SYLLABUS
March 24, 2008
The course is to introduce some interesting theorems and useful tools in real analysis and some new point of view for Sobolev spaces.
To understand rigourously and systematically the theory, you need to know Lebesgue measure on Rn, Sobolev spaces on Rn . Nevertheless, with small background of the theory of integration, you can understand almost important ideas of the theory. This is the goal of the course.
1 Covering theorems
1.2 Besicovitch’s covering theorem
2 Maximal functions
3 Some fine properties of functions
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3.2 Approximate continuity, approximate differentiability, and Lp differentiability.
3.3 Lp∗ differentiability a.e. for W 1,p(Rn) (1 ≤ p<n)
4 Connection between Poincare’s inequality and Sobolev’s inequality
4.1 Whitney’s decomposition
4.2 Calderon and Zygmund’s decomposition
4.3 Sharp functions: Definition and properties
4.4 Connection between Sobolev and Poincare inequalities. Embedding theorems.
Summary and Indication for Bibliography
In the first chapter, the fundamental covering theorems of Vitali and of Besicovitch will be presented and proved in details. One of the useful applications of these results is the theory of maximal functions. This theory will be presented in the first section of Chapter 2. The last two sections of this chapter present some interesting applications of this theory: the Lebesgue differentiation theorem and a characterization of Sobolev spaces due to J. Bourgain, H. Brezis, and P. Mironescu. Chapter 3 will introduce some fine properties of functions. In the first section of this chapter, we prove Rademacher’s theorem which says that every Lipschitz function is differentiable a.e. on Rn . The goal of last two sections is to obtain some useful properties of functions in W 1,p(Rn). In Chapter 4, we present a connection between Poincare’s inequality and Sobolev’s inequality. Under this point of view we present a proof of Sobolev’s embedding (this proof also works for the fractional Sobolev spaces !!!). The presentation of Chapter 4 depends on time and the motivation of the class.
Reference for Measure theory: [7] and [5, Chapter 1]. Reference for Sobolev spaces:
[2] and [4]. Reference for Chapter 1: [5, Chapter 1]. Reference for Chapter 2: [8, Chapter 1] see also in [9]. The characterization of J. Bourgain, H. Brezis, and P. Mironescu is in
[1] and [3]. Reference for Chapter 3: [5, Section 1 of Chapter 3] and [5, Sections 1 and 2 of Chapter 6]. Reference for Chapter 4: Theory of sharp functions: [9, Section 3 of Chapter 4] and [9, Chapter 5] for more advanced topics. Connection between the Sobolev inequality and the Poincare inequality is in [6].
References
[1] J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations (J. L. Menaldi, E. Rofman, and A. Sulem, eds.) a volume in honour of A. Bensoussan’s 60th birthday (2001), pp. 439–455.
[2] H. Brezis, Analyse Fonctionnelle. Th´eorie et applications, Math´ematiques appliqu´ees pour la maˆıtrise, Dunod, 2002.
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[3] , How to recognize constant functions. A connections with Sobolev spaces, Volume in honor of M. Vishik, Uspekhi Mat. Nauk 57 (2002), 59–74; English translation in Russian Math. Surveys 57 (2002), 693–708.
[4] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
[5] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
[6] P. Hajlasz and P. Koskela, Sobolev met Poincar´e, Memoirs Amer. Math. Soc. 145 (2000), 1–101.
[7] W. Rudin, Real and Complex Analysis, third ed. McGraw-Hill Book Co., NewYork, 1987.
[8] E. Stein, Singular integrals and differentiability propeties of functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, 1970.
[9] , Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, vol. 43, Princeton University Press, Princeton, 1993.
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