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Real Analysis JUHA KINNUNEN Real Analysis Department of Mathematics and Systems Analysis, Aalto University 2020 Contents 1 Lp spaces1 1.1 Lp functions ................................. 1 1.2 Lp norm .................................... 4 1.3 Lp spaces for 0 p 1 ............................ 12 Ç Ç 1.4 Completeness of Lp ............................. 13 1.5 L1 space ................................... 19 2 The Hardy-Littlewood maximal function 25 2.1 Local Lp spaces ............................... 25 2.2 Definition of the maximal function .................... 27 2.3 Hardy-Littlewood-Wiener maximal function theorems ........ 29 2.4 The Lebesgue differentiation theorem .................. 37 2.5 The fundamental theorem of calculus .................. 43 2.6 Points of density ............................... 44 2.7 The Sobolev embedding .......................... 47 3 Convolutions 52 3.1 Two additional properties of Lp ...................... 52 3.2 Convolution .................................. 56 3.3 Approximations of the identity ...................... 61 3.4 Pointwise convergence ........................... 64 3.5 Convergence in Lp ............................. 68 3.6 Smoothing properties ............................ 70 3.7 The Poisson kernel .............................. 73 4 Differentiation of measures 75 4.1 Covering theorems ............................. 75 4.2 The Lebesgue differentiation theorem for Radon measures ..... 84 4.3 The Radon-Nikodym theorem ....................... 89 4.4 The Lebesgue decomposition ....................... 92 4.5 Lebesgue and density points revisited .................. 95 5 Weak convergence methods 98 CONTENTS ii 5.1 The Riesz representation theorem for Lp ................. 98 5.2 Partitions of unity ............................... 104 5.3 The Riesz representation theorem for Radon measures ........ 107 5.4 Weak convergence and compactness of Radon measures .... 114 5.5 Weak convergence in Lp. ......................... 118 5.6 Mazur’s lemma ................................ 123 The Lp spaces are probably the most important function spaces in analysis. This section gives basic facts about Lp spaces for general measures. These include Hölder’s inequality, Minkowski’s inequality, the Riesz-Fischer theo- rem which shows the completeness and the corresponding facts for the L1 space. 1 Lp spaces In this section we study the Lp spaces in order to be able to capture quantitative information on the average size of measurable functions and boundedness of operators on such functions. The cases 0 p 1, p 1, p 2, 1 p and Ç Ç Æ Æ Ç Ç 1 p are different in character, but they are all play an important role in in Æ 1 Fourier analysis, harmonic analysis, functional analysis and partial differential equations. The space L1 of integrable functions plays a central role in measure and integration theory. The Hilbert space L2 of square integrable functions is important in the study of Fourier series. Many operators that arise in applications p 1 are bounded in L for 1 p , but the limit cases L and L1 require a special Ç Ç 1 attention. 1.1 Lp functions Definition 1.1. Let ¹ be an outer measure on Rn, A Rn a ¹-measurable set and ½ f : A [ , ] a ¹-measurable function. Then f Lp(A), 1 p , if ! ¡1 1 2 É Ç 1 µZ ¶1/p p f p f d¹ . k k Æ A j j Ç 1 T HEMORAL : For p 1, f L1(A) if and only if f is integrable in A. For Æ 2 j j 1 p , f Lp(A) if and only if f p is integrable in A. É Ç 1 2 j j Remark 1.2. The measurability assumption on f essential in the definition. For example, let A [0,1] be a non-measurable set with respect to the one-dimensional ½ Lebesgue measure and consider f : [0,1] R, ! 8 <1, x A, f (x) 2 Æ : 1, x [0,1] \ A. ¡ 2 1 CHAPTER 1. LP SPACES 2 Then f 2 1 is integrable on [0,1], but f is not a Lebesgue measurable function. Æ n n Example 1.3. Let f : R [0, ], f (x) x ¡ and assume that ¹ is the Lebesgue ! 1 n Æ j j i 1 i measure. Let A B(0,1) {x R : x 1} and denote Ai B(0,2¡ Å ) \ B(0,2¡ ), Æ Æ 2 j j Ç Æ i 1,2,.... Then Æ Z Z np X1 np x ¡ dx x ¡ dx B(0,1) j j Æ i 1 Ai j j Æ Z X1 npi i np npi 2 dx (x Ai x 2¡ x ¡ 2 ) É i 1 Ai 2 ) j j Ê ) j j É Æ X1 npi X1 npi i 1 2 Ai 2 B(0,2¡ Å ) Æ i 1 j j É i 1 j j Æ Æ X1 npi i 1 n ­n 2 (2¡ Å ) (­n B(0,1) ) Æ i 1 Æ j j Æ X1 npi ni n n X1 in(p 1) ­n 2 ¡ Å 2 ­n 2 ¡ , Æ i 1 Æ i 1 Ç 1 Æ Æ if n(p 1) 0 p 1. Thus f Lp(B(0,1)) for p 1. ¡ Ç () Ç 2 Ç On the other hand, Z Z np X1 np x ¡ dx x ¡ dx B(0,1) j j Æ i 1 Ai j j Æ Z X1 np(i 1) i 1 np np(i 1) 2 ¡ dx (x Ai x 2¡ Å x ¡ 2 ¡ ) Ê i 1 Ai 2 ) j j Ç ) j j È Æ X1 np(i 1) n np X1 npi in 2 ¡ Ai ­n(2 1)2¡ 2 2¡ Æ i 1 j j Æ ¡ i 1 Æ Æ i 1 i ( Ai B(0,2¡ Å ) B(0,2¡ ) j j Æ j j ¡ j j ( i 1)n in n in ­n(2 ¡ Å 2¡ ) ­n(2 1)2¡ ) Æ ¡ Æ ¡ X1 in(p 1) C(n, p) 2 ¡ , Æ i 1 Æ 1 Æ if n(p 1) 0 p 1. Thus f Lp(B(0,1)) for p 1. This shows that ¡ Ê () Ê ∉ Ê f Lp(B(0,1)) p 1. 2 () Ç n i i 1 If A R \ B(0,1), then we denote Ai B(0,2 ) \ B(0,2 ¡ ), i 1,2,..., and a Æ Æ Æ similar argument as above shows that f Lp(Rn \ B(0,1)) p 1. 2 () È 1 1 n n Observe that f L (B(0,1)) and f L (R \ B(0,1)). Thus f (x) x ¡ is a 62 62 Æ j j borderline function in Rn as far as integrability is concerned. T HEMORAL : The smaller the parameter p is, the worse local singularities an Lp function may have. On the other hand, the larger the parameter p is, the more an Lp function may spread out globally. CHAPTER 1. LP SPACES 3 Example 1.4. Assume that f : Rn [0, ] is radial. Thus f depends only on x ! 1 j j and it can be expressed as f ( x ), where f is a function defined on [0, ). Then j j 1 Z Z 1 n 1 f ( x ) dx !n 1 f (r)r ¡ dr, (1.1) Rn j j Æ ¡ 0 where 2¼n/2 !n 1 ¡ Æ ¡(n/2) is the (n 1)-dimensional volume of the unit sphere ÇB(0,1) {x Rn : x 1}. ¡ Æ 2 j j Æ Let us show how to use this formula to compute the volume of a ball B(x, r) n n Æ {y R : y x r}, x R and r 0. Denote ­n m(B(0,1)). By the translation 2 j ¡ j Ç 2 È Æ and scaling invariance, we have n n r ­n r m(B(0,1)) m(B(x, r)) m(B(0, r)) Æ Æ Æ Z Z ÂB(0,r)(y) d y Â(0,r)( y ) d y Æ Rn Æ Rn j j Z r n n 1 r !n 1 ½ ¡ d½ !n 1 . Æ ¡ 0 Æ ¡ n In particular, it follows that !n 1 n­n and ¡ Æ 2¼n/2 rn ¼n/2 m(B(x, r)) rn. Æ ¡(n/2) n Æ ¡( n 1) 2 Å Let r 0. Then È Z 1 Z 1 dx  n (x) dx ® ® R \B(0,r) Rn\B(0,r) x Æ Rn x j j Zj j n 1 r  n (rx) dx ® R \B(0,r) Æ Rn rx Z j j n ® 1 r  n (x) dx ¡ ® R \B(0,1) Æ Rn x j j Z 1 rn ® dx , ® n, ¡ ® Æ Rn\B(0,1) x Ç 1 È j j and, in a similar way, Z 1 Z 1 dx rn ® dx , ® n. ® ¡ ® B(0,r) x Æ B(0,1) x Ç 1 Ç j j j j Observe, that here we formally make the change of variables x ry. Æ On the other hand, the integrals can be computer directly by (1.1). This gives Z 1 Z dx ! 1 ½ ®½n 1 d½ ® n 1 ¡ ¡ Rn\B(0,r) x Æ ¡ r j j ¯ !n 1 ® n¯1 !n 1 ® n ¡ ½¡ Å ¯ ¡ r¡ Å , ® n Æ ® n ¯ Æ ® n Ç 1 È ¡ Å r ¡ and Z 1 Z r dx ! ½ ®½n 1 d½ ® n 1 ¡ ¡ B(0,r) x Æ ¡ 0 j j ¯r !n 1 ® n¯ !n 1 n ® ¡ ½¡ Å ¯ ¡ r ¡ , ® n. Æ ® n ¯ Æ ® n Ç 1 Ç ¡ Å 0 ¡ CHAPTER 1. LP SPACES 4 Remarks 1.5: Formula (1.1) implies following claims: ® 1 (1) If f (x) c x ¡ in a ball B(0, r), r 0, for some ® n, then f L (B(0, r)). j j É j j È Ç 2 ® On the other hand, if f (x) c x ¡ in B(0, r) for some ® n, then f j j Ê j j È ∉ L1(B(0, r)).
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