Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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 $p,p^{\ast}$ are such that MATH The $p^{\ast}$ is called "Sobolev conjugate" of $p$ .

Note that if an estimate of the form MATH holds with a constant $C$ independent of $u$ then $q=p^{\ast}$ . Such conclusion follows from the substitution of MATH in place of the $u$ .

Proposition

(Gagliardo-Nirenberg-Sobolev inequality). Let $1\leq p<n$ . There exists a constant $C$ depending only on $n$ and $p$ such that MATH

Proof

Since MATH has compact support we state MATH Hence, MATH Consequently, MATH The $i$ -th term of the product is independent of $x_{1}$ , hence, we continue MATH Note that according to the formula ( Holder inequality 2 ),

MATH (HI1)
We apply the last relationship with MATH and obtain MATH

Next, we integrate the above relationship with respect to the $x_{2}$ : MATH

We apply the formula ( HI1 ) to the $dx_{2}$ -integral with MATH and MATH and obtain MATH

Next, we integrate with respect to the $x_{3}$ : MATH and apply the formula ( HI1 ) to the $dx_{3}$ -integral with MATH , MATH and MATH : MATH We continue the integration with respect to the variables $x_{4}$ ,..., $x_{n}$ and arrive to the estimate MATH The above is the desired estimate for $p=1$ .

To obtain the estimate for $1<p<\infty$ it is enough to apply the last result to the function MATH with MATH .

Proposition

( $L^{p^{\ast}}$ vs $W^{1,p}$ estimate). Let $U$ be a bounded open subset of $\QTR{cal}{R}^{n}$ and $\partial U$ admits a locally continuously differentiable parametrization. Let $1\leq p<n$ and MATH . Then MATH and MATH where the constants $C_{1},C_{2}$ depend only on $p,n$ and $U$ .

Proof

First we produce an extension $\bar{u}$ of $u$ to $\QTR{cal}{R}^{n}$ 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 MATH , any MATH and any radius $0<r<\infty$ MATH

Proof

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

Proposition

(Morrey inequality). Let $n<p\leq\infty$ . There exists a constant $C$ , depending only on $p$ and $n$ , such that MATH where MATH .

Proof

For MATH we write MATH We estimate the first term using the proposition ( Average difference lemma ): MATH We apply the formula ( Holder inequality ) with MATH . Note that MATH because $n<p$ . MATH for a $C_{2}$ depending only on $p$ and $n$ .

We estimate the second term using the formula ( Holder inequality ) with $v=1$ : MATH where the $C_{3}$ depends only on $p$ and $n$ . We conclude MATH for any MATH and the constant MATH depending only on $p$ and $n$ . Therefore,

MATH (C to W1p estimate)

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

MATH (Cg to W1p estimate)

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

Proposition

(C0gamma vs W1p estimate). Let $U$ be a bounded open subset of $\QTR{cal}{R}^{n}$ and $\partial U$ admits a locally continuously differentiable parametrization. Let $n<p\leq\infty$ and MATH . Then there exists a MATH such that $u=u^{\ast}$ a.s. and MATH where the constant $C$ depends only on $p,n$ and $U$ .

Proof

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





Notation. Index. Contents.


















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