HÖLDER’S INEQUALITY FOR LORENTZ SPACES

TERENCE TAO

Abstract. We prove Hölder’s inequality for Lorentz spaces.

1. Introduction

The purpose of this note is to prove the estimate

∥fg∥_{L^{p,q}} ≲ ∥f∥_{L^{p_{1},q_{1}}} ∥g∥_{L^{p_{2},q_{2}}}  (1)

for all 0 < p, q, p_{1}, q_{1}, p_{2}, q_{2} ≤ ∞ such that 1/p = 1/p_{1}+1/p_{2} and 1/q = 1/q_{1}+1/q_{2}. As far as we know this result first appears in work of O’Neil [1], and nowadays could be proven by real bilinear interpolation. However, we shall give a direct proof here.

The main engine will be the following characterization of L^{p,q}:

Lemma 1.1. A function f is in L^{p,q} iff one can find for each integer j a set E_{j} of measure O(2^{j}) and a coefficient a_{j} such that (a_{j}) ∈ l^{q} and we have the pointwise estimate

|f(x)| ≲ ∑_{j} a_{j}2^{−j/p}χ_{E_{j}}(x).

In fact, the L^{p,q} norm is comparable to the best possible l^{q} norm of the a_{j}.

We leave this easy lemma to the reader; it is best proven by using the non-decreasing rearrangement of |f|.

Now we prove (1). We may write

|f| ≲ ∑_{j} a_{j}2^{−j/p_{1}}χ_{E_{j}}, |g| ≲ ∑_{k} b_{k}2^{−k/p_{2}}χ_{F_{k}}

where (a_{j}) ∈ l^{q_{1}}, (b_{k}) ∈ l^{q_{2}}, |E_{j}| = O(2^{j}), and |F_{k}| = O(2^{k}). Thus we may estimate fg by

∑_{j} ∑_{k} a_{j}b_{k}2^{−j/p_{1}}2^{−k/p_{2}}χ_{E_{j}∩F_{k}}.

Let us consider the contribution when k ≥ j. We can rewrite this as

∑_{s=0}^{∞} ∑_{j} a_{j}b_{j+s}2^{−j/p}2^{−s/p_{2}}χ_{E_{j}∩F_{j+s}}.

We can estimate this by

∑_{s=0}^{∞} 2^{−s/p_{2}} ∑_{j} a_{j}b_{j+s}2^{−j/p}χ_{E_{j}}.

By classical Hölder, the sequence a_{j}b_{j+s} is in l^{q}, and so the inner sum is in L^{p,q} uniformly in s. The claim then follows by the following

Lemma 1.2. Let X be a quasi-Banach space, and let f_{s} be a sequence of elements of X for non-negative integers s such that ∥f_{s}∥_{X} ≲ 2^{−εs}. Then ∥ ∑_{s≥0} f_{s}∥_{X} ≲ 1.

Proof By refining f by a finite factor and re-labelling s we may improve the decay to ∥f_{s}∥_{X} ≤ A2^{−C_{1}s} for some large C_{1} to be chosen later. By backwards induction on n one can then show that (if C_{1} is sufficiently large)

∥ ∑_{s=n}^{m} f_{s}∥_{X} ≤ CA2^{−C_{1}n}

for all integers n, m. The claim follows.

One can relax the hypothesis on the X norms of ∥f_{s}∥_{X} from exponential decay to rapid decrease, by summing dyadic blocks of functions using a binary tree, but we shall not do so here.

References

[1] R. O’Neil, Convolution operators and L(p, q) spaces, Duke Math. J. 30, 129-143 (1963).

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

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