On a weak type (1, 1) inequality for a maximal conjugate function

Nakhlé H. Asmar and Stephen J. Montgomery-Smith

1 Introduction

Throughout this paper, N denotes a fixed but arbitrary positive integer, T denotes the circle group, and T^{N} denotes the product of N copies of T. The normalized Lebesgue measure on T^{N} will be symbolized by P. For a measurable function f, we let ∥f∥_{1}^{∗} = sup_{y>0} yλ_{f}(y) where λ_{f}(y) = P ({x ∈ T^{N} : |f(x)| > y}). The integers will be denoted by Z and the complex numbers by C.

Let F_{n} = σ(e^{iθ_{1}}, e^{iθ_{2}}, …, e^{iθ_{n}}) denote the σ-algebra on T^{N} generated by the first n coordinate functions. For f ∈ L^{1}(T^{N}), the conditional expectation of f with respect to F_{n} will be denoted E(f|F_{n}). Let

d_{0}(f) = E(f|F_{0}) = ∫_{T_{N}} fdP,

and for j = 1, …, N, let d_{j}(f) = E(f|F_{j}) − E(f|F_{j−1}). We have the martingale difference decomposition

f = ∑_{j=0}^{N} d_{j}(f).  (1)

Consider the maximal function corresponding to (1)

D(f) = sup_{1≤n≤N} |∑_{j=0}^{n} d_{j}(f)| = sup_{1≤n≤N} |E(f|F_{n})| .  (2)

A well-known weak type (1, 1) maximal inequality due to Doob states that there is a constant a independent of f and N such that

∥Df∥_{1}^{∗} ≤ a∥f∥_{1}.  (3)

Now we recall the conjugate function operator f ↦ f̃, defined for all f ∈ L^{2}(T) by the multiplier relation

ˆf̃(n) = −isgn(n)f̂(n), for all n ∈ Z.

By Kolmogorov’s Theorem [8, Chap. IV, Theorem (3.16)], the operator f ↦ f̃ is of weak type (1, 1).

Denote an element of T^{N} by (θ_{1}, θ_{2}, …, θ_{N}). Let H_{j} denote the one-dimensional conjugate function operator defined for functions on T^{N} with respect to the θ_{j} variable. As an operator on L^{2}(T^{N}), H_{j} is given by the multiplier relation Ĥ_{j}(f)(z_{1}, z_{2}, …, z_{N}) = −isgn(z_{j})f̂(z_{1}, z_{2}, …, z_{N}), for all (z_{1}, z_{2}, …, z_{N}) ∈ Z^{N}. Plainly, the operators H_{j}, j = 1, …, N, are of weak type (1, 1) on L^{1}(T^{N}) with the same constant as in Kolmogorov’s theorem for L^{1}(T). The conjugate function that we consider is defined for all f ∈ L^{1}(T^{N}) by

H(f) = ∑_{j=1}^{N} H_{j}(d_{j}(f)).  (4)

Since both H_{j} and d_{j} are multipliers, they commute. We have

H(f) = ∑_{j=1}^{N} d_{j}(H(f)).  (5)

The maximal function that we are interested in is defined by

M(f) = sup_{1≤n≤N} |∑_{j=1}^{n} d_{j}(H_{j}(f))| = D(H(f)),  (6)

where D is as in (2). Thus M is the composition of two operators of weak type (1, 1). (The fact that H is of weak type (1, 1) is known, and will not be needed in the proofs. See Remarks 1.2 (a), below. This fact will also follow from our main theorem.) Our goal is to prove the following result. Theorem 1.1 There is a constant A independent of N such that for all f ∈ L^{1}(T^{N}) we have

∥Mf∥_{1}^{∗} ≤ A∥f∥_{1},  (7)

where M is the maximal operator given by (6). The proof of this theorem is presented in the following section, and is of independent interest. We will show that by changing the time in the Brownian motion that Burkholder, Gundy, and Silverstein used in [3] from a continuous range [0, ∞) to a semi-continuous range {1, 2, …} × [0, ∞), the proofs in [3] can be carried out on T^{N}, yielding inequalities which are independent of N (e.g., the “good λ” inequality).

We end this section with some remarks concerning the operator H that will not be used in the sequel. Remarks 1.2 (a) The operator f ↦ Hf that we defined in (5) is a conjugate function operator of the kind that was introduced and studied by Helson [6]. Helson’s definition is in terms of orders on the dual group Z^{N}. In our case, the operator H can be recast in terms of a lexicographic order on Z^{N}. As shown in [6], the operator H is bounded from L^{1}(T^{N}) into L^{p}(T^{N}), for any 0 < p < 1. Indeed it is of weak type (1, 1) (see [1, Theorem 4.3]).

(b) We proved in [1, Theorem 5.4] that the square function Sf = (∑_{j=1}^{N} |H_{j}(d_{j}(f))|^{2})^{1/2} is of weak type (1, 1). It is known that under certain conditions on the martingale, the weak type estimates for the square function and the maximal function are equivalent (see, for example, [2], Assumptions A1-A3). The martingales that we are studying do not satisfy these conditions, and so (7) does not follow from the weak (1, 1) estimates for the square function, by using general facts from probability theory.

2 Proof of Theorem 1.1

For clarity’s sake, we start with an outline of the proof, setting in the process our notation, and describing our generalization of the methods in [3].

It is enough to prove (7) with f ∈ S(T^{N}), the space of trigonometric polynomials on T^{N}. We may also assume that f is real-valued and that d_{0}(f) = 0. Write

f(θ_{1}, …, θ_{N}) = ∑ a_{j_{1},...,j_{N}} θ_{1}^{j_{1}} … θ_{N}^{j_{N}} ,

and extend f to a function on C^{N} that is harmonic in each variable as follows

f(r_{1}θ_{1}, …, r_{N}θ_{N}) = ∑ a_{j_{1},...,j_{N}} r_{1}^{|j_{1}|}θ_{1}^{j_{1}} … r_{N}^{|j_{N} |}θ_{N}^{j_{N}}

where r_{n} is a nonnegative real number, and |θ_{n}| ∈ T = {z : |z| = 1}. In this notation, the n-th term in the martingale difference decomposition of f becomes

d_{n}(f) = ∑_{j_{n}≠0^{j_{1},j_{2},...,jn}} a_{j_{1},...,j_{n},0,...,0}θ_{1}^{j_{1}} … θ_{n}^{j_{n}}.

Since by assumption d_{0}(f) = 0, it follows that

d_{n}(f)(r_{1}θ_{1}, …, r_{n−1}θ_{n−1}, 0) = 0  (8)

for all n = 0, 1, …, N.

The approach that we take is to consider a martingale on a time structure that is part continuous and part discrete. Our notion of time is T = {1, 2, …, N} × [0, ∞[ with the order (m, s) < (n, t) if and only if m < n or m = n and s < t. Construct N independent complex Brownian motions c_{n,t} = a_{n,t} + ib_{n,t} (1 ≤ n ≤ N, t ≥ 0) each one starting at 0. Define stopping times τ_{n} = inf{ t : |c_{n,t}| ≥ 1}.

Define an increasing family of sigma fields (A_{(n,t)} : (n, t) ∈ T ), where A_{(n,t)} is the sigma field generated by the functions c_{m,s} for (m, s) ≤ (n, t). Then we define a process over our new time structure by:

F_{n,t} = f(c_{1,τ_{1}}, …, c_{n−1,τ_{n−1}}, c_{n,τ_{n}∧t}, 0, …, 0)
= ∑_{k=0}^{n−1} d_{k}(f)(c_{1,τ_{1}}, …, c_{k,τ_{k}}) + d_{n}(f)(c_{1,τ_{1}}, …, c_{n,t∧τ_{n}}).  (9)

Since τ_{n} < ∞ a.s., it follows that a.s., for sufficiently large (n, t), we have F_{n,t} = F_{∞}, where

F_{∞} = ∑_{k=0}^{N} d_{k}(c_{1,τ_{1}}, …, c_{k,τ_{k}}) = f(c_{1,τ_{1}}, …, c_{N,τ_{N}}).

We will show that the family of functions (F_{n,t}) is a martingale relative to A_{(n,t)}. To be able to use results from the classical theory of martingales, it is convenient to label the family (F_{n,t}) by a continuous time parameter. This can be done by forming an order preserving bijection between T ∪ {∞} and [0, N] as follows:

ϕ(n, t) = n − 1 + t/(t + 1), and ϕ(∞) = N.

Because c_{n,t} is a.s. continuous in t, and also τ_{n} < ∞ a.s., it follows that F_{ϕ^{−1}(t)} is a continuous time martingale on [0, N]. Let F̃_{n,t} be constructed from Hf as in (9). Define the Brownian maximal function

F^{∗} = sup_{0≤t≤N} |F_{ϕ^{−1}(t)}|,

and let F̃^{∗} be defined similarly by using F̃_{n,t}. The proof of the desired inequality (7) will proceed in four steps. Step 1: ∥F_{∞}∥_{1} = ∥f∥_{1}; Step 2: ∥F^{∗}∥_{1,∞}^{∗} ≤ ∥F_{∞}∥_{1}; Step 3: ∥F̃^{∗}∥_{1,∞}^{∗} ≤ c ∥F^{∗}∥_{1,∞}^{∗};

Step 4: ∥Mf∥_{1,∞}^{∗} ≤ ∥F̃^{∗}∥_{1,∞}^{∗}.

We now proceed with the proofs. Suppose that c_{t} = a_{t}+ib_{t} is a complex Brownian motion starting at 0. Let A_{t} be the sigma field generated by c_{s} for s ≤ t. Let τ = inf{t : |c_{t}| ≥ 1}. Suppose that v is a real-valued trigonometric polynomial on T = {|z| = 1}, and extend v to be harmonic on C. It follows from [5, Theorem 4.1] that v(c_{t}) is a martingale, and v(c_{t}) is A_{t}-measurable. The following lemma, is a simple consequence of this fact and Doob’s Optional Stopping Theorem.

Lemma 2.1 With the above notation, if μ is a stopping time such that μ ≤ τ, then

E(v(c_{μ})|A_{t}) = v(c_{t∧μ}).

[?, Theorem (3.2), p.65]. We have Using Lemma (2.1), we can establish a basic property of the functions (F_{n,t}).

Lemma 2.2 In the above notation, we have that E(F_{∞}|A_{n,t}) = F_{n,t}, and hence that (F_{n,t}) is a martingale. Consequently, (F_{ϕ^{−1}(t)}) is a continuous time martingale for t ∈ [0, N].

Proof. First, it is clear that if k < n, then

E(d_{k}(c_{1,τ_{1}}, …, c_{k,τ_{k}})|A_{n,t}) = d_{k}(c_{1,τ_{1}}, …, c_{k,τ_{k}}),

because d_{k}(c_{1,τ_{1}}, …, c_{k,τ_{k}}) is A_{n,t} measurable. Also, if k > n, then

E(d_{k}(c_{1,τ_{1}}, …, c_{k,τ_{k}})|A_{n,t}) = E(E(d_{k}(c_{1,τ_{1}}, …, c_{k,τ_{k}})|A_{k,0})|A_{n,t}) = 0,

by Lemma (2.1) and (8). Similarly, by the same lemma, it also follows that if k = n, then

E(d_{k}(c_{1,τ_{1}}, …, c_{k,τ_{k}})|A_{n,t}) = d_{k}(c_{1,τ_{1}}, …, c_{k,t∧τ_{k}})

and hence E(F_{∞}|A_{n,t}) = F_{n,t}. This proves that (F_{n,t}) is a martingale. The rest of the lemma is obvious. Proof of Steps 1, 2, 4 Because of Lemma (2.2), Step 2 follows from Doob’s Maximal Inequality for continuous time martingales (see [4, Chapter VII, Section 11]). Step 1 also follows from the uniform distribution of Brownian motion over T (see [7, Corollary 3.6.2]). Step 4 is also a consequence of the same property of Brownian motion. We give details. We have

F̃^{∗} = sup_{(n,t)} |F̃_{n,t}| ≥ sup_{n} |F̃_{n,τ_{n}}|
= sup_{n} | ∑_{m=0}^{n} H_{m}(d_{m}(f))(c_{1,τ_{1}}, …, c_{m,τ_{m}})| .

But since (c_{1,τ_{1}}, …, c_{m,τ_{m}}) is equidistributed with (θ_{1}, …, θ_{m}), the right side of the displayed inequalities is equidistributed with sup_{n} |∑_{m=0}^{n} H_{m}(d_{m}(f))(θ_{1}, …, θ_{m})| , and Step 4 follows. Proof of Step 3. The proof may be done as in [3, Theorem 4]. We provide the details to show the role of analyticity on T^{N}. Here we call a function ϕ ∈ L^{1}(T^{N}) analytic if its Fourier transform is supported in the half-space

O = {0} ⋃_{j=1}^{N}{(m_{1}, m_{2}, …, m_{N}) ∈ Z^{N} : m_{j} > 0, m_{j+1} = …, m_{N} = 0}.

The following basic properties of analytic functions on T^{N} are easy to prove.

• A function ϕ ∈ L^{1}(T^{N}) is analytic if and only if each term in its martingale difference decomposition, d_{j}(ϕ) (j = 1, …, N), is analytic in the j-th variable θ_{j} and has zero mean, i.e., d_{j}(ϕ) ∈ H_{0}^{1}(T).

• If ϕ is analytic then ϕ^{2} is also analytic. (This follows from O + O = O.)

• If ϕ is a trigonometric polynomial on T^{N}, then ϕ + iH(ϕ) is analytic.

Getting back to the proof of Step 3, let

g(r_{1}θ_{1}, …, r_{N}θ_{N}) = f(r_{1}θ_{1}, …, r_{N}θ_{N}) + iH(f)(r_{1}θ_{1}, …, r_{N}θ_{N}),

and let

h = g^{2}.

Both g and h are analytic on T^{N}. Hence the functions d_{m}(g)(θ_{1}, …, r_{m}θ_{m}) and d_{m}(h)(θ_{1}, …, r_{m}θ_{m}) are analytic in the m-th variable. Form the functions G_{n,t} and H_{n,t} as in (9). By Lemma (2.2), G_{n,t} and H_{n,t} are martingales relative to A_{n,t}. We claim that, because of analyticity, we have

H_{n,t} = G_{n,t}^{2}.  (10)

To see this, write

g(θ_{1}, …, θ_{N}) = ∑_{k=1}^{N} d_{k}(g)(θ_{1}, …, θ_{k})

and

h(θ_{1}, …, θ_{N}) = ∑_{k=1}^{N} d_{k}(h)(θ_{1}, …, θ_{k}).

Then, since all the exponents of θ_{n} are positive, we get

(∑_{k=1}^{n−1} d_{k}(g)(θ_{1}, …, θ_{k}) + d_{n}(g)(θ_{1}, …, r_{n}θ_{n}))^{2} = ∑_{k=1}^{n−1} d_{k}(h)(θ_{1}, …, θ_{k})+d_{n}(h)(θ_{1}, …, r_{n}θ_{n})

and (10) easily follows. Consequently, since the functions H_{n,t} form a martingale relative to the σ-algebra A_{n,t}, we have that G_{n,t}^{2} is a martingale relative to this σalgebra. With this fact in hands, we can now proceed with the proof of Step 3 in exactly the same way as in [3, pp. 148-149]. We need a lemma.

Lemma 2.3 Suppose that μ and ν are stopping times with μ ≤ ν a. e. Let f be a real-valued trigonometric polynomial on T^{N} with ∫ fdP = 0. Then,

∥F̃_{ν} − F̃_{μ}∥_{2} = ∥F_{ν} − F_{μ}∥_{2}.

Proof. Using the fact that G_{n,t}^{2} is a martingale, we get

0 = E(G_{0}^{2}) = E(G_{μ}^{2}).

Similarly, E(G_{ν}^{2}) = 0. Hence, EF_{μ}^{2} = EF̃_{μ}^{2} and EF_{ν}^{2} = EF̃_{ν}^{2}. Next, we show that E(F_{μ}F_{ν}) = E(F_{μ}^{2}), and E(F̃_{μ}F̃_{ν}) = E(F̃_{μ}^{2}). We start with the first equality. Using Doob’s Optional Sampling Theorem and basic properties of the conditional expectation, we see that

E(F_{ν}|F_{μ}) = F_{μ},
F_{μ}E(F_{ν}|F_{μ}) = F_{μ}^{2},

and so

E(F_{μ}F_{ν}|F_{μ}) = F_{μ}^{2}.

Integrating both sides of the last equality, we get E(F_{μ}F_{ν}) = E(F_{μ}^{2}). The second equality can be proved similarly. Thus

E(F_{μ} − F_{ν})^{2} = EF_{μ}^{2} + EF_{ν}^{2} − 2E(F_{μ}F_{ν})
= EF_{μ}^{2} + EF_{ν}^{2} − 2E(F_{μ}^{2})
= EF_{ν}^{2} − E(F_{μ}^{2})
= E(F̃_{μ} − F̃_{ν})^{2},

which completes the proof. The above lemma enables us to establish a fundamental inequality. This is our version of the ‘good λ’ inequality for conjugate functions on T^{N}.

Lemma 2.4 With the notation of the previous lemma, let α ≥ 1 and β > 1. Then there is a constant c, depending only on α and β, such that whenever λ > 0 satisfies

P (G^{∗} > λ) ≤ αP (G^{∗} > βλ),

then

P (G^{∗} > λ) ≤ c P (c F^{∗} > λ).

Proof. Define stopping times

μ = inf{ (n, t) ∈ T : |G_{n,t}| > λ}, ν = inf{ (n, t) ∈ T : |G_{n,t}| > βλ}.

If the set { (n, t) : |G_{n,t}| > λ} is empty, then we set μ = ∞. Otherwise μ is such that |G_{n,t}| ≤ λ whenever (n, t) < μ, and |G_{μ}| = λ. We define ν similarly. Also, we have that μ ≤ ν, that |G_{μ}| = λ on the set {μ ≠ ∞} = {G_{∞}^{∗} > λ}, and that |G_{ν}| = βλ on the set {ν ≠ ∞} = {G^{∗} > βλ}. Thus if λ satisfies the hypothesis of the lemma, then

E(χ_{G^{∗}>λ}(F_{ν} − F_{μ})^{2}) = ∥F_{ν} − F_{μ}∥_{2}^{2}
= \frac{1}{2} ∥G_{ν} − G_{μ}∥_{2}^{2}
≥ \frac{1}{2}(βλ − λ)^{2}P (G^{∗} > βλ)
≥ cλ^{2}P (G_{∞}^{∗} > λ).

Also

E(χ_{G^{∗}>λ}(F_{ν} − F_{μ})^{4}) ≤ ∥G_{ν} − G_{μ}∥_{4}^{4} ≤ cλ^{4}P (G_{∞}^{∗} > λ).

Thus, by a lemma of Paley and Zygmund [8, Chapter V, (8,26)],

P (G^{∗} > λ) ≤ cP (c|F_{ν} − F_{μ}| > λ).

Since |F_{ν} − F_{μ}| ≤ 2F^{∗}, the lemma follows.

Now let us finish by proving Step 3. It is sufficient to show ∥G^{∗}∥_{1,∞}^{∗} ≤ c ∥F^{∗}∥_{1,∞}^{∗}. Suppose that

∥G^{∗}∥_{1,∞}^{∗} = sup_{λ>0} λP (G^{∗} > λ) = A.

Pick λ_{0} such that 2λ_{0}P (G^{∗} > 2λ_{0}) ≥ A/2. Then λ_{0}P (G^{∗} > λ_{0}) ≤ A, and thus λ_{0} satisfies the hypothesis of the lemma with α = 4 and β = 2. Then it follows that

∥F^{∗}∥_{1,∞}^{∗} ≥ λ_{0}P (cF^{∗} > λ_{0}) ≥ cA/4,

as desired.

Acknowledgements The research of the authors was supported by grants from the National Science Foundation (U. S. A.).

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