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## Academic research paper on topic "Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms"

﻿Cent. Eur. J. Math.

DOI: 10.2478/s11533-014-0401-6

VERS ITA

Central European Journal of Mathematics

Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms

Research Article

1 Institute of Mathematics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Received 1 February 2013; accepted 20 October 2013

Abstract: We study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Levy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on Rd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures on the space of matrices of dimension 2 x 2, and some transference-type arguments.

MSC: 42B15, 60G44, 42B20

Keywords: Fourier multiplier • Singular integral • Martingale © Versita Sp. z o.o.

1. Introduction

Computing the exact norm of a singular Integral operator Is In general a very difficult task. Probably the first result In this direction is that of Pichorides , where the value of Lp norm of the Hilbert transform was determined. Recall that the Hilbert transform on the line is an operator acting on f e L1(R), given by the formula

HRf (x )=- p.v. ( -M- dt. ( ) nV Jr x — t

Pichorides proved that if 1 < p < <x, then ||HR||LP(R)^LP(R) = cot(n/2p*), where p* = max{p,p/(p — 1)}. This result was generalized to the higher dimensional setting by Iwaniec and Martin , and Banuelos and Wang . If d > 1 is a fixed integer, then the collection of Riesz transforms (cf. ) is given by

Springer

^ %) ^ J-1.2.....-

For d — 1, this family contains only one element, the HUbert transform. The aforementioned result of Iwanlec, Martin, Banuelos and Wang asserts that

II «J HiP(Rd)^iP(Rd)— C0t2^ for all d. J 6{1.2.....d}. 1 <p< m.

This result can be formulated in the language of 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 following relation between Fourier transforms: Tmf — mf. The norm of Tm on L2(Rd) is equal to ||m|Lm(Rd) and it has been long of interest to study those m for which the corresponding Fourier multiplier extends to a bounded linear operator on Lp(Rd), 1 < p < m. In general, the characterization of such symbols seems to be hopeless, but one can study various examples and their properties. For instance, one can consider the above collection of Riesz transforms on Rd: it can be shown that Rj is a Fourier multiplier with the symbol i%jl\i\, J — 1.2,....d. See Stein  for the detailed computation and related discussion.

In the present paper we consider a class of symbols which can be obtained with the use of probabilistic methods: more precisely, by the modulation of jumps of certain Levy processes. This class has been introduced and studied by Banuelos and Bogdan , and Banuelos, Bielaszewski and Bogdan . Let v be a Levy measure on Rd, i.e., a nonnegative Borel measure on Rd such that v({0}) — 0 and

/ min{\x\2,1} v(dx) < oo.

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

mtf) — (2 [(Z.9M9) v(d0) + ( [1 - cos(%.x}]flx) v(dx)) i1 [(Z.9}2v(d9) + ( [1 - cos^.x}] v(dx)) (1) \ 2 Js jRd I \ 2 Js jRd I

if the denominator is not 0, and m(%) — 0 otherwise. Here (•. •} stands for the scalar product in Rd. This class includes many important examples (cf. [2, 3]). Let BA be the Beurling-Ahlfors transform on the plane, i.e., a Fourier multiplier with the symbol m(%) — % 6 C. This operator is of fundamental importance in the study of partial differential equations and quasiconformal mappings, since it changes the complex derivative d to d. It turns out that the real and imaginary parts of BA can be represented as the Fourier multipliers with the symbols of form (1). For example, the choice d — 2, v — 5(10) + 5(0.i), p( 1. 0) — 1 — —p(0.1) and v — 0 gives rise to Tm — Re BA; furthermore, d — 2, V — ^(1/V2.1/-/2) + ^(1/y2.-MV2), P(1/V2.1/^2) — —1 — -p(1/V2. -1/V2) and v — 0 leads to Tm — Im BA. A similar choice of the parameters leads to BA/2. For a higher-dimensional example, pick a proper subset J of {1. 2..... d} and take v — + 5e2 + • • • + 5ed, v — 0 and p(ej) — Xj(J), J — 1. 2..... d, where ei. e2..... ed are the vectors in Rd. This yields the operator RJ on Rd. One of the principal results of  is the following Lp estimate.

Theorem 1.1.

Let 1 < p < m and let m — be given by (1). Then for any f 6 Lp(Rd) we have || Tmf ||Lp(Rd) < (p* — 1) || f || LP(Rd).

It turns out that the constant p* — 1 is the best possible: see Geiss, Montgomery-Smith and Saksman  or Banuelos and Os^kowski  for details.

A very interesting phenomenon is that, essentially, all sharp estimates mentioned above are proved with the use of probabilistic methods (Pichorides exploits certain special superharmonic functions, but these, in fact, lead to more general inequalities for orthogonal martingales: see Banuelos and Wang ). It turns out that martingale methods lead to more general results for Fourier multipliers (see e.g. ). The purpose of this paper is to explore further this fruitful connection. Our motivation comes from a natural question about the best constant in the weak-type estimate

\\Rff\\p_m< CpHfHp. 1 < p< m.

\\g\\p.x = sup{Ap | {x e Rd : \g(x)| > A}|}

denotes the weak p-th norm of g. In fact, our argumentation will yield the inequality for a larger class of multipliers. For 2 < p < to, let c = c(p) be the unique positive solution to the equation

cp—1 = 2c + 1

and put

Cp = -I

(2c + p — 1)p—

if 2 < p < to, if p = 2.

Theorem 1.2.

Suppose that m is a symbol given by (1), where tfr and ^ are assumed to take values in [0,1]. Then for any 2 < p < to we have

\Tmf \\p.to< Cp\\f \\p.

The inequality is sharp. More precisely, for any 2 < p < to, C < Cp, d > 2 and any proper subset J of {1, 2,.... d} there is f e Lp(Rd) such that

I~_Rjf

> C\ f\ L

To the best of our knowledge, this is one of the very few results in the literature when the exact weak-type norm of a Fourier multiplier has been determined (Davis  and Janakiraman  identified the weak-type constants for the Hilbert transform on the line). Unfortunately, we have managed to push the calculations through only in the case p > 2. The reason is that for 1 < p < 2 we did not succeed in showing an appropriate martingale bound (see Section 2 below).

Before we proceed, let us mention here the following application of Theorem 1.2, which can be of interest in the theory of elliptic differential operators and potential theory. Let u be a Cto compactly supported function on Rd. By a direct comparison of Fourier transforms, we check that RjAu = —d2u/dxJ2 for all 1 < J < d and thus we obtain the sharp bound

< Cp\\Au\\Lp(Rd), 2 < p < to,

lp,to (Rd)

whenever J is a proper subset of {1, 2,..., d}.

A few words about the organization of the paper are in order. In the next section we study a martingale inequality, which can be regarded as a probabilistic counterpart of (4). In Section 3 we combine this estimate with the representation of Fourier multipliers (1) in terms of Levy processes, and provide the proof of inequality (4). Finally, in Section 4 we show that the constant Cp cannot be replaced by a smaller number, even if we restrict ourselves to the operators ^jej Rf. This will be based on the technique of laminates, an important object in the convex integration theory, and some transference arguments.

¿-3x2

2. A martingale inequality

The key ingredient of the proof of the announced estimate (4) is an appropriate inequality for differentially subordinated martingales. We begin with introducing the necessary probabilistic background and notation. Assume that (Q, F, P) is a complete probability space, equipped with (Ft)t>0, a nondecreasing family of sub-ff-fields of F, such that F0 contains all the events of probability 0. Let X, Y be two adapted martingales taking values in a certain separable Hilbert

space (H. \ \); with no loss of generality, we may put H — (2. As usual, we assume that the processes have right-continuous trajectories with the 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 in the case when the processes are real-valued, and extend the definition to the vector setting by [X. Y] — [Xk. Yk], where Xk. Yk are the k-th coordinates of X. Y,

respectively. We will say that Y is non-symmetrically differentially subordinate to X if the process ([X. Y]( — [Y. Y]()t>0 is nonnegative and nondecreasing as a function of t (see Banuelos and Wang , Os^kowski  and Wang ). The nonsymmetric differential subordination implies many interesting inequalities comparing the sizes of X and Y; see e.g. Choi  and Burkholder . We mention here only one result by the author , which will be of importance to our further considerations. For p > 2, let Cp be given by (3). We use the notation HX||p — supt>0 HXt||p.

Theorem 2.1.

Suppose that X. Y are H-valued martingales such that Y is non-symmetrically differentially subordinate to X. Then for p > 2 we have

P (sup \ Yt \> 1) < CppHX YP (5)

and the inequality is sharp.

Now we are ready to formulate our main probabilistic result. As we will see, it can be regarded as a dual estimate to (5).

Theorem 2.2.

Assume that X. Y are H-valued martingales such that Y is non-symmetrically differentially subordinate to X. Then for any 1 < q < 2,

|| Y ^ < Cqq|(q-1)HX |M|X M—1 . (6)

For each q, the constant Cq/(q-1) is the best possible.

The proof rests on Burkholder's method: we shall deduce inequality (6) from the existence of a family {^q}q6(1m) of certain special functions defined on the set S — {(x.y) 6 H x H : \x\ < 1}. In order to simplify the technicalities, we shall combine the technique with an "integration argument", invented in  (see also ): first we introduce two simple functions v1.vm: H x H —> R for which the calculations are relatively easy;then define Vq by integrating these two objects against appropriate nonnegative kernels. Let

vi{x,y) = v^(x,y) =

\2y - x\2 - \x|2 1 - 2\x\

if |x| + |2y - x\ < 1, if |x| + |2y - x\ > 1,

0 if |x| + |2y - x\ < 1,

(|2y - x\ - 1)2 -\x\2 if \x\ + \2y - x\> 1.

We have the following fact, which appears as [20, Lemmas 2.2, 3.2].

Lemma 2.3.

For all H-valued martingales X, Y such that Y is non-symmetrically differentially subordinate to X, we have Ev1(Xt, Yt) < 0 for all t > 0. In addition, if X, Y are bounded in L2, then Ev^(Xt, Yt) < 0 for all t > 0.

Recall that S — {(x.y) 6 H x H : \x\ < 1}. For 1 <q < 2, let c — c (q/(q - 1)) be given by (2) and put b — (c + 1)-1. Define Vq : S ^ R by

Vq(x, y) = ^ I br- dr + (\ 2y - x\2 -\x\2).

An easy calculation shows that if \x\ + \2y — x\ < b, then

Vg(x,y) = 2g(^1— 1) (\x\ + \2y — x\y—1 ((q — 1)\2y — x\ — \x\),

while for \x\ + \2y — x\ > b,

,,, , q(2 — q)bq—1 I b 2\x\ \ qbq—2 _ l2 , l2)

Vq(x, y) = q( 2qq1 (b — q—1) + W (\ 2y — x \ 2 — \ x \ 2).

We also define V2 by the formula V2(x,y) = (\ 2y — x\2 — \x\2)/2; it is not difficult to check that V2 is the pointwise limit of Vq as q tends to 2 (indeed: directly from (2), we infer that b — 0 and bq—2 — 2). We shall need the following majorization property of these functions.

Lemma 2.4.

For any 1 < q < 2 we have

Vq(x,y) > \y \q — Cqnq_n \x\ for all (x,y) e S. (7)

Proof. It is convenient to split the reasoning into a few parts. Step 1: q =2. Then the inequality reads

2 (\ 2y — x \2 — \ x \2) > \ y\2 — \x\ , or, equivalently, \y — x\2 + \x\ — \x\2 > 0, which is obvious.

Step 2: some reductions. Now, assume that q =2. Note that it suffices to establish the majorization for H = R and for x, y satisfying 0 < x < 2y. To see this, let us for a moment write VqH to indicate the Hilbert space in which we work. Pick x, y e H, put x' = \x\ and y' = \x/2\ + \x/2 — y\. Then 0 < x' < 2y', 2y' — x' = \2y — x\ and y' > \y \, so

VH(x. y) — \ y \q — Cqq/(q—1)\x \ > VR(x', y') — \ y' \q — Cqq/(q — 1) \ x' \,

since the dependence of VH on x and y is through x and 2y — x only. Next, observe that for any y > 0 the functions x — (x, y) and x — \2y — x\2 — \x\2 are concave on [0,1]. Therefore, Vq also has this property and since the right-hand side of (7) is linear in x, we will be done if we prove the majorization for x e {0,1}.

Step 3: x = 0. If y < b/2, then both sides of (7) are equal. If x = 0 and y > b/2, then the majorization can be rewritten in the equivalent form

which follows immediately from the mean value property.

Step 4: x = 1. We restrict ourselves to y > 1/2 (see Step 2 above). This time, (7) takes the form

F(y) = si^1 (q — \ + ^ (V — 4y) — yyq + CV„ > 0.

We easily check that F(1 - b/2) = F'(1 - b/2) = 0 and

F-M = ^ - - 1)»-2 . ^ - ^ = (b-2 - (, - 1)).

However, the latter expression is positive: we have bq 2 > 1 and q - 1 < 1. This proves that F is convex on [1/2, to) and hence it is nonnegative on this interval. This completes the proof. □

Now we are ready to establish Theorem 2.2.

Proof of Theorem 2.2. It suffices to show that II y\\qq < Cql(q_r) \\XI11 for any X, Y as in the statement satisfying the additional condition \X< 1. Fix t > 0. The non-symmetric differential subordination implies that the process ([X,X]t - [2 Y -X, 2 Y -X] t)t>o is nonnegative and nondecreasing as a function of t; consequently, for any t we have

E 12Yt - Xt|2 = E[2Y - X, 2Y - X]t < E[X,X]t = E |Xt|2 < 1, because of the boundedness of X. Therefore, Lemma 2.3 and Fubini's theorem imply

q(2 - q) fb I Xt Yt

EVq(Xt,Yt) < ^T^ rq-1 Evi

dr < 0.

To see that Fubini's theorem is applicable, note that |vi(x, y)| < c(|x| + |y| +1) for all x, y e H and some absolute constant c; thus

Xt Yt ' dr < cE(X| + Y| +1) < TO,

where c is another universal constant. Combining (8) with (7) yields E|Yt|q < C^/(q—1)E|Xt| and it suffices to let t ^ to to get the claim. □

The optimality of the constant Cq/^q-1) will be dealt with in Section 4. It will follow immediately from the sharpness

of (4).

Remark 2.5.

Unfortunately, we have been unable to find an appropriate majorant Vq in the case q > 2. Though the "integration argument" is available and we have the dual simple function vTO ready to use, all the special Vq we managed to construct have led us to non-optimal Cq/(q-1).

3. Proof of Theorem 1.2

Let m = be a multiplier as in (1). Due to results in , we may assume that the Levy measure v satisfies the symmetry condition v(B) = v(-B) for all Borel subsets B of Rd. More precisely, there are such that V is symmetric and = Assume in addition that |v| = v(Rd) is finite and nonzero, and define "¡7 = v/^l Consider 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 Rd and has 77 as the distribution. Next, put Sn = —(T—1 + T—2 + ----h Tn) for n = —1, —2,... and let

Xs,t = ^Zj, XSit— = ^Zj, AXSit = Xs,t — XSit—, —to < s < t < 0.

s<Sj <t s<Sj<t

For a given f e LTO(Rd), define its parabolic extension Uf to (^oo, 0] x Rd by Uf (s,x) = Ef(x + Xs,o). Next, fix x e Rd, s < 0 and f e LTO(Rd). We introduce the processes F = fFX,s,f)te|s0] and G = fGXsf^)te|s0] by

Ft = Uf (t,x + Xs,t),

Gt = Yi iAFu • ^(AXS,„)] - ff [Uf(v,x + Xs,v- + z) - Uf(v,x + Xs-]4>(z) v(dz) dv.

Js JRd

Note that the sum in the definition of G can be seen as the result of modulating of jumps of F by (j>, and the subsequent double integral can be regarded as an appropriate compensator. We have the following statement, proved in .

Lemma 3.1.

For any fixed x,s,f as above, Fx,s,f,Gx,s,f^ are martingales with respect to (ff({Xu : u < t}))te[s0]. Furthermore, if

< 1, then Gx,s,fis differentially subordinate to Fx,s,f

Now, fix s < 0 and define the operator S = §s,^,v by the bilinear form

[ Sf(x)g(x) dx = \ E[Gx,s,f,^g(x + X^)] dx, (10)

Jvd Jvd

where f, g e C0TO(Rd). We have the following fact, proved in . It constitutes the crucial part of the aforementioned representation of Fourier multipliers in terms of Levy processes.

Lemma 3.2.

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

M(Z) = Msj4,jV (f) =

( (1 — cos(f.z))4>(z) V(dz)

1 — exp 2 J (1 — cos(f.z)) v(dz) ^-

\ jRd n (1 — cos(f,z)) V(dz)

if f (1 — cos (f,z)) v(dz) = 0, and M(f) = 0 otherwise. We are ready to establish the following dual version of (4).

Theorem 3.3.

Assume that 1 < q < 2 and m: Rd —> C be a symbol given by (1), where ^ and ^ are assumed to take values in [0,1]. Then for any function f e L1(Rd) n Lto(Rd) we have

\\Tmf \\qq (Rd) < Cq,/(q—1) \\f \\ L1 (Rd) \\f W^jd). (11)

Proof. By homogeneity, it suffices to establish the bound for f bounded by 1. Furthermore, we may and do assume that at least one of the measures p, 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

f (1 — costf.zMz) v(dz)

M^V (f) = ^-. (12)

/ (1 — cos(f.z)) v(dz)

Assume that 0 < v(Rd) < to, so that the above machinery using Levy processes is applicable. Fix s < 0 and functions f,g e CTO(Rd) such that f is bounded by 1; of course, then the martingale Fx,s,f also takes values in the unit ball of C. By Holder's inequality, Fubini's theorem and (6), we have

1 f E[Gx0s,f,^g(x + Xs,0)] dx < ( f E|G0x,s,f,^|qdx] " I f E|g(x + Xs,0)|pdx) U Rd \ Jvd I \ Jvd I

^/q ( r . \1/q (13)

^E|Gx0^,f,*|q dx) q Wg^Rd) < (^Vd^ E|Fx,s,f|dx) s \\g\\LP{i

= (Cq/(q—1) \\f \\1)Vq WgWlpRd).

Here p = q/(q — 1) Is the harmonic conjugate to q. Plugging this Into the definition of S, we obtain

Mss^vr||q < £q

^ Cq/(q —1) ||f IkW)'

Now if we let s — — to, then Mspv converges pointwise to M<p v given by (12). The symbols are bounded in absolute value by 1, so, by Lebesgue's dominated convergence theorem, we have Sspvf — TM^vf in L2(Rd). By Plancherel's theorem, we also have Ss,p,vf — TM^vf in L2(Rd) and hence there is a sequence (sn)TO= converging to —oo such that gsn,<P,vf — TM^vf almost everywhere as n — to. Thus Fatou's lemma yields the desired bound for the multiplier TM^v.

Step 2. Now we deduce the result for the general multipliers as in (1) and drop the assumption 0 < v(Rd) < to. For a given e > 0, define a Levy measure ve in polar coordinates (r, 9) e (0, to) x S by

(dr d9) = e—25e(dr) j(d9).

Here 5e denotes the Dirac measure on {e}. Next, consider a multiplier as in (12), in which the Levy measure

is 1{|x|>e} v + ve and the jump modulator is given by 1{|x|>e}p(x) + 1{M=e}P(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

i [1 — cos{Z,x>] Jx) Ve(dx)= [{Z, 9)2p(9) 1 -c/s9;2e9> v(d9) — 1 i{Z,9)2p(9)j(d9) jRd \ \x\ I JS {Z,e9>2 2 JS

and, consequently, Me^ipjJiV — pointwise. This yields the claim by the similar argument as above, using

Plancherel's theorem and the passage to the subsequence which converges almost everywhere. □

Now we shall apply duality to deduce (4).

Proof of Theorem 1.2. Observe that the class (1) is closed under the complex conjugation: we have ~m = mp>pJV. Fix f e Lp(Rd) and put , , ,

_ ' m • *

9 = \Tf\ '{xeKd:|Tmf(x)|>1}.

By Holder's inequality and Parseval's identity,

|{x e Rd : \Tmf(x)|> 1}|< ( Tmf(x)9(x)dx = \ TJ(x)9(x)dx = [ ?(x) 7^9(x) dx

Jrnd Jrnd Jrnd

= f (x) Tw9(x) dx < II f llLP(Rd) ^^^(Rd) < || f |LP(Rd)(Cqq/(q-1( ¡9

Here in the latter passage we have used (11) and the fact that 9 takes values in the unit ball of C. However, I9I |{x e Rd : \Tmf(x)| > 1}| and Cq/(q—1 = Cp. This completes the proof of the weak type estimate. □

In the remainder of this section we discuss the possibility of extending the assertion of Theorem 1.2 to the vector-valued multipliers. For any bounded function m = (mi, m2,..., mn): Rd — Cn, we may define the associated Fourier multiplier acting on complex valued functions on Rd by the formula Tmf = (Tm1 f, Tm2 f,..., Tmn f). As we shall see, the reasoning presented above can be easily modified to yield the following statement.

Theorem 3.4.

Let v, j be two measures on Rd and S, respectively, satisfying the assumptions of Theorem 1.2. Assume further that p, p are two Borel functions on Rd taking values in the unit ball of Cn and let m: Rd — Cn be the associated symbol given by (1). Then for any Borel function f: Rd —> C we have

11 Tm f |qq(Rd;Cn) < Cq/(q—1) 11 f ||L1(Rd) |f 11 qo°(Rd), 1 < q < 2, 11 Tm f 1 LP,TO(Rd;Cn) < Cp ||f ||LP(Rd), p > 2

Proof. Suppose first that v is finite. For a given function f e CTO(Rd) bounded by 1, we introduce the martingales F and G = (G1, G2,.... G") by (9). It is not difficult to check that Lemma 3.1 is also valid in the vector-valued setting (repeat the reasoning from ). Applying the representation (10) to each coordinate of G separately, we obtain the associated multiplier S = (S1, S2,..., S"), where S' has symbol M^jjV defined in (12). Now we repeat the reasoning from (13) with a vector-valued function g: Rd — C" (the expression Gx0s,f,^g(x + Xs0) under the first integral is replaced with the corresponding scalar product). An application of (6) gives

\\Ss,^,Vf \\ qq(Rd;C") < Cq \\f \\ L1 (Rd),

which extends to general f by standard density arguments. The passage to general m as in (1) is carried over in the same manner as in the scalar case;this yields the vector version of Theorem 3.3. The duality argument explained in (14) extends to the vector-valued setting with no difficulty (one only has to replace appropriate multiplications by scalar products) and thus Theorem 1.2 holds true for the multipliers on C". □

4. Sharpness

In the final part of the paper we show the second half of Theorem 1.2: the constant Cp in (4) is the best possible, even for the special multipliers R2, where J is a proper subset of {1, 2,...,d}. This, of course, will immediately imply that the constant in (6) is also optimal (otherwise, its improvement would lead to a smaller constant in (4)). Our approach will be based on the properties of certain special probability measures, the so-called laminates. For the sake of convenience and clarity, we have decided to split this section into an number of separate parts.

4.1. Necessary definitions

Let Rmx" denote the space of all real matrices of dimension m x " and let R"ym" be the class of all real symmetric " x " matrices.

Definition 4.1.

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

Next, let P = P(Rmx") stand for the class of all compactly supported probability measures on Rmx". For v e P we denote by v = fRmx" X dv(X) the center of mass or barycenter of v.

Definition 4.2.

We say that a measure v e P is a laminate (and denote it by v e L), if

f(V) < [ fdv

for all rank-one convex functions f. The set of laminates with barycenter 0 is denoted by L0(Rmx").

Laminates arise naturally In several applications of convex Integration: they can be used to produce Interesting counterexamples, see e.g. [1, 9, 16, 17, 25]. We will be particularly interested in the case of 2 x 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 4.6 below. Let us briefly explain this;detailed proofs of the statements below can be found for example in [15, 17, 25].

Definition 4.3.

Let U C R2x2 be a given set. Then PL(U) denotes the class of prelaminates generated in U, i.e., the smallest class of probability measures on U, contained in L, which

(i) contains all measures of the form A5a + (1 — A)5B with A e [0,1] and rank (A — B) = 1 (here 5A, 5B stand for the Dirac measures concentrated on A and B);

(ii) is closed under splitting in the following sense: if A5a + (1 — A)77 belongs to PL(U) for some 77 e P(R2x2) and A e [0,1], and y belongs to PL(U) with y = A, then Ay + (1 — A)77 also belongs to PL(U).

It might be helpful to provide here an alternative, inductive definition of the class PL(U). We start from Dirac measures concentrated on the elements of U, and then allow the following modification of these. Namely, if a measure y = A5a + (1 — A)77, A e [0,1], is a laminate, we replace it with A—5A— + A+5a+ + (1 — A)77, where A± > 0, A— + A+ = A, A—A— + A+A+ = AA and rank(A+ — A—) = 1. Then one easily checks that the new object is also a laminate;actually, PL consists of all measures which can be obtained from Dirac measures after a finite number of the above splittings. It follows immediately from the above inductive definition that the class PL(U) contains atomic measures only. Also, by a successive application of Jensen's inequality, we have the inclusion PL C L. Let us state two well-known facts (see [1, 15, 17, 25]). In what follows, B denotes the unit ball of R2.

Lemma 4.4.

Let v = ^^ At5Al e PL(R2uxm2) with v = 0. Moreover, let 0 < r < min | At — Aj| /2 and 5 > 0. For any bounded domain Q C R2 there exists u e (Q) such that \\u\\C1 <5 and for all i = 1, 2.....N,

|{x e Q: |D2u(x) — Ail <r}|= AIQI.

Lemma 4.5.

Let K C R2uxm2 be a compact convex set and v e L(R2uxm ) with supp v C K. For any set U C R2xm, relatively open with respect to Rx and satisfying K C U, there exists a sequence Vj e PL(U) of prelaminates with Vj = v and Vj v.

Combining these two lemmas and using a simple mollification, we obtain the following statement, proved by Boros, Szekelyhidi Jr. and Volberg . It links laminates supported on symmetric matrices with second derivatives of functions, and will play a crucial role in our argumentation below.

Corollary 4.6.

Let v e L0(R2ux2). Then there exists a sequence uj e Cq°(B) with uniformly bounded second derivatives, such that

BI HD'u M) „„ — HV

for all continuous \$: R2yxr2 — R.

This corollary reveals the idea behind the proof of the sharpness of (4), at least in the two-dimensional case. Roughly speaking, the reasoning is as follows. Suppose that u e Qj^R2) is given and put f = Au. If are appropriately

chosen continuous functions on R2,^ (\$i(A) ^ 1{|A11|>1} and \$2(A) = IA+ A22|p for

A = A„ A12

A = A21 A22

for the precise definition of see below), then

|{x e R2:|fljf(x)| > 1}| = |{x e R2:^)| > 1}| „ Jr2 *1(Du(x)> dx

\ \f (x)\p dx \ |d21u(x) + д222u(x)|p dx \ fo(D2u(x)) dx

JR2 JR2 Jr2

Hence, by Corollary 4.6, it is enough to construct, for any e > 0, a laminate v e L0(R2Xm2) for which the ratio

J dv j j p2dv is larger than Cp

4.2. Biconvex functions and a special laminate

Let us start by introducing an auxiliary notion. A function Z: R x R — R is said to be biconvex if for any fixed z e R, the functions x — Z(x,z) and y — Z(z,y) are convex. First we show the following inequality for biconvex functions in the plane. This estimate may seem unexpected, some arguments which suggest its use are presented in subsection 4.5.

Lemma 4.7.

Suppose that Z: R x R — R is biconvex. Then for any numbers k0, k1 satisfying 0 < k0 < k1 < 1, we have

Z(—K0,K0) < Kp 1

(p — 1)(1 +2k0) + (3 — p)K1

Z—1,1 +2K0)

(1+2k0 — K1)(p — 1)k\—p i p — 3

Z —K1 , -T K1

(p — 1)(1 +2K0) + (3 — p) K1

^ f \z (p—p

p — 1 iK1 ^ s,s) + zl —s,p—4 s

p — 1 p — 3 p — 1

Proof. By a standard regularization argument, it suffices to show the inequality for Z e C1(R2). Fix s e (k0,k1) and a small positive 5. Using biconvexity of Z, we may write

Z(—s,s) Z ( —s, p—3 s| +

2 + 5(p — 1p\ p — 11 2 + 5 (p — 1)

Z(—s,s + 5s)

x , 2(5 + 1) — 5(p — 1)„ _ 5(p —1)^/3 — p, _

Z (—s,s + 5s) < ( 2(5 + 1(p ) Z(—s — 5s,s + + 2(5+1) Z( p— (s + 5s),s + 5s|.

Plugging the latter estimate into the former, subtracting Z(—s — 5s, s + 5s) from both sides and dividing throughout by 5 gives

Z(—s,s) — Z(—s — 5s, s + 5s) < I p — 1 p — 1 + -5- < ( — 2 + (p — 1)5 — 2(5+^1 Z(—s —5s,s + 5s)

+ (2 + (p — —)^)(5 + 1) Z( p—p(s + 5s),s + 5s) + 2T—^ Z(—s,p— s)

Therefore, letting 5 — 0 implies

-s ^—s,s) < (1 —p) Z(—s,s) +

^ p—p ») + Zhtf s

Multiply both sides by s p to get the equivalent bound

sl—PC(—s.s)

1 — p.

Z| ^ s.s) + z( — s.^3 s

p — 1

p — 1

Integrating over s from k0 to * gives

K^Z—^,^) — к„-PZ(—кo, K0) > „--P f

It suffices to combine this inequality with the following consequence of the biconvexity of Z:

3 — p

p — 3

Z —K1.K,) <

n 1 1 9 , , (p — 1)(1+2K0 — K1) r[ p — 3 Z(—k, 1 +2K0) + ——„w„ , ^ , , ,,,—TV— Z| — K1, T—^ *

(p — 1)(1 +2k0) + (3 — p)K1

(1 +2K0)(p — 1) + (3 — p)K1 and the claim is established.

Let y = yK0.K1 e P(R2x2) be defined by the right-hand side of (15);that is, for any f e C(R2x2), let

p — 1

f djrn ,K, = K

(p — 1)(1 +22K10)p+(3 — p) * f (diag(—K1,1+2K0))

, (1+2K0 — K1)(p — 1) K1- f (diag(— K,p—3 *

(p — 1)(1 +2k0) + (3 — p)* p

p — 1

^ [f( dia^3—4 -)) + f( diagf —s,^ s

p — 1

p — 1

Then yK0,K1 is a probability with barycenter yK0,K1 = diag(—k0, k0). Observe that if f is rank-one convex, then (x, y) 1—> f(diag(x, y)) is biconvex. Therefore, using Lemma 4.7 we see that yK0,K1 is a laminate. Hence, if we introduce a probability measure v by putting v(A) = y(A + diag(—k0, k0)) for any Borel subset A of R2x2, then v e L0(R2xm'). Next, consider a continuous function : R^m — [0, j), which satisfies \$i(A) = 0 when A22 < 1, \$i(A) = 1 when A22 > 1 + k0 and \$i(A) e [0,1] for remaining A. In addition, let \$2: R2u>im — [0, j) be given by \$2(A) = |A-n + A22\p. Directly from the definition of v, we have

>1 dv = Kp 1

J \$2 d v = J \$2 dy

(p — 1)(1 +2k0) + (3 — p) *

p-1 2Ki—p

(p — 1)(1 +2k0) + (3 — p) k

(1 +2K0 — K1)p

(1+2K0 — K1)(p — ^ / 2 V + — 1) Vds

(p — 1)(1+2K0) + (3 — p)K^ p — 1 (p )J,0 \p — 1 / sp

Now let k0 go to 0. Then

\$2 d v \$1 dv

/ 2 \p— (1 — K1)p + p + (2 — p)* ).

If we put = (p — 1)/(2c + p — 1), then some straightforward computations show that the latter limit is equal to Cf, p.

4.3. Sharpness, case d = 2

Fix p > 2. The weak p-th norms of R2 and R2 are the same, so it suffices to prove that 11Rf Mlp(r2)—• Lp,TO(R2) > Cp. By the above reasoning, if e > 0 is a given number, then we can pick k0 > 0 such that the ratio f p2 dv// dv is smaller than Cpp + e. Therefore, an application of Corollary 4.6 yields the existence of a CTO function u, supported on B, such that

[ h(D2u(x)) dx

&(D2u(x)) dx p

< Cp p + 2e.

However, by the very definition, we have p1(A) < X{a22>1}. Thus, the above inequality implies

|{x e R2: d22U(x) > 1}| f |d21u(x)+ д222u(x)|p dx.

Cp + 2e Jb

Therefore, if we put f = Au, we obtain

||R22f Mlp,to(r2) > (C—p +2e)—1/pp Hf ||lp(r2). Since e was arbitrary, this proves the desired sharpness of (4).

4.4. Sharpness, case d > 3

We use a well-known argument, see e.g. . Fix p > 2. Let J be a proper subset of {1,2,..., d} and write T = Rj. We should prove that || T||Lp(Rd)—Lp,TO(Rd) > Cp. It suffices to consider only those J, which satisfy 1 e J and 2 e J: for any J' e {1, 2,..., d} of the same cardinality as J the weak norms of T and ^jeJi Rj are the same. So, suppose that T is of that special form and assume that for some positive constant C we have

|{x e Rd :\Tf(x)|> 1}|< CHf!pppm (16)

for all f e Lp(Rd). For t > 0, define the dilation operator 5t as follows: for any function 9: R2 x Rd—2 R, we let

5t9(Z,Z) = 9(Z, tZ); for any A c R2 x Rd—2, let 5tA = {(Z, tZ) : (Z, Z) e A}. By (16), the operator Tt = 5—1 o To 5t satisfies

|{x e Rd :\Ttf(x)\ > 1}| = t^^x e Rd :\To5tf(x)\ > 1}| < Ct^ff^ = CHf||ip,Rd). (17)

Now fix f e Lp(Rd) n L2(Rd). It is straightforward to check that the Fourier transform F satisfies the identity F = td—25t o F o 5t; since 2 (¿J, the operator Tt has the property

T\f(Z,Z) = — Z]+2+t2Z\2Z f(Z,Z), (Z,Z) e R2 X Rd—2,

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

llm TTf(Z, Z) = TTf(Z, Z)

in L2(Rd), where T0f(Z,Z) = —Z2f(Z, Z)/|Z|2. By Plancherel's theorem, the passage to a subsequence which converges almost everywhere and Fatou's lemma, we see that (17) implies

|{x e Rd : \T0f(x)| > ^|< CHfHPLl

Now pick an arbitrary function g: R2 — R and define f: R2 x Rd—2 -> R by f(£, Z) = g^^d—2(Z). Denoting by R2 the planar Riesz transform, we have T0f(%, Z) = R2g(i) 1 [0 i]d—2(Z), because of the identity

T0f (Z,Z) = — J, g(i)1[0,1]^ 2 (Z).

Plugging this into (18) gives

|{x e R2: |R2g(x)| > 1} | < C\|gHpp(R2)-Fix e> 0 and apply the above inequality to the function (1 + e) • g. We obtain

|{x e R2:|R2g(x)|> 1}| < |{x e R2 : |R^g(x)| > (1 + e)—1}| < C(1 + e)WgWfp(R2).

Since e was arbitrary, the reasoning from the previous subsection gives C > Cp. The proof is complete.

4.5. On the search of an appropriate laminate

Let us present here some heuristic arguments which lead to the laminate studied in subsection 4.3. It is strictly related to the extremal example in the martingale inequality (5). Suppose that d = 2 and let us look at the estimate

|{x e R2:|R2f(x)|> 1}|< Cp i ^(x)f dx

or, since R2 + R| = Id,

|{x e R2 : |R22f(x)| > 1}| < Cp f |R2f(x) + R22f(x)|pdx.

On the other hand, a slightly weaker form of inequality (5) can be rewritten as P(|G(| > 1) < C^E^f + Gf|p, or, more or less,

E\$1(diag(Ff, Gf)) — Cp\$z(diag(Ff, Gt)) < 0, (19)

where \$1,\$2 are as in subsection 4.3, and the martingale G is non-symmetrically differentially subordinate to F + G. Thus Corollary 4.6 suggests the following approach: find the extremal martingale pair (F, G) (for which the equality in (19) is attained, or almost attained);then the distribution of the random variable diag(Ff, Gf) is the desired laminate.

The sharpness of (19) can be obtained by the use of the following example (the argument in  is slightly different and exploits the properties of the underlying boundary value problem). Fix a small number 5 > 0 such that

5(1 + 5)n = p — 1— for some integer N (20)

2c + p — 1

(where c = c(p) is given by (2)). Consider the discrete-time Markov martingale (f, g) whose transition function is uniquely determined by the following conditions:

(i) (f, g) starts from (—5, 5);

(ii) for 5 < s < (p — 1)/(2c + p — 1), the state (—s.s) leads to (—s, (p — 3)s/(p — 1)) or to (—s.s + 5s);

(iii) for 5 < s < (p — 1)/(2c + p — 1), the state (—s.s + 5s) leads to (—s — 5s,s + 5s) or to ((p — 1)(s + 5s)/(3 —p),s + 5s);

(iv) the state (— (p — 1)/(2c + p — 1), (p — 1)/(2c + p — 1)) leads to (—(p — 1)/(2c + p — 1), 1) or to (— (p — 1)/(2c + p — 1), (p — 3)/(2c + p — 1));

(v) all the remaining states are absorbing.

The technical assumption (20) guarantees that the Markov process reaches the state studied in (iv). It is not difficult to check that if we let 5 —> 0, then the distribution of diag(fj,gj) is close to the laminate v exploited in subsection 4.2. This explains the use of this particular probability measure. See also  for a similar discussion.

Acknowledgements

The author would like to express his slncerest gratitude to anonymous referees for the careful reading of the paper and many helpful suggestions. The research was partially supported by Polish Ministry of Science and Higher Education (MNiSW) grant IP2011 039571 luventus Plus.

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