SUBORDINATION BY ORTHOGONAL MARTINGALES IN L p, 1 < p Introduction: Orthogonal martingales and the Beurling-Ahlfors transform

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1 SUBORDINATION BY ORTHOGONAL MARTINGALES IN L, 1 < PRABHU JANAKIRAMAN AND ALEXANDER VOLBERG 1. Introduction: Orthogonal martingales and the Beurling-Ahlfors transform We are given two martingales on the filtration of the two dimensional Brownian motion. One is subordinated to another. We want to give an estimate of L -norm of a subordinated one via the same norm of a dominating one. In this setting this was done by Burkholder in [Bu1] [Bu8]. If one of the martingales is orthogonal, the constant should dro. This was demonstrated in [BaJ1], when the orthogonality is attached to the subordinated martingale and when <. This note contains an (almost obvious) observation that the same idea can be used in the case when the orthogonality is attached to a dominating martingale and 1 <. Two other comlementary regimes are considered in [BJVLa]. When both martingales are orthogonal, see [BJVLe]. In these two aers the constants are shar. We are not sure of the sharness of the constant in the resent note. A comlex-valued martingale Y = Y 1 + iy is said to be orthogonal if the quadratic variations of the coordinate martingales are equal and their mutual covariation is 0: Y 1 = Y, Y 1, Y = 0. In [BaJ1], Bañuelos and Janakiraman make the observation that the martingale associated with the Beurling-Ahlfors transform is in fact an orthogonal martingale. They show that Burkholder s roof in [Bu3] naturally accommodates for this roerty and leads to an imrovement in the estimate of B. Theorem 1. (One-sided orthogonality as allowed in Burkholder s roof) ((i)) (Left-side orthogonality) Suose <. If Y is an orthogonal martingale and X is any martingale such that Y X, then Y X. (1.1) 1

2 PRABHU JANAKIRAMAN AND ALEXANDER VOLBERG ((ii)) (Right-side orthogonality) Suose 1 < <. If X is an orthogonal martingale and Y is any martingale such that Y X, then Y X. (1.) It is not known whether these estimates are the best ossible. Remark. [BaJ1]. The result for right-side orthogonality is stated in [JVV] and not in But [JVV] has a comlicated (though funny and interesting) roof by construction a family of new Bellman functions very different from the original Burkholder s function. The goal of this small note is to demonstrate how one adat the idea of [BaJ1] to the right-orthogonality and 1 < regime. We use just a well-known Burkholder s function here, exactly along the lines of [BaJ1]. If X and Y are the martingales associated with f and Bf resectively, then Y is orthogonal, Y 4 X and hence by (1), we obtain Bf ( ) f for. (1.3) By interolating this estimate ( ) with the known B = 1, Bañuelos and Janakiraman establish the resent best estimate in ublication: B 1.575( 1). (1.4). New Questions and Results Since B is associated with left-side orthogonality and since we know B = B, two imortant questions are ((i)) If <, what is the best constant C in the left-side orthogonality roblem: Y C X, where Y is orthogonal and Y X? ((ii)) Similarly, if 1 < <, what is the best constant C orthogonality roblem? in the left-side We have searated the two questions since Burkholder s roof (and his function) already gives a good answer when. It may be (although we have now some doubts about that) the best ossible as well. However no estimate (better than 1) follows from analyzing Burkholder s function when 1 < <. Perhas, we may hoe, C < when 1 < = 1 <, which would then imly a better estimate for B. This aer answers this hoe in the negative by finding C ; see Theorem. We also ask and answer the analogous question of right-side orthogonality when < <. In the sirit of Burkholder [Bu8], we believe these

3 ORTHOGONAL MARTINGALES 3 questions are of indeendent interest in martingale theory and may have deeer connections with other areas of mathematics. Remark. The following shar estimates are roved in [BJVLa], they cover the left-side orthogonality for the regime 1 < and the right-side orthogonality for the regime <. Notice that two comlementary regimes have the estimates: for < and left-side orthogonality in [BaJ1], for 1 < in this note and in [JVV], but the sharness is dubious. Theorem. Let Y = (Y 1, Y ) be an orthogonal martingale and X = (X 1, X ) be an arbitrary martingale. ((i)) Let 1 <. Suose Y X. Then the least constant that always works in the inequality Y C X is C = 1 z 1 z (.1) where z is the least ositive root in (0, 1) of the bounded Laguerre function L. ((ii)) Let <. Suose X Y. Then the least constant that always works in the inequality X C Y is C = 1 z z (.) where z is the least ositive root in (0, 1) of the bounded Laguerre function L. The Laguerre function L solves the ODE sl (s) + (1 s)l (s) + L (s) = 0. These functions are discussed further and their roerties deduced in section (??); see also [?], [?], [?]. As mentioned earlier, (based however on numerical evidence) we believe in general < C < 1 and that these theorems cannot imly better estimates for B. However based again on numerical evidence, the following conjecture is made.

4 4 PRABHU JANAKIRAMAN AND ALEXANDER VOLBERG Conjecture. For 1 < = 1 <, C = C, or equivalently, 1 z 1 z = 1 z z. It is conjecture relating the roots of the Laguerre functions. Notice that such a statement is not true with the constants from Theorem 1, and < for all >. So this conjecture (if true) suggests some distinct imlications for the two settings. Note on the other hand, that the form of the two sets of constants are very analogous. 3. Right-side orthogonality, 1 < regime, Burkholder s function We just reeat the aroach of [BaJ1]. Let ( α := 1 1 ) 1, 1 <. For x R, y R we define following Burkholder: We consider Burkholder s function Then (1 < ) v(x, y) := y ( 1) x. u(x, y) := α ( y ( 1) x )( x + y ) 1. ( 1)u(x, y) = α ( x ( 1) y )( x + y ) 1. So if we denote G(t) := u(x + ht, y + kt) we have where And C 0. G (0) = α (A + B + C), A = ( 1)( h k )( x + y ) 1, B = ( )( h ( x x, h) ) x 1 ( x + y ) 1. Also ( 1)u(x, y) 0 if y x. Now let temorarily X t = (X 1 t, X t ), Y t = (Y 1 t, Y t ) denote two R valued martingales on the filtration of Brownian motion, and let d X 1, X = h 1 1h 1 + h 1 h = 0. (3.1) d X 1, X 1 = (h 1 1) + (h 1 ) = d X, X = (h 1) + (h ). (3.)

5 ORTHOGONAL MARTINGALES 5 And let us have the following subordination by the orthogonal martingale assumtion: or d Y, Y (k 1 1) + (k 1 ) + (k 1) + (k ) We write Itô s formula for E u(x t, Y t ): where (see above) E u(x t, Y t ) = E u(x 0, Y 0 ) α E d X, X, (3.3) ( 1) ( 1) ((h1 1) + (h 1 ) + (h 1) + (h ) ), (3.4) t 0 (A(t) + B(t) + C(t)) dt, A(t) = ( 1)(d X, X t d Y, Y t )( X t + Y t ) 1, B = ( )(d X, X t [( X t X t, H 1 ) ) + ( X t X t, H ) )] X t 1 ( X t + Y t ) 1. And C(t) 0. Here we denote H 1 = (h 1 1, h 1), H = (h 1, h ), or we can say that H 1 is a vector of x stochastic derivatives of vector rocess X and H is a vector of y stochastic derivatives of vector rocess X. By (3.1) and (3.) we get that the exression in [ ] is [( X t X t, H 1 ) ) + ( X t X t, H ) )] = 1 d X, X. Hence, if Y 0 X 0 we get (as X t 1 ( X t + Y t ) 1 ( X t + Y t ) ) or E u(x t, Y t ) α E t E u(x t, Y t ) α E 0 {( 1)(d X, X d Y, Y + t 0 d X, X )} dt, ( 1) ( ) {( 1) d X, X d Y, Y } dt 0, ( 1) because the integrand is ositive: see the assumtion of subordination (3.3). Therefore, using Burkholdr s discovery that v(x, y) = y ( 1) x u(x, y) we get E ( Y t ( 1) X t E u(x t, Y t ) 0,

6 6 PRABHU JANAKIRAMAN AND ALEXANDER VOLBERG and we obtain Y ( 1) X. Consider X := ( 1) X. Then (3.1) means ortogonality d X 1, X = 0. Assumtion (3.) means d X 1, X 1 = d X, X, and (3.3) means d Y, Y d X, X. Changing X to X we see that we roved Theorem 3. Let 1 <, let X t, Y t be two martingales on the filtration of dimensional Brownian motion. d X 1, X = 0 and d X 1, X 1 = d X, X. X: Then Let X be an orthogonal martingale, namely d Y, Y d X, X. Y X. References Suose that Y is subordinated to [BaJ1] [Bu1] [Bu] [Bu3] [Bu4] [Bu5] [Bu6] [Bu7] [Bu8] [BJVLa] R. Banuelos, P. Janakiraman, L bounds for the Beurling Ahlfors transform. Trans. Amer. Math. Soc. 360 (008), no. 7, D. Burkholder, Boundary value roblems and shar estimates for the martingale transforms, Ann. of Prob. 1 (1984), D. Burkholder, An extension of classical martingale inequality, Probability Theory and Harmonic Analysis, ed. by J.-A. Chao and W. A. Woyczynski, Marcel Dekker, D. Burkholder, Shar inequalities for martingales and stochastic integrals, Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987), Astérisque No (1988), D. Burkholder, Differential subordination of harmonic functions and martingales, (El Escorial 1987) Lecture Notes in Math., 1384 (1989), 1 3. D. Burkholder, Exlorations of martingale theory and its alications, Lecture Notes in Math (1991), D. Burkholder, Strong differential subordination and stochastic integration, Ann. of Prob. (1994), D. Burkholder, A roof of the Peczynski s conjecture for the Haar system, Studia Math., 91 (1988), D. Burkholder, Martingales and Singular Integrals in Banach Saces, Handbook of the Geometry of Banach Saces, Vol. 1, Ch. 6., (001), A. Borichev, P. Janakiraman, A. Volberg, Subordination by orthogonal martingales in L and zeros of Laguerre olynomials Prerint, 009,. 1 7, sashavolberg.wordress.com

7 References 7 [BJVLe] [JVV] A. Borichev, P. Janakiraman, A. Volberg, On Burkholder function for orthogonal martingales and zeros of Legendre olynomials, arxiv: , Prerint, 009,. 1 36, sashavolberg.wordress.com P. Janakiraman, V. Vasyunin, A. Volberg, Some new Bellman functions and subordination by orthogonal martingales in L, 1 <, Prerint, 009,. 1 30, sashavolberg.wordress.com Prabhu Janakiraman, Det. Math., Purdue Univ., janaki@math.urdue.edu Alexander Volberg, Det. of Math., Mich. State Univ., volberg@math.msu.edu