the functions are smooth and estimate note that , Hence, Since the support of the extended function lies strictly within , we can drop the small shift in the integral without changing the integral: Hence, According to the proposition ( Approximation by smooth functions ) this estimate extends to the functions .

Finally, we use the boundedness of (and ) and the formula ( Holder inequality ) with : Hence, Thus we established that

Next we intend to the other values of using the formula ( Lp interpolation ):

We aim to establish that dominates . Hence, we match , and . Then Hence, The sequences are bounded in . Hence, by the proposition ( Gagliardo-Nirenberg-Sobolev inequality ) these are bounded in . Therefore, the term in the last estimate is bounded and, consequently,

To see the second part of the statement we verify that and are bounded for every :

Proposition

(Rellich-Kondrachov compactness theorem). Let be a bounded open subset of and admits a locally continuously differentiable parametrization. Suppose . Then

Proof

Let be a -bounded sequence. We utilize the sequence from the proposition ( Uniformly smooth approximation lemma ): and, according to the proposition ( Arzela-Ascoli compactness criterion ),

Since is bounded, it follows that Let , . For we find such that and for such we find the subindexing such that Then for we find such that and for such we find the subindexing of such that and we continue so indefinitely.

Then Hence, the is the desired subindexing that turns into an -convergent sequence.

Proposition

(W1p embedding). Let be a bounded open subset of and admits a locally continuously differentiable parametrization ( is ). Then for

Proof

For and we have . Hence, by the proposition ( Rellich-Kondrachov compactness theorem )

For we use the proposition ( C0gamma vs W1p estimate ) and note that the structure of norm implies that a -bounded sequence satisfies conditions of the proposition ( Arzela-Ascoli compactness criterion ). We then follow the proof of the proposition ( Rellich-Kondrachov compactness theorem ) and arrive to To see the embedding for note from the proposition ( Rellich-Kondrachov compactness theorem ) that if . The is bounded, hence for Therefore, if is a -bounded sequence then it is also a -bounded sequence and we can choose so that . Then the proposition ( Rellich-Kondrachov compactness theorem ) provides existence of -convergent subsequence.

 Notation. Index. Contents.