- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 646012, 14 pages

http://dx.doi.org/10.1155/2013/646012

## Sharp Inequalities for the Haar System and Fourier Multipliers

Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Received 27 May 2013; Accepted 30 September 2013

Academic Editor: Kehe Zhu

Copyright © 2013 Adam Osȩkowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A classical result of Paley and Marcinkiewicz asserts that the Haar system on forms an unconditional basis of provided . That is, if denotes the projection onto the subspace generated by ( is an arbitrary subset of ), then for some universal constant depending only on . The purpose of this paper is to study related restricted weak-type bounds for the projections . Specifically, for any we identify the best constant such that for every and any Borel subset of . In fact, we prove this result in the more general setting of continuous-time martingales. As an application, a related estimate for a large class of Fourier multipliers is established.

#### 1. Introduction

Our motivation comes from a very natural question about , the Haar system on . Recall that this collection of functions is given by
and so on. Here we have identified a set with its indicator function. A classical result of Schauder [1] states that the Haar system forms a basis of , (with the underlying Lebesgue measure). That is, for every there is a unique sequence of real numbers satisfying . For any subset of nonnegative integers, we will denote by the projection onto the space generated by the subcollection . Let be the unconditional constant of , that is the least such that
for any and any . Using Paley’s inequality [2], Marcinkiewicz [3] proved that if and only if . This remarkable and beautiful fact and its various extensions have influenced several areas of mathematics, including the theory singular integrals, stochastic integrals, the structure of Banach spaces, and many others. As an example, let us consider the martingale version of (2), which was obtained by Burkholder in [4]. Assume that is a probability space, filtered by , a nondecreasing family of sub--fields of . Let be a real-valued martingale with the difference sequence given by and for . Let be a transform of by a predictable sequence with values in : that is, we have for all and by predictability we mean that each term is measurable with respect to . Then (cf. [4]) for there is a universal constant for which
Here we have used the notation . Let (2) and (3) denote the optimal constants in (2) and (3), respectively. The Haar system is a martingale difference sequence with respect to its natural filtration (on the probability space being Lebesgue’s unit interval) and hence so is , for given fixed real numbers (sometimes such special martingales are called *Haar martingales*, *Paley-Walsh martingales,* or *dyadic martingales*). In addition, the deterministic 0-1 coefficients are allowed in the transforming sequence, so (2) (3) for all . It follows from the results of Burkholder [5] and Maurey [6] that the constants actually coincide: (2) (3) for all . The question about the precise value of (2) was answered by Choi in [7]: the description of the constant is quite complicated, so we will not present it here and refer the interested reader to that paper.

Our objective will be to study a certain sharp version of (2), Let us provide some defnitions. Assume that is a given measure space. A linear (or sublinear) operator defined on and taking values in is said to be of *restricted weak type *, if there is a constant such that, for every measurable set of finite measure,
Here
is the usual weak quasinorm on the Lorenz space . One of the reasons for considering restricted weak-type estimates is that usually these bounds are easier to obtain than other types of inequalities: indeed, the functions involved are bounded and two-valued instead of arbitrary measurable. On the other hand, by means of standard interpolation arguments (see, e.g., Corollary 1.4.21 in Grafakos [8]), a pair of restricted weak-type estimates implies various estimates on intermediate spaces. We will establish a sharp version of restricted weak type bounds for the projections . Introduce the constants by
Furthermore, if is a discrete-time martingale, we define its weak th quasinorm by . Here is one of our main results.

Theorem 1. *Let be a martingale taking values in , terminating at (i.e., satisfying almost surely), and let be its transform by a predictable sequence with values in . Then for any one has
**
The constant is the best possible. It is already optimal in the estimate
**
for the projections associated with the Haar system.*

We will also provide a version of this result for the case in which the space is endowed with a different norming. As we will see, this new version of restricted weak-type estimates will be more convenient for applications (cf. Remark 11 below). Namely, for put where the supremum is taken over all measurable with . Unfortunately, under these norms, we have managed to prove sharp restricted bounds in the case only (and we do not know the corresponding sharp bounds for ). In analogy with the above definitions, if is a discrete-time martingale, we let .

Theorem 2. *Let be a martingale taking values in , terminating at , and let be its transform by a predictable sequence with values in . Then for any one has
**
The constant is the best possible. It is already optimal in the estimate
**
for the projections associated with the Haar system.*

So, for the best constant is the same for both norms and .

All the results discussed above can be formulated in the more general setting of continuous-time martingales. Furthermore, instead of transforms with values in , one can work under the less restrictive assumption of nonsymmetric differential subordination of martingales (for the necessary definitions and the precise statement of our results, we refer the reader to Section 2). This setting has the advantage of being more convenient for applications, which constitute the second half of the paper. Specifically, we will apply the aforementioned martingale estimates in the study of the corresponding bounds for Fourier multipliers. This will be done in Sections 3 and 4.

#### 2. A Martingale Inequality

##### 2.1. Background and Main Results

Assume that is a complete probability space, equipped with , a nondecreasing family of sub--fields of , such that contains all the events of probability . Suppose that , are two adapted real-valued martingales, whose paths are right continuous and have limits from the left. The symbol will stand for the quadratic covariance process of and (see, e.g., Dellacherie and Meyer [9] for details). Following Bañuelos and Wang [10] and Wang [11], we say that is differentially subordinate to , if the process is nonnegative and nondecreasing as a function of . For example, assume that is a discrete-time martingale and let denote its transform by a certain predictable sequence with values in . Let us treat these two sequences as continuous-time processes, via , , . Then the required condition on is equivalent to saying that which is the original definition of differential subordination due to Burkholder [5, 12]. Obviously, this condition is satisfied for the above setting of martingale transforms.

As exhibited in [13, 14], martingales , satisfying the differential subordination arise naturally in the martingale study of Fourier multipliers. In this paper, we will work with pairs , satisfying a slightly different condition: which can be understood as “nonsymmetric differential subordination.” For instance, this holds in the above setting of martingale transforms, if we assume that the sequence takes values in (and hence the continuous-time setup does form an extension of the discrete-time case described in the previous section). Inequalities for such martingales were studied by several authors: see, for example, Burkholder [15], Choi [7], and the author [16, 17]. We refer the interested reader to those papers and mention here only result, which will be needed later. It was proven for martingale transforms by Burkholder [15] and in the general continuous-time case by the author in [17]. Throughout, we use the notation , , and , .

Theorem 3. *Let , be two real-valued martingales satisfying (13). Then for any one has
**
For each the inequality is sharp. Therefore, and the constant cannot be improved.*

We turn our attention to the formulation of the main result of this section. We will use the notation

Theorem 4. *Suppose that is a martingale taking values in and is a real valued martingale such that (13) is satisfied.*(i)*For any one has
* *The bound on the right-hand side of (16) is the best possible for each , even in the following version for the Haar system: for any and ,
*(ii)*For any one has
* *The bound on the right-hand side is the best possible for each , even in the following version for the Haar system: for any and ,
*

Some comments on the above statement are in order. At the first glance, part (ii) may seem a little artificial, but this is not the case. As we will see (consult Remark 11), the inequality (18) is very convenient for our applications. The second remark concerns the proof of Theorem 4. Namely, the main difficulty lies in showing the assertion for . Indeed, when , then (16) is an immediate consequence of (14), and its sharpness follows from simple examples. Furthermore, having proved (18) for , we deduce the case by a standard limiting argument. Finally, note that , which implies that the inequality (18) is stronger than (16). Putting all these facts together, we see that we will be done if we establish the second estimate of Theorem 4 in the case and prove the sharpness of (17) for .

##### 2.2. Special Function and Their Properties

The proof of the inequality (18) will be based on Burkholder’s method. This technique reduces the problem of proving a given martingale inequality to that of constructing a special function, which possesses certain convexity and majorization properties. For the detailed description of the approach, we refer the interested reader to Burkholder’s survey [18] and to the recent monograph [19] by the author.

The purpose of this subsection is to introduce special functions corresponding to (18) and present their basic properties, which will be needed later. We assume that is a fixed parameter. First, consider the following subsets of : Now we introduce a function by and extend it to the whole strip by the condition

Let us provide some information on this object. In what follows, the symbol denotes the interior of a set .

Lemma 5. *The function enjoys the following properties.*(i)*It is of class on and of class in , , , and .*(ii)*There is a Borel function with the following property: for any and any such that ,
*(iii)*For any and one has the majorization
*(iv)*For any and one has
*

*Proof. *(i) This is straightforward. The fact that is of class on each is evident, and to show that is of class in the strip, one needs to check that the partial derivatives match appropriately at the common boundaries of , , , and . We leave the necessary calculations to the reader.

(ii) If , then the left-hand side of (23) equals , so we may take . If belongs to the interior of , the expression on the left of (23) is equal to
But , because ; therefore satisfies the desired bound. Next, assume that lies in . We compute the left-hand side of (23) and obtain
This time we have and hence works fine. Finally, if lies in the interior of , then
so we may take .

Before we proceed, let us observe that, by (22), the inequality (23) holds also in the interiors of the “reflected” domains , , , and given by
with given by .

(iii) Directly from (i) and (ii), the function has the following property: for a fixed , the function is concave on (simply plug in (23)). Since the right-hand side of (24) does not depend on , it suffices to verify the majorization for only. Furthermore, because of (22), we may restrict ourselves to two cases and ; and . If and , then the right-hand side vanishes, while the left-hand side is nonnegative. If and , then we must prove that , which is equivalent to the obvious estimate:
Next, if and , then both sides of (24) are equal to . Finally, if and , the majorization reads
or, equivalently, .

(iv) Since , we can rewrite the bound in the form
It follows from (i) and (ii) that, for any , the function is concave (if we put in (23), the right-hand side of this bound is nonpositive). Consequently, we will be done if we show that , the one-sided derivative of at , does not exceed . But this is simple: we have
This completes the proof of the lemma.

##### 2.3. Proof of (18) for

It is convenient to split the reasoning into a few separate parts.

*Step 1 (a mollification argument). *The proof of (18) rests on Itô’s formula. Since is not of class , this enforces us to modify so that it has the required smoothness. Consider a function , supported on the unit ball of and satisfying . For a given , let be defined on by the convolution
The function is of class in the interior of its domain and inherits the crucial properties from . Namely, we have the following version of (24):
for all . Next, by Lemma 5 (i) and the integration by parts, we get
Similar identities hold for and , so we see that satisfies (23) for all , with
(the function constructed above is locally bounded, so there is no problem with the integration).

*Step 2 (application of Itô’s formula). *Take martingales , as in the statement and consider the processes , , and for . Observe that the pair still satisfies (13). Furthermore, takes values in the strip , so an application of Itô’s formula to the process yields
where
Here denotes the jump of at time , and is the unique continuous part of the bracket (cf. Dellacherie and Meyer [9]). Let us analyze each of the terms separately. We have , by the properties of stochastic integrals. By straightforward approximation argument (see, e.g., Wang [11]), the inequality (23) and the domination (13) imply that . Finally, each term in the sum is also nonpositive. To see this, observe first that for each we have
since otherwise the condition (13) would not be satisfied. Now, applying the mean-value property, we get that
where is a certain point in . Using (23), this can be bounded from above by . Thus (40) gives .

*Step 3 (the final part). *If we combine all the above facts and take expectation of both sides of (38), we obtain the estimate . By (35), this implies . If we let , then , and , so we get
by Fatou’s lemma and Lebesgue’s dominated convergence theorem (we have for all ). It remains to use (25): by (13), we have and hence . Taking the supremum over completes the proof.

##### 2.4. Proofs of Inequalities of Theorems 1 and 2

We start with the following important auxiliary fact.

Corollary 6. *Suppose that is a martingale taking values in and let be a real-valued martingale such that (13) holds true. Then for any , , and one has
*

*Proof. *We have , where and . By (18),
and, obviously, . Adding the two inequalities above yields the claim.

Equipped with the above statement, we turn to Theorems 1 and 2.

*Proof of (7) and (10). *We prove these estimates in the more general continuous-time setting described above. Suppose that is a martingale taking values in and terminating at , and let be a real-valued martingale such that (13) is satisfied. Multiplying both sides of (16) by gives
Let us optimize the right-hand side over . If , then the maximal value is attained at , and we get
where in the last passage we have used the fact that . On the other hand, if , then the right-hand side of (45) is maximized for the choice . Substituting this value of gives , and hence (7) follows. The inequality (10) can be proven in a similar manner, with an additional help of (43). Namely, fix appropriate , , and of positive probability. Assume first that and optimize the right-hand side of (43) over . A straightforward analysis of the derivative shows that the maximum is attained for . Plugging this value of gives the bound
(here is the harmonic conjugate to ). But the expression in the square brackets, considered as a function of , does not exceed . Thus, it suffices to divide by , take the supremum over , and note that to get the desired bound. It remains to consider the case when . An application of Schwarz inequality, Burkholder’s bound (which follows from the chain ), and the fact that terminates at the set imply
But we have : indeed, this can be rewritten in the form and follows from the equality and the bounds , . Combining this with the above estimate, the inequality , and the equation , give
Now (10) follows immediately.

##### 2.5. Sharpness of (8), (11), (17), and (19)

By an application of the results of Burkholder (see Section 10 in [12]) and Marcinkiewicz [3], the best constants in the inequalities for the Haar system are the same as those in the corresponding estimates for discrete-time martingales (roughly speaking, any martingale pair , where is a transform of , can be appropriately embedded into a pair consisting of a dyadic martingale and its transform). This is also closely related to the equality (2) = (3), which we have discussed at the beginning of the paper. Thus, we will be done if we provide the construction of appropriate martingales.

We start from showing that (17) and (19) are sharp. Assume first that , fix , and define the pair by the following conditions:(i),(ii), (iii) is a random variable satisfying

Then we easily see that .

We turn to the more difficult case . As we have already noted, (19) is stronger than (17), so it suffices to focus on the latter estimate. Let be a fixed number and let , where is a large positive integer. Consider a sequence of independent mean-zero random variables with the distribution uniquely determined by the following conditions.(i) takes values in .(ii)For , takes values in the set .(iii)For , takes values in the set .(iv) takes values .

Next, let , with the convention . It is easy to check that is an almost surely finite stopping time (with respect to the natural filtration of ). Since are centered, the process is a martingale. Let denote the transform of by the deterministic sequence . To gain some intuition about , let us take a look at its dynamics. The pair starts from the point and, at the first move, it goes to or to . If it went to , it stays there forever; if it jumped to , then it moves horizontally either to (and stops) or to . If the latter possibility occurs, the movement continues: the pair goes to (and terminates) or to . In the latter case, it moves horizontally to or to and so on. During the first steps, the pair either hits one of the lines , (and stops) or visits the point on th step. If the latter takes place, then jumps to or to .

Directly from the above analysis, we see that takes values in and terminates at , and However, recall that . Therefore, if we let go to infinity, the latter expression converges to . This shows that the constant cannot be replaced in (17) by a smaller number.

The examples analyzed above can be also used to prove the sharpness of (8) and (11). First, suppose that and consider the above example for . Then and hence is the best in (8). On the other hand, if , we take the above example corresponding to and a large . Then and hence cannot be replaced in (8) by a smaller number. This also proves the sharpness of (11), since this bound is stronger than (8) (we easily check that for all ).

#### 3. Applications to Fourier Multipliers

For the sake of convenience, we have split this section into three parts. The first of them contains the necessary definitions, an overview of related facts from the literature and the description of our contribution. The second subsection explains very briefly the martingale representation of a certain class of Fourier multipliers, which will be of importance to us; the material is taken from [13, 14], and we have included it here for completeness. The final subsection contains the proof of our main result.

##### 3.1. Background, Notation and Results

It is well known (cf. [10, 13, 14, 20–23] and numerous other papers) that the martingale theory forms an efficient tool to obtain various bounds for many important singular integrals and Fourier multipliers. Recall that, for any bounded function , there is a unique bounded linear operator on , called *the Fourier multiplier with the symbol *, given by the equality . The norm of on is equal to and a classical problem of harmonic analysis is to study/characterize those , for which the corresponding Fourier multiplier extends to a bounded linear operator on , . This question is motivated by the analysis of the classical example, the collection of Riesz transforms on (see Stein [24]). Here, for any , the transform is a Fourier multiplier corresponding to the symbol , . An alternative definition of involves the use of singular integrals:
It is well known that singular integral operators play a distinguished role in the theory of partial differential equations and have been used, in particular, in the study of the higher integrability of the gradient of weak solutions. The exact information on the size of such operators (e.g., on the -norms) provides the insight into the degrees of improved regularity and other geometric properties of solutions and their gradients. This gives rise to another classical problem for Fourier multipliers: for a given , provide tight bounds for the size of the multiplier in terms of some characteristics of the symbol.

We will extend the aforementioned restricted weak-type estimates to this new setting. We will consider a certain subclass of symbols which are particularly convenient from the probabilistic point of view. Namely, they can be obtained by the modulation of jumps of certain Lévy processes. This class has appeared for the first time in the papers by Bañuelos and Bogdan [13] and Bañuelos et al. [14]. To describe it, let be a Lévy measure on , that is, a nonnegative Borel measure on such that , and Assume further that is a finite Borel measure on the unit sphere of and fix two Borel functions on and on which take values in the unit ball of . We define the associated multiplier on by if the denominator is not , and otherwise. Here stands for the scalar product in . This class includes many important examples, including the real and the imaginary parts of Beurling-Ahlfors operator (cf. [13, 14]). We will only present here one type of multipliers, which will be of importance later. Pick a proper subset of and take , , and , . Here are the versors in . This choice of parameters gives the operator on .

One of the principal results of [14] is the following estimate (consult also the earlier paper [25] of Nazarov and Volberg which was devoted to the version for Beurling-Ahlfors operator).

Theorem 7. *Let and let be given by (56). Then for any one has
**
where .*

It turns out that the constant is the best possible: see Geiss et al. [22] or Bañuelos and Osȩkowski [26] for details.

Our work will concern a certain subclass of (56), corresponding to those and , which take values in . There are many interesting examples of this type (cf. [26]); for instance the operator introduced above is of this form. We will prove the following result.

Theorem 8. *Suppose that is a symbol given by (56), where and are assumed to take values in . Then for any and any measurable with ,
**
The inequality is sharp. More precisely, for any , any , any , and any proper subset of there is of finite measure such that
*

Following Stein and Weiss [27], we can give the following application of the above result. Let be a Fourier multiplier as in the above statement. Then for any real-valued function , , we have To see this, assume first that , where and are pairwise disjoint subsets of of finite measure. Let and , . Then can be rewritten in the form , where , and By standard approximation, the above inequality extends to any nonnegative . To pass to general real-valued functions, it suffices to use the decomposition and the inequality .

##### 3.2. The Martingale Representation of the Fourier Multipliers (56)

By the reasoning from [14], we are allowed to assume that the Lévy measure satisfies the symmetry condition for all Borel subsets of . To be more precise, for any there is a symmetric which leads to the same multiplier. Furthermore, assume for a while that is finite and nonzero, and introduce the probability measure . Consider the independent random variables such that, for each has exponential distribution with parameter and takes values in and has as the distribution. Next, put for and let for . Next, if is a given function, define its parabolic extension to by Now, fix , and let . We introduce the processes and by These processes are martingales adapted to the filtration (see [13, 14]). The key fact is the following.

Lemma 9. *If takes values in , then the pair satisfies (13).*

*Proof. *The assertion follows immediately from the identities
which can be established by repeating the reasoning from [13].

Now we introduce a family of multipliers. Fix , a function on taking values in the unit ball of , and define the operator by the bilinear form: where . We have the following fact, proven in [13].

Lemma 10. *Let and . The operator is well defined and extends to a bounded operator on , which can be expressed as a Fourier multiplier with the symbol
**
if , and otherwise. Furthermore, (66) holds true for all and all belonging to for some .*

##### 3.3. Proof of (58)

We may and do assume that at least one of the measures , is nonzero. It is convenient to split the reasoning into two parts.

*Step 1. *First we show the estimate for the multipliers of the form
Assume that , so that the above machinery using Lévy processes is applicable. Fix and functions such that takes values in , while takes values in and is supported on a certain set of finite Lebesgue measure. Of course, then the martingale takes values in . By Fubini’s theorem and (43), for any we have
Plugging this into the definition of and taking the supremum over all as above, we obtain
Now if we let , then converges pointwise to the multiplier given by (68). By Plancherel’s theorem, in and hence there is a sequence converging to such that almost everywhere. Thus Fatou’s lemma combined with (70) yields the bound
Now we repeat word by word the optimization arguments used in Section 2 in the proof of (10) (we need to consider the cases and separately). As the result, we obtain the bound
Finally, using some standard approximation arguments, we see that (72) can be applied to (where is a measurable subset of , satisfying ), and we get the estimate:
This is precisely the desired claim (but for the above special multipliers).

*Step 2. *Now we deduce the result for the general multipliers as in (56) and drop the assumption . For a given , define a Lévy measure in polar coordinates by
Here denotes Dirac measure on . Next, consider a multiplier as in (68), in which the Lévy measure is and the jump modulator is given by . Note that this Lévy measure is finite and nonzero, at least for sufficiently small . If we let , we see that
and, consequently, pointwise. Thus (73) yields (7). Indeed, using Plancherel’s theorem as above, we see that there is a sequence converging to such that almost everywhere. It suffices to apply Fatou’s lemma, and the proof is complete.

*Remark 11. *An important comment is in order. The above proof rests on the estimate (43), which we have managed to prove in the case only; this is the reason why the restricted bound (58) holds only for . To get a sharp bound for , we would require a version of (43) for small ; unfortunately, the bound (16) does not seem to be powerful enough to yield any result of this type.

#### 4. On the Lower Bound for the Constant in (58)

We turn to the final section of the paper in which we will show that the constant in (58) is the best possible. The proof will be a combination of various analytic and probabilistic facts, and it is convenient to split the reasoning into a several separate parts. Throughout this section, denotes the ball of center and radius .

##### 4.1. Laminates: Necessary Definitions

Assume that denotes the space of all real matrices of dimension and let be the subclass of which consists of all real symmetric matrices.

*Definition 12. *A function is said to be *rank-one convex*, if is convex for all with rank .

Let stand for the class of all compactly supported probability measures on . For , we denote by the *center of mass* or *barycenter* of .

*Definition 13. *We say that a measure is a *laminate* (and write ), if
for all rank-one convex functions . The set of laminates with barycenter is denoted by .

Laminates can be used to obtain lower bounds for solutions of certain PDEs, as was first noticed by Faraco in [28]. Furthermore, laminates arise naturally in several applications of convex integration, where they can be used to produce interesting counterexamples; see, for example, [29–33]. We will be particularly interested in the case of symmetric matrices. The important fact is that laminates can be regarded as probability measures that record the distribution of the gradients of smooth maps; see Corollary 17. Let us briefly explain this; detailed proofs of the statements below can be found, for example, in [32–34].

*Definition 14. *Let be a given set. Then denotes the class of prelaminates in , that is, the smallest class of probability measures on which(i)contains all measures of the form with and satisfying rank ;(ii)is closed under splitting in the following sense: if belongs to for some and also belongs to with , then also belongs to .

By the successive application of Jensen’s inequality, we have the inclusion . Let us state two well-known facts (see [29, 32–34]).

Lemma 15. *Let with . Moreover, let and . For any bounded domain there exists such that and for all *

Lemma 16. *Let be a compact convex set and with . For any relatively open set with there exists a sequence of prelaminates with and .*

These two lemmas, combined with a simple mollification, yield the following statement proven originally by Boros et al. [35]. It exhibits the connection between laminates supported on symmetric matrices and second derivatives of functions and will play a crucial role below.

Corollary 17. *Let . Then there exists a sequence with uniformly bounded second derivatives, such that
**
for all continuous .*

Let us stress here that the corollary works for laminates of barycenter . This will give rise to some small technical difficulties, as “natural” laminates do not have this property; see below.

##### 4.2. Biconvex Functions and a Special Laminate

In the next step in our analysis, we introduce a certain special laminate. To do this, we need some additional notation. A function is said to be *biconvex* if, for any fixed , the functions and are convex. Now, for a given , pick and let , be martingales of Section 2, which exhibit the sharpness of (10) and (11) (actually, there is a whole family of examples, corresponding to different choices of and —these two parameters will be specified later). Consider the -valued martingale: