Properties of BMO functions whose reciprocals are also BMO


 Loren Parker
 3 years ago
 Views:
Transcription
1 Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and that an arbitrary u BMO can be written as u = w /w, for w as above. This leads then to some observations concerning the JohnNirenberg distribution inequality for F u, u BM O and F Lip α. Key words: Bounded mean oscillation, A p weights AMS subject classifications: 42B25. Introduction We will consider the question of when a function w and its reciprocal /w are in BMO. If we assume that w : R n R + and consider this question for various spaces X, we obtain distinct results. The answer for L p (R n ) is that if w, /w L p (R n ), then p = while w, /w L implies that w which is also equivalent to the fact that w, /w A (for the precise definition of the A p classes see below). It is known that BMO is the right space to consider in place of L p as p in a number of situations and we will give the answer to this question for BMO in this paper. The definition of BMO is that f BMO if sup f(x) f dx = f < + where f = f(x)dx, and is a cube with sides parallel to the coordinate axes. important to know that the L norm can be replaced by the L p norm for 0 < p <, ( /p sup f(x) f dx) p = f,p f. It is We need also to recall the JohnNirenberg lemma, the reason for the above result, for functions of bounded mean oscillation. If f BMO, there are constants c, c 2 > 0 independent of f and such that t : f(t) f > λ} c e c 2λ/ f, for all λ > 0. Of course, bounded functions are in BMO and ln / x is an unbounded function in BMO. The precise space we will study is BMO = w : R n R + : w, /w BMO}.
2 We need to recall the A p weights which are defined by the condition A p (w) = sup ( ) ( ) p w w p < +, where is again a cube. The A p weights solve the problem of characterizing when the Hardy Littlewood maximal function maps L p w into L p w, where Mf(x) = sup x f(y) dy, and the result is Mf(x) p w(x) dx C p f(x) p w(x) dx w A p. We will also need to consider A = w Mw(x) Cw(x)}, with the smallest such C being denoted A (w) and A = p> A p. Since the A p constants decrease by Hölder s inequality, we can set A (w) = lim p A p (w). We have the set inclusions A A p A q A, where p q. The A p weights also solve the corresponding problem for the Hilbert transform f(y) Hf(x) = lim ɛ 0 ɛ< x y </ɛ x y dy. It is known that if w, /w A p, then w A 2, and we may limit our study to the case p 2 by the inclusion properties of A p. It is also known that [, p. 474] w, /w p> A p ln w clos BMO L. () We say that w RH p0 (reverse Hölder) if ( ) /p0 w p C 0 w, and we abbreviate by RH p0 (w) the infimum of all such C. We will use the fact, due to Strömberg and Wheeden, that w RH p0 if and only if w p 0 A. An alternate proof of this fact can be found in [3, Lemma 3.]. 2. Preliminary results Our first result shows that Hölder continuous functions operate on BM O. Lemma : If F is Hölder continuous of order α, where 0 < α and f BMO, then F f BMO and F f 2 F Lip α f α. Proof. If there is a constant c such that f(x) c dx A, then it is well known that f 2A. We compute ( /p ( /p F (f(x)) F (f ) dx) p F p Lip α f(x) f dx) αp. 2
3 Thus we obtain with p = /α, F f 2 F Lip α f α. This has been, at least partially, observed by many people. If f BMO, then f α BMO, for 0 < α and maxf, g} and minf, g} are in BMO if f, g are in BMO. We haven t noticed the converse observed, but it is true. If F f A f α, then F Lip α. The proof may be found in [2], but as this is not generally available, we give the proof here. Without loss of generality, we may assume F (0) = 0 and consider only cubes centered at the origin since BMO is translation invariant. Suppose that = [ d 2, d 2 ]n and that x on the double of f(x) = 0 outside the double of. One checks that F (f(x)) = F (x ) for x 2, 0 outside the double of. and since f f d 2, one finds d d 2 d 2 F (x ) F dx Ad α, where is the onedimensional cube [ d 2, d 2 ], and by the CampanatoMeyer theorem [4], this proves the result. We can use the lemma to show that there is a close connection between BMO and BMO. Theorem : A real valued function u is in BMO there exists a w BMO such that u = w /w and w + /w u. Proof. If u admits the decomposition, it is clear that u BMO. If we are given a u BMO, it is easy to see that the equation for w leads to a quadratic equation with a solution of w = 2 (u + u 2 + 4). The function F (x) = 2 (x + x 2 + 4) is everywhere differentiable with derivative bounded by. By Lemma, w BMO. Remark. We note that the same proof proves the corresponding result for functions of vanishing mean oscillation, which are defined as is BMO but when the sup is taken over cubes of side r, and the resulting sup goes to 0 as r 0+. Another application of Lemma is to the determination of conditions under which the square of a function belongs to BMO. By Lemma with F (x) = x, it follows that such a function belongs to BMO. We show that more is true. Lemma 2: If f = F (u), F Lip α, u BMO, then x : f(x) F (u ) > λ} c e c 2λ /α / F /α Lip α u. Proof. Because u BMO, by the JohnNirenberg lemma, there are constants c and c 2 such that t : u(t) u > λ} c e c 2λ/ u. Hence, since ( ) /α t : f(t) F (u ) > λ} t : u(t) u λ > F Lip α, we have the inequality t : f(t) F (u ) > λ} c e c 2( λ ) F /α / u Lip α, 3
4 which is the desired result. Corollary : For any ɛ < c 2, e ( ) Lipα u c2 ɛ dx c. ɛ (c 2 ɛ) f(x) F (u ) /α F /α Proof. Let φ(x) = e Ax/α, which is increasing with φ (x) = A α x/α e Ax/α. As long as A is positive, e A f(x) F (u) /α dx = A x : f(x) F (u ) > λ} λ /α e Aλ/α dλ α 0 ( ) A α c 0 e c 2 F /α Lip α u A If we choose A less than the fraction, we can use the fact that α to obtain the above estimate. 0 e ɛλ/α ɛλ /α dλ = 0 e u du = λ /α λ /α dλ. If we modify the choice of φ slightly by putting ψ(x) = e Ax/α, we see that for α, ψ is convex and we can apply Jensen s formula to, p =, f = f(x) F (u ) and if we note that f F (u ) = (f(x) F (u )) f(x) F (u ), we can make the estimate ( ) ψ(2 f F (u ) ) ψ 2 f(x) F (u ) e A2/α f(x) F (u ) /α. We now combine this with Corollary and obtain e A f(x) f /α = e A f(x) F (u )+F (u ) f /α e A2/α f(x) F (u ) /α +A2 /α f F (u ) /α = e A2/α f F (u ) /α e A2/α f(x) F (u ) /α. If we choose A2 /α = (c 2 ɛ)/( F /α Lip α u ), we can estimate this and ( ) e A f(x) f /α c ( c2 ɛ ɛ 4 ) + ψ(2 f F (u ) )
5 by using Corollary and now we apply Jensen s inequality to get ( ( ) ) e A f(x) f /α c2 ɛ c + ɛ ( ( ) c2 ɛ 2 c + ). ɛ We can now state and prove the following. e A2/α f(x) F (u ) /α Theorem 2: Consider the set of f = F (u), u BMO, 0 < α. The following two statements are equivalent. (i) F Lip α (ii) there exists 0 < c, c 2 <, 0 < A <, independent of, u BMO such that and then A F /α Lip α. x : f(x) f > λ} c e c 2 λ /α A u. Proof. We will first prove that (i) implies (ii). By restricting the range of integration in the inequality derived after Corollary, we see that E λ x : f(x) f > λ} e Aλ/α e A f(x) f /α dx, since e Aλ/α e A f(x) f /α > on E λ. This is the desired result if we choose ɛ = c 2 2 and A as above. We next show that (ii) implies (i). We first observe that (ii) implies that for some constants 0 < c 3, c 4 < ( c3 f(x) f /α ) exp c 4. A u This implies that Hölder now gives us, since /α, Hence L c 3 f(x) f /α c 4. A u L ( c α 3 f(x) f ) /α A α u α. f(x) f CA α u α. The proof is now completed by an application of [2]; see the argument after Lemma. Corollary 2: If b k BMO, then x : b(x) b > λ} c e c 2λ k / b k. 5
6 Proof. Apply the above theorem with u(x) = b k, F (x) = x /k which is Lipschitz continuous of order /k with Lipschitz constant. Remark. The argument actually shows that if x : u(x) u > λ} c e c 2λ k, then x : f(x) f > λ} (c + ) 2 e c 2 2/α F /α Lip α λ k/α. Our main result connects the behavior of functions in BMO with the A p classes. Theorem 3: The set of nonnegative functions which are BMO along with their reciprocals is contained in the intersection of all the A p classes for p >, i.e. BMO p> A p. Remarks.() Of course, if b BMO, then /b BMO p> A p and () above implies ln b clos BMO L. (2) The class BMO is nonempty. For example, b (x) = max(ln / x, e) BMO and /b L BMO. Moreover, if we take b 2 (x) = max(ln / x, / ln( x e 2 )) we get an example of a function which is unbounded and whose inverse is unbounded, yet both b 2, /b 2 BMO. (3) The result is sharp in the sense that the function b in the theorem cannot be in A since if it were, /b would also be in A and then by a result of Johnson and Neugebauer [3, Lemma 2.2], b. (4) The converse is, however, not true because with the same function b as above, b 2 satisfies /b 2 L and ln b 2 = 2 ln b clos BMO L and therefore b 2 p> A p and b 2 p> A p, but b 2 / BMO. We will prove Theorem 3 as a special case of a more general result, but let us indicate how it can be proved directly. The first step is a lemma. Lemma 3: Let us denote by then we have f = f(x)dx, (fg) f g = (f(x) f )(g(x) g )dx. Proof. Compute and use the fact that g g has mean value zero. We are ready for the first step in this version of the proof of Theorem 3. Theorem 4: Suppose b BMO, then b is in A 2. 6
7 Proof. Apply Lemma 3 to b and /b which gives b (/b) = (b(x) b )(/b(x) (/b) )dx and allows us to make the estimate b (/b) b /b. Hölder s inequality shows that b (/b) and the above becomes b (/b) + b /b. Theorem 5: If b BMO, then b A 3/2. For the proof of this statement we have to estimate ( ) ( ) /2 b b 2. First we require another lemma. Lemma 4: With the same notation as in Lemma 3, we have (f(t) f )(g(t) g )(h(t) h )(l(t) l )dt = (fghl) f (ghl) g (fhl) h (fgl) l (fgh) + f g (hl) +f h (gl) + f l (gh) + g h (fl) + g l (fh) +h l (fg) 3f g h l. Proof. We expand the integrand and compute the resulting terms. Take f = h = b, g = l = b. We obtain b ( b ) ( b ) b b ( b ) ( b ) b + b ( b ) + (b ) 2 ( b 2 ) + b ( b ) + ( b ) b + (( b ) ) 2 (b 2 ) + b ( } b ) This allows us to estimate which means that + (b ) 2 ( b 2 ) + + (b ) 2 ( b 2 ) + In particular, b ( b 2 ) /2 3(b ) 2 (( b ) ) 2 = ( 2 (b(t) b ) ) 2 b (t) ( b ) dt. ( ( ) 2 ( b ) (b 2 ) 3(b ) 2 ( ) 2 b ) b 2,4 b 2,4 ( ( ) 2 ( b ) (b 2 ) 3(b ) 2 ( ) 2 b ) + b 2,4 b 2,4. b b + 3A 2 (b), which proves that A 3/2 (b) 3 + ( 3 + ) b b. The remainder of the direct proof of Theorem 3 proceeds like this. To prove that b, /b are in A 4/3 do the corresponding formula with 8 terms of which 4 are b and 4 are /b, etc
8 3. A p weights whose reciprocals are A p weights We will now obtain Theorem 3 as a special case of the next result. Theorem 6: Suppose < p 0 2. Then the following are equivalent. w, /w A p0 (2) L = w w p 0 w ( w ) p 0 c < +. (3) Proof. Suppose (2) holds. Let r = p 0. Note that L w r (w ) r w r ( w )r + (w r ) ( w )r + w r ( w r ) + w r ( w )r + A p0 ( w ) + A p 0 (w) + A 2 (w) r c < +, because w A p0 implies w A 2. Conversely, if (3) holds, then we first note that w A 2. This follows from the next sequence of inequalities: c /r L /r w w w ( w ) (w w )(( w ) w ) = w ( w ) w ( w ) + w ( w ). We use the fact that if r, then a r b r ar b 2 r. Write r (w w )( w ( w ) ) = w( w ) + w w (w ( w ) + ) which allows us to estimate the integrand below by (w w )( w ( w ) r ) 2 r w( w ) + r ( } w w w ( r w ) + ), 2 r w r ( w )r + } ( w r wr w ( r w ) + ). Now we take the average of this over which gives 2 r (w r ) ( w )r + ( } w r ) w r c + (A 2 (w) + ) r, and we conclude that w, w A p 0. Theorem 3 follows from this result; in fact, we obtain the estimate L w p 0,p(p 0 ) w p 0,p (p 0 ) and as BMO is characterized by f,p for any p > 0, we can have any p 0 > which proves the result. Although we proved Theorem 6 for A p, it immediately implies a result about RH r. 8
9 Theorem 7: The following statements are equivalent for r < : w, /w RH r (4) w, /w A +/r (5) w r A 2. (6) Proof. (4) (5). Since w r, /w r A, we have that w r, /w r A 2, and hence w, /w A +/r. (5) (4). w A +/r /w RH r. Similarly, w RH r. (4) (6). Since w r, /w r A, we have that w r A 2 as above. (6) (4). Since w r A 2, w A +/r /w RH r. From the fact that w r A 2, it follows that w r A 2 and this implies that we can apply the above remark to /w. Theorem 8: Suppose u BMO and α > 0. Then u 2 + α p> A p. Proof. For any λ > 0, write λu = w λ /w λ, for some w λ BMO. Then λ 2 u 2 = wλ wλ 2 By Theorem 7, wλ 2 A and since w λ p> A p, by Lemma 2.4 in [3], wλ 2 p> A p and a similar result holds for. This shows that λ 2 u wλ 2 p> A p and hence, u λ 2 p> A p. Since λ is an arbitrary positive number, the result follows. References [] GarcíaCuerva, J. and Rubio de Francia, J. L.: Weighted norm inequalities and related topics, (North Holland Math. Studies: Vol. 6), Amsterdam: NorthHolland, 985. [2] Johnson, R. L.: Behaviour of BMO under certain functional operations. Preprint University of Maryland, September 975. [3] Johnson, R. L. and Neugebauer, C. J.: Homeomorphisms preserving A p, Rev. Mat. Iberoamericana, 3(2), 987, [4] Meyers, N. G.: Mean oscillation over cubes and Hölder continuity, Proc. Amer. Math. Soc., 5(964), R. L. Johnson Supported by a grant from the NSF Department of Mathematics University of Maryland College Park, Maryland and C. J. Neugebauer Department of Mathematics Purdue University West Lafayette, Indiana June 2, 99 9
A STABILITY RESULT ON MUCKENHOUPT S WEIGHTS
Publicacions Matemàtiques, Vol 42 (998), 53 63. A STABILITY RESULT ON MUCKENHOUPT S WEIGHTS Juha Kinnunen Abstract We prove that Muckenhoupt s A weights satisfy a reverse Hölder inequality with an explicit
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationand s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space
RAL ANALYSIS A survey of MA 641643, UAB 19992000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σalgebras. A σalgebra in X is a nonempty collection of subsets
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More information5 Indefinite integral
5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationCylinder Maps and the Schwarzian
Cylinder Maps and the Schwarzian John Milnor Stony Brook University (www.math.sunysb.edu) Bremen June 16, 2008 Cylinder Maps 1 work with Araceli Bonifant Let C denote the cylinder (R/Z) I. (.5, 1) fixed
More information6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationMATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π
MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationON A GLOBALIZATION PROPERTY
APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 69 73 S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X.
More informationSOME TYPES OF CONVERGENCE OF SEQUENCES OF REAL VALUED FUNCTIONS
Real Analysis Exchange Vol. 28(2), 2002/2003, pp. 43 59 Ruchi Das, Department of Mathematics, Faculty of Science, The M.S. University Of Baroda, Vadodara  390002, India. email: ruchid99@yahoo.com Nikolaos
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationThe fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),
More information( ) ( ) [2],[3],[4],[5],[6],[7]
( ) ( ) Lebesgue Takagi,  []., [2],[3],[4],[5],[6],[7].,. [] M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan J. Appl. Math., (984), 83 99. [2] T. Sekiguchi and Y. Shiota, A
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationEXISTENCE AND NONEXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7669. UL: http://ejde.math.txstate.edu
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationCompactness in metric spaces
MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the
More informationAnalysis I. 1 Measure theory. Piotr Haj lasz. 1.1 σalgebra.
Analysis I Piotr Haj lasz 1 Measure theory 1.1 σalgebra. Definition. Let be a set. A collection M of subsets of is σalgebra if M has the following properties 1. M; 2. A M = \ A M; 3. A 1, A 2, A 3,...
More informationDouble Sequences and Double Series
Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine Email: habil@iugaza.edu Abstract This research considers two traditional important questions,
More informationCourse 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationx = (x 1,..., x d ), y = (y 1,..., y d ) x + y = (x 1 + y 1,..., x d + y d ), x = (x 1,..., x d ), t R tx = (tx 1,..., tx d ).
1. Lebesgue Measure 2. Outer Measure 3. Radon Measures on R 4. Measurable Functions 5. Integration 6. CoinTossing Measure 7. Product Measures 8. Change of Coordinates 1. Lebesgue Measure Throughout R
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationNonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results
Proceedings of Symposia in Applied Mathematics Volume 00, 1997 NonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results Freddy Delbaen and Walter Schachermayer Abstract. The
More informationPROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.
More informationTRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh
Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp . TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More information) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.
Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationError Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach
Outline Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach The University of New South Wales SPOM 2013 Joint work with V. Jeyakumar, B.S. Mordukhovich and
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More informationSolutions to Tutorial 2 (Week 3)
THE UIVERSITY OF SYDEY SCHOOL OF MATHEMATICS AD STATISTICS Solutions to Tutorial 2 (Week 3) MATH3969: Measure Theory and Fourier Analysis (Advanced) Semester 2, 2016 Web Page: http://sydney.edu.au/science/maths/u/ug/sm/math3969/
More informationNotes on weak convergence (MAT Spring 2006)
Notes on weak convergence (MAT4380  Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationk=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =
Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem
More informationAnalysis via Uniform Error Bounds Hermann Karcher, Bonn. Version May 2001
Analysis via Uniform Error Bounds Hermann Karcher, Bonn. Version May 2001 The standard analysis route proceeds as follows: 1.) Convergent sequences and series, completeness of the reals, 2.) Continuity,
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationChapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces
Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z
More informationMath 317 HW #5 Solutions
Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES  CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS
ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute
More informationA characterization of trace zero symmetric nonnegative 5x5 matrices
A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationHouston Journal of Mathematics. c 2012 University of Houston Volume 38, No. 4, Communicated by Charles Hagopian
Houston Journal of Mathematics c 2012 University of Houston Volume 38, No. 4, 2012 ABSOLUTE DIFFERENTIATION IN METRIC SPACES W LODZIMIERZ J. CHARATONIK AND MATT INSALL Communicated by Charles Hagopian
More informationLet H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationAn optimal transportation problem with import/export taxes on the boundary
An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint
More informationContinuity of the Perron Root
Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationIrreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients
DOI: 10.2478/auom20140007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca
More informationSets and functions. {x R : x > 0}.
Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.
More informationGeneral Theory of Differential Equations Sections 2.8, 3.13.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.13.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationMath 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko
Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. GacovskaBarandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. GacovskaBarandovska Smirnov (1949) derived four
More informationTheorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.
Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,
More information5. Factoring by the QF method
5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationto appear in Birkäuser Advanced courses in Mathematics C.R.M. Barcelona A COURSE ON SINGULAR INTEGRALS AND WEIGHTS
to appear in Birkäuser Advanced courses in Mathematics C.R.M. Barcelona A COURSE ON SINGULAR INTEGRALS AND WEIGHTS CARLOS PÉREZ Abstract. This article is an expanded version of the material covered in
More informationProblem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS
Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection
More informationLecture Notes on Elasticity of Substitution
Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationCollege of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions
College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use
More informationInfinite product spaces
Chapter 7 Infinite product spaces So far we have talked a lot about infinite sequences of independent random variables but we have never shown how to construct such sequences within the framework of probability/measure
More informationLabeling outerplanar graphs with maximum degree three
Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics
More informationRieszFredhölm Theory
RieszFredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A AscoliArzelá Result 18 B Normed Spaces
More informationQuasi Contraction and Fixed Points
Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa00168, 6 Pages doi:10.5899/2012/jnaa00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationProf. Girardi, Math 703, Fall 2012 Homework Solutions: Homework 13. in X is Cauchy.
Homework 13 Let (X, d) and (Y, ρ) be metric spaces. Consider a function f : X Y. 13.1. Prove or give a counterexample. f preserves convergent sequences if {x n } n=1 is a convergent sequence in X then
More information