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 jf(y)j dy; r>0 Br (x) 1 p defined for f 2 Lloc. We would like to obtain L boundedness of this operator for 1 ≤ p ≤ 1, but alas, this is not the case: although it is trivially bounded from L1 onto itself, it is not bounded on L1: Proposition 1.1. If f 2 L1(Rd) and f is not identically 0 then C Mf(x) ≥ d jxjd 1 d for some constant Cd dependent on f and the dimension, in particular f2 = L (R ). Proof. Since f is not identically 0, there exists an R > 0 such that Z jfj ≥ δ > 0: BR(0) Now, if jxj > R, BR(0) ⊆ B(x; 2jxj), so 1 Z δ Mf(x) ≥ jfj ≥ Cd d : jB(x; 2jxj)j BR(0) jxj 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 2 L1 kfk 1 m(fx 2 d : Mf(x) > λg) ≤ 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(fx 2 X : jf(x)j > λg) ≤ L (X,µ) : λp More precisely, define the weak-Lp space Lp;1(X; µ) to be the set of all measurable functions f for which its Lp;1 quasi-norm 1=p (3) kfkLp;1 = sup λµ(fx : jf(x)j > λg) λ>0 is finite. To abbreviate we will use the notation df (λ) = µ(fx 2 X : jf(x)j > λg) when there is no confusion as to what measure we are using. Also, we define by convention L1;1 to be L1. The following lemma is a simple application of the monotone convergence lemma: Lemma 1.3. Let ffngn2N be a sequence of measurable functions such that jfn(x)j ≤ jfn+1(x)j µ-almost everywhere and for all n 2 N. If we have jfnj ! f µ-almost everywhere for some function f then dfn (λ) % df (λ) when n ! 1 for all λ > 0. As we have seen, Chebychev's inequality gives us the inclusion Lp ,! Lp;1 for p > 0. This inclusion is proper as the following example shows: − d Example 1.4. If f(x) = jxj p then Z Z jf(x)jp dx = jxj−d dx = 1; d d R R however 1=p 1=p sup λdf (λ) = νd < 1; λ>0 d where νd is the measure of the unit ball in R . As we have noted, k · kLp;1 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;1 = sup λdf+g(λ) λ>0 1=p ≤ sup λ (df (λ/2) + dg(λ/2)) λ>0 ≤ 2Cp (kfkLp;1 + kgkLp;1 ) : 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 < 1 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) = jf(x)j dµ(x) X Z Z jf(x)j p dλ = pλ λ dµ(x) X 0 Z Z p1 dλ = p λ fs: s<jf(x)jg λ dµ(x) X + Z R Z p 1 dλ = p λ fy:jf(y)j>λg dµ(x) λ (Fubini) + X ZR p dλ = p λ df (λ) λ : + R So we have: 1=p kfkLp;1 = kλdf (λ) k 1 + 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 < 1 and 0 < q ≤ 1, 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 ffngn2N be a sequence of measurable functions such that jfnj % jfj µ-almost everywhere, then kfkLp;q = lim kfnkLp;q n!1 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 = 1 then the problem is reduced to the easier one of showing that 1=p 1=p sup kλd (λ) k 1 + = k sup λd (λ) k 1 + ; fn Lλ (R ) fn Lλ (R ) n≥1 n≥1 which is easily seen to hold true in the more general case sup kgnkL1 = k sup gnkL1 : n≥1 n≥1 4 GUILLERMO REY Suppose now that q < 1, then Z 1 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 ffngn2N 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(fx : lim inf jfn(x)j > λg) ≤ lim inf µ(fx : jfn(x)j > λg); 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 < 1, 0 < q ≤ 1, 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;1 satisfies the quasi-triangle inequality in Proposition 1.5, the proof for Lp;q when q 6= 1 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;1 norm: Z 1 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;1 .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 ffng be a Cauchy sequence in the L quasi-norm. Estimate (4) allows us to deduce that ffng 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 ff'(k)gk. Call f the µ-almost everywhere defined limit of f'(k) when k ! 1, 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 ff'(k)g so that we have −k kf'(k) − f'(k−1)kLp;q ≤ (2C) p;q using that fn is an L -Cauchy sequence and defining f'(0) = 0.