LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH

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1 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH GUILLERMO REY. Introduction If an operator T is bounded on two Lebesgue spaces, the theory of coplex interpolation allows us to deduce the boundedness of T between the two spaces. Real interpolation (of L p spaces) provides, in a way, a refineent of coplex interpolation, this will be shown as we progress. We will follow Tao s notes [6]: in particular, we clai no originality over any arguent here. Consider the (centered) Hardy-Littlewood axial operator: () Mf(x) = r>0 f(y) dy, defined for f L loc. We would like to obtain Lp boundedness of this operator for p, but alas, this is not the case: although it is trivially bounded fro L onto itself, it is not bounded on L : B r(x) Proposition.. If f L (R d ) and f is not identically 0 then Mf(x) C d x d for soe constant C d dependent on f and the diension, in particular f / L (R d ). Proof. Since f is not identically 0, there exists an R > 0 such that f δ > 0. Now, if x > R, B R (0) B(x, 2 x ), so Mf(x) B R (0) B(x, 2 x ) B R (0) δ f C d x d. Thus M is indeed unbounded as an operator fro L onto itself. However, We do have the following weaker for of L boundedness; proved by Hardy-Littlewood in [] for d = and by Wiener in [7] for d > : Theore.2. There exists a constant C > 0 such that for all f L where is the Lebesgue easure on R d. ({x R d : Mf(x) > λ}) C f L λ,

2 2 GUILLERMO REY Observe that, by Chebychev s inequality, we would have had this bound if M had been bounded on L, so it is indeed a weaker stateent. This leads us to study the space of functions satisfying Chebychev s inequality: (2) µ({x : f(x) > λ}) f p L p (,µ) λ p. More precisely, define the weak-l p space L p, (, µ) to be the set of all easurable functions f for which its L p, quasi-nor (3) f L p, = λµ({x : f(x) > λ}) /p λ>0 is finite. To abbreviate we will use the notation d f (λ) = µ({x : f(x) > λ}) when there is no confusion as to what easure we are using. Also, we define by convention L, to be L. The following lea is a siple application of the onotone convergence lea: Lea.3. Let {f n } n N be a sequence of easurable functions such that f n (x) f n+ (x) µ-alost everywhere and for all n N. If we have f n f µ-alost everywhere for soe function f then d fn (λ) d f (λ) when n for all λ > 0. As we have seen, Chebychev s inequality gives us the inclusion L p L p, for p > 0. This inclusion is proper as the following exaple shows: Exaple.4. If f(x) = x d p then f(x) p dx = R d x d dx =, R d however where ν d is the easure of the unit ball in R d. λd f (λ) /p = ν /p d <, λ>0 As we have noted, L p, is not a priori a nor, but only a quasi-nor: Proposition.5. We can easily see that d f+g (λ) d f (λ/2) + d g (λ/2), which is just the fact that when two ters su ore than λ, at least one of the has to be greater than λ/2. Using this fact we have: f + g L p, = λd f+g (λ) /p λ>0 λ (d f (λ/2) + d g (λ/2)) /p λ>0 2C p ( f L p, + g L p, ). We will generalize the weak-l p spaces a bit further, bit first we need the following well-known identity:

3 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 3 Proposition.6. Let 0 < p < and f be a easurable function, then ( f L p = p /p λ p d f (λ) dλ ) /p. R λ + Proof. and So we have: f p L p (,µ) = f(x) p dµ(x) f(x) Fro this, we could tentatively define the = pλ p dλ λ dµ(x) 0 = p λ p dλ {s: s< f(x) } λ dµ(x) R + = p λ p {y: f(y) >λ} dµ(x) dλ λ (Fubini) R + = p λ p d f (λ) dλ λ. R + f L p, = λd f (λ) /p L (R +, dλ λ ) f L p = p /p λd f (λ) /p L p (R +, dλ λ ). Definition.7 (Lorentz quasi-nor). Let (, µ) be a easure space and let 0 < p < and 0 < q, we define the Lorentz space quasi-nor L p,q (,µ) as f L p,q (,µ) = p /q λd f (λ) /p Lq (R +, dλ λ ). Observe that, with this definition, L p,p (isoetrically) coincides with L p. We have the following useful lea, which is a kind of Monotone Convergence Theore. Lea.8. Let {f n } n N be a sequence of easurable functions such that f n f µ-alost everywhere, then f L p,q = li f n L p,q n Proof. We can obviously assue f to be positive so, by definition, we only have to verify that li n λd f n (λ) /p L q (R +, dλ λ ) = λd f (λ) /p L q (R +, dλ λ ). Note that, since f n is non-decreasing, d fn ( ) is also increasing (in n), so in particular by Lea.3 d f (λ) = d fn (λ) n for all fixed λ > 0. If q = then the proble is reduced to the easier one of showing that n λd fn (λ) /p L λ (R + ) = λd fn (λ) /p L λ (R + ), n which is easily seen to hold true in the ore general case g n L n = g n L. n

4 4 GUILLERMO REY Suppose now that q <, then n λd fn (λ) /p L q (R +, dλ λ ) = ( n 0 λ q d fn (λ) q/p dλ λ ) /q, and since, as we noted before, we have λ q d fn (λ) q/p λ q d f (λ) q/p for all fixed λ > 0, we can apply the onotone convergence theore with the easure dλ λ and the desired identity. The following Fatou-like lea shows that the Lorentz space quasi-nor is lower seicontinuous. Lea.9. Let {f n } n N be any sequence of easurable functions, then li inf f n L p,q n li inf f n L p,q n Proof. Let f = li inf n f n, then, by definition, we have to check that λd f (λ) /p L q (R +, dλ li inf λd fn (λ) /p λ ) n L q (R +, dλ λ ), which would follow fro the lower-seicontinuity of the usual L q spaces provided that we have This is equivalent to showing d f (λ) li inf d fn (λ). n µ({x : li inf f n (x) > λ}) li inf µ({x : f n (x) > λ}), n n which follows fro Fatou s lea. Finally, we can prove that the L p,q spaces are quasi-banach spaces; which follows the steps of the classical proof in the L p case. Theore.0. Let (, µ) be a easure space and let 0 < p <, 0 < q, then L p,q (, µ) is a quasi-banach space, that is, it is coplete and it satisfies the quasi-triangle inequality. Proof. We have already proven that L p, satisfies the quasi-triangle inequality in Proposition.5, the proof for L p,q when q is alost identical so we will oit it. We are left to prove that it is coplete. To this end first note that the L p,q nor always controls the L p, nor: ( λd f (λ) /p L q (R +, dλ λ ) = λ q d f (λ) q/p dλ λ 0 ( s λ q d f (λ) q/p dλ λ 0 d f (s) /p ( s 0 = sd f (s) /p q /q, for any s > 0. Thus, taking the reu on s, we obtain: f L p, p,q f L p,q. λ q dλ λ ) /q ) /q ) /q In particular we have the following generalization of Chebychev s inequality: (4) d f (λ) p,q f p L p,q λ p. Now let {f n } be a Cauchy sequence in the L p,q quasi-nor. Estiate (4) allows us to deduce that {f n } is Cauchy in easure, and hence that there exists a µ-alost everywhere convergent

5 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 5 subsequence {f ϕ(k) } k. Call f the µ-alost everywhere defined liit of f ϕ(k) when k, we will show that f n f in the L p,q quasi-nor. Let C be the constant in the quasi-triangle inequality for the L p,q quasi-nor and refine the subsequence {f ϕ(k) } so that we have f ϕ(k) f ϕ(k ) L p,q (2C) k using that f n is an L p,q -Cauchy sequence and defining f ϕ(0) = 0. Observe that we have the following telescoping su identity: n (5) f ϕ(n) = f ϕ(k) f ϕ(k ). k= Recall that we want to estiate f f n L p,q and that, using the quasi-triangle inequality, we have f f n L p,q C ( f f ϕ(n) L p,q + f ϕ(n) f n L p,q). The second ter can be ade arbitrarily sall using the fact that f n is an L p,q -Cauchy sequence, so we choose n and N so large that f ϕ(n) f n L p,q < ɛ 2C. For the first ter we can use (5) to obtain: N f f ϕ(n) L p,q = (f ϕ(k) f ϕ(k ) ) (f ϕ(k) f ϕ(k ) ) (6) k= = li M M k=n+ li M M k=n+ li inf M M k=n+ k= ( fϕ(k) f ϕ(k ) ) Lp,q f ϕ(k) f ϕ(k ) f ϕ(k) f ϕ(k ) L p,q L p,q, L p,q where in the last line we have used Proposition.9. By induction, we can iteratively use the quasi-triangle inequality to exchange and L p,q and bound the last ter in (6) by li inf M M k=n+ Choosing N so large that (2C) N C k N f ϕ(k) f ϕ(k ) L p,q < ɛ 2C li inf M copletes the proof. M k=n+ = C N k=n+ (2C) N. 2. The Dyadic Decoposition C k N (2C) k It turns out that we will be able to decopose functions in L p,q into sipler functions: Sub-step functions: A sub-step function of height H and width W is any easurable function f ported on a set of easure W and which satisfies f H. 2 k

6 6 GUILLERMO REY Quasi-step function: A quasi-step function of height H and width W is any easurable function f ported on a set of easure W and which satisfies f H in its port. Exaple 2.. Let f be a quasi-step function of height H and width W, then f L p,q HW /p ( p q ) /q. We will give two different decopositions: a vertically-dyadic decoposition; where we decopose the range of our function into dyadic pieces, and a horizontally-dyadic one; where we dyadically decopose its port. Let {γ } Z be a sequence in R +, we say that it is of lower exponential growth if we have C γ γ + for soe constant C >. We will say that it is of exponential growth if we additionally have γ + C 2 γ for soe C 2 >. Note that the condition of being of lower exponential growth can easily be tested by the equivalent condition of γ + inf >, Z γ while that of being of exponential growth is in turn equivalent to being of lower exponential growth and γ + <. Z γ We can now state the ain theore of this section: Theore 2.2. Let f 0 be a easurable function, {γ } Z be a sequence of exponential growth and 0 < p <, q. Then the following stateents are equivalent: (a) There exists a decoposition f = Z f, where each f is a quasi-step function of height γ and width W, the f s have pairwise disjoint ports and we have (7) γ W /p l q p,q. (8) (b) We have the estiate f L p,q p,q. (c) There exists a decoposition f = Z f, where each f is a sub-step function of width γ and height H, the f s have pairwise disjoint ports, H is non-increasing, H + f H in the port of f and we have H γ /p l q p,q. Proof. Observe that, since {γ } Z is of lower exponential growth, we have (9) γ k C k γ and γ +k C k γ for all Z and k 0. This iplies in particular that γ 0 when and γ when. Let us first prove that (a) = (b): By onotonicity we can assue that f = Z γ E

7 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 7 where E is the port of f. Since the sequence γ is (strictly) increasing, we have d f (γ ) = µ(e +k ). k= Let us first pose that q =, we have to prove that To this end observe that λd f (λ) /p = λ>0 Z λd f (λ) /p p. λ>0 λ (γ,γ +] γ + d f (γ ) /p Z λd f (λ) /p ( ) /p = γ + W +k k= ( ) /p γ p + W +k Z k= ( = γ p + k W ( Z k= Z γ p W p γ W /p Z. k= ) /p C p(k ) Now pose q <, then we can do soething siilar: ) /p (9) f q L p,q p,q λd f (λ) /p q L q (R +, dλ λ ) = Z γ+ γ λ q d f (λ) q/p dλ λ γ+ d f (γ ) q/p Z γ q d f (γ ) q/p γ q + Z = df (γ )γ p q/p + l q/p = γ p + W +k k= We would like to exchange l p/q with, but this is not possible in general unless q/p. We could fix this using the q/p-convexity of l q/p for 0 < q/p <, but we will treat it in a unified way using Lea 5.: q/p l q/p. λ q dλ λ

8 8 GUILLERMO REY Fro (9) and (7), we have: γ p + W +k l q/p = γ p (k ) W l q/p C p(k ) γ p W l q/p C C pk γw p q/p l = C pk γ W /p p C pk. Hence, using Lea 5., we obtain: q/p γ p + W +k C,p,q. l q/p k= Let us now see that (b) = (c): Define p l q H = inf{λ > 0 : d f (λ) γ }, Clearly H is non-increasing and, by definition, we have λ < H = d f (λ) > γ. Observe also that, fro the onotone convergence theore, this infuu is achieved and so d f (H ) = γ. Since f L p,q is finite, d f (λ) has to be finite for λ > 0 (otherwise d f (s) would be infinite for 0 < s < λ). Thus, if H 0, then the sets E = {x : f(x) > H } are of finite easure. Using the onotone convergence theore again we conclude that the set has easure 0. By the definition of H E = {x : f(x) H } d f (λ) γ = H λ, thus if d f (ɛ) γ (which is true for sufficiently large since γ when ), then H ɛ, that is H 0 when. These rearks show that where f = f H+<f H. If q < then f = Z f, λd f (λ) /p q L q ((H +k+,h +k ], dλ λ ) = H+k H +k+ λ q d f (λ) q/p dλ λ γ q/p +k q γ q/p +k H+k H +k+ λ q dλ λ ( H q C,p,q C kq/p γ q/p +k Hq +k+ ) ( H q +k Hq +k+).

9 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 9 Since H 0 when, we can su telescopically as follows: ( ) /q H γ /p = Hγ q q/p ( ) /q = (H q +k Hq +k+ )γq/p so taking l q nors k=0 C,p,q ( H γ /p q l q C,p,q = = k=0 k=0 k=0 C,p,q. C kq/p λd f (λ) /p q L q ((H +k+,h +k ], dλ λ ) ) /q, Z k=0 C kq/p C kq/p λd f (λ) /p q L q ((H +k+,h +k ], dλ λ ) λd f (λ) /p q L q ((H +k+,h +k ], dλ λ ) Z C kq/p λd f (λ) /p q L q (R +, dλ λ ) Lastly, let q =. Then we iediately have so if λ > γ /p d f (λ) λ p,, then d f (λ) γ, that is: H γ /p. This iplies H γ /p l, which is what we wanted. We now prove the iplications in the reverse order. Let s start with (c) = (b): By hypothesis and the estiate (9) we have d f (H ) γ k γ k= k= C k C γ. Observe also that it follows fro (7) that H 0 when. Additionally: So, if q < : λ < H = λ [H +, H ). Z Z H λd f (λ) /p q = λ q d L q (R +, dλ λ ) f (λ) q/p dλ λ Z H + C,q γ q/p ( + H q H q ) + C2,p Z Z = γ /p H q l q. γ q/p H

10 0 GUILLERMO REY If q = then we can siilarly estiate: λd f (λ) /p = λ>0 Z The final stage of the proof is showing that Define λ [H +,H ) H d f (H + ) /p Z C,p H γ /p + Z C2,p H γ /p Z. (b) = (a). f = f γ< f γ +, then clearly W = d f (γ ) d f (γ + ) d f (γ ). If q < we proceed as follows: λd f (λ) /p q L q (R +, dλ λ ) γ+ λd f (λ) /p = λ q d f (λ) q/p dλ λ Z γ d f (γ ) q/p ( γ q + ) γq Z C2 If q =, then we have to show that but by hypothesis d f (λ) /p λ, so and the proof follows. d f (γ ) q/p γ q Z W q/p γ q Z = γ W /p q. l q W /p γ W /p, d f (γ ) /p γ Reark 2.3. Observe that the proof of (a) = (b) works verbati when the sequence {γ } Z is only of lower exponential growth. 3. Quantitative Properties of L p,q spaces As a first application of the dyadic decoposition theores of the last section we can prove the following generalization of Hölder s inequality, originally proven by O Neil in [4]. Theore 3.. Let 0 < p, p, p 2 < and 0 < q, q, q 2 exponents satisfying p = p + p 2 and q = q + q 2,

11 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH then for every easurable function we have fg L p,q p,q,p 2,q 2 f L p,q g L p 2,q 2. Proof. First observe that we can assue f L p,q = g L p 2,q 2 = and that f, g 0, so we only have to prove fg L p,q. We will oit the dependence on the exponents for the rest of the proof. Decopose f and g using the horizontally dyadic decoposition with γ = 2, i.e.: f = f, g = where f and g are sub-step functions of height H and H respectively and have ports of easure 2. To estiate the fg we proceed as follows: fg f g n n = f g +k k = f g +k + f g +k, k<0 k 0 since the L p,q quasi-nor satisfies the quasi-triangle inequality it suffices to estiate the k < 0 and k 0 sus separately. We will only show f g +k L p,q k 0 since the other su is estiated in the sae way. Observe that f g +k is a sub-step function of height H H +k and width in(2, 2 +k ), so f g +k H H +k E, where E is a set of easure 2. We can now eploy the iplication (c) = (b) in Theore 2.2 to obtain: g f g +k L p,q H H +k2 /p l q = 2 /p H 2 /p2 H +k l q 2 /p H l q 2 /p 2 H +k 2 k/p2, where we have used Hölder s inequality in the last line. To get L f g +k p,q we just use Lea 5.. k 0 l q 2 The following dual characterization of the L p, space will be useful. Theore 3.2. Let (, µ) be a σ-finite easure space, < p < and f L p,. Then

12 2 GUILLERMO REY (i) If f L p, then (0) f L p, p 0<µ(E)< µ(e) /p f E dµ. (ii) If f E is integrable for all sets E of finite easure and M p = f E dµ <, 0<µ(E)< then f L p, and f L p, p M p. µ(e) /p Proof. Let s prove (i) first. The p part is a siple consequence of Hölder s inequality for Lorentz spaces, so we will concentrate on showing the reverse inequality. Let us first assue f to be non-negative. Let E λ = {x : f(x) > λ}, the astion f L p, guarantees that µ(e λ ) <. Then, if µ(e λ ) 0: λµ(e λ ) /p = µ(e λ ) /p λµ(e λ ) µ(e λ ) /p f Eλ dµ 0<µ(E)< µ(e) /p f E dµ, taking the reu in λ > 0 yields (0). Now pose f is not necessarily non-negative. We can decopose f = u + 0 u 0 + i(u+ u ) where u ± i 0 for i {0, }, so by the quasi-triangle inequality: It thus suffices to show that but since u ± i that f L p, p u + 0 L p, + u 0 L p, + u+ L p, + u L p,. u ± i Lp, 0<µ(E)< µ(e) /p f E dµ, is non-negative, we can use what we already know to reduce the proble to showing u ± i E dµ f E dµ. 0<µ(E)< Finally, observe that u ± i we have that if µ(e F ) = 0 and µ(e) /p µ(e) /p 0<µ(E)< µ(e) /p = e iθ f F for soe F, so for any given set E with 0 < µ(e) < µ(e) /p u ± i E dµ = 0 0<µ(E )< µ(e ) /p f E dµ u ± i E dµ = µ(e) e iθ f E F dµ /p µ(e F ) f E F dµ /p µ(e ) f E /p dµ 0<µ(E )< if µ(e F ) > 0. Taking the reu over these sets yields the proof. Let us now prove (ii). The idea is to approxiate f by functions f n which are a priori in L p,, use (i) on these functions and then try to pass to the liit. Indeed, since (, µ) is σ-finite we can find an increasing sequence of sets {F n } n N of finite easure and such that = F n. n

13 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 3 We now define f n = f En with E n = F n {x : f(x) n}. These functions are qualitatively in L p, : f n L p, nµ(e n ) /p, and also f n f, so we can use Proposition.9 to get: () f L p, li inf n f n L p,. We would have finished if the right hand side of () were finite, because then we would have f L p, a priori. But observe that we can use (i) to bound each of these functions: f n L p, p 0<µ(E)< µ(e) f n E dµ. /p Now we use a procedure siilar to that in the proof of (i): 0<µ(E)< µ(e) f E En dµ /p 0<µ(E)< µ(e E n ) f E En dµ /p = µ(e )>0, E E n µ(e ) f E /p dµ µ(e ) f E /p dµ 0<µ(E )< = M p We can also prove a duality theore for L p,q spaces: Theore 3.3. Let < p < and q, then for all f Σ + : { } (2) f L p,q p,q fg dµ : g Σ +, g L p,q =: M p,q (f), where Σ + is the set of all non-negative siple functions with port of finite easure. Proof. As before, the p,q inequality follows fro Hölder s inequality for Lorentz spaces. Also, if q =, then the result follows fro Theore 3.2. We will thus prove the inequality p,q restricted to q <. Hoogeneity allows us to noralize f L p,q =, we thus have to show that Using Theore 2.2 we can decopose f as M p,q (f) p,q. f = Z f, where each f is a quasi-step function of height 2 and width W, and such that 2 W /p l q. Observe that it follows fro the proof of Theore 2.2 that each f ust also be a siple function and, also, that f is 0 for sufficiently large. Define the function g by g = Z g

14 4 GUILLERMO REY where g = a r f p, a = 2 W /p and r = ax(0, q p). Clearly g Σ +, and furtherore, since the f s have disjoint ports E : fg dµ = a r f p dµ = a r f p dµ. The functions f are quasi-step functions of height 2 and width W, thus f p dµ p 2 p W = a p, that is fg dµ p Z a r+p ( ) a r+p l q p,q f L p,q =, where in ( ) we have used that l p l p 2 when p p 2 and that r + p q. So we only have to show that g L p,q. Observe that g p,q a r 2 (p ) E with µ(e ) = W, so a direct application of Theore 2.2 would yield the result. The sequence a r 2 (p ), however, can fail to be of lower exponential growth (at least a priori). We can fix this by defining γ = 2 p 2 a r k2 k p 2 = 2 (p ) a r p k 2 k2. k k 0 Obviously γ a r 2 (p ), and furtherore: γ + = 2 p 2 2 p 2 a r k2 k p 2 k + 2 p 2 2 p 2 a r k2 k p 2 k = 2 p 2 γ, that is, {γ } Z is of sub-exponential growth with constant C = 2 p 2 >. Now we can use Theore 2.2 and the reark that follows it to reduce the proble to showing γ W /p l q p,q. But W = 2 p a p, so γ W /p l q = γ a p 2 (p ) l q = a p = k 0 k 0 2 (p ) 2 (p ) a p a r k2 a p a r k2 2 k=0 p k 2 a p p k 2 l q a r p k 2 k2 l q k 0 p k 2 l q a r k. l q

15 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 5 If q p = r = 0, but then q p = l q lp, so a p l q a p = a l p p l a p p l = 2 W /p p q l q p,q, which copletes the proof if q p. If q > p, then r = q p and so using Hölder s inequality: This finishes the proof. a p p q a q p k l q + q p q a p = q, a l q p q p k = a p l a q k q p l q p,q. l q q p We can now strengthen this theore to eliinate the qualitative astions of being siple: Corollary 3.4. Let < p < and q, then { } (3) f L p,q p,q fg dµ : g Σ, g L p,q for all f L p,q and where Σ is the set of all finite cobinations of characteristic functions of sets of finite easure. Furtherore, if the easure space is σ-finite and if f is not a priori in L p,q but the right hand side is well defined and finite, t hen the left hand side is also finite and we have (3). Proof. We will only show this in the case f L p,q and f 0, the coplete proof follows the sae pattern as in the proof of Theore 3.2. Fro Hölder s inequality we can reduce to showing only the p,q inequality. Approxiate (a.e.) f by an increasing sequence {f n } n N of positive siple functions of finite port (this can be achieved if the easure space is σ-finite or if f is a priori in L p,q, we oit the details). Using the Monotone Convergence Theore for Lorentz spaces.8: f L p,q = f n L p,q n { } p,q f n g dµ : g Σ +, g L p,q n { } = f n g dµ : g Σ +, g L p,q n { = fg dµ : g Σ +, g L p,q }, where we have used the (usual) onotone convergence theore in the last line. This shows that there is a nor (the one defined using the dual characterization) which is equivalent to the L p,q quasi-nor defined at the beginning when < p <. 4. Marcinkiewicz Interpolation Theore In this section we will present a proof of the Marcinkiewicz interpolation theore using the decoposition introduced in section 2. We will use duality fro the outset, indeed, we will need the following extension of Theore 3.2:

16 6 GUILLERMO REY Theore 4.. Let (, µ) be a easure space and let 0 < p <, the following stateents are equivalent for f L p, : () f L p, p. (2) For every set E of finite easure there exists a subset E E such that µ(e) 2 µ(e) such that f E dµ p µ(e ) /p. Furtherore, the theore holds without the a priori astion of f L p, provided (, µ) is σ-finite. Proof. We will assue 0 < p, we will also assue f 0 and f L p, = without loss of generality. The proof of (2) = () is alost identical to the one in Theore 3.2, let us show the other one. Let E be a set of finite (and positive) easure and let α = ( 2 /p. µ(e)) Define E = E \ {x : f(x) > α}, then, since we have the estiate µ({x : f(x) > α}) µ(e) 2, we can bound µ(e) µ(e ) 2 µ(e). Now we can estiate the integral ( ) /p 2 f E dµ µ(e ) µ(e) 2 /p µ(e ) /p. The Marcinkiewicz interpolation theore, as in coplex interpolation, uses the boundedness of an operator in two extreal spaces to provide boundedness in the interediate spaces. However, while coplex interpolation of L p spaces requires the extreal spaces to also be L p spaces, the Marcinkiewicz Interpolation Theore only needs the to be weak-l p spaces. It actually requires a bit less, the following definitions will ake the exposition easier: Definition 4.2. Let T be an operator defined on a subset of the set of easurable functions of a easure space (, µ) and that takes values in the easurable functions of another easure space (Y, ν). We will say that T is of strong type (p, q) if T extends to a bounded operator fro L p (, µ) to L q (Y, ν). We will say that T is of weak type (p, q) if T extends to a bounded operator fro L p (, µ) to L q, (Y, ν). In what follows we will always assue T to be an operator defined on a set D closed under addition, ultiplication by scalars and containing Σ c (, µ): the set of finite linear cobinations of characteristic functions of sets of finite easure of (, µ). We will call T sublinear if T (f + g) T (f) + T (g), T (λf) = λ T (f) for all f, g in the doain of T and λ C. We say that a sublinear operator T is of restricted weak type (p, q) if (4) T f L q, HW /p for all sub-step functions f D of height H and width W.

17 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 7 The next result shows that, usually, an operator which is of restricted weak type on characteristic functions is also of restricted weak type on Σ c. More precisely: Proposition 4.3. Let 0 < p, 0 < q, A > 0 and T be a sublinear operator defined on Σ c. Then the following stateents are equivalent: () T satisfies T f L q, AHW /p for all sub-step functions f of height H and width W in Σ c. (2) For every set F Y of finite easure there exists a subset F F with ν(f ) 2 ν(f ) and such that for all E of finite easure T E F dν Aµ(E) /p ν(f ) /q. Y Proof. That () = (2) is an easy consequence of Theore 4. and Hölder s inequality for Lorentz spaces, so we will prove the reverse iplication. First assue that f takes the for f N = N 2 j Ej j= where µ(e j ) W. In this case T f N L q, AW /p (the constant being independent of N). To show this first observe that using sublinearity and Lea 5. it suffices to show (5) T Ej L q, AW /p. Now let F Y be a set of finite easure. By hypothesis there exists a subset F F which satisfies ν(f ) 2 ν(f ) and Y F T ( Ej ) dν Aµ(E j ) /p ν(f ) /q ν(f ) /q AW /p. So by Theore 4. we obtain (5). Now let 0 f be an arbitrary function in Σ c of width W : f = M a j Ej, j= the E j s disjointly ported. If d j (x) denotes the j-th digit in the binary expansion of f(x) and we define N N f N = 2 j d j = 2 j Fj, then we easily see that j= f f N = j= M b j,n Ej j=

18 8 GUILLERMO REY with 0 b j,n 2 N. So: T f L q, T f N L q, + T (f f N ) L q, M AW /p + b j,n T Ej j= M AW /p + 2 N T Ej j= L q, L q,, since the second ter can be ade arbitrarily sall we conclude that T f L q, AW /p. By hoogeneity and sublinearity we obtain the sae bound (with a possibly larger constant) for arbitrary f Σ c. To prove the Marcinkiewicz interpolation theore we first need a weaker result: Proposition 4.4. Let 0 < p 0, p, q 0, q, A 0, A > 0 and pose that T is a sublinear operator defined on Σ c which is of restricted weak type (p i, q i ) with constants A i for i = 0,. Then it is of restricted weak type (p θ, q θ ), i.e.: (6) T f L q θ, p0,q 0,p,q A θ HW /p θ f Σ c, where and p θ = θ p 0 + θ p, A θ = A θ 0 A θ = θ + θ q θ q 0 q Proof. Theore 4.3 reduces atters to showing that for every set F Y of finite easure there exists a subset F F with ν(f ) 2ν(F ) and such that for all sets E of finite easure: F T ( E ) dν p0,q 0,p,q A θ µ(e) /p θ ν(f ) /q θ, Y but, using Theore 4.3 and the hypotheses, we have that there exists a subset F of F with ν(f ) 2ν(F ) and that for all sets E of finite easure F T ( E ) dν p0,q 0,p,q A θ µ(e) /pi ν(f ) /q i Y for i = 0,. The result now follows fro the trivial scalar interpolation inequality: Y Z = Y θ Z θ for, Y, Z 0 and 0 θ, and an application of Theore 4.3 once ore. We need one last lea: Lea 4.5. Let then Λ λ (x, y) = ( λ)x + λy, Λ α (Λ β (x, y), Λ γ (x, y)) = Λ Λα(β,γ)(x, y). We are now ready to state and prove the ain theore of this section:

19 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 9 Theore 4.6. Let T be a sublinear operator defined on Σ +. Suppose that 0 < p i, q i and q 0 q. If T satisfies T E L q i, A i E L p i, i = 0,, for all E of finite easure, then for all r and 0 < θ < such that q θ > we have for all f in Σ c. T f L q θ,r p0,p,q 0,q,r,θ A θ f L p θ,r Proof. Using Theore 4.3 we can assue that T is of restricted weak type (p i, q i ) with constants A i. We will pose first that q i. Let f Σ +, which by hoogeneity we ay assue to be noralized so that f L p θ,r =. By duality (Theore 3.3), it will be enough to verify the estiate T f g dν p0,q 0,p,q,r,θ A θ Y for all g Σ + with g L q θ =. In what follows all constants will be assued to depend on,r p 0, q 0, p, q, r and θ and will be oitted. By hypothesis and Hölder s inequality we have ( ) (7) T u v dν A i HH W /pi W /q i Aθ HH in W /pi W /q i i=0, Y for all sub-step functions u Σ + (, µ) of height H and width W and for all sub-step functions v Σ + (Y, ν) of height H and width W. By Theore 2.2 we can decopose f and g as f = f and g = n g n where each f is a sub-step function of height H and width 2 and where each g n is a sub-step function of height H n and width 2 n. These decopositions also satisfy H 2 /p θ H n 2 n/q θ. l r l r Recall that, since f and g are siple, only a finite nuber of f s and g n s are non-zero. Using (7) and sublinearity we have T f g dν A θ H H n ( in 2 /p i ) 2 n/q i. i=0, Y Call a = H 2 /p θ and b n = H n2 n/q θ, we then have to obtain the estiate ( (8) a b n in 2 (/p i /p θ ) 2 n(/q θ /q ) i), i=0,,n where a l r = b n l r =. After reorganizing ters we can write the third factor as ( ( in ) ( + n ) (, ) ( n )), p 0 p θ q θ q 0 p θ p q q θ which can be further siplified to ( in αθ,n ( βα n ), α( θ) ( βα n )), where α = p 0 p and β = q 0 q, which is easily seen to be true. Let ϕ( γn) = in (αθ ( γn), α( θ) ( γn))

20 20 GUILLERMO REY where γ = β α, then we have to show that a b n ϕ( γn).,n We can change variables and let + γn and reduce to showing a + γn b n ϕ( + γn γn).,n Observe that γn γn, so we easily see that thus we end up with ϕ( + γn γn) ϕ(), ϕ() a + γn b n l a + γn b n n n since ϕ is in l. Applying Hölder s inequality with exponents r and r we reduce to showing that a + γn l r n uniforly in, but this is just the observation that the su of a r + γn is bounded by a constant depending on γ ties the su of a r +n, which is of course bounded by a constant independent of. Finally, we show how to drop the astion of q i >. Observe that we ay interpolate using Theore 4.4 to obtain restricted weak type (p θ, q θ ) boundedness of T for 0 θ. Since we are iposing q θ >, we ay assue that q i > for at least one i {0, } (otherwise the theore is void), thus, there ust be a τ (0, ) such that < q τ < q θ and such that T is of restricted weak type (p τ, q τ ). We can now interpolate using what we have just proved fro (p τ, q τ ) to the (p i, q i ) which has q i > and use Lea 4.5 to ensure that (p θ, q θ ) is in between (p τ, q τ ) and (p i, q i ). Corollary 4.7 (Marcinkiewicz). Let T be a sublinear operator that is of weak type (p i, q i ) with constants A i > 0 and such that p i p i for i {0, }, pose also that q 0 q. Then, for 0 < θ <, T is of strong type (p θ, q θ ) with constant p0,q 0,p,q,r,θ A θ = A θ 0 A θ where = θ + θ and p θ p 0 p = θ + θ. q θ q 0 q Proof. One just needs to recall that the L p,q are nested and that L p,p coincides with L p, we leave the easy details to the reader. 5. Appendix Lea 5.. Let be a vector space endowed with a quasi-nor, that is, there exists a constant C > 0 such that (9) f + g C ( f + g ). Then, for any sequence {f n } n N of eleents of verifying (20) f j AC j 2 for soe A > 0 and C 2 >, we have (2) N f j AC 4 j=

21 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 2 where C 3 does not depend on N or A. Proof. Choose κ N so large that C C κ 2 < and write N as N = κd + r where d N and 0 r < κ. Observe that κ does not depend on N. We can now split the su as follows N κd r f j = f j + f kd+j. j= j= The nor of the second su is trivially bounded by A ties soe constant depending on C and C 2 (since r < κ,) so we can center our efforts on the first su. To this end we factorize the su as κd j= f j = d =0 j= j= κ f κ+j. Now observe that, by induction, we have the following estiate: n n g j C j g j j= for any sequence {g j } of eleents of. Using this we can estiate the first su as follows: κd r d κ f j + f kd+j C + f κ+j j= j= A = A A =0 d =0 d =0 =0 C,C 2 j= A C,C 2 A. C + j= κ j= ( C C κ 2 C j C (κ+j) 2 ) κ ( C C2 κ ) κ j= References j= C j+ C j 2 κ j= C j+ C j 2 C j+ C j 2 [] G. Hardy and J. Littlewood. A axial theore with function-theoretic applications. Acta Matheatica, 54():8 6, 930. [2] R. A. Hunt. On L(p, q) spaces. Enseigneent Math. (2), 2: , 966. [3] M. Keel and T. Tao. Endpoint Strichartz estiates. Aer. J. Math., 20(5): , 998. [4] R. O Neil. Convolution operators and L(p, q) spaces. Duke Matheatical Journal, 30:29 42, 963. [5] E. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 990. [6] T. Tao. Lecture notes for 245C: Interpolation of L p spaces. [7] N. Wiener. The ergodic theore. Duke Matheatical Journal, 5(): 8, 939.