SHARP INEQUALITIES FOR THE HAAR SYSTEM AND FOURIER MULTIPLIERS

A classical result of Paley and Marcinkiewicz asserts that the Haar system h = (hk)k≥0 on [0, 1] forms an unconditional basis of L p(0, 1) provided 1 < p <∞. That is, if PJ denotes the projection onto the subspace generated by (hj)j∈J (J is an arbitrary subset of N), then ||PJ ||Lp(0,1)→Lp(0,1) ≤ βp for some universal constant βp depending only on p. The purpose of this paper is to study related restricted weak-type bounds for the projections PJ . Specifically, for any 1 ≤ p <∞ we identify the best constant Cp such that ||PJχA||Lp,∞(0,1) ≤ Cp||χA||Lp(0,1) for every J ⊆ N and any Borel subset A of [0, 1]. 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.


Introduction
Our motivation comes from a very natural question about ℎ = (ℎ  ) ≥0 , the Haar system on [0, 1].Recall that this collection of functions is given by ℎ 0 = [0, 1) , , ) , ) , ) , ) − [ 7 8 , 1) , 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   =   (0, 1), 1 ≤  < ∞ (with the underlying Lebesgue measure).That is, for every  ∈   there is a unique sequence  = (  ) ≥0 of real numbers satisfying ‖ − ∑  =0   ℎ  ‖   (0,1) → 0. For any subset  of nonnegative integers, we will denote by P  the projection onto the space generated by the subcollection (ℎ  ) ∈ .Let   (ℎ) be the unconditional constant of ℎ, that is the least  ∈ [1, ∞] such that     P        (0,1) ≤            (0,1) , for any  ⊆ N and any  ∈   (0, 1).Using Paley's inequality [2], Marcinkiewicz [3] proved that   (ℎ) < ∞ if and only if 1 <  < ∞.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 (Ω, F, P) is a probability space, filtered by (F  ) ≥0 , a nondecreasing family of sub--fields of F. Let  = (  ) ≥0 be a real-valued martingale with the difference sequence (  ) ≥0 given by  0 =  0 and   =   −  −1 for  ≥ 1.Let  be a transform of  by a predictable sequence V = (V  ) ≥0 with values in [0, 1]: that is, we have   = V    for all  ≥ 0 and by predictability we mean that each term V  is measurable with          . (3) Here we have used the notation ‖‖  = sup  ‖  ‖   (Ω) .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 (  ℎ  ) ≥0 , for given fixed real numbers  0 ,  1 ,  2 , . . .(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 1 <  < ∞.It follows from the results of Burkholder [5] and Maurey [6] that the constants actually coincide:   (2) =    (3) for all 1 <  < ∞.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,           ,∞ () ≤             () .
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 P  .
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.
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 weaktype estimates will be more convenient for applications (cf.Remark 11 below).Namely, for  > 1 put where the supremum is taken over all measurable  ⊆  with 0 < () < ∞.Unfortunately, under these norms, we have managed to prove sharp restricted bounds in the case  ≥ 4 only (and we do not know the corresponding sharp bounds for 1 <  < 4 for the projections associated with the Haar system.
All the results discussed above can be formulated in the more general setting of continuous-time martingales.Furthermore, instead of transforms with values in [0, 1], 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.

Background and Main
Results.Assume that (Ω, F, P) is a complete probability space, equipped with (F  ) ≥0 , a nondecreasing family of sub--fields of F, such that F 0 contains all the events of probability 0. 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 ([, ]  − [, ]  ) ≥0 is nonnegative and nondecreasing as a function of .For example, assume that  is a discretetime martingale and let  denote its transform by a certain predictable sequence V with values in [−1, 1].Let us treat these two sequences as continuous-time processes, via   =  ⌊⌋ ,   =  ⌊⌋ ,  ≥ 0. 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 V takes values in [0, 1] (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 Theorem 3. Let ,  be two real-valued martingales satisfying (13).Then for any  > 0 one has For each  the inequality is sharp.Therefore, ‖‖ 1,∞ ≤ ‖‖ 1 and the constant 1 cannot be improved.
We turn our attention to the formulation of the main result of this section.We will use the notation 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  > 1.Indeed, when  ≤ 1, then ( 16) is an immediate consequence of ( 14), and its sharpness follows from simple examples.Furthermore, having proved (18) for  > 1, we deduce the case  = 1 by a standard limiting argument.Finally, note that  {||≥} ≤ 4(|| −  + 1/4) + , 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  > 1 and prove the sharpness of (17) for  > 0.

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  > 1 is a fixed parameter.First, consider the following subsets of [0, 1] × R: Now we introduce a function   by and extend it to the whole strip [0, 1] × R 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.
(iii) For any  ∈ [0, 1] and  ∈ R one has the majorization (iv) For any  ∈ [0, 1] and  ∈ [0, ] one has Proof.(i) This is straightforward.The fact that   is of class  ∞ on each    is evident, and to show that   is of class  1 in the strip, one needs to check that the partial derivatives match appropriately at the common boundaries of  1 ,  2 ,  3 , and  4 .We leave the necessary calculations to the reader.
(iv) Since   (0, 0) = 0, we can rewrite the bound in the form It follows from (i) and (ii) that, for any  ∈ [0,1], the function   :   →   (, ) is concave (if we put  = ℎ in (23), the right-hand side of this bound is nonpositive).Consequently, we will be done if we show that    (0+), the onesided derivative of   at 0, does not exceed  4−4 .But this is simple: we have This completes the proof of the lemma.(18) for  > 1.It is convenient to split the reasoning into a few separate parts.

Proof of
Step 1 (a mollification argument).The proof of (18) rests on Itô's formula.Since   is not of class  2 , this enforces us to modify   so that it has the required smoothness.Consider a  ∞ function  : R 2 → [0, ∞), supported on the unit ball of R 2 and satisfying ∫ R 2  = 1.For a given  ∈ (0, 1/4), let  ()  be defined on (, 1 − ) × R 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 (, ) ∈ (, 1 − ) × R. 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 (, ) ∈ (, 1 − ), 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  ≥ 0. Observe that the pair (, ) still satisfies (13).Furthermore,  takes values in the strip [, 1 − ] × R, so an application of Itô's formula to the process ( ()  (  )) ≥0 yields where Here Δ  =   −  − 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  1 - 3 separately.We have E 1 = 0, by the properties of stochastic integrals.By straightforward approximation argument (see, e.g., Wang [11]), the inequality ( 23) and the domination (13) imply that  2 ≤ 0. Finally, each term in the sum  3 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 (, 1 − ) × R. Using (23), this can be bounded from above by . Thus (40) gives  3 ≤ 0.
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 turn to the more difficult case  > 1.As we have already noted, (19) is stronger than (17), so it suffices to focus on the latter estimate.Let  ∈ (0, 1/2) be a fixed number and let  = ( − 1)/, where  is a large positive integer.Consider a sequence (  ) 2+1 =1 of independent mean-zero random variables with the distribution uniquely determined by the following conditions.(iv)  2+1 takes values ±1/2.

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.

Background, Notation and Results
. It is well known (cf.[10,13,14,[20][21][22][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  : R  → C, there is a unique bounded linear operator   on  2 (R  ), called the Fourier multiplier with the symbol , given by the equality T  =  f.The norm of   on  2 (R  ) is equal to ‖‖  ∞ (R  ) 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   (R  ), 1 <  < ∞.This question is motivated by the analysis of the classical example, the collection of Riesz transforms {  }  =1 on R  (see Stein [24]).Here, for any , the transform   is a Fourier multiplier corresponding to the symbol () = −  /||,  ̸ = 0.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 R  , that is, a nonnegative Borel measure on R  such that ]({0}) = 0, and Assume further that  is a finite Borel measure on the unit sphere S of R  and fix two Borel functions  on R  and  on S which take values in the unit ball of C. We define the associated multiplier  =  ,,,] on R  by if the denominator is not 0, and () = 0 otherwise.
It turns out that the constant  * −1 is the best possible: see Geiss et al. [22] or Bañuelos and Ose ¸kowski [26] for details.
Our work will concern a certain subclass of (56), corresponding to those  and , which take values in [0, 1].There are many interesting examples of this type (cf.[26]); for instance the operator ∑ ∈  2  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 [0, 1].Then for any 4 ≤  < ∞ and any measurable The inequality is sharp.More precisely, for any 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  ∈  ,1 (R  ),  ≥ 4, we have To see this, assume first that  = ∑  =1      , where  1 >  2 > ⋅ ⋅ ⋅ >   > 0 and   are pairwise disjoint subsets of R  of finite measure.Let  0 = 0 and   =  1 ∪  2 ∪ ⋅ ⋅ ⋅ ∪   ,  = 1, 2, . . ., .Then  can be rewritten in the form  = ∑  =1 (  −  +1 )   , where  +1 = 0, and By standard approximation, the above inequality extends to any nonnegative  ∈  ,∞ (R  ).To pass to general real-valued functions, it suffices to use the decomposition  =  + − − and the inequality

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 R  .To be more precise, for any ] there is a symmetric ] which leads to the same multiplier. for Now, fix  ∈ R  ,  < 0 and let ,  ∈  ∞ (R  ).We introduce the processes  = ( These processes are martingales adapted to the filtration F  = ( , :  ∈ [, 0]) (see [13,14]).The key fact is the following.
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  < 0, a function  on R  taking values in the unit ball of C, and define the operator T = T  by the bilinear form: where ,  ∈  ∞ 0 (R  ).We have the following fact, proven in [13].

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 0 < ](R  ) < ∞, so that the above machinery using Lévy processes is applicable.Fix  < 0 and functions This is precisely the desired claim (but for the above special multipliers).

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, B ⊂ C denotes the ball of center 0 and radius 1.

4.1.
Laminates: Necessary Definitions.Assume that R × denotes the space of all real matrices of dimension  ×  and let R × sym be the subclass of R × which consists of all real symmetric  ×  matrices.Definition 12.A function  : R × → R is said to be rankone convex, if   → ( + ) is convex for all ,  ∈ R × with rank  = 1.
Let P = P(R × ) stand for the class of all compactly supported probability measures on R × .For ] ∈ P, we denote by ] = ∫ R × ]() the center of mass or barycenter of ].Definition 13.We say that a measure ] ∈ P is a laminate (and write ] ∈ L), if for all rank-one convex functions .The set of laminates with barycenter 0 is denoted by L 0 (R × ).
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][30][31][32][33].We will be particularly interested in the case of 2 × 2 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][33][34].Definition 14.Let  ⊂ R 2×2 be a given set.Then PL() denotes the class of prelaminates in , that is, the smallest class of probability measures on  which (i) contains all measures of the form   + (1 − )  with  ∈ [0, 1] and satisfying rank ( − ) = 1; (ii) is closed under splitting in the following sense: if   + (1 − )] belongs to PL() for some ] ∈ P(R 2×2 ) and  also belongs to PL() with  = , then also  + (1 − )] belongs to PL().
By the successive application of Jensen's inequality, we have the inclusion PL ⊂ L. Let us state two well-known facts (see [29,[32][33][34]).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.Let us stress here that the corollary works for laminates of barycenter 0. This will give rise to some small technical difficulties, as "natural" laminates do not have this property; see below.