AN ATOMIC DESCRIPTION OF ORLICZ SPACES, AND SOME EXTRAPOLATION THEORY

TERENCE TAO

Abstract. We give an atomic description of the spaces L log^{r} L.

1. Notation

Let X be a measure space, and r ≥ 0 be a number. We define the unit ball of L log^{r} L to consist of those functions f such that

∫_{X} |f| log^{r}(2 + |f|) ≤ 1;

this defines a Banach space. Define an L log^{r} L atom to be any function f supported on a set E of dyadic size such that

∥f∥_{∞} ≤ |E|^{−1} log(2 + \frac{1}{|E|})^{r}.

Proposition 1.1. L log^{r} L is the atomic space generated by its atoms. In fact, one can use just a single atom for each dyadic size of |E|.

Proof Clearly each atom is in L log^{r} L. Conversely, for any function f in the unit ball of L log^{r} L, we need to find co-efficients c_{k} and atoms a_{k} such that f = ∑_{k} c_{k}a_{k} and ∑_{k} |c_{k}| ≲ 1.

We may assume that f is non-negative. Since L log^{r} L is rearrangement invariant, we may assume that X is the positive real line with Lebesgue measure, and that f is non-increasing. For each integer k, we define

c_{k} = 2^{k}f(2^{k}) log(2 + 2^{−k})^{−r}

and

a_{k}(x) = c_{k}^{−1}f(x)χ_{[2^{k},2^{k+1})}(x).

Clearly f = ∑_{k} c_{k}a_{k} and the a_{k} are L log^{r} L atoms. To finish the proof, we have to show that the c_{k} are summable, that is

∑_{k} 2^{k}f(2^{k}) log(2 + 2^{−k})^{−r} ≲ 1.

We first consider the contribution of the positive k. In this case we may estimate the left-hand side by the L^{1} norm of f, which is bounded.

Thus we may restrict ourselves to negative k. Since f is non-decreasing, we may bound this expression by

C ∫_{0}^{1} f(t) log(2 + 1/t)^{−r} dt.

The portion of the integral where f(t) ≤ t^{−1/2} is clearly acceptable, so we may assume that f(t) ≥ t^{−1/2}. But then we may estimate the above by

C ∫_{0}^{1} f(t) log(2 + f(t))^{−r} dt ≲ 1.

A similar claim holds for L^{p} log^{r} L functions. If we define an L^{p} log^{r} L atom as above but with

∥f∥_{∞} ≤ |E|^{−1/p} log(2 + \frac{1}{|E|})^{r/p}

then every f in the unit ball of L^{p} log^{r} L can be written as f = ∑_{k} c_{k}a_{k} where the a_{k} are L^{p} log^{r} L atoms with |E| = 2^{k} and the c_{k} are summable in l^{p}. We leave this as an exercise to the reader.

If X is of bounded measure, then of course we only need to deal with atoms whose support also has bounded measure.

2. Extrapolation

We can now easily prove Yano’s extrapolation theorem:

Lemma 2.1. Let X be a space of finite measure, and suppose that the L^{p} operator norm of T grows like O((p − 1)^{−r}) as p → 1. Then T maps L log^{r} L to L^{1}.

Proof We have

∥T ∥_{L^{p}→L^{p}} ≲ \frac{1}{(p − 1)^{r}} .

To show boundedness on L log^{r} L to L^{1} it suffices to show it for atoms. So let f be an atom on a set E, with |E| = 2^{k}; we may assume k ≤ −C. A calculation shows that

∥f∥_{p} ≲ 2^{−k/p′}|k|^{−r};

from this and Hölder’s inequality we have

∥T f∥_{1} ≲ ∥T f∥_{p} ≲ \frac{2^{−k/p′}}{(|k|(p − 1))^{−r}} .

The claim now follows by choosing p′ = |k|.

The converse is not quite true, even if we assume that T is bounded on, say, L^{2}; this can be seen by considering the rank 1 operator

T f = n^{r}2^{n}⟨f, χ_{E}⟩χ_{F}

where E, F are fixed sets such that |E| = 2^{−n} and |F | = n^{2r}2^{−n}, and n is a large number. The best we can say is that T maps L^{p} log^{q} L to L^{p} for all 1 < p < 2 and q = r(2/p − 1).

However, if we assume some sort of translation invariance, then the implication can be reversed:

Proposition 2.2. [1] Let T be a translation-invariant operator on the torus which is bounded on L^{2} and maps L log^{r} L to L^{1}. Then the L^{p} operator norm of T grows like O((p − 1)^{−r}) as p → 1.

The proof is a little tricky. Here is a weaker result which contains one of the two core ideas, namely using factorization theory to exploit translation invariance. (The other core idea is to replace the dyadic decomposition by a more widely spaced decomposition based on p).

Proposition 2.3. Let T be a translation-invariant operator on the torus which is bounded on L^{2} and maps L log^{r} L to L^{1}. Then the L^{p} operator norm of T grows like O((p − 1)^{−r−1}) as p → 1.

Proof By duality it suffices to show that

|⟨T f, g⟩| ≲ (p − 1)^{−r−1}

for all 1 < p < 2 and all functions f, g normalized in L^{p}, L^{p′}. By the atomic decomposition we may write f = ∑_{k} c_{k}a_{k} and g = ∑_{l} d_{l}b_{l} where k, l range over the negative integers, a_{k} and b_{l} are L^{p} and L^{p′} atoms adapted to sets of measure 2^{k} and 2^{l} respectively. It then suffices to show that

∑_{k} ∑_{l} c_{k}d_{l}|⟨T a_{k}, b_{l}⟩| ≲ (p − 1)^{−r−1}.

We shall show that

|⟨T a_{k}, b_{l}⟩| ≲ 2^{−(k−l)( \frac{1}{p} − \frac{1}{2} )}

when l ≤ k + C and

|⟨T a_{k}, b_{l}⟩| ≲ 2^{−(l−k)/p′}(l − k)^{r}

when l > k + C. The claim then becomes

⟨c ∗ K, d⟩ ≲ (p − 1)^{−r−1}

where K(t) = 2^{t( \frac{1}{p} − \frac{1}{2} )} for t < C and K(t) = 2^{−t/p′}t^{r} for t ≥ C. Since ∥K∥_{1} ∼ (p − 1)^{−r−1}, the claim follows from Young’s inequality and Hölder’s inequality.

The first bound follows immediately from Cauchy-Schwarz and the L^{2} boundedness of T. To show the second bound, let E and F denote the support of a_{k} and b_{l} respectively, and define A by

∫_{F} |T a_{k}| = A2^{k/p′}

where F is the support of b_{l}. It suffices to show that A = O((l − k)^{r}). If l = O(1) this would follow directly from the L log^{r} L → L^{1} mapping property of T, but for l large and negative one must invoke some factorization theory.

Let M = c2^{−l} for some constant c ≪ 1, and Ω_{1}, …, Ω_{M} be a collection of translations on the torus such that

∫_{Ω_{J} (E)} ∑_{j<J} χ_{E} ∘ Ω_{j} ≲ 2^{k}J2^{k}

and

∫_{Ω_{J} (F)} ∑_{j<J} |χ_{F} T a_{k}| ∘ Ω_{j} ≲ 2^{l}JA2^{k/p′}  (1)

for all 1 ≤ J ≤ M. Such a family can be constructed inductively, since once Ω_{1}, …, Ω_{J−1} is determined, there is a positive probability that Ω_{J} satisfies the

4 TERENCE TAO

above two properties. From the first property we have

∥ ∑_{j=1}^{M} χ_{E} ∘ Ω_{j}∥_{2} ≲ 2^{−l/2}2^{k/2}

so that

∥ ∑_{j=1}^{M} ±a_{k} ∘ Ω_{j}∥_{2} ≲ 2^{−l/2}2^{−k/p}2^{k/2}.

Since the function inside the norm is supported on a set of measure O(2^{k−l}), we thus have (by some distribution function estimates)

∥ ∑_{j=1}^{M} ±a_{k} ∘ Ω_{j}∥_{L log^{r} L} ≲ 2^{−l}2^{k/p′}(l − k)^{r}.

Thus we have

∥ ∑_{j=1}^{M} ±T a_{k} ∘ Ω_{j}∥_{1} ≲ 2^{−l}2^{k/p′}(l − k)^{r}.

and thus by randomizing the signs that

∥(∑_{j=1}^{M} |χ_{F} T a_{k} ∘ Ω_{j}|^{2})^{1/2}∥_{1} ≲ 2^{−l}2^{k/p′}(l − k)^{r}.

However, from (1) and (??) we have (if c is sufficiently small)

∥(∑_{j=1}^{J} |χ_{F} T a_{k} ∘ Ω_{j}|^{2})^{1/2}∥_{1} − ∥(∑_{j=1}^{J−1} |χ_{F} T a_{k} ∘ Ω_{j}|^{2})^{1/2}∥_{1} ≳ A2^{−k/p′}

for all 1 ≤ J ≤ M, hence

∥(∑_{j=1}^{M} |χ_{F} T a_{k} ∘ Ω_{j}|^{2})^{1/2}∥_{1} ≳ 2^{−l}A2^{k/p′}.

The desired estimate on A then follows by combining the two estimates.

References

[1] T. Tao, A converse extrapolation theorem for translation invariant operators, to appear. Department of Mathematics, UCLA, Los Angeles, CA 90024

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