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 | Hajłasz P. Weakly differentiable mappings between manifolds / Hajł
asz P., Iwaniec T., Malý J., Onninen J. - Providence: AMS, 2008. - 72 p. - (Memoirs of the American mathematical society; N 899). - ISSN 0065-9266; ISBN 0821840797
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Chapter 1. Introduction ......................................... 1
Chapter 2. Preliminaries Concerning Manifolds ................... 7
2.1. Manifolds ................................................ 7
2.2. The Sobolev space W1,p(X,Y) ............................... 8
2.3. Differential forms ....................................... 9
2.4. Mollifiers and smoothing operator ....................... 15
2.5. Maximal operators ....................................... 16
Chapter 3. Examples ............................................ 21
3.1. The longitude projection ................................ 21
3.2. Spherical coordinates ................................... 22
3.3. Winding around the longitude circles .................... 22
3.4. A mapping of infinite degree ............................ 23
Chapter 4. Some Classes of Functions ........................... 25
4.1. Marcinkiewicz space Lpweak(X) ............................ 25
4.2. The space Lαp(X) ........................................ 26
4.3. The Orlicz space Lp ..................................... 28
4.4. Grand GLp-space ......................................... 31
4.5. Relations between spaces ................................ 32
4.6. Sobolev classes ......................................... 35
Chapter 5. Smooth Approximation ................................ 37
5.1. Web like structures ..................................... 37
5.2. Vanishing web oscillations .............................. 37
5.3. Statements of the results ............................... 38
5.4. Proof of Theorem 5.1 .................................... 39
5.5. Spinning a web on X ..................................... 42
5.6. Proof of Theorems 1.1 and 1.2 ........................... 44
5.7. Proof of Theorem 5.2 .................................... 45
5.8. Proof of Theorem 1.3 .................................... 45
Chapter 6. L1-Estimates of the Jacobian ........................ 47
6.1. Weak wedge products ..................................... 48
6.2. Distributional Jacobian ................................. 49
6.3. Proof of Theorem 6.5 .................................... 51
Chapter 7. H1-Estimates ........................................ 55
7.1. The Hausdorff content ................................... 55
7.2. The H1-Theorem .......................................... 56
Chapter 8. Degree Theory ....................................... 63
8.1. Definition of the degree via weak integrals ............. 63
8.2. Weak integrals .......................................... 64
8.3. Stability of the degree ................................. 66
8.4. The degree in Orlicz and grand Sobolev spaces ........... 67
Chapter 9. Mappings of Finite Distortion ....................... 69
Acknowledgements ............................................. 70
Bibliography ................................................... 71
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