Notes on Lorentz Spaces and Interpolation

LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH

GUILLERMO REY

1. Introduction If an operator T is bounded on two Lebesgue spaces, the theory of complex interpolation allows us to deduce the boundedness of T “between” the two spaces. Real interpolation (of Lp spaces) provides, in a way, a reﬁnement of complex interpolation, this will be shown as we progress. We will follow Tao’s notes [6]: in particular, we claim no originality over any argument here. Consider the (centered) Hardy-Littlewood maximal operator: Z (1) Mf(x) = sup |f(y)| dy, r>0 Br (x) 1 p deﬁned for f ∈ Lloc. We would like to obtain L boundedness of this operator for 1 ≤ p ≤ ∞, but alas, this is not the case: although it is trivially bounded from L∞ onto itself, it is not bounded on L1:

Proposition 1.1. If f ∈ L1(Rd) and f is not identically 0 then C Mf(x) ≥ d |x|d

1 d for some constant Cd dependent on f and the dimension, in particular f∈ / L (R ). Proof. Since f is not identically 0, there exists an R > 0 such that Z |f| ≥ δ > 0. BR(0)

Now, if |x| > R, BR(0) ⊆ B(x, 2|x|), so 1 Z δ Mf(x) ≥ |f| ≥ Cd d . |B(x, 2|x|)| BR(0) |x| Thus M is indeed unbounded as an operator from L1 onto itself. However, We do have the following weaker form of L1 boundedness; proved by Hardy-Littlewood in [1] for d = 1 and by Wiener in [7] for d > 1: Theorem 1.2. There exists a constant C > 0 such that for all f ∈ L1

kfk 1 m({x ∈ d : Mf(x) > λ}) ≤ C L , R λ where m is the Lebesgue measure on Rd. 1 2 GUILLERMO REY

Observe that, by Chebychev’s inequality, we would have had this bound if M had been bounded on L1, so it is indeed a weaker statement. This leads us to study the space of functions satisfying Chebychev’s inequality: p kfk p (2) µ({x ∈ X : |f(x)| > λ}) ≤ L (X,µ) . λp More precisely, deﬁne the weak-Lp space Lp,∞(X, µ) to be the set of all measurable functions f for which its Lp,∞ quasi-norm 1/p (3) kfkLp,∞ = sup λµ({x : |f(x)| > λ}) λ>0 is ﬁnite. To abbreviate we will use the notation

df (λ) = µ({x ∈ X : |f(x)| > λ}) when there is no confusion as to what measure we are using. Also, we deﬁne by convention L∞,∞ to be L∞. The following lemma is a simple application of the monotone convergence lemma:

Lemma 1.3. Let {fn}n∈N be a sequence of measurable functions such that |fn(x)| ≤ |fn+1(x)| µ-almost everywhere and for all n ∈ N. If we have |fn| → f µ-almost everywhere for some function f then

dfn (λ) % df (λ) when n → ∞ for all λ > 0. As we have seen, Chebychev’s inequality gives us the inclusion Lp ,→ Lp,∞ for p > 0. This inclusion is proper as the following example shows:

− d Example 1.4. If f(x) = |x| p then Z Z |f(x)|p dx = |x|−d dx = ∞, d d R R however 1/p 1/p sup λdf (λ) = νd < ∞, λ>0 d where νd is the measure of the unit ball in R .

As we have noted, k · kLp,∞ is not a priori a norm, but only a quasi-norm: Proposition 1.5. We can easily see that

df+g(λ) ≤ df (λ/2) + dg(λ/2), which is just the fact that when two terms sum more than λ, at least one of them has to be greater than λ/2. Using this fact we have:

1/p kf + gkLp,∞ = sup λdf+g(λ) λ>0 1/p ≤ sup λ (df (λ/2) + dg(λ/2)) λ>0

≤ 2Cp (kfkLp,∞ + kgkLp,∞ ) . We will generalize the weak-Lp spaces a bit further, bit ﬁrst we need the following well-known identity: LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 3

Proposition 1.6. Let 0 < p < ∞ and f be a measurable function, then Z 1/p 1/p p dλ kfkLp = p λ df (λ) . + λ R Proof. Z p p kfkLp(X,µ) = |f(x)| dµ(x) X Z Z |f(x)| p dλ = pλ λ dµ(x) X 0 Z Z p1 dλ = p λ {s: s<|f(x)|} λ dµ(x) X + Z R Z p 1 dλ = p λ {y:|f(y)|>λ} dµ(x) λ (Fubini) + X ZR p dλ = p λ df (λ) λ . + R So we have: 1/p kfkLp,∞ = kλdf (λ) k ∞ + dλ L (R , λ ) and 1/p 1/p kfkLp = p kλdf (λ) k p + dλ . L (R , λ ) From this, we could tentatively define the Definition 1.7 (Lorentz quasi-norm). Let (X, µ) be a measure space and let 0 < p < ∞ and 0 < q ≤ ∞, we define the Lorentz space quasi-norm k · kLp,q (X,µ) as 1/q 1/p kfkLp,q (X,µ) = p kλdf (λ) k q + dλ . L (R , λ ) Observe that, with this definition, Lp,p (isometrically) coincides with Lp. We have the following useful lemma, which is a kind of Monotone Convergence Theorem.

Lemma 1.8. Let {fn}n∈N be a sequence of measurable functions such that |fn| % |f| µ-almost everywhere, then kfkLp,q = lim kfnkLp,q n→∞ Proof. We can obviously assume f to be positive so, by deﬁnition, we only have to verify that 1/p 1/p lim kλdfn (λ) kLq ( +, dλ ) = kλdf (λ) kLq ( +, dλ ). n≥1 R λ R λ

Note that, since fn is non-decreasing, dfn (·) is also increasing (in n), so in particular by Lemma 1.3

df (λ) = sup dfn (λ) n≥1 for all ﬁxed λ > 0. If q = ∞ then the problem is reduced to the easier one of showing that 1/p 1/p sup kλd (λ) k ∞ + = k sup λd (λ) k ∞ + , fn Lλ (R ) fn Lλ (R ) n≥1 n≥1 which is easily seen to hold true in the more general case

sup kgnkL∞ = k sup gnkL∞ . n≥1 n≥1 4 GUILLERMO REY

Suppose now that q < ∞, then Z ∞ 1/q 1/p q q/p dλ sup kλdfn (λ) k q + dλ = sup λ dfn (λ) , L (R , λ ) λ n≥1 n≥1 0 q q/p q q/p and since, as we noted before, we have λ dfn (λ) % λ df (λ) for all ﬁxed λ > 0, we can dλ apply the monotone convergence theorem with the measure λ and the desired identity. The following Fatou-like lemma shows that the Lorentz space quasi-norm is lower semi- continuous.

Lemma 1.9. Let {fn}n∈N be any sequence of measurable functions, then

k lim inf fnkLp,q ≤ lim inf kfnkLp,q n n

Proof. Let f = lim infn fn, then, by deﬁnition, we have to check that 1/p 1/p kλdf (λ) k q + dλ ≤ lim inf kλdfn (λ) k q + dλ , L (R , λ ) n L (R , λ ) which would follow from the lower-semicontinuity of the usual Lq spaces provided that we have

df (λ) ≤ lim inf df (λ). n n This is equivalent to showing

µ({x : lim inf |fn(x)| > λ}) ≤ lim inf µ({x : |fn(x)| > λ}), n n which follows from Fatou’s lemma. Finally, we can prove that the Lp,q spaces are quasi-Banach spaces; which follows the steps of the classical proof in the Lp case. Theorem 1.10. Let (X, µ) be a measure space and let 0 < p < ∞, 0 < q ≤ ∞, then Lp,q(X, µ) is a quasi-Banach space, that is, it is complete and it satisfies the quasi-triangle inequality. Proof. We have already proven that Lp,∞ satisfies the quasi-triangle inequality in Proposition 1.5, the proof for Lp,q when q 6= ∞ is almost identical so we will omit it. We are left to prove that it is complete. To this end first note that the Lp,q norm always controls the Lp,∞ norm: Z ∞ 1/q 1/p q q/p dλ kλdf (λ) k q + dλ = λ df (λ) L (R , λ ) λ 0 Z s 1/q q q/p dλ ≥ λ df (λ) λ 0 Z s 1/q 1/p q dλ ≥ df (s) λ λ 0 1/p −1/q = sdf (s) q , for any s > 0. Thus, taking the supremum on s, we obtain:

kfkLp,∞ .p,q kfkLp,q . In particular we have the following generalization of Chebychev’s inequality: p kfk p,q (4) d (λ) L . f .p,q λp p,q Now let {fn} be a Cauchy sequence in the L quasi-norm. Estimate (4) allows us to deduce that {fn} is Cauchy in measure, and hence that there exists a µ-almost everywhere convergent LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 5 subsequence {fϕ(k)}k. Call f the µ-almost everywhere defined limit of fϕ(k) when k → ∞, we p,q will show that fn → f in the L quasi-norm. Let C be the constant in the quasi-triangle inequality for the k · kLp,q quasi-norm and refine the subsequence {fϕ(k)} so that we have −k kfϕ(k) − fϕ(k−1)kLp,q ≤ (2C) p,q using that fn is an L -Cauchy sequence and defining fϕ(0) = 0. Observe that we have the following telescoping sum identity: n X (5) fϕ(n) = fϕ(k) − fϕ(k−1). k=1

Recall that we want to estimate kf − fnkLp,q and that, using the quasi-triangle inequality, we have kf − fnkLp,q ≤ C kf − fϕ(N)kLp,q + kfϕ(N) − fnkLp,q . p,q The second term can be made arbitrarily small using the fact that fn is an L -Cauchy sequence, so we choose n and N so large that kf − f k p,q < . ϕ(N) n L 2C For the ﬁrst term we can use (5) to obtain: ∞ N X X kf − fϕ(N)kLp,q = (fϕ(k) − fϕ(k−1)) − (fϕ(k) − fϕ(k−1)) p k=1 k=1 L ,q M X = lim fϕ(k) − fϕ(k−1) M→∞ p,q k=N+1 L M X ≤ lim |fϕ(k) − fϕ(k−1)| M→∞ p,q k=N+1 L M X (6) ≤ lim inf |fϕ(k) − fϕ(k−1)| , M→∞ p,q k=N+1 L where in the last line we have used Proposition 1.9. By induction, we can iteratively use the P quasi-triangle inequality to exchange and k · kLp,q and bound the last term in (6) by M M X k−N X k−N −k lim inf C kfϕ(k) − fϕ(k−1)kLp,q ≤ lim inf C (2C) M→∞ M→∞ k=N+1 k=N+1 ∞ X = C−N 2−k k=N+1 ≤ (2C)−N .

1 Choosing N so large that (2C)N < 2C completes the proof.

2. The Dyadic Decomposition It turns out that we will be able to decompose functions in Lp,q into simpler functions: • Sub-step functions: A sub-step function of height H and width W is any measurable function f supported on a set of measure W and which satisﬁes |f| ≤ H. 6 GUILLERMO REY

• Quasi-step function: A quasi-step function of height H and width W is any measurable function f supported on a set of measure W and which satisﬁes |f| ∼ H in its support. Example 2.1. Let f be a quasi-step function of height H and width W , then 1/q 1/p p kfk p,q ∼ HW . L q We will give two diﬀerent decompositions: a “vertically-dyadic” decomposition; where we decompose the range of our function into dyadic pieces, and a “horizontally-dyadic” one; where we dyadically decompose its support. + Let {γm}m∈Z be a sequence in R , we say that it is of lower exponential growth if we have

C1γm ≤ γm+1 for some constant C1 > 1. We will say that it is of exponential growth if we additionally have

γm+1 ≤ C2γm for some C2 > 1. Note that the condition of being of lower exponential growth can easily be tested by the equivalent condition of γ inf m+1 > 1, m∈Z γm while that of being of exponential growth is in turn equivalent to being of lower exponential growth and γ sup m+1 < ∞. m∈Z γm We can now state the main theorem of this section:

Theorem 2.2. Let f ≥ 0 be a measurable function, {γm}m∈Z be a sequence of exponential growth and 0 < p < ∞, 1 ≤ q ≤ ∞. Then the following statements are equivalent: P (a) There exists a decomposition f = fm, where each fm is a quasi-step function of m∈Z height γm and width Wm, the fm’s have pairwise disjoint supports and we have 1/p (7) γmWm `q .p,q 1. (b) We have the estimate kfkLp,q .p,q 1. P (c) There exists a decomposition f = fm, where each fm is a sub-step function of m∈Z width γm and height Hm, the fm’s have pairwise disjoint supports, Hm is non-increasing, Hm+1 ≤ |fm| ≤ Hm in the support of fm and we have 1/p (8) Hmγm k`q .p,q 1.

Proof. Observe that, since {γm}Z is of lower exponential growth, we have −k k (9) γm−k ≤ C1 γm and γm+k ≥ C1 γm for all m ∈ Z and k ≥ 0. This implies in particular that γm → 0 when m → −∞ and γm → ∞ when m → ∞. Let us ﬁrst prove that (a)= ⇒ (b): By monotonicity we can assume that X 1 f = γm Em m∈Z LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 7 where Em is the support of fm. Since the sequence γm is (strictly) increasing, we have

∞ X df (γm) = µ(Em+k). k=1 Let us ﬁrst suppose that q = ∞, we have to prove that

1/p sup λdf (λ) .p 1. λ>0 To this end observe that

1/p 1/p sup λdf (λ) = sup sup λdf (λ) λ>0 m∈Z λ∈(γm,γm+1] 1/p ≤ sup γm+1df (γm) m∈Z ∞ !1/p X = sup γm+1 Wm+k m k=1 ∞ !1/p X p ≤ sup γm+1Wm+k m∈ k=1 Z ∞ !1/p X p = sup γm+1−kWm m∈ k=1 Z ∞ !1/p p X −p(k−1) ≤ sup γmWm C1 (9) m∈ Z k=1 1/p .p sup γmWm m∈Z . 1. Now suppose q < ∞, then we can do something similar:

q 1/p q kfkLp,q ∼p,q kλdf (λ) k q + dλ L (R , λ ) Z γm+1 X q q/p dλ = λ df (λ) λ γ m∈Z m Z γm+1 X q/p q dλ ≤ df (γm) λ λ γ m∈Z m X q/p q .q df (γm) γm+1 m∈Z p q/p = df (γm)γm+1 `q/p ∞ q/p X p = γm+1Wm+k . q/p k=1 ` P We would like to exchange k · k`p/q with , but this is not possible in general unless q/p ≥ 1. We could ﬁx this using the q/p-convexity of `q/p for 0 < q/p < 1, but we will treat it in a uniﬁed way using Lemma 5.1: 8 GUILLERMO REY

From (9) and (7), we have: p p kγ Wm+kk q/p = kγ Wmk q/p m+1 `m m−(k−1) `m −p(k−1) p ≤ C kγ Wmk q/p 1 m `m −pk p C1 C kγ Wmk q/p . 1 m `m −pk 1/p p = C kγmW k q 1 m `m −pk .p C1 . Hence, using Lemma 5.1, we obtain: ∞ q/p X p γm+1Wm+k .C1,p,q 1. q/p k=1 ` Let us now see that (b)= ⇒ (c): Deﬁne

Hm = inf{λ > 0 : df (λ) ≤ γm},

Clearly Hm is non-increasing and, by deﬁnition, we have

λ < Hm =⇒ df (λ) > γm. Observe also that, from the monotone convergence theorem, this infumum is achieved and so df (Hm) = γm. Since kfkLp,q is finite, df (λ) has to be finite for λ > 0 (otherwise df (s) would be infinite for 0 < s < λ). Thus, if Hm 6= 0, then the sets

Em = {x : f(x) > Hm} are of ﬁnite measure. Using the monotone convergence theorem again we conclude that the set

E = {x : f(x) ≥ sup Hm} m has measure 0. By the deﬁnition of Hm

df (λ) ≤ γm =⇒ Hm ≤ λ, thus if df () ≤ γm (which is true for m suﬃciently large since γm → ∞ when m → ∞), then Hm ≤ , that is Hm → 0 when m → ∞. These remarks show that X f = fm, m∈Z 1 where fm = f Hm+1

Z Hm+k 1/p q q q/p dλ kλdf (λ) k q dλ = λ df (λ) λ L ((Hm+k+1,Hm+k], λ ) Hm+k+1 Z Hm+k q/p q dλ ≥ γm+k λ λ Hm+k+1 q/p q q ∼q γm+k Hm+k − Hm+k+1 kq/p q/p q q &C1,p,q C1 γm Hm+k − Hm+k+1 . LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 9

Since Hm → 0 when m → ∞, we can sum telescopically as follows: 1/q 1/p q q/p Hmγm = Hmγm

∞ !1/q X q q q/p = (Hm+k − Hm+k+1)γm k=0 ∞ !1/q X −kq/p 1/p q .C1,p,q C1 kλdf (λ) k q dλ , L ((Hm+k+1,Hm+k], λ ) k=0 so taking `q norms ∞ 1/p q X X −kq/p 1/p q kHmγm k`q .C1,p,q C1 kλdf (λ) k q dλ L ((Hm+k+1,Hm+k], λ ) m∈Z k=0 ∞ X −kq/p X 1/p q = C1 kλdf (λ) k q dλ L ((Hm+k+1,Hm+k], λ ) k=0 m∈Z ∞ X −kq/p 1/p q = C1 kλdf (λ) k q + dλ L (R , λ ) k=0

.C1,p,q 1. Lastly, let q = ∞. Then we immediately have −p df (λ) . λ , −1/p −1/p so if λ > γm , then df (λ) . γm, that is: Hm . γm . This implies 1/p kHmγm k`∞ . 1, which is what we wanted. We now prove the implications in the reverse order. Let’s start with( c)= ⇒ (b): By hypothesis and the estimate (9) we have ∞ ∞ X X −k df (Hm) ≤ γm−k ≤ γm C1 .C1 γm. k=1 k=1

Observe also that it follows from (7) that Hm → 0 when m → ∞. Additionally: [ λ < sup Hm =⇒ λ ∈ [Hm+1,Hm). m∈Z m∈Z So, if q < ∞: X Z Hm kλd (λ)1/pkq = λqd (λ)q/p dλ f Lq ( +, dλ ) f λ R λ H m∈Z m+1 X q/p q q .C1,q γm+1 Hm − Hm+1 m∈Z X q/p .C2,p γm Hm m∈Z 1/p q = kγm Hmk`q . 1. 10 GUILLERMO REY

If q = ∞ then we can similarly estimate: 1/p 1/p sup λdf (λ) = sup sup λdf (λ) λ>0 m∈Z λ∈[Hm+1,Hm) 1/p ≤ sup Hmdf (Hm+1) m∈Z 1/p .C1,p sup Hmγm+1 m∈Z 1/p .C2,p sup Hmγm m∈Z . 1. The ﬁnal stage of the proof is showing that (b)= ⇒ (a). Deﬁne 1 fm = f γm<|f|≤γm+1 , then clearly Wm = df (γm) − df (γm+1) ≤ df (γm). If q < ∞ we proceed as follows: 1/p q 1 & kλdf (λ) k q + dλ L (R , λ ) Z γm+1 X q q/p dλ = λ df (λ) λ γ m∈Z m X q/p q q ≥ df (γm) γm+1 − γm m∈Z X q/p q &C2 df (γm) γm m∈Z X q/p ≥ Wm γq m∈Z 1/p q = γmWm `q . If q = ∞, then we have to show that 1/p sup γmWm . 1, m 1/p −1 but by hypothesis df (λ) . λ , so 1/p 1/p −1 Wm ≤ df (γm) . γm and the proof follows.

Remark 2.3. Observe that the proof of( a)= ⇒ (b) works verbatim when the sequence {γm}m∈Z is only of lower exponential growth.

3. Quantitative Properties of Lp,q spaces As a ﬁrst application of the dyadic decomposition theorems of the last section we can prove the following generalization of H¨older’sinequality, originally proven by O’Neil in [4].

Theorem 3.1. Let 0 < p, p1, p2 < ∞ and 0 < q, q1, q2 ≤ ∞ exponents satisfying 1 1 1 1 1 1 = + and = + , p p1 p2 q q1 q2 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 11 then for every measurable function we have

p,q p ,q p ,q kfgkL .p1,q1,p2,q2 kfkL 1 1 kgkL 2 2 .

Proof. First observe that we can assume kfkLp1,q1 = kgkLp2,q2 = 1 and that f, g ≥ 0, so we only have to prove kfgkLp,q . 1. We will omit the dependence on the exponents for the rest of the proof. m Decompose f and g using the horizontally dyadic decomposition with γm = 2 , i.e.: X X f = fm, g = gm m m 0 where fm and gm are sub-step functions of height Hm and Hm respectively and have supports of measure ≤ 2m. To estimate the fg we proceed as follows: X X fg ≤ fmgn m n X X = fmgm+k k m X X X X = fmgm+k + fmgm+k, k<0 m k≥0 m since the Lp,q quasi-norm satisﬁes the quasi-triangle inequality it suﬃces to estimate the k < 0 and k ≥ 0 sums separately. We will only show X X k fmgm+kkLp,q . 1 k≥0 m since the other sum is estimated in the same way. 0 m m+k Observe that fmgm+k is a sub-step function of height HmHm+k and width ≤ min(2 , 2 ), so X X 0 1 fmgm+k ≤ HmHm+k Em , m m m where Em is a set of measure 2 . We can now employ the implication( c)= ⇒ (b) in Theorem 2.2 to obtain: X 0 m/p p,q k fmgm+kkL . HmHm+k2 `q m

m/p1 m/p2 0 = 2 Hm2 Hm+k `q ≤ 2m/p1 H 2m/p2 H0 m `q1 m+k `q2 −k/p . 2 2 , where we have used H¨older’sinequality in the last line. To get

X X fmgm+k . 1 p,q k≥0 m L we just use Lemma 5.1. The following dual characterization of the Lp,∞ space will be useful. Theorem 3.2. Let (X, µ) be a σ-ﬁnite measure space, 1 < p < ∞ and f ∈ Lp,∞. Then 12 GUILLERMO REY

(i) If f ∈ Lp,∞ then Z 1 (10) kfk p,∞ ∼ sup f1 dµ . L p 1/p0 E 0<µ(E)<∞ µ(E) X

(ii) If f1E is integrable for all sets E of ﬁnite measure and 1 Z M = sup f1 dµ < ∞, p 1/p0 E 0<µ(E)<∞ µ(E) X p,∞ then f ∈ L and kfkLp,∞ .p Mp.

Proof. Let’s prove (i) first. The &p part is a simple consequence of Hölder’sinequality for Lorentz spaces, so we will concentrate on showing the reverse inequality. Let us first assume f to be non-negative. Let Eλ = {x ∈ X : f(x) > λ}, the assumption p,∞ f ∈ L guarantees that µ(Eλ) < ∞. Then, if µ(Eλ) 6= 0: 1 1 Z 1 Z λµ(E )1/p = λµ(E ) ≤ f1 dµ ≤ sup f1 dµ , λ 1/p0 λ 1/p0 Eλ 1/p0 E µ(Eλ) µ(Eλ) X 0<µ(E)<∞ µ(E) X taking the supremum in λ > 0 yields (10). + − + − Now suppose f is not necessarily non-negative. We can decompose f = u0 − u0 + i(u1 − u1 ) ± where ui ≥ 0 for i ∈ {0, 1}, so by the quasi-triangle inequality: + − + − kfkLp,∞ .p ku0 kLp,∞ + ku0 kLp,∞ + ku1 kLp,∞ + ku1 kLp,∞ . It thus suffices to show that Z ± 1 ku k p,∞ ≤ sup f1 dµ , i L 1/p0 E 0<µ(E)<∞ µ(E) X ± but since ui is non-negative, we can use what we already know to reduce the problem to showing that 1 Z 1 Z sup u±1 dµ ≤ sup f1 dµ . 1/p0 i E 1/p0 E 0<µ(E)<∞ µ(E) X 0<µ(E)<∞ µ(E) X ± iθ 1 Finally, observe that ui = e f F for some F ⊆ X, so for any given set E with 0 < µ(E) < ∞ we have that Z Z 1 ± 1 u 1 dµ = 0 ≤ sup f1 0 dµ 1/p0 i E 0 1/p0 E µ(E) X 0<µ(E0)<∞ µ(E ) X if µ(E ∩ F ) = 0 and 1 Z 1 Z u±1 dµ = eiθf1 dµ 1/p0 i E 1/p0 E∩F µ(E) X µ(E) X 1 Z ≤ f1 dµ 1/p0 E∩F µ(E ∩ F ) X Z 1 ≤ sup f1 0 dµ 0 1/p0 E 0<µ(E0)<∞ µ(E ) X if µ(E ∩ F ) > 0. Taking the supremum over these sets yields the proof. Let us now prove (ii). The idea is to approximate f by functions fn which are a priori in Lp,∞, use (i) on these functions and then try to pass to the limit.

Indeed, since (X, µ) is σ-finite we can find an increasing sequence of sets {Fn}n∈N of finite measure and such that [ X = Fn. n LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 13

1 We now deﬁne fn = f En with En = Fn ∩ {x : |f(x)| ≤ n}. These functions are qualitatively in Lp,∞: 1/p kfnkLp,∞ . nµ(En) , and also |fn| % |f|, so we can use Proposition 1.9 to get:

(11) kfkLp,∞ ≤ lim inf kfnkLp,∞ . n→∞ We would have ﬁnished if the right hand side of (11) were ﬁnite, because then we would have f ∈ Lp,∞ a priori. But observe that we can use (i) to bound each of these functions: Z 1 kf k p,∞ sup f 1 dµ . n L .p 1/p0 n E 0<µ(E)<∞ µ(E) X Now we use a procedure similar to that in the proof of (i): 1 Z 1 Z sup f1 dµ ≤ sup f1 dµ 1/p0 E∩En 1/p0 E∩En 0<µ(E)<∞ µ(E) X 0<µ(E)<∞ µ(E ∩ En) X Z 1 1 0 = sup 0 f E dµ 0 0 0 1/p µ(E )>0,E ⊆En µ(E ) X Z 1 ≤ sup f1 0 dµ 0 1/p0 E 0<µ(E0)<∞ µ(E ) X

= Mp We can also prove a duality theorem for Lp,q spaces: Theorem 3.3. Let 1 < p < ∞ and 1 ≤ q ≤ ∞, then for all f ∈ Σ+: Z + (12) kfkLp,q ∼p,q sup fg dµ : g ∈ Σ , kgkLp0,q0 ≤ 1 =: Mp,q(f), X where Σ+ is the set of all non-negative simple functions with support of ﬁnite measure.

Proof. As before, the &p,q inequality follows from H¨older’sinequality for Lorentz spaces. Also, if q = ∞, then the result follows from Theorem 3.2. We will thus prove the inequality .p,q restricted to q < ∞. Homogeneity allows us to normalize kfkLp,q = 1, we thus have to show that

Mp,q(f) &p,q 1. Using Theorem 2.2 we can decompose f as X f = fm, m∈Z m where each fm is a quasi-step function of height 2 and width Wm, and such that m 1/p 2 Wm `q ∼ 1.

Observe that it follows from the proof of Theorem 2.2 that each fm must also be a simple function and, also, that fm is 0 for |m| suﬃciently large. Deﬁne the function g by X g = gm m∈Z 14 GUILLERMO REY

r p−1 m 1/p + where gm = amfm , am = 2 Wm and r = max(0, q − p). Clearly g ∈ Σ , and furthermore, 0 since the fm s have disjoint supports Em: Z Z X r p fg dµ = amfm dµ X X m Z X r p = am fm dµ. m X m The functions fm are quasi-step functions of height 2 and width Wm, thus Z p mp p fm dµ ∼p 2 Wm = am, X that is Z (∗) X r+p r+p fg dµ ∼p am ≤ kamk`q ∼p,q kfkLp,q = 1, X m∈Z where in (∗) we have used that k · k`p1 ≤ k · k`p2 when p1 ≥ p2 and that r + p ≥ q. So we only have to show that

kgkLp0,q0 ≤ 1. Observe that r m(p−1)1 gm .p,q am2 Em with µ(Em) = Wm, so a direct application of Theorem 2.2 would yield the result. The sequence r m(p−1) am2 , however, can fail to be of lower exponential growth (at least a priori). We can ﬁx this by deﬁning p−1 p−1 p−1 m 2 r k 2 m(p−1) r −k 2 γm = 2 sup ak2 = 2 sup am−k2 . k≤m k≥0 r m(p−1) Obviously γm ≥ am2 , and furthermore: p−1 p−1 p−1 2 m 2 r k 2 γm+1 = 2 2 sup ak2 k≤m+1 p−1 p−1 p−1 2 m 2 r k 2 ≥ 2 2 sup ak2 k≤m p−1 = 2 2 γm,

p−1 2 that is, {γm}m∈Z is of sub-exponential growth with constant C1 = 2 > 1. Now we can use Theorem 2.2 and the remark that follows it to reduce the problem to showing 1/p0 γmWm `q0 .p,q 1. −mp p But Wm = 2 am, so 1/p0 p−1 −m(p−1) 0 γmWm k`q = γmam 2 `q0 p−1 p−1 −m(p−1) m(p−1) r −k 2 = am 2 2 sup am−k2 `q0 k≥0 p−1 p−1 r −k 2 = sup am am−k2 `q0 k≥0

X p−1 p−1 r −k 2 ≤ am am−k2 q0 k≥0 ` ∞ X p−1 −k 2 p−1 r ≤ 2 am am−k `q0 . k=0 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 15

0 0 If q ≤ p =⇒ r = 0, but then q ≥ p =⇒ k · k`q0 ≤ k · k`p0 , so p−1 p−1 p−1 p−1 m 1/p p−1 am `q0 ≤ am `p0 = kamk`p ≤ kamk`q = 2 Wm `q .p,q 1, which completes the proof if q ≤ p. If q > p, then r = q − p and p − 1 q − p 1 + = , q q q0 so using H¨older’sinequality: p−1 q−p p−1 q−p a a k q0 ≤ a q a q m m−k ` m ` p−1 m−k ` q−p p−1 q−p = kamk`q kam−kk`q .p,q 1. This ﬁnishes the proof. We can now strengthen this theorem to eliminate the qualitative assumptions of being simple: Corollary 3.4. Let 1 < p < ∞ and 1 ≤ q ≤ ∞, then Z

(13) kfkLp,q ∼p,q sup fg dµ : g ∈ Σ, kgkLp0,q0 ≤ 1 X for all f ∈ Lp,q and where Σ is the set of all finite combinations of characteristic functions of sets of finite measure. Furthermore, if the measure space is σ-finite and if f is not a priori in Lp,q but the right hand side is well defined and finite, t hen the left hand side is also finite and we have (13). Proof. We will only show this in the case f ∈ Lp,q and f ≥ 0, the complete proof follows the same pattern as in the proof of Theorem 3.2. From Hölder’sinequality we can reduce to showing only the .p,q inequality. Approximate (a.e.) f by an increasing sequence {fn}n∈N of positive simple functions of finite support (this can be achieved if the measure space is σ-finite or if f is a priori in Lp,q, we omit the details). Using the Monotone Convergence Theorem for Lorentz spaces 1.8:

kfkLp,q = sup kfnkLp,q n Z + ∼p,q sup sup fng dµ : g ∈ Σ , kgkLp0,q0 ≤ 1 n X Z + = sup sup fng dµ : g ∈ Σ , kgkLp0,q0 ≤ 1 n X Z + = sup fg dµ : g ∈ Σ , kgkLp0,q0 ≤ 1 , X where we have used the (usual) monotone convergence theorem in the last line. This shows that there is a norm (the one deﬁned using the dual characterization) which is equivalent to the Lp,q quasi-norm deﬁned at the beginning when 1 < p < ∞.

4. Marcinkiewicz Interpolation Theorem In this section we will present a proof of the Marcinkiewicz interpolation theorem using the decomposition introduced in section2. We will use duality from the outset, indeed, we will need the following extension of Theorem 3.2: 16 GUILLERMO REY

Theorem 4.1. Let (X, µ) be a measure space and let 0 < p < ∞, the following statements are equivalent for f ∈ Lp,∞: (1) kfkLp,∞ .p 1. 0 1 (2) For every set E of ﬁnite measure there exists a subset E ⊆ E such that µ(E) ≥ 2 µ(E) such that Z 1 0 1/p0 f E0 dµ .p µ(E ) . X Furthermore, the theorem holds without the a priori assumption of f ∈ Lp,∞ provided (X, µ) is σ-ﬁnite.

Proof. We will assume 0 < p ≤ 1, we will also assume f ≥ 0 and kfkLp,∞ = 1 without loss of generality. The proof of (2) =⇒ (1) is almost identical to the one in Theorem 3.2, let us show the other one. 2 1/p 0 Let E ⊂ X be a set of ﬁnite (and positive) measure and let α = µ(E) . Deﬁne E = E \{x : f(x) > α}, then, since we have the estimate µ(E) µ({x : f(x) > α}) ≤ , 2 0 1 we can bound µ(E) ≥ µ(E ) ≥ 2 µ(E). Now we can estimate the integral Z 1/p 2 0 f1E0 dµ ≤ µ(E ) X µ(E) 0 ≤ 21/pµ(E0)1/p .

The Marcinkiewicz interpolation theorem, as in complex interpolation, uses the boundedness of an operator in two “extremal spaces” to provide boundedness in the “intermediate spaces”. However, while complex interpolation of Lp spaces requires the extremal spaces to also be Lp spaces, the Marcinkiewicz Interpolation Theorem only needs them to be weak-Lp spaces. It actually requires a bit less, the following definitions will make the exposition easier: Definition 4.2. Let T be an operator defined on a subset of the set of measurable functions of a measure space (X, µ) and that takes values in the measurable functions of another measure space (Y, ν). • We will say that T is of strong type (p, q) if T extends to a bounded operator from Lp(X, µ) to Lq(Y, ν). • We will say that T is of weak type (p, q) if T extends to a bounded operator from Lp(X, µ) to Lq,∞(Y, ν). In what follows we will always assume T to be an operator defined on a set D closed under addition, multiplication by scalars and containing Σc(X, µ): the set of finite linear combinations of characteristic functions of sets of finite measure of (X, 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 domain of T and λ ∈ C. We say that a sublinear operator T is of restricted weak type (p, q) if 1/p (14) kT fkLq,∞ . HW for all sub-step functions f ∈ D of height H and width W . LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 17

The next result shows that, usually, an operator which is of restricted weak type on characteristic functions is also of restricted weak type on Σc. More precisely:

Proposition 4.3. Let 0 < p ≤ ∞, 0 < q ≤ ∞, A > 0 and T be a sublinear operator deﬁned on Σc. Then the following statements are equivalent: (1) T satisﬁes 1/p kT fkLq,∞ . AHW

for all sub-step functions f of height H and width W in Σc. 0 0 1 (2) For every set F ⊆ Y of finite measure there exists a subset F ⊆ F with ν(F ) ≥ 2 ν(F ) and such that for all E ⊆ X of finite measure Z 1 1 1/p 0 1/q0 |T E| F 0 dν . Aµ(E) ν(F ) . Y Proof. That (1) =⇒ (2) is an easy consequence of Theorem 4.1 and Hölder’sinequality for Lorentz spaces, so we will prove the reverse implication. First assume that f takes the form

N X −j1 fN = 2 Ej j=1

1/p where µ(Ej) ≤ W . In this case kT fN kLq,∞ . AW (the constant being independent of N). To show this ﬁrst observe that using sublinearity and Lemma 5.1 it suﬃces to show

1/p 1 q,∞ (15) kT Ej kL . AW . Now let F ⊆ Y be a set of ﬁnite measure. By hypothesis there exists a subset F 0 ⊆ F which 0 1 satisﬁes ν(F ) ≥ 2 ν(F ) and Z 1/p 0 1/q0 1 0 1 F T ( Ej ) dν . Aµ(Ej) ν(F ) Y 0 ≤ ν(F 0)1/q AW 1/p.

So by Theorem 4.1 we obtain (15). Now let 0 ≤ f ≤ 1 be an arbitrary function in Σc of width W :

M X 1 f = aj Ej , j=1 the Ej’s disjointly supported. If dj(x) denotes the j-th digit in the binary expansion of f(x) and we deﬁne N N X −j X −j1 fN = 2 dj = 2 Fj , j=1 j=1 then we easily see that M X 1 f − fN = bj,N Ej j=1 18 GUILLERMO REY

−N with 0 ≤ bj,N ≤ 2 . So:

kT fkLq,∞ . kT fN kLq,∞ + kT (f − fN )kLq,∞ M 1/p X 1 . AW + bj,N |T Ej | q,∞ j=1 L M 1/p −N X 1 . AW + 2 |T Ej | , q,∞ j=1 L since the second term can be made arbitrarily small we conclude that 1/p kT fkLq,∞ . AW . By homogeneity and sublinearity we obtain the same bound (with a possibly larger constant) for arbitrary f ∈ Σc. To prove the Marcinkiewicz interpolation theorem we ﬁrst need a weaker result:

Proposition 4.4. Let 0 < p0, p1, q0, q1 ≤ ∞, A0,A1 > 0 and suppose that T is a sublinear operator defined on Σc which is of restricted weak type (pi, qi) with constants Ai for i = 0, 1. Then it is of restricted weak type (pθ, qθ), i.e.: 1/p q ,∞ θ (16) kT fkL θ .p0,q0,p1,q1 AθHW ∀f ∈ Σc, where 1 1 − θ θ 1 1 − θ θ = + , = + pθ p0 p1 qθ q0 q1 and 1−θ θ Aθ = A0 A1 Proof. Theorem 4.3 reduces matters to showing that for every set F ⊆ Y of finite measure there 0 0 1 exists a subset F ⊆ F with ν(F ) ≥ 2 ν(F ) and such that for all sets E ⊆ X of finite measure:

Z 0 1/pθ 0 1/q 1 0 1 θ F T ( E) dν .p0,q0,p1,q1 Aθµ(E) ν(F ) , Y but, using Theorem 4.3 and the hypotheses, we have that there exists a subset F 0 of F with 0 1 ν(F ) ≥ 2 ν(F ) and that for all sets E of ﬁnite measure

Z 0 1/pi 0 1/q 1 0 1 i F T ( E) dν .p0,q0,p1,q1 Aθµ(E) ν(F ) Y for i = 0, 1. The result now follows from the trivial scalar interpolation inequality: X ≤ Y ∧ X ≤ Z =⇒ X ≤ Y 1−θZθ for X,Y,Z ≥ 0 and 0 ≤ θ ≤ 1, and an application of Theorem 4.3 once more. We need one last lemma: Lemma 4.5. Let

Λλ(x, y) = (1 − λ)x + λy, then

Λα(Λβ(x, y), Λγ (x, y)) = ΛΛα(β,γ)(x, y). We are now ready to state and prove the main theorem of this section: LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 19

+ Theorem 4.6. Let T be a sublinear operator defined on Σ . Suppose that 0 < pi, qi ≤ ∞ and q0 6= q1. If T satisfies 1 1 kT EkLqi,∞ . Aik EkLpi,1 i = 0, 1, for all E of finite measure, then for all 1 ≤ r ≤ ∞ and 0 < θ < 1 such that qθ > 1 we have

q ,r p ,r kT fkL θ .p0,p1,q0,q1,r,θ AθkfkL θ for all f in Σc.

Proof. Using Theorem 4.3 we can assume that T is of restricted weak type (pi, qi) with constants Ai. + We will suppose ﬁrst that qi ≥ 1. Let f ∈ Σ , which by homogeneity we may assume to be normalized so that kfkLpθ ,r = 1. By duality (Theorem 3.3), it will be enough to verify the estimate Z |T f|g dν .p0,q0,p1,q1,r,θ Aθ Y + for all g ∈ Σ with kgk q0 ,r0 = 1. In what follows all constants will be assumed to depend on L θ p0, q0, p1, q1, r and θ and will be omitted. By hypothesis and H¨older’sinequality we have

Z 0 0 0 1/pi 01/q 0 1/pi 01/q (17) |T u|v dν . AiHH W W i ≤ AθHH min W W i Y i=0,1 for all sub-step functions u ∈ Σ+(X, µ) of height H and width W and for all sub-step functions v ∈ Σ+(Y, ν) of height H0 and width W 0. By Theorem 2.2 we can decompose f and g as X X f = fm and g = gn m n m where each fm is a sub-step function of height Hm and width 2 and where each gn is a sub-step 0 n function of height Hn and width 2 . These decompositions also satisfy 0 m/pθ 0 n/qθ Hm2 `r ∼ Hn2 `r0 ∼ 1.

Recall that, since f and g are simple, only a ﬁnite number of fm’s and gn’s are non-zero. Using (17) and sublinearity we have

Z 0 X 0 m/pi n/qi |T f|g dν Aθ HmHn min 2 2 . . i=0,1 Y m,n

0 m/pθ 0 n/q Call am = Hm2 and bn = Hn2 θ , we then have to obtain the estimate

X m(1/pi−1/pθ ) n(1/qθ −1/qi) (18) ambn min 2 2 1, i=0,1 . m,n where kamk`r = kbnk`r0 = 1. After reorganizing terms we can write the third factor as 1 1 1 1 1 1 1 1 min m − + n − , −m − − n − , p0 pθ qθ q0 pθ p1 q1 qθ which can be further simpliﬁed to β β min αθ m − n , −α(1 − θ) m − n , α α where α = 1 − 1 and β = 1 − 1 , which is easily seen to be true. Let p0 p1 q0 q1 ϕ(m − γn) = min (αθ (m − γn) , −α(1 − θ)(m − γn)) 20 GUILLERMO REY

β where γ = α , then we have to show that X ambnϕ(m − γn) . 1. m,n We can change variables and let m 7→ m + bγnc and reduce to showing X am+bγncbnϕ(m + bγnc − γn) . 1. m,n Observe that |bγnc − γn| ≤ 1, so we easily see that ϕ(m + bγnc − γn) ∼ ϕ(m), thus we end up with X X X ϕ(m) am+bγncbn . am+bγncbn `∞ m n n m since ϕ is in `1. Applying H¨older’sinequality with exponents r and r0 we reduce to showing that

ka k r 1 m+bγnc `n . r uniformly in m, but this is just the observation that the sum of am+bγnc is bounded by a constant r depending on γ times the sum of am+n, which is of course bounded by a constant independent of m. Finally, we show how to drop the assumption of qi > 1. Observe that we may interpolate using Theorem 4.4 to obtain restricted weak type (pθ, qθ) boundedness of T for 0 ≤ θ ≤ 1. Since we are imposing qθ > 1, we may assume that qi > 1 for at least one i ∈ {0, 1} (otherwise the theorem is void), thus, there must be a τ ∈ (0, 1) such that 1 < qτ < qθ and such that T is of restricted weak type (pτ , qτ ). We can now interpolate using what we have just proved from (pτ , qτ ) to the (pi, qi) which has qi > 1 and use Lemma 4.5 to ensure that (pθ, qθ) is in between (pτ , qτ ) and (pi, qi).

Corollary 4.7 (Marcinkiewicz). Let T be a sublinear operator that is of weak type (pi, qi) with constants Ai > 0 and such that 1 ≤ pi ≤ pi ≤ ∞ for i ∈ {0, 1}, suppose also that q0 6= q1. Then, 1−θ θ for 0 < θ < 1, T is of strong type (pθ, qθ) with constant .p0,q0,p1,q1,r,θ Aθ = A0 A1 where 1 1 − θ θ 1 1 − θ θ = + and = + . pθ p0 p1 qθ q0 q1 Proof. One just needs to recall that the Lp,q are nested and that Lp,p coincides with Lp, we leave the easy details to the reader. 5. Appendix Lemma 5.1. Let X be a vector space endowed with a quasi-norm k · k, that is, there exists a constant C1 > 0 such that

(19) kf + gk ≤ C1(kfk + kgk).

Then, for any sequence {fn}n∈N of elements of X verifying −j (20) kfjk . AC2 for some A > 0 and C2 > 1, we have N X (21) fj ≤ AC4

j=1 LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH 21 where C3 does not depend on N or A. −κ Proof. Choose κ ∈ N so large that C1C2 < 1 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 sum as follows N κd r X X X fj = fj + fkd+j. j=1 j=1 j=1

The norm of the second sum is trivially bounded by A times some constant depending on C1 and C2 (since r < κ,) so we can center our efforts on the first sum. To this end we factorize the sum as κd d−1 κ X X X fj = fκm+j. j=1 m=0 j=1 Now observe that, by induction, we have the following estimate: n n X X j gj ≤ C kgjk 1 j=1 j=1 for any sequence {gj} of elements of X. Using this we can estimate the first sum as follows: κd r d−1 κ X X X m+1 X fj + fkd+j ≤ C fκm+j 1 j=1 j=1 m=0 j=1 d−1 κ X m+1 X j −(κm+j) ≤ A C1 C1 C2 m=0 j=1 d−1 κ X −κm X j+1 −j = A C1C2 C1 C2 m=0 j=1 ∞ κ X −κm X j+1 −j ≤ A C1C2 · C1 C2 m=0 j=1 κ X j+1 −j .C1,C2 A C1 C2 j=1

.C1,C2 A. References [1] G. Hardy and J. Littlewood. A maximal theorem with function-theoretic applications. Acta Mathematica, 54(1):81–116, 1930. [2] R. A. Hunt. On L(p, q) spaces. Enseignement Math. (2), 12:249–276, 1966. [3] M. Keel and T. Tao. Endpoint Strichartz estimates. Amer. J. Math., 120(5):955–980, 1998. [4] R. O’Neil. Convolution operators and L(p, q) spaces. Duke Mathematical Journal, 30:129–142, 1963. [5] E. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1990. [6] T. Tao. Lecture notes 1 for 245C: Interpolation of Lp spaces. [7] N. Wiener. The ergodic theorem. Duke Mathematical Journal, 5(1):1–18, 1939.