Sharp and Strict $L^p$-Inequalities for Hilbert-Space-Valued Orthogonal Martingales

The paper contains the proofs of sharp moment estimates for Hilbert-space valued martingales under the assumptions of differential subordination and orthogonality. The results generalize those obtained by Banuelos and Wang. As an application, we sharpen an inequality for stochastic integrals with respect to Brownian motion.


Introduction
and the constant is the best possible. It is already the best possible if Y is assumed to be real valued.
The motivation of the present paper comes from the question about the optimal constants in the estimates above in the case when there are no restrictions for the ranges of X and Y . Before we formulate the precise statement, let us first mention here a seemingly unrelated result, due to Davis [9]. Let w p be the smallest positive zero of the confluent hypergeometric function of parameter p and let z p be the largest positive zero of the parabolic cylinder function of parameter p (for the definitions of these objects, see [1] or Section 3 below). (i) For 1 < p < ∞ and ||τ 1/2 || p < ∞, where a p = z −1 p for 1 < p ≤ 2 and a p = w −1 p for 2 ≤ p < ∞.
where A p = w p for 0 < p ≤ 2 and A p = z p for 2 ≤ p < ∞.
The main result of the present paper can be stated as follows.
Theorem 1. 4. Let X , Y be two orthogonal martingales taking values in such that Y is differentially subordinate to X . Then ||Y || p ≤ C p ||X || p , 1 < p < ∞, (1. 4) where The constant C p is the best possible. Furthermore, the inequality is strict if p = 2 and 0 < ||X || p < ∞.
Therefore we see that the constants C p have quite surprising behavior. For 1 < p ≤ 3 they are the same as those in appropriate Davis' estimates (a p for 1 < p ≤ 2 and A p for 2 ≤ p ≤ 3), while for p ≥ 3 they are equal to Pichorides-Cole constants. In other words, comparing the above with Theorem 1.2, we see that the passage from real to -valued martingales affects the optimal constants if and only if 1 < p < 3.
A few words about the proof and the organization of the paper. Our argumentation is based on Burkholder's technique, which reduces the problem of proving a given martingale inequality into that of finding an appropriate special function. This transference is described in the next section.
The special function is constructed by means of confluent hypergeometric and parabolic cylinder functions, which are introduced and studied in Section 3. Then, in Section 4, we present the proof of our main result, Theorem 1. 4. The final section of the paper is devoted to some applications to stochastic integrals.

On the method of proof
We start with describing the main tool which will be exploited to establish our result. We shall use the following notation: if U : × → R and x, y, h ∈ , then provided the partial derivatives exist. The term 〈hU y y (x, y), h〉 is defined in a similar manner.
Assume further that there exists a nondecreasing sequence (M n ) n≥1 such that where the supremum is taken over all (x, y) ∈ S i such that 1/n 2 ≤ |x| 2 + | y| 2 ≤ n 2 and all i > 1. Let X and Y be -valued orthogonal martingales such that Y is differentially subordinate to X . If sup s |U(X s , Y s )| is integrable, then for any 0 ≤ s ≤ t we have This is a slight modification of Proposition 1 from [4]. Essentially, the difference is that the inequality (2.1) is replaced there by a more restrictive condition.
The proof of Theorem 2.1 is based on approximation and Itô's formula. To be more specific, one reduces the problem to the finite-dimensional case = R n and convolves U with a C ∞ function to get a smooth U on R n × R n . Then one applies Itô's formula to U(X t , Y t ), takes conditional expectation of both sides and uses the conditions (2.1) and (2.2) to control the finite-variation terms. Since similar argumentation appears in so many places (see e.g. Proposition 1 in [4], Lemma 1.1 in [3], Theorem 2.1 in [13], Lemma 3 in [16] . . .), we have decided not to include the details here.
Fix p ∈ (1, ∞) and let us now sketch the proof of (1.4). Obviously, we may and do assume that ||X || p < ∞; otherwise there is nothing to prove. By Burkholder's inequality (1.1), we obtain that ||Y || p is also finite. In consequence, all we need is to show that Here the inequality (2.4) comes into play: if we manage to find a function U p as in Theorem 2.1, satisfying the majorization U p (x, y) ≥ | y| p − C p p |x| p for all x, y ∈ , (2.5) and the condition U p (x, y) ≤ 0 for all x, y ∈ satisfying | y| ≤ |x|, (2.6) then we will be done. Indeed, we have |Y 0 | ≤ |X 0 | by the differential subordination, so the properties We search for U p in the class of functions of the form for all x, y ≥ 0 and some c p depending only on p. The latter condition immediately implies the integrability of sup s |U p (X s , Y s )|, by means of Doob's inequality. Let us rephrase the requirements (2.1), (2.2), (2.5) and (2.6) in terms of the function V p . We start from the latter condition: the inequality (2.6) reads The majorization (2.5) takes the form The condition (2.1) can be rewritten as follows: for all (x, y) ∈ i S i and h ∈ , where x = x/|x| for x = 0 and x = 0 for x = 0. To see this, observe first that by continuity, we may restrict ourselves to linearly independent vectors x and h: x + th = 0 for all t ∈ R. The estimate (2.1) is equivalent to saying that for any fixed x, y, h ∈ , the function G x, y,h : R → R given by is concave. Thus, since this function is of class C 1 , (2.1) will follow if we check that G x, y,h (t) ≤ 0 for all those t, for which (x + th, y) ∈ i S i . By the translation property G x, y,h (t + s) = G x+th, y,h (s) valid for all s, t ∈ R, we see that it suffices to check the latter inequality for t = 0 only. By (2.7), this is precisely (2.11).

Parabolic cylinder functions and their properties
In this section we introduce a family of special functions and present some of their properties, needed in our further considerations. Much more information on this subject can be found in [1].
We start with the definition of the Kummer confluent hypergeometric function M (a, b, z). It is a solution of the differential equation and its explicit form is given by The confluent hypergeometric function M p is defined by the formula If p is an even positive integer: p = 2n, then M p is a constant multiple of the Hermite polynomial of order 2n (where the constant depends on n).
The parabolic cylinder functions (also known as Whittaker's functions) are closely related to the confluent hypergeometric functions. They are solutions of the differential equation We will be particularly interested in the special case There are two linearly independent solutions of this equation, given by The parabolic cylinder function D p is defined by where Throughout the paper we will use the notation and z p will stand for the largest positive root of D p . If D p has no positive roots, we set z p = 0.
Later on, we will need the following properties of φ p .
Lemma 3.1. Let p be a fixed number. and (iv) We have z p = 0 for p ≤ 1 and z p > z p−1 for p > 1.
(iv) The first part is an immediate consequence of (iii). To prove the second, we use induction on p . When 1 < p ≤ 2, we have φ p (0) = A 1 < 0 (see (3.2)) and, by (ii), φ p (s) → ∞ as s → ∞, so the claim follows from the Darboux property. To carry out the induction step, take p > 2 and write But, by the hypothesis, z p−1 > 0: this implies that z p−1 is the largest root of φ p−1 . Therefore, by asymptotics (3.5), we obtain φ p−1 (z p−1 ) ≥ 0. Plugging this above yields φ p (z p−1 ) ≤ 0, so, again by (3.5), we have z p−1 ≤ z p . However, the inequality is strict, since otherwise, by (i), we would have φ p−n (z p ) = 0 for all integers n. This would contradict (iii).
(v) This follows immediately from (iii), (iv) and the equalities φ The further property of φ p is described in the following.
A little calculation gives that for s > 0,

which, by (3.3), can be rephrased in the form
After lengthy but simple manipulations, this can be written as The second term above is nonnegative for 0 < p ≤ 1 (see Lemma 3.1 (iii)). Furthermore, Now suppose that F p (s 0 ) > 0 for some s 0 ≥ 0. Then, by (3.10) and the above estimates, F p (s) ≥ F p (s)s for s > s 0 , which yields F p (s) ≥ F p (s 0 ) exp((s 2 − s 2 0 )/2) for s ≥ s 0 . However, by (3.5), the function F p has polynomial growth. A contradiction, which finishes the proof of (i).
(ii) It can be easily verified that we have F p (s) = pF p−1 (s) for all p and s. In consequence, by the previous part, we have that F p is nonincreasing and it suffices to prove that lim s→∞ F p (s) ≥ 0. In fact, the limit is equal to 0, which can be justified using (3.8) and (3.5): F p is of order at most s p+2 as s → ∞, and one easily checks that the coefficients at s p and s p+2 vanish.
(iii) We proceed in the same manner as in the proof of (ii). The function F p is nondecreasing and lim s→∞ F p (s) = 0, by means of (3.8) and (3.5) (this time one also has to check that the coefficient at s p−2 is equal to 0).
Before we proceed to the contruction of the special functions V p , let us mention here that the arguments presented in the proof of the above lemma (equations (3.8) and (3.9)) lead to some interesting bounds for the roots z p , 1 ≤ p ≤ 3. For example, if 1 ≤ p ≤ 2, then, as we have shown, the function F p is nonnegative: thus, putting s = z p in (3.8) and exploting (3.6) yields z 2 p ≥ 2p − 3 (which is nontrivial for p > 3/2). Furthermore, F p is nonincreasing, so taking s = z p in (3.9) gives which can be rewritten in the more explicit form Note that the bound is quite tight: we have equality for p ∈ {1, 2}. Similarly, in the case when 2 ≤ p ≤ 3 we obtain the following estimates: and we have the (double) equality for p ∈ {2, 3}.
In particular, the above inequalities yield Furthermore, we get the following bound, which, clearly, is also valid for p > 3.

Proof of Theorem 1.4
We turn to the proof of our main result. For the sake of convenience, we have decided to split it into five parts: the proof of (1.4) in the cases 1 < p ≤ 2, 2 < p < 3, p ≥ 3, the strictness and, finally, the sharpness of the estimate.

The case
We will check below in seven steps that the function V p has all the properties described in Section 2.
1 • Regularity. It is easy to see that V p is continuous, of class C 1 on the set (0, ∞) × (0, ∞) and of class C 2 on the set (0, ∞) × (0, ∞) \ {(x, y) : y = z −1 p x}. In consequence, the function U p defined by (2.7) has the required smoothness. It can also be verified readily that the first order derivative of U p is bounded on bounded sets not containing 0 ∈ × .
3 • The inequality (2.9). Note that φ p is increasing (by Lemma 3.1 (v)) and z p ≤ 1 (by Corollary 3.3). Thus, for 0 < y ≤ x, (2.10). This is obvious for y ≥ z −1 p x, so we focus on the case y < z −1 p x. Then the majorization is equivalent to Both sides are equal when s = z p , so it suffices to establish an appropriate estimate for the derivatives: α p φ p (s) ≤ −pz −p p s p−1 for s > z p . We see that again both sides are equal when s = z p ; thus we will be done if we show that the function s → φ p (s)/s p−1 is nondecreasing on (z p , ∞). After differentiation, this is equivalent to or, by Lemma 3.1 (i), φ p (s) ≤ 0 for s > z p . This is shown in the part (v) of that lemma.
Before we proceed, let us mention here a fact which will be exploited during the proof of the strictness. Namely, if p = 2, then the above reasoning gives that the majorization is strict on {(x, y) : y < z −1 p x}. Indeed, we have that φ p is negative on [z p , ∞) (see Lemma 3.1 (v)). 5 • The condition (2.11). If z p | y| > |x| > 0, then we have and so (2.11) is valid. When |x| > z p | y| > 0, we compute that where we have used the notation r = |x|/| y|. Furthermore, Adding this to (4.4) yields (2.11), since φ p (r) ≥ 0: see Lemma 3.1 (v).
6 • The condition (2.12). If z p | y| > |x| > 0, then Therefore, combining this with (4.2) and (4.3) we see that the left-hand side of (2.12) is not larger than as needed. Here in the last passage we have used the estimates z −1 p |x| < | y| and z 2 p ≤ p − 1 (see Corollary 3.3). On the other hand, if 0 < z p | y| < |x|, then recall the function F p given by (3.7). A little calculation yields which is nonpositive by means of Lemma 3.2 (here r = |x|/| y|, as before). Furthermore, by (3.3), which, combined with (4.4) and (4.5) implies that the left-hand side of (2.12) does not exceed α p | y| p−2 φ p (r)(|h| 2 − |k| 2 ).

The case
As in the previous case, we will verify that V p enjoys the requirements listed in Section 2.
1 • Regularity. Clearly, we have that V p is continuous, of class C 1 on (0, ∞) × (0, ∞) and of class C 2 on (0, ∞) × (0, ∞) \ {(x, y) : y = z p x}. Hence U p given by (2.7) has the necessary smoothness. In addition, it is easy to see that the first order derivative of U p is bounded on bounded sets.
3 • The inequality (2.9) This is obvious: (2.10). This can be established exactly in the same manner as in the previous case. In fact one can show that the majorization is strict provided y > z p x. The details are left to the reader. (2.11). Note that if 0 < | y| < z p |x|, then

• The condition
and so (2.11) follows. Suppose then, that | y| > z p |x| > 0 and recall F p introduced in Lemma 3.2. By means of this lemma, after some straightforward computations, one gets where we have set r = | y|/|x|. Moreover, by (3.3), is nonpositive; this completes the proof of (2.11).
3). By the above considerations, we are forced to take c 1 (x, y) = pz p p |x| p−2 and c 2 (x, y) = α p |x| p−2 φ p (r) and it is clear that the condition is satisfied. This establishes (1.4) for 2 < p < 3.
In the proof of (1.4) we will need the following auxiliary fact. Furthermore, the inequality is strict if β = π/(2p).
Furthermore, due to the complexity of the calculations, it is convenient to gather the bounds for the partial derivatives of V p in a separate lemma.
(ii) If y < cot(π/(2p))x, then Furthermore, so it suffices to show that the expression in the square brackets is nonpositive. This follows immediately from ≤ 1 p and we are done. Now we are ready for the proof of (1.4). As in the previous cases, we verify that V p has the properties studied in Section 2.
1 • Regularity. It is easily checked that V p is continuous, of class C 1 on (0, ∞) × (0, ∞) and of class C 2 outside {(x, y) : θ = π/2 − π/(2p)}. This guarantees the appropriate smoothness of U p given by (2.7). Furthermore, it is clear that the first order derivative of U p is bounded on bounded sets.

•
The growth condition (2.8). This is clear from the very formula for the function V p .

Strictness
Suppose that p = 2 and that X , Y are orthogonal -valued martingales such that 0 < ||X || p < ∞, Y is differentially subordinate to X and we have equality in (1.4). These conditions imply in particular that both martingales converge almost surely and in L p . Denoting the corresponding limits by X ∞ and Y ∞ , we may write ||Y ∞ || To see this, assume first that 1 < p < 2. For this range of parameters p, the inequality (2.10) is strict if | y| < C p |x|. This, by (4.22), implies P(|Y ∞ | ≥ C p |X ∞ |) = 1 and, again by (4.22), yields (4.23). In the case p > 2 the reasoning is the same.
The condition (4.23) gives U p (X ∞ , Y ∞ ) = 0. Furthermore, by (2.4) and (2.8) we see that (U p (X t , Y t )) t≥0 is a uniformly integrable supermartingale satisfying U p (X 0 , Y 0 ) ≤ 0. This gives P(U p (X t , Y t ) = 0 for all t ≥ 0) = 1. However, using the formulas for V p in the cases 1 < p < 2, 2 < p < 3 and p ≥ 3, we see that this is equivalent to For any t ≥ 0 we have that in probability (see e.g. [10] for the proof in the real case; the reasoning presented there can be easily extended to the Hilbert-space setting). Using a similar statement for Y and combining this with which contradicts the differential subordination unless C p = 1 or [X , X ] ≡ 0. However, the first possibility occurs only for p = 2 (Corollary 3.4) and the second one implies ||X || p = 0; we have excluded both these possibilities at the beginning.
This completes the proof of the strictness.
The proof of the Theorem 1.4 is complete.

Remark 4.3.
Referee asked a very interesting question whether the constant C p , 1 < p < 2, is still the best possible when we restrict ourselves to Y taking values in R 2 . The answer is affirmative and can be obtained by a modification of the above example. Let B = (B (1) , B (2) ) be a two-dimensional Brownian motion, let τ be an arbitrary stopping time of B with τ ∈ L p/2 and define X t = B (1) τ∧t for t ≥ 0. For a fixed positive integer n we introduce Y (n) using the following inductive procedure: (ii) for k = 0, 1, 2, . . . and t ∈ (k · 2 −n , (k + 1) · 2 −n ], let where ξ k is a norm-one vector in R 2 , orthogonal to Y (n) τ∧k·2 −n , not depending on t.
In other words, for t ∈ (k · 2 −n , (k + 1) · 2 −n ], the increment Y The remainder of the proof is the same as above: we easily check that X and Y (n) are orthogonal and that [X , X ] t = [Y (n) , Y (n) ] t = τ ∧ t. Letting n → ∞ we obtain that C p ≥ z −1 p due to Davis' inequality (1.2).
A similar reasoning can be carried out to show that the constant C p , 2 < p < 3, is the best possible if we restrict ourselves to X taking values in R 2 .

Inequalities for stochastic integrals
We will apply the results from the previous section and obtain some sharp inequalities for stochastic integrals. Let M = (M t ) t≥0 be an adapted martingale and let H = (H t ) t≥0 and K = (K t ) t≥0 be two predictable processes. Let X = H · M , Y = K · M denote the stochastic integrals of H and K with respect to M . The following result is due to Burkholder (see Theorem 5.1 in [7]). for all t ≥ 0. By Cauchy-Riemann equations, the integrands satisfy the domination (5.1) as well as the orthogonality. It suffices to note that (5.2) reduces to Pichorides' inequality ||v|| p ≤ cot π 2p * ||u|| p , which is known to be sharp (see [14]). This completes the proof.