THE REAL INTERPOLATION OF LORENTZ SPACES

TERENCE TAO

Abstract. We discuss the real interpolation of Lorentz spaces.

1. Lorentz spaces

Let X be a measure space. For any integer k, we define F_{k}(X) to be the space of all functions f on X which are supported on a set of measure at most 2^{k} and which satisfy ∥f∥_{∞} ≤ 1.

Definition 1.1. For any 0 < p ≤ ∞ and 1 ≤ q ≤ ∞, we define the Lorentz space L^{p,q}(X) to be the space of all functions f of the form

f = ∑_{k} c_{k}2^{−k/p}f_{k}

where f_{k} ∈ F_{k}(X) and non-negative c_{k} ∈ l^{q}, with the L^{p,q} norm being the infimum of the set of all possible ∥c∥_{q}.

For a proof that this definition corresponds to the usual definition of L^{p,q}, see [1]. The space L^{p,p} is strong L^{p}, the space L^{p,∞} is weak L^{p}, and the space L^{p,1} has a unit ball which is the convex hull of the normalized characteristic functions |E|^{−1/p}χ_{E}, if p ≥ 1. Note that L^{p,q_{1}} ⊂ L^{p,q_{2}} when q_{1} ≤ q_{2}.

Exercise: if 1 < p, q < ∞ then the spaces L^{p,q} and L^{p′,q′} are dual.

2. From restricted weak-type to strong type

Let T be a linear operator from functions on X to functions on Y; let us make the a priori assumption that T is bounded from every L^{p} to every L^{q}, but with possibly a large constant; our bounds below will not depend on this constant.

We say that T is of strong type (p, q) if T maps L^{p} to L^{q}, and is of restricted weak-type (p, q) if T maps L^{p,1} to L^{q,∞}. Suppose 1 < p, q < ∞. By duality, the former statement is equivalent to

|Λ(f, g)| ≲ ∥f∥_{p}∥g∥_{q′}

and the latter is equivalent to

|Λ(f, g)| ≲ ∥f∥_{p,1}∥g∥_{q′,1}  (1)

where Λ is the bilinear form

Λ(f, g) = ⟨T f, g⟩.

Since L^{p,1} and L^{q′,1} are atomic spaces, (1) is equivalent to the statement that

|Λ(f_{k}, g_{l})| ≲ 2^{k/p}2^{l/q′}  (2)

for all integers k, l and all f_{k} ∈ F_{k}(X), g_{l} = F_{l}(Y).

We now show how restricted weak-type estimates can be bootstrapped to strong type estimates.

Proposition 2.1 (Special case of Marcinkiewicz interpolation). Let 1 < p_{0} < ∞, and suppose T is of restricted weak-type (p, p) for all p in a neighbourhood of p_{0}. Then T is of strong type (p_{0}, p_{0}).

Proof We have to show that

|Λ(f, g)| ≲ ∥f∥_{p_{0}}∥g∥_{p′_{0}}.

By our characterization of L^{p_{0}}, it suffices to show that

|Λ(∑_{k} c_{k}2^{−k/p_{0}}f_{k}, ∑_{l} d_{l}2^{−l/p′_{0}}g_{l}⟩| ≲ 1

where f_{k} ∈ F_{k}(X), g_{l} = F_{l}(Y), ∥c∥_{p_{0}} ≲ 1 and ∥d∥_{p′_{0}} ≲ 1.

By linearity we may estimate the left-hand side by

∑_{k} ∑_{l} c_{k}d_{l}2^{−k/p_{0}}2^{−l/p′_{0}}|Λ(f_{k}, g_{l})|,

which by (2) and our hypothesis is estimated by

∑_{k} ∑_{l} c_{k}d_{l}2^{−k/p_{0}}2^{−l/p′_{0}} inf_{p} 2^{k/p}2^{l/p′}.

Optimizing this in p, we estimate this by

∑_{k} ∑_{l} c_{k}d_{l}2^{−ε|k−l|}.

We re-arrange this as

∑_{s} 2^{−ε|s|} ∑_{k} c_{k}d_{k+s}.

Using Hölder and our normalization on c, d, this is bounded by

∑_{s} 2^{−ε|s|} ≲ 1

as desired.

Interpolating from one restricted weak-type estimate to another is even easier: if T is of restricted weak-type (p_{i}, q_{i}) for i = 0, 1 and 1 < p_{i}, q_{i} < ∞ then it is also of restricted weak-type (p_{θ}, q_{θ}) for all 0 ≤ θ ≤ 1 where 1/p_{θ} = (1 − θ)/p_{0} + θ/p_{1} and similarly for q_{θ}; this follows simply by taking various geometric means of (2) applied to (p, q) = (p_{0}, q_{0}) and (p, q) = (p_{1}, q_{1}).

3. Interpolation below L^{1}.

When q ≤ 1 the equivalence between restricted weak-type and (1) breaks down, however there is an acceptable substitute.

Lemma 3.1. Suppose 1 ≤ p ≤ ∞ and 0 < q ≤ ∞, and T is of restricted weak-type (p, q). Then for every pair of integers k, l and all f_{k} ∈ F_{k}(X), g_{l} ∈ F_{l}(Y), we can find g′_{l} ∈ F_{l−1}(Y) such that

|Λ(f_{k}, g_{l} − g′_{l})| ≲ 2^{k/p}2^{l/q′}.  (3)

If q > 1 then one could simply telescope this to obtain (1), but for q ≤ 1 this is not possible as q′ is non-positive. The implication is not quite reversible for q < 1, but it is very close. (In practice, the definition (3) seems to be a better notion to use than restricted weak-type in any event). Proof Let f_{k}, g_{l} be as above. Since ∥T f_{k}∥_{q,∞} ≲ ∥f_{k}∥_{p} ≲ 2^{k/p} by hypothesis, we have

|\{|T f_{k}| > C2^{−l/q}2^{k/p}\}| ≤ \frac{1}{2}2^{l}

for sufficiently large C. The claim then follows by letting g′_{l} be the restriction of g_{l} to the above set.

Using this, we can now interpolate below L^{1}. A representative result is as follows.

Proposition 3.2. Let T be an operator such that both T and T^{∗} is of weak-type (1,1). Then T is bounded on L^{p} for every 1 < p < ∞.

Proof It suffices to show that T is of restricted weak-type (p, p) for all such p, i.e. that

|Λ(f_{k}, g_{l})| ≲ 2^{k/p}2^{l/p′}

for all f_{k} ∈ F_{k}(X), g_{l} ∈ F_{l}(Y).

Fix p, and let A be the best constant such that

|Λ(f_{k}, g_{l})| ≤ A2^{k/p}2^{l/p′}  (4)

for all k, l, f_{k}, g_{l}. By our a priori assumption A is finite.

Now suppose k ≤ l, so 2^{k} ≤ 2^{k/p}2^{l/p′}. Since T is of restricted weak-type (1,1), we can find g_{l−1} such that

|Λ(f_{k}, g_{l} − g_{l−1})| ≲ 2^{k} ≤ C2^{k/p}2^{l/p′}

Combining this with (4) with g_{l} replaced by g_{l−1}, we obtain

|Λ(f_{k}, g_{l})| ≤ (A2^{−1/p′} + C)2^{k/p}2^{l/p′}.

Applying the same type of reasoning when k > l, but using T^{∗} instead of T, we see that the above estimate holds for all k, l. Taking suprema we obtain

A ≤ A2^{−1/p′} + C.

Thus A is finite, and the claim follows.

Thanks to Loukas Grafakos for helpful comments.

References

[1] M. Keel, T. Tao, Endpoint Strichartz Estimates, Amer. Math. J. 120 (1998), 955–980.

Department of Mathematics, UCLA, Los Angeles, CA 90024

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