(Updated with endpoint )

I’ve written down an almost self contained exposition of basic properties of Lorentz spaces. I’ve found the sources on the subject to leave something to be desired, and I grew a bit confused at the beginning. Therefore this relatively short note (I might be ruining someone’s assignments out there, but I think the pros of writing down everything in one place balance the cons).

Here’s a link to the pdf version of this post: Lorentz spaces primer

**1. Lorentz spaces **

In the following take otherwise specified, and a -finite measure space with no atoms.

The usual definition of Lorentz space is as follows:

Definition 1The space is the space of measurable functions such that

where is the decreasing rearrangement [I] of . If then define instead

A few things are to be noticed here. First of all, is a quasi-norm. This follows from known properties of rearrangements: indeed, , thus

and the constants implied are bigger than 1. Another thing to notice is that , as

The spaces are the usual weak spaces: for small by definition of it follows indeed that , from which

and by taking the limit one recovers the usual weak norm; as for the opposite direction, it is , and thus

and by taking the limit again one concludes the two definitions agree. It is worth noticing that spaces are not just quasi-normed, they are indeed normed spaces.

Proposition 2For , with , define

Then is a norm, and

*Proof:* That it is a norm is trivial. As for the equivalence with the Lorentz quasi-norm, by taking one has

and by taking the supremum in one has . As for the other direction, since , one can estimate

thus proving the equivalence.

An important inequality is the so called Hardy-Littlewood inequality for rearrangements,

Theorem 3 (Hardy-Littlewood inequality)For any measurable and vanishing at infinity one has

*Proof:* By the layer cake representation (see below) one can write

and therefore

which by Fubini is

but now the sets are one contained inside the other thus we can substitute the minimum by the intersection,

and by Fubini again, since ,

From this last inequality it follows that the dual of is :

Proposition 4 (Duality of Lorentz spaces)It is

*Proof:* In one direction, notice that by H\”{o}lder it is simply

The opposite direction is a bit more complicated but we reproduce it here briefly for the sake of completeness. The interested reader can find more details in Grafakos’ Classical Fourier Analysis [Grafakos] (although a big part of it is given as an exercise, with a reasonable amount of hints). A slightly different proof is in [Bennett, Sharpley]. We need a few facts: first of all that for and

to prove it you can consider the multiplicative group and its Haar measure , take the convolution of with (w.r.t. to the group!), and finish by Young’s inequality.

Next we need to know a refinement of Hardy-Littlewood inequality: namely that

where means that and are equidistributed: for any . This can be proven by observing that, since the measure is non-atomic, for any set of finite non-zero measure we can find a set s.t. and . To prove it, fix and take with measure – which you can find, because there are no atoms. Then layer cake representation applied to proves the claim. With this one can prove (2) by applying it to the case of positive simple function. Then with all positive, and if one can write with nested, namely ; and by taking the sets as before (i.e. and ) we can form for which , and verify that

For a general use approximation, since the simple functions are dense in for .

Now we’re ready to finish the proof. By usual considerations, akin to those needed to conclude duality for spaces, we can see that by Radon-Nykodim any functional in is given by integration against a measurable function. Fix such a linear functional and call the associated function and the operator norm of as a linear functional. We need to prove . To do so, take a function whose rearrangement is which you can do (again, prove it on simple functions for a generic choice to convince yourself; notice is decreasing by definition). Then, by formula (1)

(if finite). Now notice that by formula (2), since , it is

On the other hand

It follows that , on the condition that the quasi-norm be finite. But the measure is -finite, and thus we can find a sequence of nested sets of finite measure covering all of , and on each one the quasi-norm is indeed finite and we get the result by taking the limit in .

We can state with no ambiguity that a linear (or sublinear) operator is of **strong type** if it maps to boundedly, that it is of **weak type** if it maps to boundedly, and it is of **restricted weak type** if it maps to boundedly. The first two definitions agree with the classical ones by the discussion above. To see the last definition agrees with the usual one, one has to notice that the unit ball in is the convex hull of normalized characteristic functions . Indeed , and then

Anyway, we won’t prove this here. Details can be found in [de Souza] or [Sadosky]. For our interpolation purposes, we only need to know that the usual definition of restricted weak type implies this one given here. Recall an operator is usually called of restricted weak type if for every measurable of finite measure it is

Proposition 5If is a linear operator such that

for every measurable, then maps into boundedly. The opposite is true as well, since .

*Proof:* Take to be a simple function of the form

where and (a positive simple function can always be written like this). Then its decreasing rearrangement is

since is a normed space (see Proposition 2) we have

On the other hand

The general result follows by the fact that simple functions are dense in for .

Remark 1Another common definition for restricted weak type is given in dual terms as

This definition is equivalent to the one above: take , then

which implies , and by taking the supremum in it follows . As for the opposite direction, trivially by Hardy-Littlewood inequality one has

Next we introduce a different definition of Lorentz space and prove it’s equivalent to definition 1. This definition has the advantage of making interpolation proofs easier (at the cost of proving the equivalence of the two definitions, of course).

Definition 6[Atomic decomposition of ] A function is in if it can be decomposed as

where , and the functions ‘s have supports disjoint from each other and the properties and .

Notice that by this definition we get for free that when , because of the corresponding property of the spaces. *Proof:* We prove definition 1 implies 6 first. Consider the expression for the quasi-norm and write

Now put this aside for a moment and consider the expression . By the so-called layer cake representation formula one has

notice though that we can always ignore the last two terms on the RHS by assuming that for every , and we can always do so [II]. Going back to the above expression then we have, since ,

and we define . Notice . Then, since , one also has , if our assumption that holds; then

(when not empty – something that happens only if ). Therefore

which implies that . Then we can set , and then write

and has exactly the properties we ask for.

Now we prove the opposite, namely that 6 implies 1. Suppose first that . Again

where is defined as before. By the disjointness of supports of ‘s the RHS is

and by the properties of the integral is bounded by . We then sum in first, optimizing accordingly: we get two terms, the first one being

and the second one being

and thus we’ve proved . As for the case , we proceed by duality, since then . Thus we have to verify that

and by the previous point this amounts to prove that for all with one has

(with constant depending on , of course). Then decompose as before and reduce to estimate

since for any , we can optimize choosing a particular for s.t. and a different for , in such a way that the last sum is bounded by

which proves the claim. (The case is left as an easy exercise).

Notice we have effectively proved that

where the infimum is taken over all the possible decompositions of such as in definition 6.

** 1.1. Interpolation of Lorentz spaces **

Theorem 7 (Upgrading uniform restricted weak type to strong type)Let be of restricted weak type for all in a neighbourhood of , with uniform constant. Then is of strong type .

*Proof:* We proceed by duality. By the decomposition lemma and the observation at the end of its proof, it suffices to prove

for any two functions , which can be worsened to

Now, observe that . Indeed, since then , and since also , and therefore

same holds for , namely . Therefore, by assumption (and duality) it is

with constant uniform for in a neighbourhood of . Then our expression above on the LHS is bounded by

where the infimum is taken in the aforementioned neighbourhood of . This sum looks like one we estimated before, and we optimize again the term

Since , we choose bigger or smaller than according to the sign of ; then again the sum is bounded by

which is exactly what we wanted.

Therefore we can bootstrap local restricted weak type estimates to strong estimates in the same range. An evident restriction of this is that , so we ask whether we can interpolate between restricted weak type estimates. The answer is provided by the following

Proposition 8Assume . If a linear operator is restricted weak type and restricted weak type , then it is also of restricted weak type for any , where

*Proof:* Assume first that . The proof is immediate: again by duality one has to prove

for . But then

As for the case , we prove (the result then follows by Proposition 5 and the subsequent remark). Notice in this case. Therefore, by hypothesis we have

and by taking the geometric average

The last two proofs are taken from Tao’s notes in [Tao].

Notice that if we have that an operator is restricted weak type and bounded, then by the combination of the two propositions above it follows that is of strong type for every .

[I] : That is, . This implies immediately that . Another useful equality is .

[II] : More precisely, for any given there exists a function such that a.e. on and such that . For a proof of this easy statement see [Carro, Mart\'{i}n], which also helps getting used to rearrangements.

References:

- Bennett, Sharpley: C. Bennett and R. Sharpley,
*Interpolation of operators*, Academic Press, Inc., Orlando, Florida. - Carro, Mart\'{i}n: M. J. Carro and J. Mart\'{i}n,
*A useful estimate for the decreasing rearrangement of the sum of functions*, Quart. J. Math., vol. 55 (2004), no. 1, 41-45. - Grafakos: L. Grafakos,
*Classical Fourier Analysis*, Springer, Graduate Texts in Mathematics (Book 249). - Sadosky: C. Sadosky,
*Interpolation of operators and singular integrals*, Marcel Dekker Inc., 1979 - de Souza: G. S. de Souza,
*A proof of Carleson’s theorem based on a new characterization of the Lorentz spaces {} for {} and other applications*. - Tao: T. Tao,
*Real interpolation of Lorentz spaces*, \url{http://www.math.ucla.edu/~tao/preprints/Expository/interpolation.dvi}

Reblogged this on Being simple and commented:

a great post, in the future I will post some important estimates for semilinear evolution equations on euclidian spaces and half-euclidian spaces, such as, Heat estimates, wave estimates, Yamazaki estimates, etc.