Elementary theory of L p spaces
|
|
- Juliet Morgan
- 5 years ago
- Views:
Transcription
1 CHAPTER 3 Elementary theory of L saces 3.1 Convexity. Jensen, Hölder, Minkowski inequality. We begin with two definitions. A set A R d is said to be convex if, for any x 0, x 1 2 A x = x 0 + (x 1 x 0 ) 2 A 8 2 [0, 1]. With this definition, the emty set and singletons are convex. If d = 1, A is convex if and only if A is an interval (ossibly unbounded). Thus all convex subsets of R are measurable. It is not hard to see that if A R d is convex, then all of its rojections onto the first d 1 coordinates A t = {x 0 (x 1,...,x d 1 ) : (x 0, t) 2 A}, t 2 R are also convex. By using inductively that A 2B(R d ) is Borel measurable if and only if A t 2B(R d 1 ) for all t 2 R, all convex sets are Borel sets. Let A R d be a convex set. A function f : A! R is said to be convex if f (x ) ale f (x 0 )+ ( f (x 1 ) f (x 0 )) 8 2 [0, 1]. Equivalently the set {(x, y) 2 A R : y f (x)} is convex in R d+1. Also equivalently, for all 2 (0, 1), x 0 6= x 1 f (x ) f (x 0 ) (3.1) ale f (x 1) f (x ) x x 0 x 1 x LEMMA. Let A R d be oen and convex set, and f : A! R be convex. Then f is a continuous function. Furthermore, for all v 2 R d the limits (3.2) D ± f (x + tv) f (x) f (x) := lim v t!0 ± t exist finite for all x 2 A and D v f (x) ale D + f (x). v PROOF. Let x 2 A. To show that f is continuous at x, it suffices to show that for all v 2 R d the function t 2 I x,v 7! g x,v (t)= f (x + tv), where I x,v = {t 2 R : x + tv 2 A} is an oen interval containing zero, is continuous at 0 2 R. We omit the subscrit in the notation from now on. It is easy to see that g is a convex function. Let a, s, t, b 2 I with a < s < 0 < t < c < d. From (3.1), we learn that g(0) g(a) g(s) ale g(a)+(s a), 0 a and similarly for the other air. This yields L(a) := g(0) g(a) ale L(s) := g(0) g(s) ale R(t) := g(t) g(0) ale R(b) := g(b) g(0) 0 a 0 s t 0 b 0 33
2 34 3. ELEMENTARY THEORY OF L SPACES Hence s! L(s) is increasing and bounded above, t! R(t) is increasing and bounded below, therefore the limits below exist finite and lim s"0 L(s) ale lim t#0 R(t), which establishes (3.2), and in articular, imlies that g is continuous at zero. A consequence of the above Lemma is that convex functions on R d are measurable. The next inequality is a very instructive and useful alication of convexity JENSEN S INEQUALITY. Let (X, F, µ) be a measure sace. Assume that µ(x ) < 1. Let f 2 L 1 (X, F, µ) be real-valued and : R! R be a convex function. Then 1 f dµ ale 1 f dµ. µ(x ) µ(x ) PROOF. Define t = 1 µ(x ) f dµ. Let = D (t). From the roof of Lemma one has It follows that (s) (t) (s t), 8s 2 R. ( f (x)) (t)+ ( f (x) t) and integrating both sides with resect to x in dµ yields the claimed inequality, since the integral of the second summand on the right hand side is zero REMARK. We do not know whether f 2 L 1 (X, F, µ). However, the roof shows that ( f ) 2 L 1 (X, F, µ). Thus either ( f ) + 2 L 1 (X, F, µ) or we may define the right hand side to be 1, and the inequality holds trivially EXAMPLE -YOUNG S INEQUALITY. Take for instance (t) =e t. Then, if µ(x )=1, the articular case of Jensen s inequality reads ex f dµ ale e f dµ Let us consider X = {x 1,...,x n } with F = P (X ) and µ defined via nx µ({x j })= j 2 [0, 1], j = 1,..., n, µ(x )= j = 1 Let y j be ositive numbers f : X! R be defined by f (x j )=log y j. The above inequality becomes the familiar relationshi between the arithmetic and the geometric mean, known as Young s inequality. ny nx (3.3) j=1 y j j ale j=1 j y j. j=1
3 3. ELEMENTARY THEORY OF L SPACES 35 A air (, 0 ) 2 [1, 1] 2 is called conjugate if there exists 0 ale ale 1 such that (3.4) = 1, 0 = 1 1 When we write = 1 and = 1, we mean that = HOLDER S INEQUALITYFORMEASURABLEFUNCTIONS. Let f, g : X! [0, 1] be measurable functions and (, 0 ) be a conjugate air with 1 < < 1. Then 1 1 fgdµ ale f 0 dµ g 0 dµ. PROOF. Call A and B resectively each factor in the right hand side. We may assume that A, B are both nonzero and finite, otherwise there is nothing to rove in each of the other cases. Writing F = f /A, G = g/b, note that F dµ = G 0 dµ = 1 Now, by Young s inequality alied with 1 =, 2 = 1, y 1 = F(x), y 2 = G(x) we obtain that FG ale F +(1 )G 0 so that 1 fgdµ = AB which is what we had to rove. FGdµ ale F dµ +(1 ) G 0 dµ = MINKOWSKI S INEQUALITY FOR MEASURABLE FUNCTIONS. Let f, g : X! [0, 1] be measurable and 1 ale < 1. Then ( f + g) dµ 1 ale f dµ 1 + g dµ PROOF. We already know the case = 1. So assume > 1. We can assume both summands on the right hand side are finite and nonzero. Otherwise the inequality holds trivially. In this case, since t 7! t is convex, ( f + g) ale 2 1 ( f + g ) and the left hand side is finite and nonzero as well. Write ( f + g) = f ( f + g) 1 + g( f + g) 1. Integrating both sides and alying Hölder inequality to both summands on the right hand side, we obtain 1 1! 1 ( f + g) dµ ale f dµ + g dµ ( f + g) dµ 0, and dividing by the second factor on the right hand side of the last dislay, which is finite and nonzero, we obtain the claimed inequality. 1.
4 36 3. ELEMENTARY THEORY OF L SPACES 3.2 L saces. Let (X, F, µ) be a measure sace which we assume fixed throughout unless otherwise mentioned, and let 0 < < 1. We define L (X, F, µ)= f : X! C measurable : f 2 L 1 (X, F, µ). If f 2 L (X, F, µ), the quantity (3.5) k f k := f dµ is finite, and moreover k f k = 0 () f = 0 1. We extend the definition of (3.6) to = 1 by (3.6) k f k 1 = inf M > 0:µ {x 2 X : f (x) > M} = 0. and define 1 L 1 (X, F, µ)={ f : X! C measurable : k f k 1 < 1}. A nontrivial remark is that, if k f k 1 < 1, it must be µ {x 2 X : f (x) > k f k 1 })=0 since {x 2 X : f (x) > k f k 1 } is the countable union of {x 2 X : f (x) > k f k 1 + n 1 }, each of which must have measure zero. Consequently, k f k 1 = M if and only if there exists g = f a.e.[µ] with su g(x) = M. x2x We record the fundamental Hölder s inequality, which is easily derived from (3.1.6). The missing case = 1 is certainly the easiest and is left as an exercise HÖLDER S INEQUALITY. Let (, 0 ) be a air of conjugate exonents with 1 ale ale1. Let f 2 L (X, F, µ), g 2 L 0 (X, F, µ). Then the roduct f g 2 L 1 (X, F, µ) and k fgk 1 alekf k kgk PROPOSITION. Let (X, F, µ) be a measure sace. 1. for all 0 < ale1,l (X, F, µ) is a linear sace, in the sense that f, g 2 L (X, F, µ), 2 C =) f + g 2 L (X, F, µ). 2. For 1 ale < 1, k k is a norm on L (X, F, µ), that is 2a. k f k = 0 =) f = 0; 2b. 2 C, f 2 L (X, µ, F )=)k f k = k f k 2c. f, g 2 L (X, µ, F )=)kf + gk alekf k + kgk 3. For 0 ale < 1, k k is a quasinorm on L (X, F, µ), that is roerties 2.a and 2.b above continue to hold and 2c. is relaced by (3.7) k f + gk ale k f k + kgk 4. For 0 < < 1, there holds Chebychev s inequality su "µ({x 2 X : f (x) > "}) 1 alekfk. ">0 1 Here, and everywhere else, we identify functions which are equal a.e.[µ] without exlicit mention
5 3. ELEMENTARY THEORY OF L SPACES The saces L (X, F, µ) are comlete, in the sense that if f n is a Cauchy sequence in L (X, F, µ), i.e. k f n f m k! 0, n, m!1, then there exists f 2 L (X, F, µ) such that f n L! f, i.e. k f n f k! 0 as n!1. 6. Simle functions with comact suort are dense in L (X, F, µ), 0 < < 1. PROOF. We leave the roof of oints 1. to 4. as an easy exercise. In articular, art 3. uses the inequalities (a + b) ale a + b Ä ; ä (a + b) 1 1 ale 2 1 a b. for a, b 0, 0 < ale 1, which are to be roved in Exercise 3.3. We turn to the roof of comleteness. There is nothing to rove in the case = 1. We turn to the case 0 < < 1. Let f n be a Cauchy sequence in L. Then, from Chebychev s inequality, f n is Cauchy in measure and furthermore k f n k is a bounded sequence. The main tool we will use is the following SUBLEMMA. Let f n be a sequence of measurable functions which is Cauchy in measure, in the sense that for all " > 0 µ({x 2 X : f n (x) f m (x) > "})! 0, m, n!1. Then there exists a subsequence f nj and f : X! C such that f nj! f ointwise. Moreover, f n! f in measure as well. We ostone the roof of the Sublemma to the exercises. We obtain f : X! C such that f n j! f ointwise. By Fatou s lemma so that f 2 L 1. At this oint, f nj f dµ ale lim inf j!1 f n j dµ<1 f ale 2 2 f for j large enough, thus k f n j f k! 0 by the dominated convergence theorem. It follows that f nj! f in L. But then k f n f k ale C k f n f n j k + C k f f n j k and both terms on the right hand side go to zero as n, n j!1, which imlies that f n! f in L, concluding the roof of comleteness. Density of simle functions with is roved in a totally analogous way to the case = REMARK. Point 5. of Proosition can be reformulated as comleteness of L (X, F, µ) with resect to the metric d ( f, g)=k f gk min{,1}, 0< ale1. The fact that the above is a metric is immediate when 1. In the exercises, the numerical inequality a b ale a + b 80 < ale 1
6 38 3. ELEMENTARY THEORY OF L SPACES is roved. This may be relied uon to infer that k f gk alekf hk + kh gk and conclude that d is a metric when 0 < < 1 as well. We conclude this introductory subsection with an aroximation theorem. The roof is identical to the case = 1 when 1 ale < 1, and very similar in the case 0 < < 1, where the quasi-triangle inequality relaces the triangle inequality THEOREM. Let 0 < < 1. and f 2 L (R d ) (R d with Lebesgue measure). Then there is a sequence f n 2C 0 (R d ) with f n! f in L. 3.3 Linear bounded functionals. Let (X, F, µ) be a measure sace and 1 ale ale1. A linear bounded, linear continuous or simly linear functional on L (X, F, µ) is a ma satisfying : L (X, F, µ)! C (linearity) f, g 2 L, 2 C =) ( f + g)=( f )+ (g) (boundedness) 9C 0 such that ( f ) ale Ck f k for all f 2 L. The least C > 0 such that the above holds for all f 2 L is denoted by kk. It is easy to see that ( f ) kk = su = su ( f ) f 2L :f 6=0 k f k k f k = REMARK (CONTINUITY). There is no difference between boundedness and (Lischitz) continuity for linear functionals. In other words a linear bounded functional is uniformly Lischitz continuous. This simly follows from ( f ) (g) = ( f g) alekkk f gk. A ivotal examle is as follows. Let g 2 L 0 (X, F, µ). Since f ḡ 2 L 1 whenever f 2 L we can define the ma (3.8) g : L (X, F, µ)! C, g ( f )= f ḡ dµ. It is easy to see that this is a linear ma, bounded by virtue of Hölder inequality which yields Therefore k g k alekgk 0. In fact, there holds g ( f ) alekfgk 1 alekgk 0k f k LEMMA. Let 1 ale ale1and define g as in (3.8). Then PROOF. Exercise 3.9. k g k = kgk 0 A natural question which arises from Lemma is whether all linear bounded functionals on L (X, F, µ) arise from g 2 L 0 as in (3.8). This is true if 1 ale < 1 and false for = 1. The first half of this statement goes under the name of Riesz reresentation theorem for the dual sace of L (X, F, µ). We will rove both statements in Chater 5 in a general
7 3. ELEMENTARY THEORY OF L SPACES 39 setting. In the next section, we devote ourselves to the secial case of this statement for ` saces. 3.4 The ` saces and their Riesz reresentation. In this aragrah, we consider the saces ` = L (N, P (N), ) where is the counting measure, namely, the sace of sequences ~f =(f 1,...,f j,...) such that ( Ä P 1 ä 1 1 > k f k` = f j=1 j 0 < < 1, su j2n f j = 1 The sequences b k =(b k 1,...,bk j,...), bk j = 0 j 6= k 1 j = k form an unconditional basis of `, in the sense that if f 2 `, the sequence nx f n = f k b k converges to f in `. k= THEOREM. Let 1 ale < 1 and : `! C be a linear bounded functional. Then there exists a unique g 2 `0 such that = g as in (3.8), namely and furthermore kk = kgk` 0. ( f )= g ( f )= 1X f j g j j=1 8 f 2 `, PROOF. We deal with 1 < < 1. The case = 1 follows with minor changes. The candidate g is obtained by setting g j = (b j ), j 2 N. Let f 2 `. By linearity, for all n Ç n å X ( f n )= f k b k = It follows that for n m nx j=m+1 k=1 nx f k b k = g ( f n ). k=1 f j g j = ( f n ) ( f m ) alekkk f n f m k`, which since f n is Cauchy, imlies that the series 1X g ( f )= f j g j j=1
8 40 3. ELEMENTARY THEORY OF L SPACES converges to ( f ). By Lemma 3.3.2, to finish the roof it suffices to rove that g 2 `0 and kgk 0 alekk. Define 0 g j = 0 h = {h j : j 2 N}, h j = g j 0 2 g j g j 6= 0 and let again h n = P n j=1 h j b j, g n = P n j=1 g j b j be the truncation to the first n comonents. Then h n 2 ` and kh n k = kg n k Furthermore, a comutation reveals that kg n k 0 0 = g (h n ) alekkkh n k = kkkg n k 0 1 0, and rearranging kg n k 0 alekk. The monotone convergence theorem finally yields kgk 0 ale kk as well.
9 3. ELEMENTARY THEORY OF L SPACES 41 Chater 3 Exercises 3.1. Prove Sublemma Converse of Jensen s inequality. In this roblem, X =[0, 1] with the Lebesgue measure. Suose : R! R is a function with the roerty that Ç å for all f 2 L 1 (X ). Prove that [0,1] f dµ ale [0,1] is a convex function. f dµ 3.3. Let 0 < ale 1, a, b 0 Prove the two numerical inequalities (a + b) ale a + b Ä ; ä (a + b) 1 1 ale 2 1 a b. 3.4 Log-convexity of L -norms and endoint saces. Let f 2 L \L q with 1 ale < q ale1. a. Prove that k f k r alekf k k f k1 q 8 2 [0, 1], 1 r = (1 ) + q. b. Prove that if f 2 L \ L 1 for large enough, lim!1 k f k = k f k 1. You should use a. for art of the statement. c. Assume in addition that µ(x )=1. Let f 2 L for all 0 < < 1 and such that k f k su 1ale<1 ale 17. Prove that e f 2 L 1 and ke f k 1 ale e. d. Assume again that µ(x )=1. Let f 2 L r for some 0 < r ale1. Prove that lim #0 k f k = ex with the convention that ex( 1)=0. log f dµ 3.5 Converses of Hölder s inequality. In this exercise, (X, F, µ) is a measure sace, µ(x )= 1, and 1 < q ale1is fixed. a. Let f : X! C be a measurable function with the following roerty: for all g 2 L q, the roduct fg2 L 1 and fg dµ alekgk q. Prove that f 2 L, with = q 0, and that k f k ale 1. b. Let f : X! C be a measurable function with the following roerty 2 : for all g 2 L q, the roduct fg2 L 1. Prove that f 2 L, with = q 0. 2 For this art, any solution using tools which do not fall into the scoe of the first three chaters will not be considered.
10 42 3. ELEMENTARY THEORY OF L SPACES 3.6. Here X = R with Lebesgue measure. Find a function F : X! [0, 1) such that {q 2 (0, 1) : F 2 L q } = {7}. Can there be a function G such that {q 2 (0, 1) : F 2 L q } = {e, }? 3.7. Let 0 < < 1, f 2 L (X, F, µ). Define for k 2, F k = {x 2 X : f (x) > 2 k } and Ç å 1 X [ f ] := 2 k µ(f k ). k2 Prove that there exist constants c < C such that Do the case with µ(x ) < 1 first. c [ f ] alekf k ale C [ f ] Let : [0, 1)! [0, 1) be a convex, strictly increasing function with (0) =0. This exercise is about finding a norm for the sace L = L (X, F, µ)= f : X! C measurable : x 7! ( f (x) ) 2 L 1 (X, F, µ). 1. Prove that if f : X! C I ( f ) := is a decreasing function of Ä ä f (x) dµ(x), > 0 and lim!0 + I ( f )=1 unless f = 0 a.e., in which case I ( f )=0 for all > Prove that if f 2 L (3.9) lim!1 I ( f )=0 and therefore k f k := inf{ > 0:I ( f ) ale 1} 2 [0, 1) is well defined. Furthermore rove that k f k = 0 if and only if f = 0 a.e. and that f 2 L, f 6= 0 =) I k f k ( f )=1. 3. Assume that f, g 2 L, 2 C. Prove that k f k = k f k, k f + gk alekf k + kgk and conclude that k k is a norm on L (in fact, arguing in the same way we used for L, L is comlete with resect to this norm) REMARK. In fact, using the above for (t) =t roves the triangle inequality for L saces without any aeal to Hölder s inequality. There is actually a way to derive Hölder s inequality in this framework, using the concet of conjugate function. Given : [0, 1)! [0, 1) convex, one define the conjugate : [0, 1)! [0, 1) by (s)=su {ts (t) : t 0}.
11 3. ELEMENTARY THEORY OF L SPACES 43 It turns out that is also convex and ( ) =. For instance, if (t)=t with 1 < < 1 then (s)=s 0. Using the definition of conjugate functions, it should be easy to rove the following: if f 2 L, g 2 L then fg2l 1 and k fgk 1 ale 2k f k kgk Prove Lemma Hint.
MATH 6210: SOLUTIONS TO PROBLEM SET #3
MATH 6210: SOLUTIONS TO PROBLEM SET #3 Rudin, Chater 4, Problem #3. The sace L (T) is searable since the trigonometric olynomials with comlex coefficients whose real and imaginary arts are rational form
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationIntroduction to Banach Spaces
CHAPTER 8 Introduction to Banach Saces 1. Uniform and Absolute Convergence As a rearation we begin by reviewing some familiar roerties of Cauchy sequences and uniform limits in the setting of metric saces.
More informationSums of independent random variables
3 Sums of indeendent random variables This lecture collects a number of estimates for sums of indeendent random variables with values in a Banach sace E. We concentrate on sums of the form N γ nx n, where
More informationCHAPTER 2: SMOOTH MAPS. 1. Introduction In this chapter we introduce smooth maps between manifolds, and some important
CHAPTER 2: SMOOTH MAPS DAVID GLICKENSTEIN 1. Introduction In this chater we introduce smooth mas between manifolds, and some imortant concets. De nition 1. A function f : M! R k is a smooth function if
More informationGENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS
GENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS PALLAVI DANI 1. Introduction Let Γ be a finitely generated grou and let S be a finite set of generators for Γ. This determines a word metric on
More informationB8.1 Martingales Through Measure Theory. Concept of independence
B8.1 Martingales Through Measure Theory Concet of indeendence Motivated by the notion of indeendent events in relims robability, we have generalized the concet of indeendence to families of σ-algebras.
More informationReal Analysis 1 Fall Homework 3. a n.
eal Analysis Fall 06 Homework 3. Let and consider the measure sace N, P, µ, where µ is counting measure. That is, if N, then µ equals the number of elements in if is finite; µ = otherwise. One usually
More informationJUHA KINNUNEN. Sobolev spaces
JUHA KINNUNEN Sobolev saces Deartment of Mathematics and Systems Analysis, Aalto University 217 Contents 1 SOBOLEV SPACES 1 1.1 Weak derivatives.............................. 1 1.2 Sobolev saces...............................
More informationL p Spaces and Convexity
L p Spaces and Convexity These notes largely follow the treatments in Royden, Real Analysis, and Rudin, Real & Complex Analysis. 1. Convex functions Let I R be an interval. For I open, we say a function
More informationExamples of Dual Spaces from Measure Theory
Chapter 9 Examples of Dual Spaces from Measure Theory We have seen that L (, A, µ) is a Banach space for any measure space (, A, µ). We will extend that concept in the following section to identify an
More information1 Riesz Potential and Enbeddings Theorems
Riesz Potential and Enbeddings Theorems Given 0 < < and a function u L loc R, the Riesz otential of u is defined by u y I u x := R x y dy, x R We begin by finding an exonent such that I u L R c u L R for
More informationIMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES
IMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES OHAD GILADI AND ASSAF NAOR Abstract. It is shown that if (, ) is a Banach sace with Rademacher tye 1 then for every n N there exists
More informationp-adic Measures and Bernoulli Numbers
-Adic Measures and Bernoulli Numbers Adam Bowers Introduction The constants B k in the Taylor series exansion t e t = t k B k k! k=0 are known as the Bernoulli numbers. The first few are,, 6, 0, 30, 0,
More information3 Integration and Expectation
3 Integration and Expectation 3.1 Construction of the Lebesgue Integral Let (, F, µ) be a measure space (not necessarily a probability space). Our objective will be to define the Lebesgue integral R fdµ
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationBest Simultaneous Approximation in L p (I,X)
nt. Journal of Math. Analysis, Vol. 3, 2009, no. 24, 57-68 Best Simultaneous Aroximation in L (,X) E. Abu-Sirhan Deartment of Mathematics, Tafila Technical University Tafila, Jordan Esarhan@ttu.edu.jo
More informationMollifiers and its applications in L p (Ω) space
Mollifiers and its alications in L () sace MA Shiqi Deartment of Mathematics, Hong Kong Batist University November 19, 2016 Abstract This note gives definition of mollifier and mollification. We illustrate
More informationHENSEL S LEMMA KEITH CONRAD
HENSEL S LEMMA KEITH CONRAD 1. Introduction In the -adic integers, congruences are aroximations: for a and b in Z, a b mod n is the same as a b 1/ n. Turning information modulo one ower of into similar
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A -measurable functions on it. 2. For f L 1 (, µ) set f 1 = f L 1 = f L 1 (,µ) = f dµ.
More informationReview of measure theory
209: Honors nalysis in R n Review of measure theory 1 Outer measure, measure, measurable sets Definition 1 Let X be a set. nonempty family R of subsets of X is a ring if, B R B R and, B R B R hold. bove,
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. Appearance of normed spaces. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A - measurable functions on it. 2. For f L 1 (, µ) set f 1
More informationMath 205A - Fall 2015 Homework #4 Solutions
Math 25A - Fall 25 Homework #4 Solutions Problem : Let f L and µ(t) = m{x : f(x) > t} the distribution function of f. Show that: (i) µ(t) t f L (). (ii) f L () = t µ(t)dt. (iii) For any increasing differentiable
More informationIntegration on Measure Spaces
Chapter 3 Integration on Measure Spaces In this chapter we introduce the general notion of a measure on a space X, define the class of measurable functions, and define the integral, first on a class of
More informationHEAT AND LAPLACE TYPE EQUATIONS WITH COMPLEX SPATIAL VARIABLES IN WEIGHTED BERGMAN SPACES
Electronic Journal of ifferential Equations, Vol. 207 (207), No. 236,. 8. ISSN: 072-669. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu HEAT AN LAPLACE TYPE EQUATIONS WITH COMPLEX SPATIAL
More information16.2. Infinite Series. Introduction. Prerequisites. Learning Outcomes
Infinite Series 6.2 Introduction We extend the concet of a finite series, met in Section 6., to the situation in which the number of terms increase without bound. We define what is meant by an infinite
More informationMEASURE AND INTEGRATION: LECTURE 15. f p X. < }. Observe that f p
L saes. Let 0 < < and let f : funtion. We define the L norm to be ( ) / f = f dµ, and the sae L to be C be a measurable L (µ) = {f : C f is measurable and f < }. Observe that f = 0 if and only if f = 0
More informationLEIBNIZ SEMINORMS IN PROBABILITY SPACES
LEIBNIZ SEMINORMS IN PROBABILITY SPACES ÁDÁM BESENYEI AND ZOLTÁN LÉKA Abstract. In this aer we study the (strong) Leibniz roerty of centered moments of bounded random variables. We shall answer a question
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More information2 Asymptotic density and Dirichlet density
8.785: Analytic Number Theory, MIT, sring 2007 (K.S. Kedlaya) Primes in arithmetic rogressions In this unit, we first rove Dirichlet s theorem on rimes in arithmetic rogressions. We then rove the rime
More informationAnalysis Comprehensive Exam Questions Fall 2008
Analysis Comprehensive xam Questions Fall 28. (a) Let R be measurable with finite Lebesgue measure. Suppose that {f n } n N is a bounded sequence in L 2 () and there exists a function f such that f n (x)
More information2 Asymptotic density and Dirichlet density
8.785: Analytic Number Theory, MIT, sring 2007 (K.S. Kedlaya) Primes in arithmetic rogressions In this unit, we first rove Dirichlet s theorem on rimes in arithmetic rogressions. We then rove the rime
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More information16.2. Infinite Series. Introduction. Prerequisites. Learning Outcomes
Infinite Series 6. Introduction We extend the concet of a finite series, met in section, to the situation in which the number of terms increase without bound. We define what is meant by an infinite series
More informationTHE 2D CASE OF THE BOURGAIN-DEMETER-GUTH ARGUMENT
THE 2D CASE OF THE BOURGAIN-DEMETER-GUTH ARGUMENT ZANE LI Let e(z) := e 2πiz and for g : [0, ] C and J [0, ], define the extension oerator E J g(x) := g(t)e(tx + t 2 x 2 ) dt. J For a ositive weight ν
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationBest approximation by linear combinations of characteristic functions of half-spaces
Best aroximation by linear combinations of characteristic functions of half-saces Paul C. Kainen Deartment of Mathematics Georgetown University Washington, D.C. 20057-1233, USA Věra Kůrková Institute of
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 9,. 29-36, 25. Coyright 25,. ISSN 68-963. ETNA ASYMPTOTICS FOR EXTREMAL POLYNOMIALS WITH VARYING MEASURES M. BELLO HERNÁNDEZ AND J. MíNGUEZ CENICEROS
More informationON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX)0000-0 ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS WILLIAM D. BANKS AND ASMA HARCHARRAS
More informationGENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS
International Journal of Analysis Alications ISSN 9-8639 Volume 5, Number (04), -9 htt://www.etamaths.com GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS ILYAS ALI, HU YANG, ABDUL SHAKOOR Abstract.
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationLecture 6. 2 Recurrence/transience, harmonic functions and martingales
Lecture 6 Classification of states We have shown that all states of an irreducible countable state Markov chain must of the same tye. This gives rise to the following classification. Definition. [Classification
More informationLecture 4 Lebesgue spaces and inequalities
Lecture 4: Lebesgue spaces and inequalities 1 of 10 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 4 Lebesgue spaces and inequalities Lebesgue spaces We have seen how
More informationConvergence of random variables, and the Borel-Cantelli lemmas
Stat 205A Setember, 12, 2002 Convergence of ranom variables, an the Borel-Cantelli lemmas Lecturer: James W. Pitman Scribes: Jin Kim (jin@eecs) 1 Convergence of ranom variables Recall that, given a sequence
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More informationPETER J. GRABNER AND ARNOLD KNOPFMACHER
ARITHMETIC AND METRIC PROPERTIES OF -ADIC ENGEL SERIES EXPANSIONS PETER J. GRABNER AND ARNOLD KNOPFMACHER Abstract. We derive a characterization of rational numbers in terms of their unique -adic Engel
More informationExistence Results for Quasilinear Degenerated Equations Via Strong Convergence of Truncations
Existence Results for Quasilinear Degenerated Equations Via Strong Convergence of Truncations Youssef AKDIM, Elhoussine AZROUL, and Abdelmoujib BENKIRANE Déartement de Mathématiques et Informatique, Faculté
More informationANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM
ANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM JOHN BINDER Abstract. In this aer, we rove Dirichlet s theorem that, given any air h, k with h, k) =, there are infinitely many rime numbers congruent to
More information#A37 INTEGERS 15 (2015) NOTE ON A RESULT OF CHUNG ON WEIL TYPE SUMS
#A37 INTEGERS 15 (2015) NOTE ON A RESULT OF CHUNG ON WEIL TYPE SUMS Norbert Hegyvári ELTE TTK, Eötvös University, Institute of Mathematics, Budaest, Hungary hegyvari@elte.hu François Hennecart Université
More informationLECTURE 7 NOTES. x n. d x if. E [g(x n )] E [g(x)]
LECTURE 7 NOTES 1. Convergence of random variables. Before delving into the large samle roerties of the MLE, we review some concets from large samle theory. 1. Convergence in robability: x n x if, for
More informationAdvanced Analysis Qualifying Examination Department of Mathematics and Statistics University of Massachusetts. Monday, August 26, 2013
NAME: Advanced Analysis Qualifying Examination Department of Mathematics and Statistics University of Massachusetts Monday, August 26, 2013 Instructions 1. This exam consists of eight (8) problems all
More informationStochastic integration II: the Itô integral
13 Stochastic integration II: the Itô integral We have seen in Lecture 6 how to integrate functions Φ : (, ) L (H, E) with resect to an H-cylindrical Brownian motion W H. In this lecture we address the
More informationFunctional Analysis, Stein-Shakarchi Chapter 1
Functional Analysis, Stein-Shakarchi Chapter 1 L p spaces and Banach Spaces Yung-Hsiang Huang 018.05.1 Abstract Many problems are cited to my solution files for Folland [4] and Rudin [6] post here. 1 Exercises
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationResearch Article An iterative Algorithm for Hemicontractive Mappings in Banach Spaces
Abstract and Alied Analysis Volume 2012, Article ID 264103, 11 ages doi:10.1155/2012/264103 Research Article An iterative Algorithm for Hemicontractive Maings in Banach Saces Youli Yu, 1 Zhitao Wu, 2 and
More informationApproximating min-max k-clustering
Aroximating min-max k-clustering Asaf Levin July 24, 2007 Abstract We consider the roblems of set artitioning into k clusters with minimum total cost and minimum of the maximum cost of a cluster. The cost
More informationExtremal Polynomials with Varying Measures
International Mathematical Forum, 2, 2007, no. 39, 1927-1934 Extremal Polynomials with Varying Measures Rabah Khaldi Deartment of Mathematics, Annaba University B.P. 12, 23000 Annaba, Algeria rkhadi@yahoo.fr
More informationCommutators on l. D. Dosev and W. B. Johnson
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Commutators on l D. Dosev and W. B. Johnson Abstract The oerators on l which are commutators are those not of the form λi
More informationAnalysis of some entrance probabilities for killed birth-death processes
Analysis of some entrance robabilities for killed birth-death rocesses Master s Thesis O.J.G. van der Velde Suervisor: Dr. F.M. Sieksma July 5, 207 Mathematical Institute, Leiden University Contents Introduction
More informationMultiplicity of weak solutions for a class of nonuniformly elliptic equations of p-laplacian type
Nonlinear Analysis 7 29 536 546 www.elsevier.com/locate/na Multilicity of weak solutions for a class of nonuniformly ellitic equations of -Lalacian tye Hoang Quoc Toan, Quô c-anh Ngô Deartment of Mathematics,
More informationMeasure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond
Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................
More informationApplications of the course to Number Theory
Alications of the course to Number Theory Rational Aroximations Theorem (Dirichlet) If ξ is real and irrational then there are infinitely many distinct rational numbers /q such that ξ q < q. () 2 Proof
More informationAdvanced Calculus I. Part A, for both Section 200 and Section 501
Sring 2 Instructions Please write your solutions on your own aer. These roblems should be treated as essay questions. A roblem that says give an examle requires a suorting exlanation. In all roblems, you
More information212a1214Daniell s integration theory.
212a1214 Daniell s integration theory. October 30, 2014 Daniell s idea was to take the axiomatic properties of the integral as the starting point and develop integration for broader and broader classes
More informationConvex Analysis and Economic Theory Winter 2018
Division of the Humanities and Social Sciences Ec 181 KC Border Conve Analysis and Economic Theory Winter 2018 Toic 16: Fenchel conjugates 16.1 Conjugate functions Recall from Proosition 14.1.1 that is
More informationDependence on Initial Conditions of Attainable Sets of Control Systems with p-integrable Controls
Nonlinear Analysis: Modelling and Control, 2007, Vol. 12, No. 3, 293 306 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls E. Akyar Anadolu University, Deartment
More informationRIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES
RIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES JIE XIAO This aer is dedicated to the memory of Nikolaos Danikas 1947-2004) Abstract. This note comletely describes the bounded or comact Riemann-
More informationON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE. 1. Introduction
ON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE GUSTAVO GARRIGÓS ANDREAS SEEGER TINO ULLRICH Abstract We give an alternative roof and a wavelet analog of recent results
More informationBEST CONSTANT IN POINCARÉ INEQUALITIES WITH TRACES: A FREE DISCONTINUITY APPROACH
BEST CONSTANT IN POINCARÉ INEQUALITIES WITH TRACES: A FREE DISCONTINUITY APPROACH DORIN BUCUR, ALESSANDRO GIACOMINI, AND PAOLA TREBESCHI Abstract For Ω R N oen bounded and with a Lischitz boundary, and
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationTopic 7: Using identity types
Toic 7: Using identity tyes June 10, 2014 Now we would like to learn how to use identity tyes and how to do some actual mathematics with them. By now we have essentially introduced all inference rules
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Alied Mathematics htt://jiam.vu.edu.au/ Volume 3, Issue 5, Article 8, 22 REVERSE CONVOLUTION INEQUALITIES AND APPLICATIONS TO INVERSE HEAT SOURCE PROBLEMS SABUROU SAITOH,
More informationHILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define
HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,
More informationTHE ERDÖS - MORDELL THEOREM IN THE EXTERIOR DOMAIN
INTERNATIONAL JOURNAL OF GEOMETRY Vol. 5 (2016), No. 1, 31-38 THE ERDÖS - MORDELL THEOREM IN THE EXTERIOR DOMAIN PETER WALKER Abstract. We show that in the Erd½os-Mordell theorem, the art of the region
More informationBy Evan Chen OTIS, Internal Use
Solutions Notes for DNY-NTCONSTRUCT Evan Chen January 17, 018 1 Solution Notes to TSTST 015/5 Let ϕ(n) denote the number of ositive integers less than n that are relatively rime to n. Prove that there
More informationMAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT p-adic L-FUNCTION. n s = p. (1 p s ) 1.
MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT -ADIC L-FUNCTION XEVI GUITART Abstract. We give an overview of Mazur s construction of the Kubota Leooldt -adic L- function as the -adic Mellin transform of a
More informationarxiv:math/ v4 [math.gn] 25 Nov 2006
arxiv:math/0607751v4 [math.gn] 25 Nov 2006 On the uniqueness of the coincidence index on orientable differentiable manifolds P. Christoher Staecker October 12, 2006 Abstract The fixed oint index of toological
More informationA Note on Guaranteed Sparse Recovery via l 1 -Minimization
A Note on Guaranteed Sarse Recovery via l -Minimization Simon Foucart, Université Pierre et Marie Curie Abstract It is roved that every s-sarse vector x C N can be recovered from the measurement vector
More informationSECTION 5: FIBRATIONS AND HOMOTOPY FIBERS
SECTION 5: FIBRATIONS AND HOMOTOPY FIBERS In this section we will introduce two imortant classes of mas of saces, namely the Hurewicz fibrations and the more general Serre fibrations, which are both obtained
More informationFréchet-Kolmogorov-Riesz-Weil s theorem on locally compact groups via Arzelà-Ascoli s theorem
arxiv:1801.01898v2 [math.fa] 17 Jun 2018 Fréchet-Kolmogorov-Riesz-Weil s theorem on locally comact grous via Arzelà-Ascoli s theorem Mateusz Krukowski Łódź University of Technology, Institute of Mathematics,
More information(U) =, if 0 U, 1 U, (U) = X, if 0 U, and 1 U. (U) = E, if 0 U, but 1 U. (U) = X \ E if 0 U, but 1 U. n=1 A n, then A M.
1. Abstract Integration The main reference for this section is Rudin s Real and Complex Analysis. The purpose of developing an abstract theory of integration is to emphasize the difference between the
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More information6. Duals of L p spaces
6 Duals of L p spaces This section deals with the problem if identifying the duals of L p spaces, p [1, ) There are essentially two cases of this problem: (i) p = 1; (ii) 1 < p < The major difference between
More informationSOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES
Kragujevac Journal of Mathematics Volume 411) 017), Pages 33 55. SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES SILVESTRU SEVER DRAGOMIR 1, Abstract. Some new trace ineualities for oerators in
More informationL p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by
L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may
More informationOn the minimax inequality and its application to existence of three solutions for elliptic equations with Dirichlet boundary condition
ISSN 1 746-7233 England UK World Journal of Modelling and Simulation Vol. 3 (2007) No. 2. 83-89 On the minimax inequality and its alication to existence of three solutions for ellitic equations with Dirichlet
More informationArithmetic and Metric Properties of p-adic Alternating Engel Series Expansions
International Journal of Algebra, Vol 2, 2008, no 8, 383-393 Arithmetic and Metric Proerties of -Adic Alternating Engel Series Exansions Yue-Hua Liu and Lu-Ming Shen Science College of Hunan Agriculture
More informationMath 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,
MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write
More informationBEST POSSIBLE DENSITIES OF DICKSON m-tuples, AS A CONSEQUENCE OF ZHANG-MAYNARD-TAO
BEST POSSIBLE DENSITIES OF DICKSON m-tuples, AS A CONSEQUENCE OF ZHANG-MAYNARD-TAO ANDREW GRANVILLE, DANIEL M. KANE, DIMITRIS KOUKOULOPOULOS, AND ROBERT J. LEMKE OLIVER Abstract. We determine for what
More informationA sharp generalization on cone b-metric space over Banach algebra
Available online at www.isr-ublications.com/jnsa J. Nonlinear Sci. Al., 10 2017), 429 435 Research Article Journal Homeage: www.tjnsa.com - www.isr-ublications.com/jnsa A shar generalization on cone b-metric
More informationON JOINT CONVEXITY AND CONCAVITY OF SOME KNOWN TRACE FUNCTIONS
ON JOINT CONVEXITY ND CONCVITY OF SOME KNOWN TRCE FUNCTIONS MOHMMD GHER GHEMI, NHID GHRKHNLU and YOEL JE CHO Communicated by Dan Timotin In this aer, we rovide a new and simle roof for joint convexity
More informationGroup Theory Problems
Grou Theory Problems Ali Nesin 1 October 1999 Throughout the exercises G is a grou. We let Z i = Z i (G) and Z = Z(G). Let H and K be two subgrous of finite index of G. Show that H K has also finite index
More informationStrong Matching of Points with Geometric Shapes
Strong Matching of Points with Geometric Shaes Ahmad Biniaz Anil Maheshwari Michiel Smid School of Comuter Science, Carleton University, Ottawa, Canada December 9, 05 In memory of Ferran Hurtado. Abstract
More informationOn a Markov Game with Incomplete Information
On a Markov Game with Incomlete Information Johannes Hörner, Dinah Rosenberg y, Eilon Solan z and Nicolas Vieille x{ January 24, 26 Abstract We consider an examle of a Markov game with lack of information
More informationA CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract
A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this aer is to rovide a concise way for undergraduate mathematics students to learn about how rime numbers behave
More informationHAUSDORFF MEASURE OF p-cantor SETS
Real Analysis Exchange Vol. 302), 2004/2005,. 20 C. Cabrelli, U. Molter, Deartamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and CONICET, Pabellón I - Ciudad Universitaria,
More information