Weighted maximal inequality for differentially subordinate martingales *

We establish a weighted maximal L 1 -inequality for differentially subordinate martingales taking values in R ν , ν ≥ 1 , under the assumption that the weight satisﬁes Muckenhoupt’s condition A 1 . An optimal dependence of the constant on the A 1 char- acteristics is identiﬁed.

H s dX s , t ≥ 0.
If H takes values in [−1, 1], then Y is differentially subordinate to X; this follows at once from the identity areas of mathematics. See e.g. the monograph [13] by the author and the papers [1], [2], [3], [6], [16], [17] for an overview of the results in this direction. In this paper, our particular emphasis will be put on maximal inequalities. In [4], Burkholder introduced a general method of proving such estimates in the context of stochastic integrals and exploited it to establish the following result.
The constant is the best possible.
If the martingales X and Y are assumed to have continuous trajectories, the constant changes. Here is the result of [12], under the less restrictive assumption of differential subordination and in the wider context of vector-valued processes.
See also [10], [11] and [15] for related results and extensions. We will be interested in the following weighted version of (1.1): ||Y || L 1 (W ) ≤ C||X * || L 1 (W ) . (1.2) Here W is a weight, i.e., a uniformly integrable, positive, mean-one and continuous-path martingale W = (W t ) t≥0 , and we have used the standard notation ||Y || L 1 (W ) = sup t≥0 E|Y t |W ∞ and ||X * || L 1 (W ) = EX * W ∞ for the weighted L 1 norms of Y and X * . It is not difficult to see that the above bound cannot hold with some finite C for all processes W . A natural assumption on the weight (in the context of the above L 1 estimate) is that it belongs to the class A 1 . This class was originally introduced by Muckenhoupt [9] in the analytic setting, and its probabilistic counterpart is due to Izumisawa and Kazamaki. Following [7] and [8], W satisfies the A 1 condition if there is a finite constant c such that P(W * t ≤ cW t for all t ≥ 0) = 1. The least c with this property is denoted by [W ] A1 and called the A 1 characteristics of W .
We will show that if W belongs to the class A 1 , then (1.2) holds for all martingales X, Y satisfying the differential subordination. Actually, we will additionally study the following aspect of the weighted bound. Namely, there is a very interesting question of extracting the sharp dependence of the constant C on the characteristics [W ] A1 . More precisely: what is the least exponent κ for which there exists an absolute constantC such that for all W and all X, Y satisfying the differential subordination?
The main result of this paper gives a full answer to this question.
where C = 5 + 2 ln(3/2) = 5.81093 . . .. The dependence on the A 1 characteristics of the weight is optimal in the sense that for any κ < 1 and any K > 0, there is a weight W , a real-valued martingale X and a predictable sequence H with values in {−1, 1} such that the stochastic integral Y = H · X satisfies We should emphasize here that the constant C we obtain above is not sharp, however, we believe that it is not far from the optimal one. There is a well-known method of proving maximal inequalities for stochastic integrals and differentially subordinate martingales. This method, invented by Burkholder in [4] and modified by the author in [12,13], allows to deduce a given estimate from the existence of a certain special function, enjoying appropriate majorization and concavity. However, we should stress here that all the works in which the method has been successfully implemented, concerned the unweighted setting. To the best of our knowledge, this paper contains the first example in which Burkholder's method has been successfully applied to yield a nontrivial weighted maximal bound for differentially subordinate martingales.
The inequality (1.3) is proved in the next section. The optimality of the exponent 1 is studied in Section 3. In the final part of the paper we sketch some ideas leading to the special function U on which the proof of (1.3) rests.

Proof of the maximal inequality
Let c ≥ 1 be a fixed parameter and consider the set As we have mentioned in the introduction, a crucial role in the proof of (1.3) is played by a special function. Let U = U (c) : D → R be given by where γ = 2 + ln(3/2). Let us study some crucial properties of the special function.
Recall that C is the constant appearing in (1.3).
Proof. The proof of (2.2) is very simple: since ln(1 + s) ≤ s for any positive s, we may so the preceding estimate gives Therefore, the majorization (2.3) will be established if we manage to show that But observe that The inequality (2.4) is evident once one computes the partial derivative with respect to z: To show (2.5), we derive that and hence we will be done if we show that the expression in the square brackets is nonpositive. This is elementary: substitute a = c −1 ∈ [0, 1] and consider the function It suffices to note that ξ(0) = 0 and provided a ∈ (0, 1). This yields (2.5). Finally, we turn our attention to the property (v). We easily check that M(x, y, z, w, v) equals where Id denotes the identity matrix of dimension ν × ν. Since ln( Sylvester's criterion. Most of the entries of the matrix are zero, and it is not difficult to compute the determinant. If we substitute A = (2v(ln( and use cofactor expansion along the last row, we see that Thus, we must prove that the expression in the square brackets is nonnegative. But this is immediate when one recalls the definition of γ and notes that by (2.6), ln( The proof is complete. Proof of (1.3). Fix an arbitrary ε > 0 and pick martingales X, Y , W as in the statement. Let c = [W ] A1 . We will apply Itô's formula to the composition of U (c) and the process P t = (X t , Y t , X * t ∨ ε, W t , W * t ), t ≥ 0. For any t ≥ 0 we have X * t ∨ ε > 0 and W * t ≤ cW t , by the very definition of [W ] A1 . Hence the process P takes values in the domain of U and the composition U (P ) makes sense. Furthermore, U has the necessary regularity: actually, the formula (2.1) can be used for all (x, y, z, w, v) ∈ R ν ×R ν ×(0, ∞) 3 and defines a C ∞ function there, so the use of Itô's formula is permitted. As the result, we obtain U (P t ) = I 0 + I 1 + I 2 + I 3 /2, where I 0 = U (P 0 ), Here t 0 U xx (P s )d[X] s is the shortened notation for Note that the remaining second-order terms are equal to 0, either due to vanishing of the corresponding partial derivatives, or to the fact that the processes X * ∨ ε, W * are nondecreasing (and hence of finite variation). Let us analyze the terms I 0 through I 3 separately. First, observe that due to (2.2). The term I 1 is a local martingale, by the properties of stochastic integrals. To handle I 2 , note that by the continuity of paths, the times at which the process X * ∨ ε increases are contained in the set {s : |X s | = X * s }; however, for such s we have U z (P s ) ≤ 0, by virtue of (2.4), so the first integral in I 2 is nonpositive. An analogous reasoning exploiting (2.5) shows that the second integral also has this property and hence I 2 ≤ 0. To deal with I 3 , observe that U xx = [U xixj ] 1≤i,j≤ν is a negative multiple of the identity matrix and hence, by the differential subordination of Y to X, which implies I 3 ≤ 0. Putting all the above facts together, if (τ n ) n≥1 denotes the localizing sequence for the local martingale I 1 , then EU (P τn∧t ) ≤ 0, n = 1, 2, . . . .

By (2.3), this yields E|Y
Letting n → ∞ and applying Fatou's lemma, we get E|Y t |W ∞ ≤ C[W ] A1 EX * W ∞ . Since t was arbitrary, the claim follows.

On the optimality of the exponent
Let c > 1 be a fixed parameter, take a large positive integer N and set δ = c/N . Let B be a standard, one-dimensional Brownian motion starting from 1. Consider the family (τ n ) N n=0 of stopping times given by τ 0 ≡ 0 and τ n = inf{t : B t ≤ c −1 (1 + δ) n−1 or B t = (1 + δ) n }, n = 1, 2, . . . , N.

On the search of a suitable function
The purpose of this section is to present some informal reasoning which has led us to the discovery of the special function U = U (c) satisfying the properties listed in Lemma 2.1. We would like to stress here that we have not tried to optimize the choice of various parameters that will arise below (which might lead to a slight improvement of the constant C in the statement of our main theorem). Instead, we rather focused on obtaining a relatively simple formula, for which the calculations will be not too x = z → 0 and y = 1 in the majorization). Thus in particular we see that ϕ(1) > 0, which gives a + b > 1, and ϕ(c −1 ) > 0, which is equivalent to Next, (4.1) yields which combined with a + b > 1 implies that a is positive. Let us further exploit the two inequalities above: they yield some crucial information on a and b. First, (4.4) implies that b < 1: indeed, for b = 1 the inequality does not hold, and the left-hand side is an increasing function of b. Thus, we have b = 1 − ε for some ε = ε(c) > 0; then (4.3) gives a ≥ cε, and since the left-hand side of (4.4) is an increasing function of a, we get that Let us try to extract some information on the size of ε from this estimate. We must have ε = o(c −1 ) as c → ∞, since otherwise letting c → ∞ above gives a contradiction. Now, transform the latter bound into the equivalent form Letting c → ∞, we see that the left-hand side converges to −1/2 and therefore we have lim sup c→∞ (c − 1) 2 ε ≤ 2. This suggests to take ε = c −2 and, in the light of (4.3), a = αc −1 for some α = α(c) ≥ 1. We assume that α is a constant function and come back to (4.4), obtaining that α must satisfy log(αc −1 + 1 − c −2 ) − αc −1 αc −1 + 1 − c −2 ≤ 0 for c ≥ 1, or, if we substitute a = c −1 ∈ [0, 1], then ξ(a) := log(αa + 1 − a 2 ) − αa αa + 1 − a 2 ≤ 0 (a similar function, also denoted by ξ, has already appeared in Section 2). Since ξ(0) = 0, we see that the above inequality enforces ξ (a) ≤ 0 for a sufficiently close to 0. However, since ξ (a) = a(α 2 − 2 − 4αa + 2a 2 ) (αa + 1 − a 2 ) 2 , the latter requirement will hold if α ≤ √ 2. This leads us to the choice α = √ 2. Finally, it remains to choose γ. The second inequality in (4.2) can be transformed into γ − log(as + b) ≥ 2, and this requirement is most restrictive when s = 1: γ ≥ 2 + log( √ 2c −1 + 1 − c −2 ). One easily checks that the function c → √ 2c −1 + 1 − c −2 , c ∈ [1, ∞), attains its maximal value 3/2 for c = √ 2. This leads to our final choice γ = 2 + log(3/2), which produces the function U used in Section 2.