I l = [a, a + b 2 ), I r = [ a + b

Transcription

1 1. Lecture 1 n the first lecture we discuss representation of functions on the real line by Haar expansions, with particular focus on the Hilbert space of square summable expansions and Bourgain s jump inequality. Classically a function is understood as an assignment of a value, taken from a set called target range, to every point in a set called domain. For functions on the real line, this classical concept is unnatural: the cardinality of the set of real numbers is uncountable, and set of functions from R into a set containing at least two elements has at least cardinality of the power set of the continuum. This is too large for us: note that well known alternatives to the set of functions such as the space of tempered distributions, or the space of Lebesgue measurable functions have only cardinality of the continuum. An interval in R is of the form [a, b). Focusing on half open intervals will have certain technical advantages, the special choice of left closed and right open intervals is arbitrary. A dyadic interval is one of the form [2 k n, 2 k n + 1) with n Z and k Z. The parameter k (or sometimes 2 k, as will be clear form the context) is called the scale of the interval. Lemma 1.1. (1) The class of dyadic intervals is invariant under the scaling x 2x of the real line, more specifically, [a, b) [2a, 2b) is a sef-bijection of the set of dyadic intervals. (2) The translation x x + 1 on the real line, i.e. [a, b) [a + 1, b + 1) is a bijection of the set of dyadic intervals of scale at most 0. (3) For every k Z and every x R, x is contained in a unique dyadic interval of scale k. (4) For every dyadic interval, the left and right halves defined by l = [a, a + b 2 ), r = [ a + b 2, b). They are again dyadic and called the children of the parent. (5) f and J are two dyadic intervals, then either J = or one of the two intervals is contained in the other. (6) Let be some collection of dyadic intervals with scale bounded above, and let max be the collection of maximal intervals in with respect to set inclusion, then the intervals in are pairwise disjoint and cover the union of all intervals in. 1

2 2 Proof: 1) This bijection amounts to the obvious bijection (translation) (k, n) (k + 1, n) in the parameter range Z 2 of dyadic intervals. 2) This transformation amonts to the obvious bijection (nonlinear shearing) (k, n) (k, n + 2 k ) of the parameter space Z Z 0 of dyadic intervals of scale at most 0. 3) By 1), it suffices to prove this for k = 0. Let n be the greatest integer smaller or equal to x, then n x < n + 1 and clearly n is the only integer satisfying these two inequalities. 4) By 1) and 2) it suffices to prove this for = [0, 1), where it follows from direct inspection. 5) Without loss of generality we may assume J. f and J have the same scale, the claim follows from 3). nductively, assume the claim is proven for = 2 k J and consider a pair of intervals with = 2 k+1 J. We need to show that if there exists x J. then J. Let x J. Let Ĩ be child of that contains x. By induction J Ĩ and hence J. This completes the proof. 6) Given any two different intervals in max, since neither is contained in the other by maximality, they are disjoint by 5). For each point in the union of intervals in, there is a maximal dyadic interval containing this point (use the upper bound on the scale). Hence the point is in the union of elements in max. Given an interval and a Lebesgue integrable function f on, we may form the average of f on : [f] = 1 f(x) dx. Our approach to functions is that we will take the collection of numbers [f] as the more fundamental object than the function itself. We will thus define a function by a collecrtion of numbers b. Note that in the example of such collections coming from averages of a function f, there are many dependencies of the form 2b = b l + b r t is thus technically more convenient to work with differences of averages of sibling intervals. This is encoded in the following. Define the characteristic function of an interval as 1 (x) = 1, if x

3 1 (x) = 0, if x The Haar function on an interval is defined as h = 1/2 (1 l 1 r ). Usually one considers the collection of Haar functions of dyadic intervals. This collection has an improtant orthogonality property: Lemma 1.2. For any two dyadic intervals and J h (x)h J (x) = 1 if = J, h (x)h J (x) = 0 if J. Proof: f = J, then h (x)h j (x) = 1 1 (x) dx = 1. f J and J, then either J l or J r. The cases are analogous, assume that J l. Then 1 h (x)h J (x) = 1 l (x) 1 r (x) dx = 0 J 3 f J = 0, then trivially h (x)h J (x) = 0 These are all cases to be considered. Note that for the definition of Haar functions and the orthogonality property of Haar functions we only need a very primitive notion of the integral, that for finite step functions (finite linear combinations of characteristic functions of intervals). Linear combinations of Haar functions are a basic and generic example for martingales. The word martingale was used in the French culture for a strategy to play a zero sum game. Note that we can identify the interval [0, 1) up to a set of measure zero with an infinte sequence of coin flips, each coin flip deciding on a digit of the binary expansion. n other words, the first coin flip decides whether we are in left or right child of the dyadic interval [0, 1), the next coin flip decides on the next generation of dyadic subinterval and so on. Up to a set of measure zero, the unit interval uniquely parameterizes teh possible outcomes of an infinite sequence of coin flips. A multiple of the Haar function represents a zero sum value for a coin flip, one either looses or wins a fixed amount determined by the Haar

4 4 function, with equal probability. A strategy is a value (bet) attached to every coin flip, which may depend on what happened in the previous flips. The value of every event is then simply determined by evaluation of the proper linear combination of Haar functions. A famous example is the doubling strategy: one bets 1 unit on the first coin flip. f one wins, one is satisfied and bets zero from there on, or else one has lost 1 units and one bets 2 on the next coin flip to offset the loss and win the additional one unit. Whenever one wins, one quits the game (chooses 0 for all future bets) with a total gain of 1, while until one wins the first time one keeps doubling the previous bet. The corresponding linear combination of Haar functions, truncated after n rounds of the game, is 1 on most of the interval [0, 1), but 1 2 n on the interval [1 2 n, 1). While one is likely to win one unit, if one is to not go bankrupt with probability p one can at most bet approximately the fraction p of the begining total wealth. Martingales nowadays are an important concept of stochastic analysis. We form the pre Hilbert space of all finite linear combinations of Haar functions with inner product f, g = f(x)g(x)dx Here for future reference we have written the complex conjugation of g, though currently we shall be satisfied considering the real valued case and omit the conjugation bar. The Haar functions are an orthonormal basis of this pre Hilbert space. n particular, if we have the expansions f = a h g = b h where only finitely many a and b are non-zero, we have (1) f, g = a b and we note f, f = f 2 2 = a 2 Call a sequence (a ) parameterized by dyadic intervals finite if only finitely many entries are non zero. Theorem 1.3 (Bourgain s jump inequality). Assume a is a finite sequence. For each x R, denote by N λ half of the longest possible

5 5 chain k 1 > k 2 k 3 > k 4 k 2Nλ such that for every odd index i we have a h (x) > λ. 2 k i >2 k i+1 Then λ 2 N(x) dx < a 2 < Here the symbol stands for less than or equal a constant times where the constant is universal: in the present case one may choose 100 for example. Note also that the conclusion of the Theorem may be written as λn 1/2 λ 2 f 2 where f = a h. Let be the set of intervals for which a 0. t suffices to show for each max λ 2 N(x) dx < a J 2 J since the desired inequality follows upon summing over J. Fix J max. Let J be the collection of maximal dyadic intervals J such that for all x a h (x) > λ J = Note that the left hand side is constant for x, so that the specific choice of x is not important. Let T be the collection of all dyadic intervals contained in J but not contained in for any J. The set T is called a stopping time region, or a tree. We shall write T for the top (largest) interval J of the tree. f x and J, then we have { T : x } = { : J} To see equality of these sets, note that the inclusion is trivial from definition whiel the inclusion follows from the fact that the intervals in J are disjoint. We have λ 2 1 (x) dx < a h (x) 2 dx = a 2. J T T

6 6 We then iterate the procedure on each stopping time interval J, declaring this interval a new tree top and proceeding as before. After iterating on all stopping times obtained in the process, we have partitioned the set of dyadic intervals contained in J into a collection T of trees. Summing the previous display over all stopping times we have λ 2 Ñ λ (x) dx a 2 = a 2 T T T where Ñλ(x) denotes the number of times x lies in a stopping time interval. To obtain the conclusion of the Theorem, we apply the above inequality with λ/2 instead of λ and note that N λ (x) Ñλ/2(x) for all x. Namely, if a h (x) > λ 2 k i >2 k i+1 then there has to be a stopping time interval with 2 k i > 2 k i+1. Namely, let J be the smallest tree top containing x that is larger than or equal to 2 k i. Then a h (x) J >2 k i 2 k i >2 k i+1 a h (x) + J >2 k i+1 a h (x). One of the terms on the right hand side has to be greater than λ/2, hence There must be a stopping time interval in the tree with top J that is of scale at most 2 k i+1 and at least 2 k i by choice of J. This proves N λ (x) Ñλ/2(x) and completes the proof of Theorem 1.3 Define the r- variation norm of a sequence b k, k Z to be (b k ) V r = N 1 sup ( b ki+1 b ki r ) 1/r N,k 1 <k 2 <k 3 <k N Bourgain s jump inequality is almost a 2-variation norm of the sequence of partial sums of a h. Note that if a sequence (b k ) has finite r- variation norm then it converges as k. We have the following r-variation norm variant of Bourgain s jump inequality. i=1

7 Corollary 1.4 (Weak type Lépingle). Let 2 r. Then {x : M 1 sup ( M,k 1 > >k M i=1 The implicit constant in the symbol 2 k i >2 k i+1 a h (x) r ) 1/r µ} < µ 2 < depends on r. a 2 Proof: By multiplying a by µ 1, it suffice to prove this for µ = 1. For each x let M(x), k 1 (x),...k M (x) be choices of jump points that come within a factor of 2 of the supremum over all choices. Fix x and let N l (x) be the number of indices i for which 2 l. Then we have for some 2 < r < r: M(x) 1 {x : ( = {x : i=1 2 k i (x) >2 k i+1 (x) a h (x) < 2 1 l M(x) 1 i=1 2 k i(x) >2 k i+1(x) 2 k i(x) >2 k i+1(x) {x : l 0 N l (x)2 rl c r } < {x : N l (x)2 r l c} l 0 a h (x) r ) 1/r 1} a h (x) r 1} 7 < 2 (2 r )l l 0 a 2 where in the last inequlaity we have used Bourgain s jump inequality. Summing the geometric series completes the proof. So far we have worked with the pre Hilbert space of finite sequences a. We appeal to abstract Hilbert space technique to pass to the completion of this pre Hilbert space. t consists of all abstract linear combinations f = a h with a 2 < with the obvious vector operations on the sequences a and the inner product defined by (1).

8 8 Note that by abstract Hilbert space technique the sum a h converges in L 2 unconditionally (regardless of the order of summation). This is because for any order of the sum N N a n h n 2 = a n 2 The latter as positive series converges to a 2. By the Pythagorean theorem, the difference M N M N a n h n a n h n 2 2 = a n 2 a n 2 satisfies a Cauchy condition, easily establishing convergence of the sum a n h n. At every point x, the formal sum f(x) = a h (x) is a two sided infinite sequence ordered by scale. We shall split the sum formally into (2) f(x) = a h (x) + a h (x) >1 1 and discuss convergence separately. The first sum we rewrite formally a h (x) = a h (x) >1 n 1 =2 n The sum =2 n a h (x) converges absolutely and unconditionally, because there is only one nonzero term for every x, The sum in n converges uniformly, because for = 2 n a h (x) a 2 2 n/2. Hence the first sum in (2) is well defined as a pointwise limit.

9 Note that the function 1 [0,1) may be obtained as the pointwise limit of the sequence 2 n/2 h [0,2 n ) n This function does not have integral zero, although all Haar functions do. This is not a contradiction, since the integral over all of R is not a bounded linear functional on L ( R). (n the language of martingales, this function has the meaning that desaster has struck after infinitely long playing of the doubling strategy). We now consider the second sum of (2). The integer intervals [n, n+ 1) are independent in this sum, and we may restrict attention to the interval [0, 1). We consider the truncations a [L] = a if 2 L < and a [L] = 0 otherwise. The variation norm expressions in Lépingle estimate M 1 sup ( a L h (x) r ) 1/r M,k 1 > >k M i=1 2 k i >2 k i+1 are clearly increasing in L, but remain less than Cδ 1/2 a 2 outside a set of measure δ by Lépingle s estimate, for any choice of δ. This implies convergence of a [L] h (x) lim L outside this set of δ. Letting δ tend to zero and taking the intersections of the sets for each δ, we obtain a set of measure 0 outside which the series converges. Proposition 1.1. The Hilbert space span of Haar functions is the space L 2 (R), the Haar expansions of a function f converge pointwise almost everywhere to the function f. Note that prior to this definition we have not used Lebesgue integration, instead only integration of functions constant on small dyadic intervals (finitle linear combinations of Haar functions). Thie proposition can therfore be seen as a natural appearance of the space L 2 (R) independent of Lebesgue measure theory itself. t is an improtant reason to consider Lebesgue integration. The proposition also provides a means of make sense of functions in L 2 (R) pointwise almost everywhere. 9

10 10 Exercise 1.5. Prove that if the series a h (x) in teh above defined order converges to 0 almost everywhere (outside a set of arbitrarily small prescribed measure), then a = 0 for all dyadic intervals.