Weak-type estimates for martingale maximal functions

The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given 1 < p < ∞ and a pair ( x, y ) of nonnegative numbers satisfying x p ≤ y , we identify the optimal upper bounds for (cid:107)| sup n f n |(cid:107) p, ∞ , for nonnegative martingales f = ( f n ) n ≥ 0 satisfying (cid:107) f (cid:107) 1 = x and (cid:107) f (cid:107) pp = y .


Introduction
As evidenced in numerous works, maximal inequalities play a distinguished role in harmonic analysis and probability theory. The purpose of this paper is to present a refined study of certain weak-type estimates arising in the context of martingales.
We start with the description of the background and notation used throughout the text. In what follows, (Ω, F, P) will denote the probability space, equipped with the discrete-time filtration (F n ) n≥0 . For an adapted martingale f = (f n ) n≥0 , the symbol f * := sup n≥0 |f n | will stand for the associated maximal function, we will also use the notation f * n := max 0≤k≤n |f k | for its truncated version (n = 0, 1, 2, . . .). Estimates between f and f * (equivalent to the boundedness of the maximal operator on various function spaces) are of fundamental importance to the martingale theory and form the base for stochastic integration. For example, we have the classical weak-and strong-type estimates (cf. [2], see also [1,8] for a different perspective) where f * p,∞ := sup λ>0 λ p P(f * ≥ λ) 1/p stands for the weak-L p quasinorm of f * and f p := sup n≥0 E|f n | p 1/p is the L p norm of a martingale f . Both inequalities are sharp: the constants 1 and p/(p − 1) cannot be decreased without additional conditions on f . The purpose of this paper is to study a certain modification and extension of the weak-type bound. Consider the following alternative norming of the Lorentz space L p,∞ : for 1 < p < ∞ and an arbitrary random variable ξ, put |ξ| p,∞ := sup P(A) 1 where the supremum is taken over all events A of positive probability. It is well-known that the quantities · p,∞ and | · | p,∞ are equivalent for 1 < p < ∞ (cf. [3]): we have ξ p,∞ ≤ |ξ| p,∞ ≤ c p ξ p,∞ for some constant c p depending only on p. We will identify the optimal constant in the weak-type estimate under this new norming.
Note that it is enough to study the above estimate for nonnegative martingales f only. Indeed, given an arbitrary, real-valued L p -bounded martingale f , let us denote its pointwise limit by f ∞ . Then the passage from f to the nonnegative martingale (E(|f ∞ | |F n )) n≥0 does not change the right-hand side of (1.1), while the left-hand side can only increase. Thus, from now on, we will restrict ourselves to nonnegative martingales. We will be able to study the following much more precise version of (1.1). Namely, fix 1 < p < ∞ and suppose that f = (f n ) n≥0 is a nonnegative, L p bounded martingale satisfying f 1 = x and f p p = y. Here x, y are arbitrary positive numbers with x p ≤ y (it is easy to see that for such x and y, there is at least one nonnegative martingale satisfying the norm requirements). What is optimal upper bound for |f * | p,∞ ? Of course, (1.1) will give |f * | p,∞ ≤ Γ 2p−1 p−1 1−1/p y 1/p , but this does not have to be sharp: for example, if x p = y, then f must be a constant martingale: f 0 = f 1 = f 2 = . . . ≡ x and hence |f * | p,∞ = x.
Our main result can be formulated as follows. Suppose that x, y are arbitrary positive numbers with x p ≤ y. Introduce the function where the function γ and c * = c * (x, y) are defined in Section 2.1 below.
By a standard approximation, the above result extends to the continuous-time context.
That is, if (X t ) t≥0 is a nonnegative, continuous-time cádlág martingale satisfying X 1 = x and X p p = y, then its maximal function X * = sup t≥0 |X t | satisfies Furthermore, the constant on the right cannot be decreased: for each x, y there is a martingale X with prescribed first and p-th norms, for which both sides above are equal.
For related results involving the standard weak L p -norm · p,∞ in the place of | · | p,∞ , see the paper [4] by Melas and Nikolidakis, which makes use of combinatorial and analytic arguments.
The advantage of | · | p,∞ over · p,∞ lies in the fact that the former is actually a norm on L p,∞ (see [3]), which enables an additional averaging procedure. This, for example, leads to the following extension of our result to the multivariate case. Suppose that (f (j) ) j≥0 is a sequence of p-integrable random variables satisfying f (j) 1 = x and f (j) p p = y for each j (e.g., take an arbitrarily dependent sequence of random variables having the same distribution). Let (F (j) n ) n≥0 be a family of filtrations and let Q be some probability measure on Z + = {0, 1, 2, . . .}, independent of P. With a slight abuse of notation, let f (j) = (E(f (j) |F (j) n )) n≥0 be the martingale generated by f (j) and the filtration F (j) . Then the "average" maximal operator Mf = Z+ (f (j) ) * dQ satisfies This bound is sharp: if all the variables f (j) are the same and the filtrations coincide, we recover our main result. There is a natural question whether the above estimate holds if we pass to the quasinorm · p,∞ (and consider the corresponding value of B p (x, y)).
We strongly believe that this is not the case and some multiplicative constant in front of B p (x, y) is necessary. As an indication, note that the first bound (i.e., Mf p,∞ ≤ E Q (f (j) ) * p,∞ ) does not hold in general. Indeed, consider the probability space [0, 1] equipped with its Borel subsets and Lebesgue's measure, with a single filtration at least for small p. For related phenomena involving · p,∞ and | · | p,∞ , arising in the context of martingale study of Fourier multipliers, see e.g. [6,7].
Let us say a few words about our approach towards Theorem 1.2. A natural idea is to apply Burkholder's method (sometimes referred to in the literature as the Bellman function technique). This approach relates a given martingale inequality to the existence of a certain special function, enjoying appropriate size and concavity requirements: convenient references on this subject are [1] and [5]. However, a direct application of the method requires the invention of a complicated special function of four variables (which control the first norm of f , the p-th norm of f , the size of the maximal function and the size of the event A which appears in the definition of the weak norm, respectively). To overcome this technical difficulty, we propose an alternative novel approach which is of independent interest. Namely, appropriate optimization and homogenization arguments allow to reduce the problem to the investigation of a much simpler martingale inequality. To prove this inequality, we will still use Burkholder's method, but this time the special functions will involve two variables only.
The paper is organized as follows. In the next section we introduce the technical background needed for our investigation and establish the simple martingale estimate discussed above. The final part of the paper contains the proofs of Theorem 1.1 and 1.2.

An auxiliary estimate
In this section, we assume that 1 < p < ∞ is a fixed parameter. Throughout, the monotonicity of functions is understood in the strict sense: by saying that a given function is increasing or decreasing, we mean that it is strictly monotone.

Some technical facts
We start our analysis with the introduction of a certain special function of one variable.
We will need the following properties of this object.
Lemma 2.1. The function g is increasing and satisfies g(s) > s, lim s→∞ g(s)/s = 1.
Proof. The asymptotics lim s→∞ g(s)/s = 1 follows easily by de l'Hospital rule. The estimate g(s) > s is equivalent to It is enough to note that the left-hand side vanishes at infinity and its derivative at s equals − exp(−ps p−1 ) < 0. Finally, the monotonicity of g is a direct consequence of the equation and the estimate g(s) > s we have just established.
and let γ : [λ 0 , ∞) → [0, ∞) be the inverse to g. Then the estimate g(s) > s implies that γ(t) < t for t ≥ λ 0 . Furthermore, plugging s := γ(t) into (2.1) and noting that g (γ(t))γ (t) = 1, we see that γ satisfies the differential equation We extend γ to a continuous function on the whole half-line [0, ∞), setting γ(t) := 0 for t < λ 0 . Later on, we will need the following property of γ. Proof. The equality lim x→0 ξ(x) = ∞ is evident, the identity lim x→∞ ξ(x) = 1 follows directly from the asymptotics lim s→∞ g(s)/s = 1 established in the previous lemma. Since γ is continuous, so is ξ and hence, to show the monotonicity of ξ, it is enough to prove it separately on (0, λ 0 ] and (λ 0 , ∞). The property holds on the interval x ∈ (0, λ 0 ], because γ(x) = 0 there. For x ∈ (λ 0 , ∞), we make the substitution x := g(y); since g is increasing, we see that we must prove that the function is decreasing on (0, ∞). By the direct differentiation of this function and (2.1), it suffices to show that g(y) 2 < p(g(y) − y)((p − 1)y p + g(y)).  Finally, we will need the following statement. y ≤ x p < y. Consider the equation in the variable c > 0. This equation has a unique root c * = c * (x, y). This root satisfies c * ≤ x/λ 0 . Furthermore, the function considered on (0, x/λ 0 ], attains its minimum for c = c * (x, y).
Proof. The equation (2.5) is equivalent to ξ(x/c) = y/x p , so the existence and the uniqueness of the root follows at once from the previous lemma. To show that c * (x, y) ≤ x/λ 0 (that is, x/c * (x, y) ≥ λ 0 ), we use the monotonicity of ξ together with the estimate which is assumed in the statement of the lemma. The second part of the assertion follows from differentiation. Indeed, the derivative of the function in question is precisely the left-hand side of (2.5); obviously, this derivative is a continuous function and, as we have just proved, it has a unique zero. Thus it suffices to note that its value at c = x/λ 0 is nonnegative and its limit as c → 0 is negative. The first inequality has already been analyzed above, the negativity of the limit follows at once from observing that and recalling that y > x p and γ(t) < t for all t.

Two martingale inequalities
We are ready to introduce a family (U λ ) λ≥0 of special functions, defined on the angular domain D := {(x, y) : 0 ≤ x ≤ y}, which will play a central role in this paper. First we need to consider appropriate subdomains D λ i of D. We consider two cases. If λ ≥ λ 0 , we introduce three domains D λ 0 , D λ 1 and D λ 2 , given by ECP 27 (2022), paper 52.
See Figure 1 below. Figure 1: The subdomains D λ i in the case λ ≥ λ 0 (left) and λ < λ 0 (right). On the right picture, the common boundary between D λ 0 and D λ 1 is the graph of the function g.
To define U λ : D → R, we also consider two cases. For λ ≥ λ 0 , we let It is easy to check that for any fixed y > 0, the function U λ (·, y) is of class C 1 on [0, y], in particular, the partial derivative ∂ x U λ (x, y) exists for any x ∈ [0, y]. In the lemma below we study two further important properties of the above functions.
(i) The claim is trivial for (x, y) ∈ D λ 0 (both sides are equal). If (x, y) ∈ D λ 1 or (x, y) ∈ D λ 2 , the estimate (2.6) is equivalent to respectively, which follows at once from Young's inequality.
(ii) It is obvious from the formulas on D λ 0 , D λ 1 and D λ 2 that for each y, the function U λ (·, y) : [0, y] → R is concave. Therefore, the estimate (2.7) holds true for x + h ≤ y and we may restrict ourselves to x + h > y. Exploiting the concavity of U λ (·, y) again, we may To this end, we make three observations. First, the function U λ is of class C 1 in some neighborhood of the set {(x, y) ∈ D : x = y} (straightforward); second, the function y → ∂ x U λ (y, y) is nonincreasing (this is also very simple); finally, we have ∂ y U λ (y, y) = 0 for any y > 0: this is clear if (y, y) ∈ D λ 0 ∪ D λ 2 , and follows from the differential equation (2.2) for (y, y) ∈ D λ 1 . Putting these observations together, we obtain that the function y → U λ (y, y) is concave and hence The proof is complete.
We are ready to prove the main results of this section.

Theorem 2.5.
Suppose that (f n ) n≥0 is an arbitrary nonnegative martingale bounded in L p . Then for any λ ≥ 0 we have the estimate Proof. Let us extend the filtration (F n ) n≥0 by setting F −1 := {∅, Ω}; this adds the variable f −1 ≡ Ef 0 to the martingale (f n ) n≥0 (and possibly increases its maximal function, but this will not affect the proof). The key observation is that the composition (U λ (f n , f * n )) n≥−1 is a supermartingale. Indeed, the integrability of U λ (f n , f * n ) follows from the estimate U λ (x, y) ≤ c p (1 + x p + y p ), valid for some constant c p depending only on p, and the assumed L p -boundedness of f . Furthermore, for any n ≥ −1 we have where the inequality follows directly from (2.7), applied with x := f n , y := f * n and h := df n+1 . Consequently, for any n we have which combined with (2.6) yields It remains to let n → ∞ and apply Lebesgue's monotone convergence theorem.

Proofs of Theorems 1.1 and 1.2
We start with Theorem 1.2. If y = x p , then the claim is trivial: the only martingale which satisfies the conditions f 1 = x and f p p = x p is the constant one: f ≡ x, for which |f * | p,∞ = x. Hence, from now on, we will assume that y > x p .

Proof of the upper bound for B p
We consider two major cases: (p − 1)λ p−1 0 y ≥ x p and (p − 1)λ p−1 0 y < x p . In the first case, we apply the estimate (2.8) with λ = λ 0 and the martingale f /c, where c is a positive parameter which will be specified in a moment. Since U λ0 (s, s) ≤ 0 for all s ≥ 0, Now pick an arbitrary event A of positive probability. We may write The latter expression, considered as a function of c, attains its minimal value for c := (p − 1)yλ −1 0 P(A) −1 1/p . Plugging this choice above, we get Since A was arbitrary, the estimate follows. In the case (p − 1)λ p−1 0 y < x p the reasoning is similar, but we apply Theorem 2.6 instead. Namely, we take an arbitrary event A with P(A) > 0 and argue as above, obtaining The expression on the right, considered as a function of λ, attains its minimum for λ := K(x, y)P(A) −1/p . Plugging this special λ above, we obtain the claim.

Proof of the lower bound for B p
We will proceed directly and construct appropriate examples. It is enough to consider continuous-time martingales: as we have mentioned in the introductory section. As previously, we consider two cases.
y ≤ x p Let c * = c * (x, y) be the number defined in (2.5). Consider the probability space equal to the interval [0, 1], equipped with its Borel subsets and ECP 27 (2022), paper 52.