Sharp L p bounds for a perturbation of Burkholder s Martingale Transform

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1 Sharp L p bounds for a perturbation of Burkholder s Martingale Transform Nicholas Boros a, Prabhu Janakiraman b Alexander Volberg c a Department of Mathematics, Michigan State University, East Lansing, MI 4884, USA b Department of Mathematics, Michigan State University, East Lansing, MI 4884, USA c Department of Mathematics, Michigan State University, East Lansing, MI 4884, USA Received *****; accepted after revision Presented by Gilles Pisier Abstract Let {d k } k 1 be a real martingale difference in L p [0, 1), where 1 < p <, and {ε k } k 1 {±1}. We obtain the following generalization of Burkholder s famous result. If τ [ 1, 1 ] and n Z+ then n k=1 (ε kd k, τd k ) L p ([0,1),R ) ((p 1) + τ ) 1 n k=1 d k, where L p ([0,1),R) ((p 1) + τ ) 1 is sharp and p 1 1 = max{p 1, }. To cite p 1 this article: A. Nicholas Boros, A. Prabhu Janakiraman, Alexander Volberg, C. R. Acad. Sci. Paris, Ser. I. Résumé Estimations optimales dans L p pour des transformations de martingale perturbées. Le calcul de la norme des opérateurs du type intégrales singulières est connu seulement dans très peu de cas. Le cas le plus célèbre est celui de la transformation de martingale dans L p trouvée par Burkholder égale à p 1. Outre des résultats de Pichorides [6], on peut signaler un résultat de Choi [3] et un calcul récent de R1 R p par Nazarov Volberg [5], Banuelos Janakiraman [1], et par Geiss Montgomery-Smith Saksman [4]. Les transformées de Riesz sur R sont notées R 1 et R. La note est consacrée à un résultat qui donne la norme d une certaine perturbation de R1 R. Pour citer cet article : Nicholas Boros, A. Prabhu Janakiraman, Alexander Volberg, C. R. Acad. Sci. Paris, Ser. I. Version française abrégée Le calcul de la norme des opérateurs du type intégrales singulières est connu seulement dans très peu de cas. Le cas le plus célèbre est celui de la transformation de martingale dans L p trouvée par Burkholder addresses: borosnic@msu.edu (Nicholas Boros), pjanakir1978@gmail.com (Prabhu Janakiraman), volberg@math.msu.edu (Alexander Volberg). Preprint submitted to the Académie des sciences 5 mars 011

2 égale à p 1. Ici on obtient une formule pour la norme d une certaine perturbation de la transformation de martingale de Burkholder. Comme conséquence, on peut obtenir une formule précise pour la norme d une petite perturbation de l opérateur R 1 R qui joue un rôle important parce qu il est la partie réelle de la transformation de Ahlfors Beurling. Voici notre estimation d une perturbation de la transformation de martingale : n k=1 (ε kd k, τd k ) L p ([0,1),R ) ((p 1) +τ ) 1 n k=1 d k Lp ([0,1),R), où ((p 1) +τ ) 1 est optimale et p 1 1 = max{p 1, p 1 }. Cela est vrai pour tous les petits τ s. Et voici notre estimation de la perturbation de la partie réelle de l opérateur de Ahlfors Beurling (ici encore pour tous les petits τ s). : ( R 1 R, τ I ) Lp(C,R) Lp(C,R) = ((p 1) + τ ) 1. Notre résultat est valable pour les fonctions réelles. Pour des fonctions complexes, nous pouvons le démontrer pour 1 < p et pour tous les p pairs. Comme la preuve utilise la théorie des intégrales stochastiques on rencontre dans le cas général les mêmes difficultés relatives aux martingales complexes que l on rencontre dans des tentatives pour résoudre un célèbre problème p 1 de Iwaniec. Néanmoins, nous pensons qu ici on pourra géneraliser ce résultat au cas des fonctions complexes. Notre approche est basée sur l utilisation de l équation de Monge Ampère. Plus précisement, on commence par la recherche d une fonction M(y 1, y, y 3 ) définie dans Ω := {(y 1, y, y 3 ) : y 3 y 1 y p } et telle que M est bi-concave par rapport à (y 1, y 3 ) et (y, y 3 ) et telle que sur Ω on a M = ((y + y 1 ) + τ (y 1 y ) ) p/. L équation de Monge Ampère apparait parce que maintenant nous voulons trouver M satisfaisant cette équation dans les variables y, y 3 pour chaque y 1 fixé dans l intersection de Ω avec y 1 y > 0, y 1 + y > Introduction Let I be an interval and α ± R + such that α + + α = 1. These α ± generate two subintervals I ± such that I ± = α ± I and I = I I +. We can continue this decomposition indefinitely as follows. For any sequence {α n,m : 0 < α n,m < 1, 0 m < n, 0 < n <, α n,k + α n,k+1 = 1}, we can generate the sequence I := {I n,m : 0 m < n, 0 < n < }, where I n,m = In,m I n,m + = I n+1,m+1 I n+1,m+1 and α = α n+1,m, α + = α n+1,m+1. Note that I 0,0 = I. For any J I we define the Haar function α h J := + α J χ J + α α + J χ J +. Burkholder defined a martingale transform [], MT ε, as MT ε ( n k=1 d k) := n k=1 ε kd k. Using this definition of the martingale transform, we can state one of the main results as ( ( n ) ( n MT ε d k, τ d k) ((p k=1 k=1 ) L 1) + τ ) 1 n d k, (1) p ([0,1),R ) k=1 L p ([0,1),R) where τ [ 1, 1 ]. We will define the martingale transform slightly differently. Burkholder in [] showed that the norm in L p has to be the same for this more general definition. Fix I = [0, 1) and I = D as the dyadic subintervals of I. Denote by f J the average 1 J J f, J D. Then f Lp [0, 1) can be decomposed in terms of the Haar system as f = f [0,1) χ [0,1) + I D (f, h I)h I. We define the martingale transform, g of f, as g := g [0,1) χ [0,1) + I D ε I (f, h I )h I,

3 where ε I {±1}. Requiring that (g, h J ) = (f, h J ) for all J D is equivalent to g being the martingale transform of f. Now we define the Bellman function as B(x 1, x, x 3 ) := sup{ (g + τ f ) p I : x 1 = f I, x = g I, x 3 = f p I, (f, h J ) = (g, h J ), J D} f,g on the domain Ω = {x R 3 : x 3 0, x 1 p x 3 }. Note that the Bellman function is a sort of the supremum of the left side of the inequality that we would like to prove (with MT ε ( n k=1 d k) replaced with g and n k=1 d k replaced with f) with all other quantities fixed and x 1 p x 3 in the domain is just Hölder s inequality. Our goal is to find B. It can be found in [] that proving (1) is equivalent to showing that B(0, 0, x 3 ) ((p 1) + τ ) p x 3. For p =, B(x) = x x 1 + (1 + τ )x 3 is a trivial calculation. However, for p it is much more difficult to find B, so we need some properties. We start with simple properties that followed from Minkowski s inequality combined with famous Burkholder s result [], where he proved (1) for τ = 0: 1) If p <, B(0, 0, x 3 ) ((p 1) + τ ) p x 3 ; if 1 < p, then B(0, 0, x 3 ) ((p 1) + τ ) p x 3. These inequalities are valid for any τ. However, the converse inequalities seem to be far from being simple. We can prove them only for relatively small τ (see the explanation in Section 5), and we know that the best constant ((p 1) + τ ) 1 in (1) changes its form if τ becomes large ( phase transition ) in the case 1 < p <.. Bellman function properties Since B is independent of the initial choice of I (of finite length) and {α m,n } m,n, we will return them to being arbitrary. Lemma.1 (Weak Concavity) Suppose x ± Ω such that x = α + x + +α x, α + +α = 1. If x + 1 x 1 = x + x then B(x) α+ B(x + ) + α B(x ). Observe that x + 1 x 1 = x+ x is just the condition (f, h J) = (g, h J ). This condition can be made more manageable but making the following change of coordinates: y 1 := x+x1, y := x x1 and y 3 := x 3. Also, let M(y 1, y, y 3 ) := B(x 1, x, x 3 ) = B(y 1 y, y 1 +y, y 3 ) and Ξ := {y R 3 : y 3 0, y 1 y p y 3 } denote the Bellman function and domain in the y-variable. Note that x + 1 x 1 = x+ x is equivalent to either y 1 or y being fixed. Therefore an equivalent way to check weak concavity is the following. Lemma. Fix j i {1, } and y i as y + i = y i. Then M y jy j 0, 0 and D j = M yjy j (M yjy 3 ) 0 is equivalent to M being a concave function of y j, y 3. The remaining Bellman function properties can now be easily derived. Proposition.1 Suppose M is C 1 (R 3 ), then M has the following properties: (i) Symmetry: M(y 1, y, y 3 ) = M(y, y 1, y 3 ) = M( y 1, y, y 3 ); (ii) Dirichlet boundary data: M(y 1, y, (y 1 y ) p ) = ((y 1 +y ) +τ (y 1 y ) ) p ; (iii) Neumann conditions: M y1 = M y on y 1 = y and M y1 = M y on y 1 = y ; (iv) Homogeneity: M(ry 1, ry, r p y 3 ) = r p M(y 1, y, y 3 ), r > 0; (v) Homogeneity relation: y 1 M y1 + y M y + py 3 M y3 = pm. An added assumption simplifies the problem Lemma. introduces a partial differential inequality into our problem: M yjy j (M yjy 3 ) 0. Rather than dealing with this difficulty, we add an assumption. Assumption 1 Fix i j {1, } and y i. Then the Hessian matrix {M yk y l } k,l=j,3 is degenerate. The degeneracy of the above matrix gives us the well known Monge Ampère Equation equation, M yjy j (M yjy 3 ) = 0, which has a solution. Theorem.3 (Pogorelov 1956)For j = 1 or, M yjy j (M y3y j ) = 0 has the solution M(y) = y j t j + y 3 t 3 + t 0 on the characteristics y j dt j + y 3 dt 3 + dt 0 = 0, which are straight lines in the y j y 3 plane. 3

4 Furthermore, t 0, t j, t 3 are constant on characteristics with the property M yj = t j, M y3 = t Bellman function candidate The characteristics for Monge Ampère solution, in Theorem.3 can only behave one of four possible ways in the y j y 3 plane. Using the Bellman function properties shown in Proposition.1 and checking that the weak concavity in Lemma. is satisfied we can deduce the following Bellman function candidate. Refer to [7] and [8] for more details of this method. ( ) 1 Proposition 3.1 Denote G(z 1, z ) = (z 1 + z ) p 1 [z 1 (p 1)z ], ω = M(y) p y 3 and γ = 1 τ 1+τ. Then, for τ 1, the Monge Ampère solution that satisfies all Bellman function properties is the following. ( ) p (A) For p (1, ], M(y) = (1+τ ) p [y1+γy 1 y +y] p 1 + (p 1) + τ [y 3 (y 1 y ) p ], y [ p p y 1, y 1 ) and M is given by implicit equation G(y 1 y, y 1 + y ) = y 3 G(1, ω τ ), y ( y 1, p p y 1]. (B) For p [, ), M(y) = (1 + τ ) p [y1 + γy 1 y + y] p + ((p 1) + τ ) p [y 3 (y 1 y ) p ], y ( y 1, p p y 1] and M is given implicitly by G(y 1 + y, y 1 y ) = y 3 G( ω τ, 1), y [ p p y 1, y 1 ). Remarks. 1) For p (1, ) one gets easily M( x1+x, x x1, x 3 ) B(x 1, x, x 3 ); ) If p [, ) one gets easily B(x 1, x, x 3 ) M( x1+x, x x1, x 3 ). Both relationships are easy consequences of Minkowski s inequality. Both of them are true for any τ. We will see that for relatively small τ the converse inequalities hold (and so we have equalities). This breaks down for large τ. 3) Equality B = M( x1+x, x x1, x 3 ) obtained in the next section obviously implies the calculation of the best constant in B(0, 0, x 3 ) c(τ, p)x 3, it turns out that c(τ, p) = ((p 1) + τ ) p, which gives (1). 4. Our Bellman function candidate is actually the Bellman function Since B, B, M, M actually depend on τ, we will use B τ, B τ, M τ, M τ. We will now revert to x rather than y for the variable, Ω rather than Ξ for the domain and B rather than M for the the Bellman function. In Proposition 3.1 we obtained a Bellman function candidate that we think is the Bellman function. Let B(x 1, x, x 3 ) = M( x1+x, x x1, x 3 ) denote the Bellman function candidate from Proposition 3.1. We must now show that B = B. This is the price that we must pay for trading the partial differential inequality for the Monge Ampère equation. Proposition 4.1 If 1 < p and τ 1/ then B τ is bi-concave in Ω, that is B τ is concave in (x 1 + x, x 3 ) and in (x x 1, x 3 ). Proposition 4. If some function B is bi-concave with B(x, y, x p )) (τ x + y ) p, then B B τ. In particular, B τ B τ, τ 1/, 1 < p. Also B τ B τ for all p [, ) just by Minkowski s inequality. This follows in exactly the same way as in [8]. Notice that for p < the inequality B τ B τ is true for any τ (this can be proved just by Minkowski s inequality). Proposition 4.3 For 1 < p <, B τ B τ. We already mentioned that for 1 < p the use of Minkowski s inequality will show this result for any τ. For p <, the use of the same extremal function sequence as in [7] will give the result. 4

5 5. Main result V p,τ (x, y) := (τ x + y ) p ((p 1) + τ ) p x p : y (p 1) x Let U p,τ (x, y) = p (1 1p ) p 1 ( τ ) p 1 + ( x + y ) p 1 (p 1) [ y (p 1) x ] : y (p 1) x and < p <. Then the Bellman function in Proposition 3.1 can be more compactly written as, the unique positive solution to ( ) U p,τ (x 1, x ) = U p,τ x 1 p 3, B p τ τ x p 3. () One can show that U p,τ is the least biconcave majorant of V p,τ. Similarly for 1 < p, U p,τ (with the two pieces of the above function interchanged) is the least biconcave majorant of V p,τ and the Bellman function satisfies the relation in (). This allows us to prove the next theorem. But first let us explain the role of smallness of τ. Comparing B τ with Burkholder s function B = B 0 we see that Bτ /p = B /p + τ x /p 3. We wish B τ to be bi-concave in (x 1 + x, x 3 ) and (x x 1, x 3 ), B has these properties. Let us think that /p > 1, take the second derivative of B τ in x 3. When τ is small, it is easy to see that this second derivative is still negative (as this is so for B). But if τ is not small the convexity of τ x /p 3 may (and will) start to win over the concavity of B. Theorem 5.1 Let τ [ 1, 1 ], I = [0, 1], f I = x 1, g I = x, g be the martingale transform of f, in terms of the Haar expansion, and x x 1. Then (τ f + g ) p I ((p 1) + τ ) p f p I with sharp constant, where p 1 1 = max{p 1, p 1 }. For p [, ) the estimate from above is true for any τ. For p (1, ) the estimate from above is no longer true for large τ. For p (1, ) the sharpness of the constant (the estimate from below) is true for any τ. We have the application for the Riesz transforms. The lower bound is a generalization of the lower bound technique in [4] to vector-valued multipliers. The upper bound is a generalization of the result in [1] to the vector setting. Theorem 5. Let R i, i = 1, be the Riesz transform on the plane and p (1, ). Then for τ 1/ we can calculate the norm in L p (C): ( R1 R, τi = ((p 1) + τ ) 1, ) Lp(C,R) Lp(C,R) where the estimate from above for the norm is true even for all τ if p [, ). Remark. The estimate in this theorem is on real valued functions. We expect that the same is true for complex valued functions. References [1] R. Bauelos, P. Janakiraman, L p -bounds for the Beurling-Ahlfors transform, Trans. Amer. Math. Soc. 360 (008), [] D. Burkholder, Boundary value problems and sharp estimates for the martingale transforms, Ann. of Prob. 1 (1984), [3] K.P. Choi, Some sharp inequalities for martingale transforms, Trans. Amer. Math. Soc. 307 (1988), [4] S. Geiss, S. Montgomery-Smith, E. Saksman, On singular integral and martingale transforms, Trans. Amer. Math. Soc. 36 (010),

6 [5] F. Nazarov, A. Volberg, Heating the Beurling operator and estimates of its norms, St. Petrsburg Math. J. 14 (003) no. 3 [6] S. Pichorides, On the best value of the constants in the theorems of Riesz, Zygmund, and Kolmogorov, Studia Mathematica 44 (197), [7] V. Vasyunin, A. Volberg, The Bellman function technique in Harmonic Analysis, sashavolberg.wordpress.com [8] V. Vasyunin, A. Volberg, Burkholder s function via Monge Ampère equation, arxiv: v1,