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.

## Traces of Sobolev spaces.

roposition

(Trace theorem). Assume that is a bounded set and admits a locally continuously differentiable parametrization. The is restricted . There exists a bounded linear operator such that for any

Proof

The statement is true for a flat boundary and . If the boundary is not flat then there is a change of variables that makes it locally flat. Then procedure extends globally by partition of unity (see the proof of the proposition ( Global approximation by smooth functions ) for an example of the technique). The procedure extends to by closure of in .

Proposition

(Trace-zero functions in ) Assume that is a bounded set and admits a locally continuously differentiable parametrization. The is restricted . Let . Then

 Notation. Index. Contents.