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Math 212A - Real Analysis Math 212a - Real Analysis Taught by Yum-Tong Siu Notes by Dongryul Kim Fall 2017 This course was taught by Yum-Tong Siu. The class met on Tuesdays and Thursdays at 2:30{4pm, and the textbooks used were Real analysis: measure theory, integration, and Hilbert spaces by Stein and Shakarchi, and Partial dif- ferential equations by Evans. There were weekly problem sets, a timed take- home midterm, and a take-home final. There were 6 students enrolled, and detailed lecture notes and problem sets might be found on the course website. Contents 1 August 31, 2017 4 1.1 Convergence of the Fourier series . .4 2 September 5, 2017 6 2.1 Measure of a set . .6 2.2 More properties of the measure . .8 3 September 7, 2017 9 3.1 Integration . .9 3.2 Egorov's theorem . 10 3.3 Convergence theorems . 10 4 September 12, 2017 12 4.1 Differentiability of nondecreasing dunctions . 12 4.2 Fundamental theorem of calculus . 14 5 September 14, 2017 15 5.1 Fundamental theorem of calculus II . 15 5.2 High-dimensional analogue . 17 6 September 19, 2017 19 6.1 The averaging problem . 19 6.2 Convergence of Fourier series . 22 1 Last Update: January 15, 2018 7 September 21, 2017 23 7.1 Dirichlet test for Fourier series . 23 7.2 Approximation to identity . 24 7.3 Fourier transform . 26 8 September 26, 2017 27 8.1 Fej´er{Lebesgue theorem . 27 8.2 Fourier inversion formula . 29 8.3 Fubini's theorem . 29 9 September 28, 2017 30 9.1 Mean value theorem . 30 9.2 H¨olderand Minkowski inequalities . 31 9.3 Hilbert space . 32 10 October 3, 2017 34 10.1 Riesz representation theorem . 34 10.2 Adjoint operator . 35 11 October 5, 2017 37 11.1 Approximating by convolutions . 38 12 October 10, 2017 40 12.1 Compact operators . 40 12.2 Spectral theorem . 41 13 October 12, 2017 43 13.1 Sturm{Liouville equation . 43 13.2 Fredholm's alternative . 44 13.3 Banach spaces . 45 14 October 17, 2017 46 14.1 Useful facts about Banach spaces . 46 14.2 Hahn{Banach theorem . 47 15 October 19, 2017 49 15.1 Calculus on manifolds . 49 15.2 Poincar´elemma . 50 15.3 Rellich's lemma . 50 16 October 24, 2017 52 16.1 L2 space of differential forms . 52 16.2 G˚arding'sinequality . 53 17 October 26, 2017 55 17.1 Solving the equation . 55 17.2 Regularity of harmonic forms . 57 2 18 October 31, 2017 58 18.1 Smearing out by polynomials . 58 18.2 Differential equations with constant coefficients . 59 19 November 2, 2017 61 19.1 Distributions . 61 19.2 Tempered distributions . 62 20 November 7, 2017 64 20.1 Malgrange{Ehrenpreis on unbounded region . 64 21 November 9, 2017 67 21.1 Division problem for distributions . 68 21.2 Changing the domain of integration . 68 22 November 14, 2017 70 22.1 Elliptic regularity . 70 23 November 16, 2017 73 23.1 Division problem of tempered distributions by polynomials . 73 23.2 Sobolev embedding . 74 24 November 21, 2017 76 24.1 Nirenberg's trick of integrating in a rectangular direction . 76 24.2 Moser's iteration trick . 78 25 November 28, 2017 79 25.1 Radon{Nikodym derivative . 79 25.2 Ergodic measure theory . 80 26 November 30, 2017 83 26.1 Hausdorff measure . 83 26.2 Radon transform . 84 3 Math 212a Notes 4 1 August 31, 2017 We are going to use two books: Stein{Shakaarchi, Real Analysis and Lawrence Evans, Partial Differential Equations. The motivation studying real analysis is solving differential equations. Fourier in 1822 wrote a book about the solving the heat equation. His idea was to con- vert the differential equation to algebraic equations. Solving differential equa- tions is finding eigenfunctions for differential operators. For instance, erx is an eigenfunctions for d=dx with eigenvalue r. So if we let X rj x f(x) = cje ; j P rj x then df=dx = j cjrje and so this becomes an algebraic equation in the coefficients. A special case is when f(x) is on R with period 2π. Then the eigenfunctions are einx with n 2 Z. P inx But one needs to argue that there is a unique representation f(x) = cne . n2Z In case of a finite dimensional vector space, with an inner product, this is easy. You can easily choose an orthonormal basis of Cn, and it is easy to check that n X ~v = cj~ej; where cj = h~v;~eji: j=1 p People tried to do a similar thing. It is easy to check that einx= 2π are or- thonormal with respect to the inner product Z 2π (f; g) = f(x)g(x)dx: x=0 So it would be reasonable that p Z 2π inx inx e γn = (f; e = 2π) = f(x)p dx: x=0 2π Now the question is whether 1 X f(x) = (f; einx)einx 2π n2Z is true. Lebesgue answered this question in 1902, using Lebesgue theory. Ac- tually this didn't solve the problem entirely even in constant coefficients, and it remained unsolved until 1955 when Malgrange and Ehrenpreis solve it in constant coefficients. 1.1 Convergence of the Fourier series We need to start with replacing the function f by (cn)n2Z. Consider the Nth partial sum N X inx sN = cne : n=−N Math 212a Notes 5 We want to know if limN!1 sN = f. It can be computed that n n X X 1 Z 2π s (x) = c eikx = f(y)e−ikydyeinx n k 2π k=−n k=−n y=0 n 1 Z 2π X = f(y) eik(x−y) dy: 2π y=0 k=−n This sum in the integral is called the Dirichlet{Dini kernel. We can sum them as n X e−inx − ei(n+1)x sin(n + 1 )x D (x) = eikx = = 2 : n 1 − eix sin 1 x k=−n 2 Then we can write 1 Z 2π sn(x) = f(y)Dn(x − y)dy; 2π 0 as the convolution. To argue about the limit of sn(x) as n ! 1, we need to say about the commutation of limits. This is not very true of most functions, for instance characteristic functions of bad sets. Lebesgue had this big idea of partitioning the range instead of the domain. The base of the rectangles may be strange, though. If we change the period from [0; 2π] to [−L; L], we can rescale the length and use einxπ=L instead. In the limiting case L ! 1, we will get another theory. Then Z 1 Z 1 f^(ξ) = f(x)e−2πiξxdx; f(x) = f^(ξ)e2πiξxdξ: x=−∞ ξ=−∞ These are called the Fourier transform and the inversion formula. This still doesn't solve the differential equations with constant coefficients. Let me tell you why. Suppose you have a differential equation like @2 @2 2 + ··· + 2 f = g: @x1 @xn If you take the Fourier transform, we get 2 2 2 ^ −4π (ξ1 + ··· + ξn)f =g ^ and then divide by the polynomial. But this is a problem because there might be zeros in the polynomial. This was a big problem, and it took a long time to handle these. Math 212a Notes 6 2 September 5, 2017 So we start out with the details. We want to solve differential equations by eigenfunctions, but there are two problems here. The first is the convergence of P1 inx the series like f = n=−∞ cne . The second is the effect of differentiation. But here, it is easier to do integration, because it will be easier to exchange sums. So the tool we need here is the fundamental theorem of calculus. For Riemann integration, we need \uniformity", but this is too strong and so we will use Lebesgue theory. We are going to study the size of the set when a certain property holds or fails. 2.1 Measure of a set We start with the reals R. For a set E ⊆ R, we are going to study this. Theorem 2.1 (Structre theorem). Let O be an open subset of R. Then O is the disjoint union of a countable number of open intervals, possibly of infinite length. Proof. The key is that a point disconnects an open set in R. Then any connected component of O is an open interval. This is because, if x 2 O, then Ax = fy 2 R : y > x and [x; y] ⊆ Og is of the form (x; b). If you do the same thing on the other side, the connected component of x in O is (a; b). Countability follows from the fact that the rational numbers is dense and countable. S This means that we can write O = k2J (ak; bk) as a disjoint union. This allows us to define its measure as X m(O) = (bk − ak): k2J Now given arbitrary set E ⊆ R, Lebesgue tried to approximate E from the outside by open set, and from the inside by closed sets. But we are going to do something different. Definition 2.2. Given a set E ⊆ R, we define its outer measure as m∗(E) = inf m(O): O⊇E open Lebesgue's original definition for the inner measure was also like mi(E) = supF ⊆E m(F ) and call E measurable if the outer and inner measures coincide. But we are not doing this. Instead, we define Definition 2.3. Given E ⊆ R, E is measurable if for each > 0, there exists an open set O ⊇ E such that m∗(O − E) < .
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