ORNSTEIN’S COUNTEREXAMPLE TO AN L^{1} MAXIMAL INEQUALITY

TERENCE TAO

Abstract. We present Ornstein’s example showing that the L^{p} semigroup maximal inequality of Stein cannot be extended to a weak (1, 1) estimate.

1. Introduction

Let (X, B, μ) be a finite measure space, and let P : L^{1}(X) → L^{1}(X) be a self-adjoint Markov operator (thus P is linear and self-adjoint, is a contraction on both L^{1} and L^{∞}, preserves non-negative functions, and maps 1 to itself). We refer to the quadruplet (X, B, μ, P) as a Markov system. The semigroup maximal inequality of Stein ^{\underline{stein}} [3] asserts that

∥ sup_{n≥0} |P^{n}f|∥_{L^{p}(X)} ≤ C_{p}∥f∥_{L^{p}(X)}.

for all 1 < p ≤ ∞, where C_{p} depends only on p. An alternate proof of this inequality was later given by Rota ^{\underline{rota}} [2] (the key idea being that one could lift X up to the space of random walks using the process P, at which point one could recast P^{2n} as the composition of two conditional expectation operators and use the martingale maximal inequality.).

It is natural to ask whether a weak L^{1} version of this inequality holds, especially in view of the Dunford-Schwartz-Hopf maximal inequality

∥ sup_{N>0} |\frac{1}{N} ∑_{n=1}^{N} P^{n}f|∥_{L^{1,∞}(X)} ≤ ∥f∥_{L^{1}(X)}.

Unfortunately, the answer is negative, as shown by Ornstein ^{\underline{ornstein}} [1]:

Theorem 1.1. There does not exist a constant C such that

∥ sup_{n≥0} |P^{n}f|∥_{L^{1,∞}(X)} ≤ C∥f∥_{L^{1}(X)}  (1.1) pnf

for all self-adjoint Markov operators on finite measure spaces.

Indeed, Ornstein constructed an operator which was the composition of two conditional expectation operators for which the analogous maximal function was not of weak-type (1, 1) (we will not present this slightly stronger result here). Note that one can also make P self-adjoint by the simple trick of replacing P with P^{2n} and using the pigeonhole principle (partitioning n into the odd and even numbers).

We will present Ornstein’s construction in a contrapositive format, in order to more efficiently conceal some of the technicalities of the argument. Thus we shall assume for contradiction that there is an absolute constant C for which (^{\underline{pnf}} 1.1) holds. We shall then bootstrap this inequality into increasingly ridiculous statements, until we finally reach one which is demonstratably absurd.

1.2. Delay. One can think of P^{n} as the operation of evolving f forward by n steps of a Markov process. It turns out that if (^{\underline{pnf}} 1.1) holds, it is possible to break f arbitrarily into components and delay the evolution of any of these components by an arbitrary amount. The basic device for doing so is formalised as follows.

unit Lemma 1.3 (Unit delay lemma). Let (X, B, μ, P) be a Markov system, let f_{0}, f_{1} ∈ L^{1}(X), and let ε > 0. Then there exists a Markov system (X′, B′, μ′, P′) and f′ ∈ L^{1}(X′) with the following properties:

(i) The measure space (X, B, μ) can be identified with a subspace of (X′, B′, μ′). (ii) ∥f′∥_{L^{1}(X′)} = ∥f_{0}∥_{L^{1}(X)} + ∥f_{1}∥_{L^{1}(X)}. (ii) For all g ∈ L^{1}(X) ⊂ L^{1}(X′) and n ≥ 0 we have

∥(P′)^{n}g − P^{n}g∥_{L^{1}(X′)} ≤ 2nε∥g∥_{L^{1}(X)}.

(iv) For all n ≥ 0 we have

∥(P′)^{n}(f′) − [P^{n}f_{0} + 1_{n≥1}P^{n−1}f_{1}]∥_{L^{1}(X)} ≤ 2nε∥f′∥_{L^{1}(X′)}.

Proof. The idea is to add a small “antechamber” to X which can hold the mass of f_{1} for a single unit of time, before it gets injected into X. The self-adjointness of Markov operators will cause a little bit of blowback into the antechamber, but the effect of this can be minimised by making the measure of this antechamber small.

We turn to the details. We make the following definition:

• X′ := X × {0, 1}, with X × {0} identified with X;

• B′ is the product of B with the discrete σ-algebra on {0, 1};

• μ^{P (1)} is the product of μ with the measure δ_{0} + εδ_{1} which assigns mass ε to 1 and 1 to 0;

• P′ is the Markov operator

P′(g_{0}, g_{1}) := ((1 − ε)P g_{0} + εg_{1}, g_{0})

for all g_{0}, g_{1} : X → ℝ, where (g_{0}, g_{1}) is the function which equals g_{0} on the copy X × {0} of X and equals g_{1} on the copy X × {1} of X.

• f′ := (f_{0}, \frac{1}{ε}f_{1}). The properties (i) and (ii) are clear. Now we verify (iii). The cases n = 0, 1 are easily verified. The higher cases then follow by induction on n, using the triangle inequality

∥(P′)^{n+1}g − P^{n+1}g∥_{L^{1}(X′)} ≤ ∥P′[(P′)^{n}g − P^{n}g]∥_{L^{1}(X′)} + ∥P′(P^{n}g) − P (P^{n}g)∥_{L^{1}(X′)},

the induction hypothesis, and the fact that P′ and P are contractions on L^{1}(X′) and L^{1}(X) respectively.

To prove (iv), observe that the claim is trivial for n = 0. For n = 1 we in fact have the slightly stronger statement

∥P′f′ − [P f_{0} + f_{1}]∥_{L^{1}(X′)} ≤ 2ε∥f′∥_{L^{1}(X′)}.

The claim for higher n then follows by (iii). □

Informally, the above lemma allows us to “delay” any portion f_{1} of an L^{1} Markov evolution process by a unit time step. We can iterate this to obtain

multi Lemma 1.4 (Multiple delay lemma). Let (X, B, μ, P) be a Markov system, let f_{0}, f_{1}, …, f_{k} ∈ L^{1}(X) for some k ≥ 1, and let ε > 0. Then there exists a Markov system (X′, B′, μ′, P′) and f′ ∈ L^{1}(X′) with the following properties:

(i) The measure space (X, B, μ) can be identified with a subspace of (X′, B′, μ′). (ii) ∥f∥_{L^{1}(X′)} = ∥f_{0}∥_{L^{1}(X)} + … + ∥f_{k}∥_{L^{1}(X)}. (iii) For all g ∈ L^{1}(X) ⊂ L^{1}(X′) and n ≥ 0 we have

∥(P′)^{n}g − P^{n}g∥_{L^{1}(X′)} ≤ 2nkε∥g∥_{L^{1}(X)}.

(iv) For all n ≥ 0 we have

∥(P′)^{n}(f′) − ∑_{j=0}^{k} 1_{n≥j}P^{n−j}f_{j}∥_{L^{1}(X)} ≤ 4nkε∥f′∥_{L^{1}(X′)}.

Proof. We induct on k. The case k = 0 is trivial, and the case k = 1 is just Lemma ^{\underline{unit}} 1.3. Now suppose k ≥ 2 and the claim has already been proven for k −1. Then by induction hypothesis (using the functions f_{1}, …, f_{k}) we can find a Markov system (X′′, B′′, μ′′, P′′) and f′′ ∈ L^{1}(X′′) such that (X, B, μ) is a subspace of (X′′, B′′, μ′′), that

∥f′′∥_{L^{1}(X′′)} = ∥f_{1}∥_{L^{1}(X)} + … + ∥f_{k}∥_{L^{1}(X)}

and

∥(P′′)^{n}g − P^{n}g∥_{L^{1}(X′′)} ≤ 2n(k − 1)ε∥g∥_{L^{1}(X)}  (1.2) 3a

for all g ∈ L^{1}(X) and n ≥ 0 and

∥(P′′)^{n}(f′′) − ∑_{j=1}^{k} 1_{n≥j−1}P^{n−j+1}f_{j}∥_{L^{1}(X)} ≤ 4n(k − 1)ε∥f′′∥_{L^{1}(X′′)}  (1.3) tweeto

for all n ≥ 0. We then apply Lemma ^{\underline{unit}} 1.3 (with the functions f_{0} (extended by zero to all of X′′) and f′′) to obtain a Markov system (X′, B′, μ, P′) and f′ ∈ L^{1}(X′) such that (X′′, B′′, μ′′) is a subspace of (X′, B′, μ′), that

∥f′′∥_{L^{1}(X′′)} = ∥f_{0}∥_{L^{1}(X)} + ∥f′′∥_{L^{1}(X′′)}

such that

∥(P′)^{n}g − (P′′)^{n}g∥_{L^{1}(X′′)} ≤ 2nε∥g∥_{L^{1}(X′′)}  (1.4) 3b

for all g ∈ L^{1}(X′′) and n ≥ 0, and

∥(P′)^{n}(f′) − (P′′)^{n}f_{0} − 1_{n≥1}(P′′)^{n−1}f′′∥_{L^{1}(X)} ≤ 2nε∥f′∥_{L^{1}(X′)}.  (1.5) tweet

The claims (i), (ii) are clear, and the claim (iii) follows from (^{\underline{3a}} 1.2), (^{\underline{3b}} 1.4), and the triangle inequality. Now we prove (iv). The n = 0 case already follows from (^{\underline{tweet}} 1.5), so take n ≥ 1. From (iii) we have

∥(P′′)^{n}f_{0} − P^{n}f_{0}∥_{L^{1}(X)} ≤ 2nε∥f_{0}∥_{L^{1}(X)}.

Combining this with (^{\underline{tweet}} 1.5) and (^{\underline{tweeto}} 1.3) (for n − 1) we see that

∥(P′)^{n}(f′) − P^{n}f_{0} − ∑_{j=1}^{k} 1_{n≥j}P^{n−j}f_{j}∥_{L^{1}(X)} ≤ [4(k − 1)nε + 2nε + 2nε]∥f′∥_{L^{1}(X′)}

thus closing the induction. □

This lemma implies that the estimate (^{\underline{pnf}} 1.1) bootstraps to a version of itself with arbitrary delays.

Corollary 1.5 (Delayed maximal inequality). Suppose that (^{\underline{pnf}} 1.1) holds for some absolute constant C and all Markov systems (X, B, μ, P). Then for any f_{0}, f_{1}, f_{2}, … ∈ L^{1}(X) we have

∥ sup_{n≥0} ∑_{j=0}^{n} |P^{n−j}f_{j}|∥_{L^{1,∞}(X)} ≤ C ∑_{j=0}^{∞} ∥f_{j}∥_{L^{1}(X)}.  (1.6) pnf-2

Proof. We may take all the f_{j} to be non-negative. By monotone convergence it suffices to show that

∥ sup_{0≤n≤N} ∑_{j=0}^{N} 1_{n≥j}P^{n−j}f_{j}∥_{L^{1,∞}(X)} ≤ C ∑_{j=0}^{N} ∥f_{j}∥_{L^{1}(X)}

uniformly for all N ≥ 0.

Fix N, and let ε > 0 be small. By Lemma ^{\underline{multi}} 1.4 we can find a Markov system (X′, B′, μ′, P′) with (X, B, μ) a subspace of (X′, B′, μ′), as well as f ∈ L^{1}(X) such that

∥f∥_{L^{1}(X)} = ∑_{j=0}^{N} ∥f_{j}∥_{L^{1}(X)}

and

∥(P′)^{n}(f′) − ∑_{j=0}^{N} 1_{n≥j}P^{n−j}f_{j}∥_{L^{1}(X)} ≤ 4N^{2}ε∥f′∥_{L^{1}(X′)}

for all 0 ≤ n ≤ N. Taking suprema and using the triangle inequality we see that

∥ sup_{0≤n≤N} ∑_{j=0}^{N} 1_{n≥j}P^{n−j}f_{j}∥_{L^{1}(X)} ≤ ∥ sup_{0≤n≤N}(P′)^{n}(f′)∥_{L^{1}(X)} + 4N^{3}ε∥f′∥_{L^{1}(X′)}.

Applying the hypothesis (^{\underline{pnf}} 1.1) we conclude that

∥ sup_{0≤n≤N} ∑_{j=0}^{N} 1_{n≥j}P^{n−j}f_{j}∥_{L^{1}(X)} ≤ (C + 4N^{3}ε) ∑_{j=0}^{N} ∥f_{j}∥_{L^{1}(X)}.

Since ε > 0 is arbitrary, the claim follows. □

Informally, what is going on is that we have a rapidly decreasing sequence of antechambers, each of which is injecting material into the next larger antechamber, so that different portions of the L^{1} mass emerge into the main area X at different times.

One consequence of (^{\underline{pnf-2}} 1.6) is the following.

Definition 1.6. A real number α > 0 is said to be good if for any σ > 0 there exists a (X, B, μ, P) is a Markov system and non-negative f_{0}, f_{1}, f_{2}, …, f_{N} ∈ L^{1}(X) is such that

sup_{0≤n≤N} ∑_{j=0}^{n} P^{n−j}f_{j} ≥ 1 − σ  (1.7) fillip

almost surely on X, but

∑_{j=0}^{N} ∥f_{j}∥_{L^{1}(X)} = α.  (1.8) al

Thus for instance any α ≥ 1 is easily seen to be good. From (^{\underline{pnf-2}} 1.6) we see however that sufficiently small values of α are not good.

2. Mixing

In the previous section we took advantage of the ability to delay insertion of mass into a system by any given amount of time. Now we take advantage of a different feature of Markov systems, namely the ability to wait until one portion of mass has fully diffused before inserting the remainder of the mass, which allows one to use less mass in the later portion to achieve the same height in the supremum. This gives

Lemma 2.1 (Mixing bootstrap). Suppose that 0 < α ≤ 1 is good. Then α − α^{2}/4 is also good.

Iterating this we see that arbitrarily small α can be good, a contradiction.

Proof. Let σ > 0 be arbitrary. Fix (X, B, μ, P) and f_{0}, f_{1}, …, f_{N} obeying (^{\underline{fillip}} 1.7) and (^{\underline{al}} 1.8), but with 1 − σ replaced by 1 − σ/2. By a simple limiting argument we may assume the f_{j} to all be bounded.

Let ε > 0 be small (compared with N, f_{0}, …, f_{N}, σ), and consider the Markov system (X′, B′, μ′, P′) defined as follows:

• X′ = X × {0, 1}.

• B′ is the product of B with the discrete σ-algebra on {0, 1}.

• μ′ is the product of μ with the uniform probability measure on {0, 1}.

• P′ is the Markov operator P′(f_{0}, f_{1}) := ((1 − ε)f_{0} + ε𝔼(f_{1}), (1 − ε)f_{1} + ε𝔼(f_{0})), where 𝔼(f) = \frac{1}{μ(X)} ∫_{X} f dμ.

Thus we have split X into two identical halves and allowed an epsilon of diffusion between the two.

Let M > N be a huge integer (sufficiently large depending on ε, α, σ, and N). We apply the hypothesis (^{\underline{pnf-2}} 1.6) on X′ with f′_{j} := (f_{j}, 0) for 0 ≤ j ≤ N, f′_{j+M} := (0, (1 − α/2)f_{j}) and f′_{j} = 0 otherwise. Then one easily computes

∑_{j=0}^{N+M} ∥f′_{j}∥_{L^{1}(X′)} = (α − α^{2}/4).

Now, for 0 ≤ j ≤ n ≤ N, we see that (P′)^{n−j}f′_{j} can be made arbitrarily close to P^{n−j}f_{j} on X × {0} in the uniform metric by making ε sufficiently small depending on N, f_{0}, …, f_{N}. In particular we can ensure that

sup_{0≤n≤N} ∑_{j=0}^{n}(P′)^{n−j}f′_{j} ≥ 1 − σ

on X × {0}. Similar arguments give

sup_{M≤n≤M+N} ∑_{j=M}^{n}(P′)^{n−j}f′_{j} ≥ (1 − α/2)(1 − σ)

on X × {1}. On the other hand, if M is sufficiently large, we see from the mixing properties of P′ that we have the pointwise estimate

(P′)^{n−j}f′_{j} ≥ (1 − σ)𝔼(f_{j})/2

whenever n ≥ M and 0 ≤ j ≤ N. This implies that

inf_{M≤n≤M+N} ∑_{j=0}^{N}(P′)^{n−j}f′_{j} ≥ (1 − σ) ∑_{j=1}^{N} 𝔼(f_{j})/2 = (1 − σ)α/2.

Adding this with the preceding inequality we obtain (^{\underline{fillip}} 1.7), and the claim follows. □

References

ornstein [1] D. Ornstein, On the pointwise behavior of iterates of a self-adjoint operator, J. Math. Mech. 18

(1968/1969), 473–477.

rota [2] G.-C. Rota, An “Alternierende Verfahren” for general positive operators, Bull. Amer. Math. Soc.

68 (1962), 95–102.

stein [3] E.M. Stein, On the maximal ergodic theorem, Proc. Nat. Acad. Sci. USA 47 (1961), 1894–1897.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA.

E-mail address: tao@math.ucla.edu