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.

## Sobolev inequalities.

efinition

(Sobolev conjugate). Let are such that The is called "Sobolev conjugate" of .

Note that if an estimate of the form holds with a constant independent of then . Such conclusion follows from the substitution of in place of the .

Proposition

(Gagliardo-Nirenberg-Sobolev inequality). Let . There exists a constant depending only on and such that

Proof

Since has compact support we state Hence, Consequently, The -th term of the product is independent of , hence, we continue Note that according to the formula ( Holder inequality 2 ),

 (HI1)
We apply the last relationship with and obtain

Next, we integrate the above relationship with respect to the :

We apply the formula ( HI1 ) to the -integral with and and obtain

Next, we integrate with respect to the : and apply the formula ( HI1 ) to the -integral with , and : We continue the integration with respect to the variables ,..., and arrive to the estimate The above is the desired estimate for .

To obtain the estimate for it is enough to apply the last result to the function with .

Proposition

( vs estimate). Let be a bounded open subset of and admits a locally continuously differentiable parametrization. Let and . Then and where the constants depend only on and .

Proof

First we produce an extension of to according to the proposition ( Extension theorem ). Then we approximate the extension with smooth functions according to the proposition ( Local approximation by smooth functions ). Finally, we apply the proposition ( Gagliardo-Nirenberg-Sobolev inequality ) to the sequence of the smooth approximations and pass it to the limit using the estimate part of the propositions ( Local approximation by smooth functions ) and ( Extension theorem ).

Proposition

(Average difference lemma). For any , any and any radius

Proof

We switch to the -dimensional polar coordinates: Hence, the becomes and the area element of a unit sphere is We substitute the change into the integral here the denotes all values of spanning the unit sphere , the is the direction vector, . We proceed to estimate the integral: We change the order of integration: Hence, the constitutes the volume element , , : We conclude

Proposition

(Morrey inequality). Let . There exists a constant , depending only on and , such that where .

Proof

For we write We estimate the first term using the proposition ( Average difference lemma ): We apply the formula ( Holder inequality ) with . Note that because . for a depending only on and .

We estimate the second term using the formula ( Holder inequality ) with : where the depends only on and . We conclude for any and the constant depending only on and . Therefore,

 (C to W1p estimate)

Fix . Let and . We have We follow the same path that lead to the formula ( C to W1p estimate ): Acting similarly, we obtain Therefore, Since we divide by and obtain Since , we conclude

 (Cg to W1p estimate)

The formulas ( C to W1p estimate ) and ( Cg to W1p estimate ) conclude the proof.

Proposition

(C0gamma vs W1p estimate). Let be a bounded open subset of and admits a locally continuously differentiable parametrization. Let and . Then there exists a such that a.s. and where the constant depends only on and .

Proof

The proof is similar to the proof of the proposition ( Lp vs W1p estimate ).

 Notation. Index. Contents.
 Copyright 2007