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## Academic research paper on topic "Sharp Inequalities for the Haar System and Fourier Multipliers"

﻿Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 646012,14 pages http://dx.doi.org/10.1155/2013/646012

Research Article

Sharp Inequalities for the Haar System and Fourier Multipliers

Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland Correspondence should be addressed to Adam Osekowski; ados@mimuw.edu.pl Received 27 May 2013; Accepted 30 September 2013 Academic Editor: Kehe Zhu

A classical result of Paley and Marcinkiewicz asserts that the Haar system h = (hk)k>0 on [0,1] forms an unconditional basis of Lp(0,1) provided 1 < p < m. That is, if p denotes the projection onto the subspace generated by (hj) (J is an arbitrary subset of N), then \\Pj |lP(0 j) ^Lp(01) < ßp for some universal constant ßp 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 < m we identify the best constant Cp such that \\P}Xa\\lp,<^(0 1) < ^p\\Xa\\lp^0 1) for every J Q 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.

system forms a basis of If = Lp(0,1), 1 < p < ot (with the underlying Lebesgue measure). That is, for every f e Lp there is a unique sequence a = (an)n>0 of real numbers satisfying \\f - £"=0 akhk\\LP{0 l) ^ 0. For any subset J of nonnegative integers, we will denote by Pj the projection onto the space generated by the subcollection (hj)^j. Let fip(h) be the unconditional constant of h, that is the least p e [1, ot] such that

£||/L(0,1)> (2)

for any J c N and any f e Lp(0,1). Using Paley's inequality , Marcinkiewicz  proved that fip(h) < ot if and only if 1 < p < ot. 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 . Assume that (Q., F, P) is a probability space, filtered by (Fk)k>0, a nondecreasing family of sub-a-fields of F. Let f = (fk)k>0 be a real-valued martingale with the difference sequence (dfk)k>0 given by df0 = f0 and dfk = fk - fk-1 for k> 1. Let g be a transform of f by a predictable sequence v = (vk)k>0 with values in [0,1]: that is, we have dgk = vkdfk for all k > 0 and by predictability we mean that each term vk is measurable with

1. Introduction

Our motivation comes from a very natural question about h = (hn)n>0, the Haar system on [0,1]. Recall that this collection of functions is given by

ho = [0,1),

2' 4 1 8 1 3

1 5 2' 8 3 7 8

1 1 L 8 4

3 1 8' 2

and so on. Here we have identified a set with its indicator function. A classical result of Schauder  states that the Haar

respect to F(fc-1)V0. Then (cf. ) for 1 < p < to there is a The constant Cp is the best possible. It is already optimal in the

estimate

for the projections associated with the Haar system.

universal constant cp for which

II, < C,H

Here we have used the notation WfWp = supJ|/J|Lf,(n). Let Cp (2) and c^ (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 (akhk)k>0, for given fixed real numbers a0,a1,a2,... (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 cp (2)< c'p (3) for all 1 < p < to. It follows from the results of Burkholder  and Maurey  that the constants actually coincide: cp (2) = c'p (3) for all 1 < p < to. The question about the precise value of cp (2) was answered by Choi in : 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 (M,p) is a given measure space. A linear (or sublinear) operator T defined on LP(M) and taking values in Lp'm(M) is said to be of restricted weak type (p, p), if there is a constant C such that, for every measurable set Ac M of finite measure,

.AllLp-m(M)

■AlLf(M)'

iilp-™ (m)

:= sup ({xeM: \ f (x)\ > A})}

is the usual weak quasinorm on the Lorenz space Lp'm(M). One of the reasons for considering restricted weak-type estimates is that usuallythese 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 ), 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 Pj. Introduce the constants Cp by

1 if 1<p <4,

Pe(4-P)/P if p>4. 4 y

Furthermore, if g is a discrete-time martingale, we define its

weak pth quasinorm by one of our main results.

= supJ#Jlp.~(n). Here is

Theorem 1. Let f be a martingale taking values in [0,1], terminating at {0,1} (i.e., satisfying limn^mfn e {0,1} almost surely), and let g be its transform by a predictable sequence with values in [0,1]. Thenfor any 1 < p < to one has

IU <CPII

We will also provide a version of this result for the case in which the space Lp'm 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 p > 1 put

Lp • ~(M)

:= sup

1-1 /p

where the supremum is taken over all measurable E c M with 0 < p(E) < to. Unfortunately, under these norms, we have managed to prove sharp restricted bounds in the case p > 4 only (and we do not know the corresponding sharp bounds for 1 < p < 4). In analogy with the above definitions, if g is a discrete-time martingale, we let

sup„>olll#JH

Theorem 2. Let f be a martingale taking values in [0,1], terminating at {0,1}, and let g be its transform by a predictable sequence with values in [0,1]. Thenfor any p > 4 one has

^ <cMIp-

The constant Cp is the best possible. It is already optimal in the estimate

i \\pjXaihlp.~(0,1) < c,ixailp(0,1)' for the projections associated with the Haar system.

So, for p > 4 the best constant is the same for both norms

1 ' and 111 ' |||p,m.

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.

2. A Martingale Inequality

2.1. Background and Main Results. Assume that (Q, F, P) is a complete probability space, equipped with (Ft)t>0, a nondecreasing family of sub-u-fields of F, such that F0 contains all the events of probability 0. Suppose that X, Y are two adapted real-valued martingales, whose paths are right continuous and have limits from the left. The symbol

[X, Y] will stand for the quadratic covariance process of X and Y (see, e.g., Dellacherie and Meyer  for details). Following Banuelos and Wang  and Wang , we say that Y is differentially subordinate to X, if the process ([X,X]t - [Y,Y]t)t>0 is nonnegative and nondecreasing as a function of t. For example, assume that f is a discrete-time martingale and let g denote its transform by a certain predictable sequence v with values in [-1,1].Letustreatthese two sequences as continuous-time processes, via Xt = f^tj, Yt = g^tj, t > 0. Then the required condition on [X, X] - \Y, Y] is equivalent to saying that

(i) For any X > 0 one has

\dgk\2 < \dfk\2,

k = 0,1,2,...,

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 X, Y satisfying the differential subordination arise naturally in the martingale study of Fourier multipliers. In this paper, we will work with pairs X, Y satisfying a slightly different condition:

([X, Y]t - \Y, Y]t)t>0 is nondecreasing and nonnegative

as a function of t, (13)

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 , Choi , 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  and in the general continuous-time case by the author in . Throughout, we use the notation \\X\\^ =

supt>o\\xJ supt>o\\xJ

supt>olHXt III lp'<™(q), 1 <P<ot.

Theorem 3. Let X, Y be two real-valued martingales satisfying (13). Then for any X > 0 one has

X sup P (\Yt\ > 1) < \\X\\!.

For each X the inequality is sharp. Therefore, ||Y||1œ < HX^ 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

P(X)=\

„4-4X

if 0 < X < 1, if X > 1.

Theorem 4. Suppose that X is a martingale taking values in [0,1] and Y is a real valued martingale such that (13) is satisfied.

sup P (|Yt| >X)<P(X)\\X\\i.

The bound on the right-hand side of (16) is the best possible for each X, even in the following version for the Haar system: for any J c N and f: [0,1] ^ [0,1],

Lp(0,1)'

\{xe[0,1]:\Prf(x)\>X}\<P(X) (ii) For any X > 1 one has

4 sup EÎ\Yt\-X+1-) <P(X)\\X\\i.

f>0 \ 4) +

The bound on the right-hand side is the best possible for each X, even in the following version for the Haar system: for any J c N and f: [0,1] ^ [0,1],

(V-X+Ï)

IIlî(0,i)-

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 X > 1. Indeed, when X < 1, then (16) is an immediate consequence of (14), and its sharpness follows from simple examples. Furthermore, having proved (18) for X > 1, we deduce the case X = 1 by a standard limiting argument. Finally, note that X[\y\>x} < 4(lyl - X + 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 X > 1 and prove the sharpness of (17) for X > 0.

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  and to the recent monograph  bythe 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 X > 1 is a fixed parameter. First, consider the following subsets of [0, 1] x R:

1 1 1 1

D1 = {(x, v) : -x--< v < x--,V<-},

1 7' 2 4 7 4 7 4J

D2 = {(x,y) : x - 1 < y < x + X - 1,x < 2} ,

D3 = {(x,У):4<У<X-2,x>2],

D4 = {(x, y) : y > X - 1 or y > x + X - 1] .

Now we introduce a function Ux by

U, (x,y) =

„3-4X

(Ay2 - 4yx

+2y+4)

x exp (-4x + 4y - 4A + 4) if (x, y) e D2, (1 - x) exp (4y -4X + 2) if (x, y) e D3,

(y-x-X) (4y -4X+3) +5 (y-X)+ 4

if (x, y) e D4 (21)

and extend it to the whole strip [0,1] x R by the condition

U, (x,y) = Ux (l-x,-y) for X e [0,1], ye R.

Let us provide some information on this object. In what follows, the symbol A0 denotes the interior of a set A.

Lemma 5. The function Ux enjoys the following properties.

(i) It is of class C1 on (0,1) x R and of class Cm in D°, D02, D°, and D°°.

(ii) There is a Borel function c : D° U D° U D° U D° ^ [0, to) with the following property: for any (x,y) e D1 U DO U D'0 U D°° and any h,k e R such that lx + hl < 1,

UXxx (x,y)h2 + 2UXxy (x,y)hk

+ UXyy (x,y)k2 <c(x,y)(k2 -hk).

(iii) For any x e [0,1] and ye R one has the majorization

Ux (x,y)>4(\y\-X+4)+. (24)

(iv) For any x e [0,1] and y e [0, x] one has

Ux (x,y)<xe°-°X. (25)

Proof. (i) This is straightforward. The fact that Ux is of class Cm on each D° is evident, and to show that Ux is of class C in the strip, one needs to check that the partial derivatives match appropriately at the common boundaries of D1, D2, D°, and D°. We leave the necessary calculations to the reader.

(ii) If (x, y) e Dp then the left-hand side of (23) equals 8e°-°x(k2 - hk), so we may take c(x,y) = 8e°-°x. If (x,y) belongs to the interior of D2, the expression on the left of (23) is equal to

16(x-2)(k-h)2e4y-4x-4X+4 + 8 (k2 -hk)e4y-4x-4X+4.

But x < 1/2, because (x,y) e D°; therefore c(x,y) = 8e°y-°x-° satisfies the desired bound. Next, assume that (x, y) lies in D°. We compute the left-hand side of (23) and obtain

if (x,y) eDu 16(2- x) k2e4y-4X+2 + 8(h2 - hk) e

4y-4X+2

This time we have x > 1/2 and hence c(x,y) = 8e°y °x+2 works fine. Finally, if (x, y) lies in the interior of D°, then

Uxxx (x,y)h2 + 2Uxxy (x,y)hk

+ UXyy (x,y)k2 = 8(k2 -hk),

so we may take c(x, y) = 8.

Before we proceed, let us observe that, by (22), the inequality (23) holds also in the interiors of the "reflected" domains D[, D'2, D°, and D° given by

D't ={(x,y):(1-x,-y)eD,}, (29)

with c given by c(1 - x, -y) = c(x, y).

(iii) Directly from (i) and (ii), the function Ux has the following property: for a fixed y, the function x ^ Ux(x, y) is concave on [0,1] (simply plug k = 0 in (23)). Since the right-hand side of (24) does not depend on x, it suffices to verify the majorization for x e {0,1} only. Furthermore, because of (22), we may restrict ourselves to two cases x = 0 and y > -1/4;x = 1 andy > 1/4.Ifx = 0and-1/4 < y < X-1/4, then the right-hand side vanishes, while the left-hand side is nonnegative. If x = 0 and y > X - 1/4, then we must prove that 4(y -X+1)2 > 4(y -X + 1/4), which is equivalent to the obvious estimate:

(y-X)2 + (y-X) + 4>0.

Next, if x = 1 and 1/4 < y < X-1/4, then both sides of (24) are equal to 0. Finally, if x = 1 and y > X-1/4, the majorization reads

4(y-X+ï) ~4(y-X+\)

or, equivalently, (y-X) > 0.

(iv) Since Ux(0,0) = 0, we can rewrite the bound in the form

Ux (x,y)-Ux (0,0) <xe

It follows from (i) and (ii) that, for any a e [0,1], the function : x ^ Ux(x,ax) is concave (if we put k = ah in (23), the right-hand side of this bound is nonpositive). Consequently wewillbedoneifweshowthat ^ (0+),the onesided derivative of <;a at 0, does not exceed e4-4x. But this is simple: we have

£ (0+) = lim

a dlQ d

= lim e = e .

This completes the proof of the lemma.

2.3. Proof of (18) for X > 1. It is convenient to split the reasoning into a few separate parts.

Step 1 (a mollification argument). The proof of (18) rests on Ito's formula. Since Ux is not of class C2, this enforces us to modify Ux so that it has the required smoothness. Consider a Cm function g : R2 ^ [0, ot), supported on the unit ball of R2 and satisfying Jr2 g =1. For a given S e (0,1/4), let uf] be defined on (8, 1 - S) x R by the convolution

U^ (x, y) = \ Ux (x + 8u, y + Sv) g (u, v) du dv.

h-1,1]2

The function U^ is of class Cm in the interior of its domain and inherits the crucial properties from Ux. Namely, we have the following version of (24):

Uf (x,y)>4(\y\-X+-4-s)+, (35)

for all (x, y) e (8,1 - 8) x R. Next, by Lemma 5 (i) and the integration by parts, we get

U\L (x, y)=\ , uxxx (x + 8u,y + Sv) g (u, v) du dv.

h-1,1]2

Similar identities hold for u[s) and u[s), so we see that u[s)

AXy Xyy A

satisfies (23) for all (x, y) e (8,1 - 8), with

c® (x, y) = \ c(x + 8u, y + 8v) g (u, v) dudv > 0

h-1,1]2

(the function c constructed above is locally bounded, so there is no problem with the integration).

Step 2 (application of Ito's formula). Take martingales X, Y as in the statement and consider the processes Xt = 8 + Xt •( 125), Yt = Yt•(1-28),andZt = (Xt,Yt) for t > 0.0bservethat the pair (X, Y) still satisfies (13). Furthermore, Z takes values in the strip [5,1-5] x R, so an application of Ito's formula to the process (Uf\zt))t>0 yields

T(S) (7 ) = TT(S) (7 )+T +1! + T (38)

U? (Zt) = u? (Z0) + Ii + f +I3,

1 = f (Zs-)dXs + f (Zs-)dYs,

J0+ J0+ 7

+ uZ (zs-)d[x,x]s

+ ufl (zs-)d[xj]s

+ i0+ ^ M^l

I3 = I [uf (Zs)-U(S) (Zs-)

-U[SJ (Zs-)AXs -U(V (Zs-)AYs].

Here AXS = Xs - Xs- denotes the jump of X at time s, and [X, X]° is the unique continuous part of the bracket [X, X] (cf. Dellacherie and Meyer ). Let us analyze each of the terms I1-I3 separately. We have E/1 = 0, by the properties of stochastic integrals. By straightforward approximation argument (see, e.g., Wang ), the inequality (23) and the domination (13) imply that I2 < 0. Finally, each term in the sum I3 is also nonpositive. To see this, observe first that for each to we have

\AYs (to) I < AXs (to)AYs (to),

since otherwise the condition (13) would not be satisfied. Now, applying the mean-value property, we get that

(Zs) - uf (ZJ

-U™ (Zs-) AXs -U™ (Zs-)AYs 1 ^ (0|AXs(to)|2

(Ç)AXs (to) AY s (to)

+U™y AYs (to)

where % is a certain point in (8,1 - 8) x R. Using (23), this can be bounded from above by c(5)(£)[|AYs(to)|2 -AXs(to)A7s(to)]. Thus (40) gives I3 < 0.

Step 3 (the final part). If we combine all the above facts and take expectation of both sides of (38), we obtain the estimate EUf(Zt) < EUf](Z0). By (35), this implies 4E(\YtI - X +

— , ,\2

1/4 - S)+ < EUf(Z0). If we let 5

(X0,Y0) and Uf (Z0)

0, then Yt ^ Yt, U\(%0, ^0), so we get

4E( \Yf\ -X +

< EU, (X0,Y0),

by Fatou's lemma and Lebesgue's dominated convergence theorem (we have \Z0\ < |X01 + \Y0I < 2 for all 8). It remains to use (25): by (13), we have Y0 e [0, X0] and hence EUx(X0,Y0) < P(X)EX0 = P(A)\X\1. Takingthe supremum over t > 0 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 X is a martingale taking values in [0,1] and let Y be a real-valued martingale such that (13) holds true. Then for any E e F, t > 0, and X> 3/4 one has

E\Yt\XE <4e3-4XEXo + AP (E). (43)

Proof. We have E = E+ U E-, where E+ = En [\Yt\ > X] and

E- =En{\Yt\ < A|.By(18),

4E (\Yt\-x)xe+ <4e(\Yt\-(x+4) + 1)

< e4-4{X+1/4) EX0 = e3-4X EX0

and, obviously, 4E(\Yi\ - X)%E- < 0. 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 X is a martingale taking values in [0,1] and terminating at {0,1], and let Y be a real-valued martingale such that (13) is satisfied. Multiplying both sides of (16) by Xp gives

Xp sup P ( | Yt | >X)< ^

Xp-1\\X\\1 if \<l, Xpe4-4X\\X\\1 if X>1.

Let us optimize the right-hand side over A. If p < 4, then the maximal value is attained at X = 1, and we get

\\Y\\pp,m

where limt . _

in the

last passage we have used the fact that {0,1]. On the other hand, if p > 4, then

the right-hand side of (45) is maximized for the choice X = p/4. Substituting this value of X gives \\Y\\P^ < Cp\\X\\1 = Cp\\X\\p, and hence (7) follows. The inequality (10) can be proven in a similar manner, with an additional help of (43). Namely, fix appropriate X, Y, and E e F of positive probability. Assume first that P(E) < \\X\1 and optimize the right-hand side of (43) over A. A straightforward analysis of the derivative shows that the maximum is attained for X = 3/4 + (1/4) log(\X\1/P(£)) > 3/4. Plugging this value of X gives the bound

- log ™i

4 4 4 P(

Eta <1P №) + (3 + i log Ek) I t\AE 4 \ ) ^ 4 4 Sp (E)J

= P(£)

P(E)1/p + 1 P(E)1/p log ^ 4 8P (E)\

(here q = p/(p - 1) is the harmonic conjugate to p). But the expression in the square brackets, considered as a function of P(£), does not exceed Cp(\\X\\1)1lp. Thus, it suffices to

divide by P(E)1lq, take the supremum over E, and note that \\X\1 = \\X\\p to get the desired bound. It remains to consider the case when P(E) > \\X\1. An application of Schwarz inequality, Burkholder's bound \\Y\\2 < (which follows from the chain E\Yt\2 = E[Y,Y] t < E[X,Y]t < (E[Y,Y]t)1l2(E[X,X]t)112 < \y\2\X\2), and the fact that X terminates at the set {0, 1] imply

E | Yt\Xe <\\YtlP(E)112

< \\X\\2P(E)1/2 = \\X\\\/2P(E)1/2 = \\X\\\/2P(E)1/2.

Butwehave \\X\\1l2P(E)112 < \\X\\1lpP(E)1lq:indeed, thiscan be rewritten in the form \\X\\1l2-1lp < P(E)11^-112 and follows from the equality 1/2-1/p = 1/q-1/2 and the bounds p > 4, P(E) > \X\1. Combining this with the above estimate, the inequality (p/4)e(4-pyp > 1,andtheequation \X\1 = \\X\\P, give

E | Yt\ Xe <Cp\\X\\pP(E)1/q. Now (10) follows immediately.

(49) □

2.5. Sharpness of (8), (11), (17), and (19). By an application of the results of Burkholder (see Section 10 in ) and Marcinkiewicz , 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 (f, g), where g is a transform of f, can be appropriately embedded into a pair consisting of a dyadic martingale and its transform). This is also closely related to the equality Cp (2) = c'p (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 X < 1, fix w e (0,X), and define the pair (f, g) by the following conditions:

(i) f = g,

(ii) f0 = we(0,X),

(iii) f1 = f2 = ••• is a random variable satisfying

P (h = 0) = !-j

P (h = X) = J

Then we easily see that P(\^ \ > X) = w/X = P(X)\f\1.

We turn to the more difficult case X > 1. As we have already noted, (19) is stronger than (17), so it suffices to focus on the latter estimate. Let w e (0,1/2) be a fixed number and let S = (X - 1)/N, where N is a large positive integer. Consider a sequence

(Ul^1 of independent mean-zero random variables with the distribution uniquely determined by the following conditions.

(i) ^ takes values in {-w, S - w + 1/2].

(ii) For n = 1,2, ...,N, ^2n takes values in the set {-5,1/2-5].

(iii) For n = 1,2,... ,N - 1, ^2n+1 takes values in the set {-1/2,5].

(iv) ^2n+1 takes values ±1/2.

Next, let x = inf{n : w + ^ + + ••• + e {0,1]], with the convention inf 0 = to. It is easy to check that x is an almost surely finite stopping time (with respect to the natural filtration of £). Since are centered, the process

f=(fn)Z+i =(" + *1 +S2 + (51)

is a martingale. Let g denote the transform of f by the deterministic sequence (1,1,0,1,0,1,0,...). To gain some

intuition about (f, g), let us take a look at its dynamics. The pair starts from the point (w, w) and, at the first move, it goes to (0, 0) or to (1/2 + 8,1/2 + 8). If it went to (0, 0), it stays there forever; if it jumped to (1/2 + 8,1/2 + 8), then it moves horizontally either to (1,1/2 + 5) (and stops) or to (1/2,1/2 + 8). If the latter possibility occurs, the movement continues: the pair goes to (0,8) (and terminates) orto (1/2+8,1/2+28). In the latter case, it moves horizontally to (1,1/2 + 28) or to (1/2,1/2 + 28) and so on. During the first 2N steps, the pair either hits one of the lines x = 0, x = 1 (and stops) or visits the point (1/2, X- 1/2) on 2№h step. If the latter takes place, then (f, g) jumps to (0,X- 1) or to (1, X).

Directly from the above analysis, we see that f takes values in [0,1] and terminates at {0,1}, and

P (\d2N+1\

= P (Z1 >0,^2 <0,^3 >0,

< 0,...^2N < 0,\$2N+1 > 0)

1 + 25 V 1/2

f1/2-S\N( 1/2 (1/2 + 8

1 + 28

"-^'-TTs)

However, recall that 8 = (X- 1)/N. Therefore, if we let N go to infinity, the latter expression converges to we4-iX. This shows that the constant P(X) 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 p e [1,4] and consider the above example for X = 1. Then \\#\\^TO > P(I^I > 1) = \\f\\1 = \\f\\Fp and hence Cp = 1 is the best in (8). On the other hand, if p > 4, we take the above example corresponding to X = p/4 and a large N. Then

>A^P (\g2

= CUA = cnft

and hence Cp 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 f e Lp,m).

11 p,^

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 m : Rd ^ C, there is a unique bounded linear operator Tm on L2(Rd), called the Fourier multiplier with the symbol m, given by the equality Tmf = mf. The norm of Tm on L2(Rd) is equal to \\m\\L<»(R<j) and a classical problem of harmonic analysis is to study/characterize those m, for which the corresponding Fourier multiplier extends to a bounded linear operator on LP(Rd), 1 < p < ot. This question is motivated by the analysis of the classical example, the collection of Riesz transforms {Rj}j=1 on Rd (see Stein ). Here, for any j, the transform Rj is a Fourier multiplier corresponding to the symbol m(%) = -%j/ISI, £, = 0. An alternative definition of Rj involves the use of singular integrals:

Rjf(x) =

T((d+1)/2)

n(d+1)/2

j=1,2,..

f{y)dy,

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 p-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 m,provide tightboundsfor the size of the multiplier Tm 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 Levy processes. This class has appeared for the first time in the papers by Banuelos and Bogdan  and Banuelos et al. . To describe it, let v be a Levy measure on Rd, that is, a nonnegative Borel measure on Rd such that v({0}) = 0, and

min {M2,1} v(dx) < ot. (55)

Assume further that ^ is a finite Borel measure on the unit sphere S of Rd and fix two Borel functions 0 on Rd and f on S which take values in the unit ball of C. We define the associated multiplier m = v ^ v on Rd by

(MU (Z,d)2f(d)v(dd)

+ 1 [1 - cos (<;,x}]\$(x) v (dx)

1 f (t,e)2v(de)

+ f [1 - cos (£,%)] v (dx))

if the denominator is not 0, and m(%) = 0 otherwise. Here {■, ■) stands for the scalar product in Rd. 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 J of {1,2,. ..,d] and take p = Sei + Sei + ■■■ + Sed, v = 0, and y(e}) = Xj(j), j = 1,2,...,d. Here e1,e2,... ,ed are the versors in Rd. This choice of parameters gives the operator R2 on Rd.

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

Theorem 7. Let 1 < p < to and let m = ^ ^ v be given by (56). Then for any f e Lp (Rd) one has

\KfhR) <(p -1) 11/11

where p* = max{p, p/(p - 1)}.

It turns out that the constant p* -1 is the best possible: see Geiss et al.  or Baiiuelos and Os^kowski  for details.

Our work will concern a certain subclass of (56), corresponding to those 0 and f, which take values in [0,1]. There are many interesting examples of this type (cf. ); for instance the operator R2 introduced above is of this form. We will prove the following result.

Theorem 8. Suppose that m is a symbol given by (56), where 0 and f are assumed to take values in [0,1]. Thenfor any 4 < p < to and any measurable Ac Rd with \A\ < to,

111^mXa I Il^'^r^) < ^^aWl^r^ (58)

The inequality is sharp. More precisely, for any 4 < p < to, any C < Cp, anyd > 2, and anypropersubset J of {1,2,...,d]

there is Ac Rd of finite measure such that

ir2 xa

> c\fc

aIIlp^)'

Following Stein and Weiss , we can give the following application of the above result. Let Tm be a Fourier multiplier as in the above statement. Then for any real-valued function f eLp,1(Rd), p> 4,wehave

\ Lp-°

2-1-1/pe(2-p)/2\\

To see this, assume first that f = ajXEj, where a1 > a2 > ■■■ > aN > 0 and E: are pairwise disjoint subsets of Rd of

finite measure. Let F0 = 0 and Fj = E1 U E2 U ■ ■ ■ U Ej, j = 1,2, ...,N. Then/ canberewrittenintheform f = ^¿N=1(aj -ai+1 )Xf , where aN+1 = 0, and

I \\Tmf\\\Lp^(Rä) < Z (ai - aj+1) \\\TmXFj

}=1 Pj4-p)/p

<4/4-p)/pl (*, -*»1)mp

= Pe(4-p)/p 4

!aj+111 fj+1 j=0

1/ p 1/ p - \ Fj \

By standard approximation, the above inequality extends to any nonnegative f e Lp'm(Rd). To pass to general real-valued functions, it suffices to use the decomposition f = f+ -f- and the inequality \\f+ Wi^) + \\f-Wi^) < 21-1/p\\f\\iP-i(Rä).

3.2. The Martingale Representation of the Fourier Multipliers (56). By the reasoning from , we are allowed to assume that the Levy measure v satisfies the symmetry condition v(B) = v(-B) for all Borel subsets B of Rd. To be more precise, for any v there is a symmetric v which leads to the same multiplier. Furthermore, assume for a while that \v\ = v(Rd) is finite and nonzero, and introduce the probability measure v = v/\v\. Consider the independent random variables T-1,T-2,..., Z-1, Z-2,... such that, for each n = -1, -2, ...,Tn has exponential distribution with parameter \ v\ and Zn takes values in R and has v as the distribution. Next, put Sn = -(T-1 + T-2 + ■■■ + Tn) for n= -1,-2,... and let

Xs,t = Z Zj, Xs,t- = Z Zj

= Xs,t - Xs,t-

for -to < s <t <0. Next, if f e Lm(Rd) is a given function, define its parabolic extension U^ to (-to, 0] x Rd by

Uf (s,x) = Ef(x + Xsfi). (63)

Now, fix x e Rd, s < 0 and let e Lm(Rd). We introduce

the processes F = (F*'s^)s&t&0 and G = (Gt Ft = Uf (t,x + Xst), Gt = Z [AFu ^(^Xs,u)]

)s<t<0 by

s<u<t s

\uf (v,X + XSV- + z) -uf (v, x + Xs>V-)] 0 (z) v (dz) dv.

s<Sj<t

s<Sj<t

These processes are martingales adapted to the filtration F = a(XSJ. :t e[s,0]) (see [13,14]). The key fact is the following.

Lemma 9. If (f> takes values in [0,1], then the pair (Fx's'f, Gx's'M) satisfies (13).

Proof. The assertion follows immediately from the identities [F,G]t = £ \AFfy(AXSM),

[G,G]t= £ \AFu\2(\$(AXsm))2,

which can be established by repeating the reasoning from . □

Now we introduce a family of multipliers. Fix s < 0, a function (f> on Rd taking values in the unit ball of C, and define the operator T = Ts by the bilinear form:

Tf(x)g(x)dx = \ E \Gx0's'f4g(x + Xs0)]dx, (66)

where f,ge C™(Rd). We have the following fact, proven in .

Lemma 10. Let 1 < p < ot and d > 2. The operator Ts is well defined and extends to a bounded operator on Lp(Rd), which can be expressed as a Fourier multiplier with the symbol

m(s) = ms (\$)

1 - exp (2s I (1 - cos {£,, z) ) v (dz)

fR„ (1- cos (¡■,z))<f>(z) v (dz) JR, (1 - cos z)) v (dz)

if Jr(1 - cos{^,z))v(dz) = 0, and M(^) = 0 otherwise. Furthermore, (66) holds true for all f e C™(Rd) and all g belonging to Lq(Rd) for some 1 < q < ot.

3.3. Proof of (58). We may and do assume that at least one of the measures v 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

M (n R (1- cos (H,z))<p(z) v (dz) *v L (1- cos (S,z)) v (dz) ■

Assume that 0 < v(Rd) < ot, so that the above machinery using Levy processes is applicable. Fix s < 0 and functions f,ge C ™(Rd) such that/ takesvaluesin [0,1], while g takes values in [-1,1] and is supported on a certain set E of finite Lebesgue measure. Of course, then the martingale Fx's'f takes

values in [0,1]. By Fubini's theorem and (43), for any X > 3/4 we have

g(x + Xsfi)]dx

x,s,f,<p I

jrd 3-4X

X[x+Xs^E}dx ^'s'f\ dx

X\ P (x + xs0 eE)dx

„3-4X

IIlUr^)

+ X\E\.

Plugging this into the definition of S and taking the supre-mum over all g as above, we obtain

' \Ss'f'vf(x)\dx< E 2

IIlUr^)

+ X\E\. (70)

Now if we let s ^ -œ, then Ms ^ v converges pointwise to the multiplier M^ v given by (68). By Plancherel's theorem,

f ^ TM^ f in L2(Rd) and hence there is a sequence

(snconverging to -œ such that limn^œ Ss"'^'vf ^ Tm f almost everywhere. Thus Fatou's lemma combined with (70) yields the bound

f i i e3-u

jE\TM,J(x)\dx<—Wf\\Ll(Rd) + X\E\. (71)

Now we repeat word by word the optimization arguments used in Section 2 in the proof of (10) (we need to consider the cases ||f||li(r<j) < \E\ and ||f||li(r<j) > \E\ separately). As the result, we obtain the bound

\TM^J(x)\dx<CpWfWXd)\E\1/q.

Finally, using some standard approximation arguments, we see that (72) can be applied to f = \A (where A is a measurable subset of Rd, satisfying \A\ < œ), and we get the estimate:

\\TMé ]Xa\

lp,<x,(Rd)

* < cpua\

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 0 < v(R ) < ot. For a given £ > 0, define a Levy measure v£ in polar coordinates (r,0) e (0, ot) x S by

V£ (drdd) = e-2S£ (dr) p (dd) .

Here S£ denotes Dirac measure on {£}. Next, consider a multiplier y ^ v as in (68), in which the Levy measure is

X\\x\>c}V + v£ and the jump modulator is given by X{\x\>e}\$(x) + X{\x\=c}f(x/\x\). Note that this Levy measure is finite and nonzero, at least for sufficiently small e. If we let e ^ 0,we see that

f [1- cos v£ (dx)

Jr^ v m /

A -2— p (dd)

f (Z,e)2t(d)

Js (\$, ed)

^1 f (z,e)2\$(e)^(de)

and, consequently, ^ pointwise. Thus (73)

yields (7). Indeed, using Plancherel's theorem as above, we see that there is a sequence (en)^1 converging to 0 such that

XA ^ XA 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 X > 3/4 only; this is the reason why the restricted bound (58) holds only for p > 4. To get a sharp bound for p < 4,we would require a version of (43) for small A; 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 Cp 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 C denotes the ball of center 0 and radius 1.

4.1. Laminates: Necessary Definitions. Assume that Rmxn denotes the space of all real matrices of dimension m x n and let Rnymn be the subclass of Rnxn which consists of all real symmetric nxn matrices.

Definition 12. A function f : Rmxn ^ R is said to be rank-one convex, if t ^ f(A + tB) is convex for all A,B e Rmxn with rank B = 1.

Let P = P(Rmxn) stand for the class of all compactly supported probability measures on Rmxn. For v e P, we denote by v = JRmx„ Xdv(X) the center of mass or barycenter of v.

Definition 13. We say that a measure v e P is a laminate (and write v e L), if

f(v)<f fd v

for all rank-one convex functions f. The set oflaminates with barycenter 0 is denoted by L0(Rmxn).

Laminates can be used to obtain lower bounds for solutions of certain PDEs, as was first noticed by Faraco in . 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 2x2 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 U c R2x2 be a given set. Then PL(U) denotes the class of prelaminates in U, that is, the smallest class of probability measures on U which

(i) contains all measures of the form XSA + (1 - X)SB with A e [0,1] and satisfying rank (A - B) = 1;

(ii) is closed under splitting in the following sense: if a5a + (1 - X)v belongs to PL(U) for some v e P(R2x2) and p also belongs to PL(U) with ^ = A, then also X^+(1-X)v belongs to PL(U).

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

Lemma 15. Let v = ^ XiSAi e PL(R2ym) with v = 0. Moreover, let 0 < r < (1/2) min \At - Aj\ and S > 0. For any bounded domain Q. c R2 there exists u e W02'm(Q.) such that \\w\\ci < S and for all i = ■N

I {x e n : \D2u(x)-Ai\ < r\\ = Xi \Q\.

Lemma 16. Let K c R^ be a compact convex set and v e

L(Rsym) with supp v c K. For any relatively open set U c

R2xm2 with K cc U there exists a sequence v.- e PL(U) of

prelaminates with v.- = v and v

These two lemmas, combined with a simple mollification, yield the following statement proven originally by Boros et al. . 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 v e L0(R2x2). Then there exists a sequence Uj e C^ (B) with uniformly bounded second derivatives, such that

B\i (x))

for all continuous \$ : Rsyxm

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.

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 ( : R x R ^ R is said to be biconvex if, for any fixed z e R, the functions x ^ ((x, z) and y ^ ((z, y) are convex. Now, for a given p > 4, pick A = p/4 and let f, g 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 w and N—these two parameters will be specified later). Consider the R2-valued martingale:

(F, G) :=(g-w,f-g).

We subtract w on the first coordinate to ensure that the pair (F, G) has mean (0, 0). This sequence has the following zigzag property: for any 0 < n < 2N +1 we have Fn = Fn+1 with probability 1 or Gn = Gn+1 almost surely; that is, in each step (F, G) moves either horizontally or vertically. Indeed, this follows directly from the construction that for each n we have P(dfn = dgn) = 1 or P(dgn = 0) = 1. This property combines nicely with biconvex functions: if (is such a function, then a successive application of Jensen's inequality gives

2N+2' G2N+2)

> E((F2N+1, G

2N+1 )>•••> E((Fo,Go) = ((0,0).

Now, the martingale (F, G), or rather the distribution of its terminal variable (F2N+2,G2N+2), gives rise to a probability measure v on R^: put

v (diag (x, y)) = P ((F2N+2, G2N+2) = (x, y)),

(x, y) e R2.

Here and below, diag(x, y) denotes the diagonal matrix (o°y). The key observation is that v is a laminate of barycenter 0. To prove this, note that if y : R2x2 is a rank-one convex, then (x, y) ^ y(diag(x, y)) is biconvex and thus, by (80),

\ ydv=Ey (diag (F2N+2> G2N+2))

>f (diag (0, 0)) = f(v).

Finally, note that P(F2N+2 + G2N+2 e {-w, 1 - w}) =

P(f2N+2 in

e {0,1}) = 1, and hence the support of v is contained

K = {diag (x, y) : x + y e {-w, 1 - w}}.

4.3. Sharpness of (58): The Case d = 2. We will prove that Cp is the best in

l>:,<»(rd) < CpIUaIIl?(r^).

For the convenience of the reader, let us first sketch the idea. We start with the application Corollary 17 to the laminate

v: let (uj)j>1 be the corresponding sequence of smooth functions. As we have just observed above, the support of ] is contained in K given by (83). Since the distribution of Uj is close to v (in the sense of Corollary 17), we expect that Auj, essentially, takes only values close to -w or close to 1 - w. Thus, if we define Vj = Auj + w\B for j = 1,2,..., then Vj is close to an indicator function of a certain set A. Thus, to prove the sharpness of (84), one can try to study this estimate with \A replaced by Vj. We will look separately at the action on R^ on Auj and w\B; to handle the Laplacian, we will use the arguments from the previous two subsections, and w\B will be dealt with the aid of (57).

Step 1. We start from the specification of the parameters N and w. For a given p > 4, pick an arbitrary number M smaller than ei-iX/4 (recall that A = p/4): thus, M = (1/4)ei-iX ■ t] for some q< 1.Let we (0,1/2) be arbitrary and choose N so that E(Ig2N+2I - X)+ > MEf0 = Mw. This is possible, in view of the results of Section 2. Furthermore, let £ be an arbitrary positive number (which will eventually be sent to 0). In what follows, we will use the following convention: C1,C2,C3,... will denote constants which depend only on w and N.

Step 2. Consider a continuous function 0 : R^ ^ R given by 0(diag(x, y)) = Ix + y + wI.By Corollary 17, we have 1

p 2x2 Rsym

= E \ F2N+2 + G2N+2

+ W = w,

so for sufficiently large j we have

< w (1 + e).

Step 3. Consider a continuous function 0 : R2ym ^ [0,1], which satisfies \$(diag(x, y)) = 0 if x + y + w e {0,1} and which is 1 if the distance between x + y + w and the set {0,1} is larger than £. By Corollary 17,

^ f D2u:) \ \$dv=0,

B\ Jb v ^ W

since P(F2n+2 + G2N+2

+ we{0,1}) = 1. Consider the sets A= {x e B : | Auj (x) - 1 + w\ < £},

A= {x e B : \Auj (x) + w| < e}

Then (87) implies that

\b \ (a\ja)\

for sufficiently large j. (89)

Step 4. Next, consider a continuous function 0 : R [0,1], satisfying \$(diag(x, y)) = 1 if x + y + w = 1 and \$(diag(x, y)) = 0 if Ix + y + w - 1I > £. Then

\A\ > f \$(d2u) f (pdv

)b Jr»?

= \B\ P (F2N+2 + G2N+2 =1-w) =

Thus, for large j,

\A\ is bounded from below by C1. Consequently, for any 1 < q < to and large j,

lb - Xa\

L«(r2)

= \\Au: + w-xa\

f \ AUj + w- xa \q + | + w- xa \'

+ 1 _| AUj + w-xa \

Jb\(AuA) 1

< £q \A\ +£q Ul + £ \

sup \AU:\ + w

Here in the last passage we have used the definition of A, A, and (89). Combining this with (91) (and the fact that the second-order partial derivatives of Uj are uniformly bounded by C2; see Corollary 17), we get that, for sufficiently large j,

\\vj ^aIIl«(R2)

<C3e\A\.

In other words, the function Vj is close to the indicator function of A.

Step 5. Next, consider the function 0 : R

R given by

<p(diag(x, y)) = (\x+w\-X)+. By the choice of w, N,and (86),

BiL t^^L>*dv = E(^2N+2\-X) +

> Mw >

1 + e \B\ J,

\B\ Jb

1 + e \B\ V ' J,

Now multiply throughout by \B\ and apply (93) with q = 1; we get that, for sufficiently large j,

f (p(D2Uj)>1^(1-C3e)\A\. jb 1 + e

However, observe that

0 (D2Uj) = (|9n«j + - A)+

= (\R21Auj + w\ -\)+

= (\r2ivj -wr\xb + w\-x)+

on B. Therefore, the preceding considerations yield that, for large j,

1- C3e) \A| < f (^j - wR2ixb + w\- X)+ 1 + e Jb 1 +

< f (¡^Xa -wR2iX.b +H~A)+ (97)

Jb 1 1 +

+ f i*2 (vj -xa)\. jb

However, the norm of R2 as an operator on L2(R2) is bounded by 1: see (57). Consequently, by Schwarz inequality, (93), and then (91),

I \r21 (vj -XA)\<\\Vj -Xa\\

jb 1 11 11

<C4e1/2\A\1/2 < C5e1/2\A|.

Plugging this into the above inequality, we get that if j is sufficiently large, then

f (\r1xa - wR2xb + >rr(1- c6tl2) w.

Jb 1 + 1 + £ v '

Therefore, if we let E = {x e B : \r2ixa - w^1xB + w\ > and recall that M = (1/4)e4-4X ■ q, then

22 | \r1xa -wR1 xb +w\

-^(1-C6e1/2)\A\+X\E\ 1 + £ v '

(1-C6£1/2 )n

\A\+X\E\

However, we have X = p/4. Plugging this above and applying the Young inequality, we see that the right-hand side is not smaller than

(1-C6£1/2)n p

1 + £

^p(4-p)/p\\ y II 4e IIaa|IlP(R2)

\£| 1-1/p. (101)

On the other hand, \\-R2\\LP(R2)^Lf.(R2) < p - 1 (see (57)), so

by the Schwarz inequality and the bound \A\ > w (see the estimate above (91),

| \wr2iXb -w\

<w(\E\ + (p-1)\B\1lp\E\1-1lp) <w1-1lp\A\ 1/p (102)

x(\E\ + (p-1)\B\1lp\E\1-1lp)

< pn1lpw1-1lpWxa\\l,(r2)\E\ 1-1lp.

Combining this with the previous estimate, we get

f \r1xa\

(1-c+mh ■ 4e(4-p)lp - pn1lpw1-1lp) (103)

x\\*a\ur2)\£i 1-1lp.

Using the fact that q < 1, e > 0 and w > 0 were arbitrary, we obtain that the constant (p/4)e(4-p^lp is indeed the best possible in (58).

vi - Xa

4.4. Sharpness of (58): The Case d > 3. Let J be a proper subset of {1,2,. ..,d} and write T = R2. It suffices to consider only those J, which satisfy 1 e J and 2 i J: for any J' e {1,2,. ..,d} of the same cardinality as J, the restricted weak constants of T and Xjej' Rj are the same. So, suppose that T is of that special form and assume that for some positive constant C we have

£ \r2^Xa (x)| dx < C\\XA\\LP(Kd)\E\1-1/P (104)

for all measurable subsets A, E of Rd of positive and finite Lebesgue measure. For t > 0, define the dilation operator St as follows: for any function g : R2 x R -2 ^ R, we let 8tg(\$,0 = g(%, t(); for any A c R2 x Rd-2,letStA = {(^,t() : (£,, 0) e A}. Note that the function StxA is supported on S-1 A and hence, by (104), the operator Tt := S-1 ° r1 ° St satisfies

\TtXA (x)| dx = td-2 I R o StXA (x)| dx

JE JS-1E 1 1

<c(td-2 I \StXA (x)\pdx

\ JS^A x(td-2 \S-1E\)1-1/P

Plug this into (108) with the choice E = Ex[0,1]d 2,where E is an arbitrary subset of R2 with 0 < \E\ < <x>. As the result, we obtain

= c\\x.

■AWu^)

It is straightforward to check that the Fourier transform F satisfies the identity F = td-2St o F o St; since 1 e J and 2 i J, the operator Tt has the property that

Ttf(Ç,Ç) = -

% +12 UkÇ Kl2 + i2KI2

(Ç, Ç)e R2 x R

where the set K is defined by the requirement that k e K if and only if k + 2 e J. By Lebesgue's dominated convergence theorem, we have

limTf(U) = Tf(U)

in L2(Rd), where T0f(\$, 0 = 0)/\^\2. By Plancherel's

theorem, the passage to a subsequence which converges almost everywhere, and Fatou's lemma, we see that (105) implies

^ \t0xa (x)\dx < c\\xa\\l,(r^)\e\1-1/p. (108)

Now pick an arbitrary set Ac R2 of nonzero and finite Lebesgue measure and put A = A x [0,1] -2. Denoting by R1 the first planar Riesz transform, we see that T0xA(0) = Rixa(^)x[0 1]d-2 (0), because of the identity

__ Ç2 ^ ___

toxa (Ç,Ç) = -jÇjl xâ (Ç)X{o,if-2(Ç)-

f R2xx (ï)\dï<c\\xA

JE 1 1

But we have shown in the previous subsection that this implies C > pe(-4-p^p/4. The proof is complete.

Acknowledgments

The author would like to thank the anonymous referee for the careful reading of the paper. The research was partially supported by Polish Ministry of Science and Higher Education (MNiSW) Grant IP2011 039571 "Iuventus Plus."

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