---

INTERPOLATION COUPLE L∞(dµ ; L1(dν)) , L∞(dν ; L1(dµ))

arXiv:math/9306209v1 [math.FA] 9 Jun 1993THE Kt–FUNCTIONAL FOR THEINTERPOLATION COUPLE L∞(dµ ; L1(dν)) , L∞(dν ; L1(dµ))Albrecht HeßGilles Pisier1§0. IntroductionLet (A0, A1) be a compatible couple of Banach spaces in the sense of Interpolation Theory(cf.

e.g. [BL]).

Then the Kt– functional of an element x in A0 + A1 is defined for all t > 0as followsKt(x ; A0 , A1) = inf {∥x0∥A0 + t ∥x1∥A1 | x = x0 + x1} .For instance the Kt– functional of the couple (L1(R), L∞(R)) is well known. We haveKt(x ; L1(R), L∞(R)) =Z t0 x∗(s) ds = supnZE |x(s)| ds , |E| = to.See [BL] for more details and references.Let (M, µ) and (N, ν) be measure spaces.

In this paper, we study the Kt– functionalfor the couple(0.1)A0 = L∞(dµ ; L1(dν)) ,A1 = L∞(dν ; L1(dµ)) .Here, and in what follows the vector valued Lp– spaces Lp(dµ ; Lq(dν)) are meant inBochner’s sense.One of our main results is the following, which can be viewed as a refinement of alemma due to Varopoulos [V].Theorem 0.1. Let (A0, A1) be as in (0.1).

Then for all f in A0 + A1 we have12 Kt(f; A0 , A1) ≤supµ(E) ∨t−1ν(F)−1 ZE×F |f| dµ dν≤Kt(f; A0 , A1) ,where the supremum runs over all measurable subsets E ⊂M , F ⊂N with positive andfinite measure and u∨v denotes the maximum of the reals u and v.1 Partially supported by the N.S.F.1

This result is a particular case of Theorem 3.2. below and its corollaries, which givean analogous estimate for the coupleA0 = Lp,∞(dµ ; Lq(dν)) ,A1 = Lp,∞(dν ; Lq(dµ)) ,when 1 ≤q < p ≤∞. The preceding Theorem 0.1 corresponds to the case p = ∞, q = 1.The paper is organized as follows.

In §1, we prove Theorem 0.1 in the finite discretecase, i.e. when M an N are finite sets equipped with discrete measures.

In §2, we discussthe extension to Lp,∞(dµ ; Lq(dν)) , Lp,∞(dν ; Lq(dµ)) , again in the discrete case and treatthe case of general measure spaces in §3. Finally in §4 we give some applications.To describe the main one, let us denote by B1 (resp.

B0) the space of all boundedoperators from L1(dν) to L1(dµ) (resp. from L∞(dν) to L∞(dµ)).

Our results yield adescription of the space (B0, B1)θ,q, 0 < θ < 1, 1 ≤q ≤∞, obtained by the real (i.e.Lions – Peetre) interpolation method. The result is particularly simple in the case q = ∞.In that case, we proveTheorem 0.2.

The space (B0, B1)θ,∞coincides with the space of all bounded regularoperators u from Lp,1(dν) to Lp,∞(dµ) with p = 1/θ.Here ”regular“ is meant in the sense of [MN]. Equivalently u belongs to (B0, B1)θ,∞iffthere is a positive (i.e.

positivity preserving) operator v from Lp,1(dν) to Lp,∞(dµ) whichdominates u, i.e. such that −v ≤u ≤v.

(Note that we implicitly use only real scalars,but this is not essential.) In particular, it follows that u belongs to (B0, B1)θ,∞iff|u| isbounded from Lp,1(dν) to Lp,∞(dµ), or equivalently iff|u| is of ”very weak type (p , p)“in the sense of [SW, chapter 3.3].

We refer to [MN] for the definition of the modulus |u|of a regular operator acting between Banach lattices. We merely recall that if u is givenby a matrix (aij) acting between two sequence spaces, then |u| corresponds to the matrix(|aij|).

A similar fact holds with ”kernels“ instead of matrices.2

§1. The Kt– functional for the interpolation couple ℓ∞m(ℓ1n) , ℓ∞n (ℓ1m)We use the following notation: for p ∈R , p ≥1 , let p∗be the conjugate exponent of p ,1/p+1/p∗= 1, the dot• denotes the pointwise multiplication of matrices a , b accordingto (a •b)(i, j) = a(i, j)b(i, j) .

For a set A let 1A be the characteristic function of A (thewhole set will always be known from the context).Let (M, µ) be the measure space consisting of the atoms {1, . .

. , m} of positivemasses µ1, .

. .

, µm and (N, ν) the measure space consisting of the atoms {1, . .

. , n} ofpositive masses ν1, .

. .

, νn. We equip M ×N with the product measure µ×ν.Besides the ℓp–norms|| a ||ℓp =X(i,j)∈M×N|a(i, j)|pµiνj1/p ,we introduce in this section for m×n–matrices a the norms|| a || = || a ||ℓ∞m (ℓ1n) = maxi∈MXj∈N|a(i, j)| νj,|| a ||⊤= || a⊤||ℓ∞n (ℓ1m) = maxj∈N Xi∈M|a(i, j)| µi.It is straightforward to see that for E ⊂M , F ⊂N ,|| 1E×F •a ||ℓ1 ≤min {µ(E)|| a ||, ν(F)|| a ||⊤} .Therefore we have for matrices b, c and any t > 0(1.1)|| 1E×F •(b + c) ||ℓ1 ≤µ(E) ∨t−1ν(F)|| b || + t || c ||⊤.If we introduce the norm||| a |||t = supE,Fµ(E) ∨t−1ν(F)−1|| 1E×F •a ||ℓ1 ,then (1.1) means that the Kt– functional for the interpolation couple ℓ∞m(ℓ1n) , ℓ∞n (ℓ1m)Kt(a) = inf{|| b || + t || c ||⊤| a = b + c }is an upper bound for this norm||| a |||t ≤Kt(a) .In the following we prove an upper estimate of Kt( . ) by the functional ||| .

|||t .3

Proposition 1.1. Let t > 0.

For any matrix a with ||| a |||t ≤1 we can split M ×N intoa disjoint union M ×N = A ∪B , A ∩B = ̸⃝, such that(1.2)|| 1A •a || ≤1 ,|| 1B •a ||⊤≤1/t ,henceKt(a) ≤2 ||| a |||t .Proof. We proceed by induction on m. For m = 1 suppose, without loss of generality,that(1.3)|a(1, 1)| ≥|a(1, 2)| ≥.

. .

≥|a(1, n)| .If t−1ν(N) ≤µ({1}) = µ1 we put k = n, if not, let k , 0 ≤k < n , be as large as possiblesuch that t−1ν({1, . .

. , k}) ≤µ1.

For E = {1} and F = {1, . .

. , k} we have in any case(1.4)µ(E) ∨t−1ν(F)−1|| 1E×F •a ||ℓ1 =Xj≤k|a(1, j)| νj ≤1.If k < n then t−1ν({1, .

. .

, k+1}) > µ1, so we obtain for E = {1} and F = {1, . .

. , k+1}µ(E) ∨t−1ν(F)−1|| 1E×F •a ||ℓ1 = µ1tPj≤k+1 |a(1, j)| νjPj≤k+1 νj≤1.By (1.3) this yields(1.5)|a(1, n)| µ1 ≤.

. .

≤|a(1, k+1)| µ1 ≤1/t .Take now A = {1}×{1, . .

. , k} , B = (M ×N)\A , and (1.2) is fulfilled.Let us assume the truth of the proposition for 1, .

. .

, m−1. Without loss of generality, wecan suppose the sums over Mσj =Xi≤m|a(i, j)| µito be in descending order σ1 ≥σ2 ≥.

. .

≥σn. If t−1ν(N) ≤µ(M) let k = n, if not, letk , 0≤k

. .

, k}) ≤µ(M). For E =M andF ={1, .

. .

, k} we have(1.6)|| 1E×F •a ||ℓ1 ≤µ(M).If we permute the rows such thatτi =Xj≤k|a(i, j)| νj4

are in descending order τ1 ≥τ2 ≥. .

. ≥τm then it follows from (1.6) that(1.7)τm ≤1.By the induction hypothesis on {1, .

. .

, m−1}×{1, . .

., k} we find a splitting{1, . .

. , m−1}×{1, .

. ., k} = A1 ∪B1 ,A1 ∩B1 = ̸⃝,such that|| 1A1•a || ≤1 ,|| 1B1•a ||⊤≤1/t .For the wanted splitting of M ×N we only have to putA = A1 ∪({m}×{1, .

. .

, k}) ,B = (M ×N)\A.Indeed, by the induction hypothesis on A1 and (1.7)|| 1A •a || ≤1 ,and we are done for k = n which implies B = B1 . If k < n we argue as in case m = 1(using E = M and F = {1, .

. .

, k+1}) and show thatσn ≤. .

. ≤σk+1 ≤1/t .This yields together with the hypothesis on B1|| 1B •a ||⊤≤1/t .Remark 1.2.

There is another “norm”|| a ||t = sup {µ(E)−1|| 1E×F •a ||ℓ1 | t−1ν(F) ≤µ(E)}closely related to ||| a |||t . This functional || a ||t has the drawback that for small values oft or for strongly varying masses the supremum may be on an empty set, in that case weput || a ||t = 0 .

In any case,|| a ||t ≤||| a |||t ≤Kt(a) .But for uniform masses, say µi = νj = 1 , and for t ≥1 , where || . ||t is indeed a norm,one can construct for a matrix a with || a ||t ≤1 almost along the same lines as above asplitting M ×N = A ∪B , A ∩B = ̸⃝, such that (denoting by [ t ] the integer part of t)|| 1A •a || ≤1 ,|| 1B •a ||⊤≤1/[ t ] ,henceKt(a) ≤(1 + t/[ t ])|| a ||t < 3 || a ||t ,an inequality due, for t=1, to Varopoulos [V], cf.

also [BF].5

§2. The Kt– functional for the interpolation couple ℓp,∞M (ℓqN) , ℓp,∞N (ℓqM)For any Bochner–measurable function f on a measure space (Ω, Σ, µ) with values in aBanach space X and for p > 0 let(2.1)|| f ||Lp,∞(X) = inf {C | tpµ{∥f∥>t} ≤Cp for all t>0} ,or more generally by using the nonincreasing, equimeasurable rearrangement f ∗of ∥f∥(2.2)|| f ||Lp,∞(X) = supt>0 t1/pf ∗(t) ,|| f ||Lp,q(X) =Z ∞0 (t1/pf ∗(t))q dtt1/q .In general, these quantities are not norms but can be replaced for p>1, q≥1 by equivalentnorms.

We do not use this and refer to [BS], Lemma IV.4.5, p. 219.We start with the case of the interpolation couple ℓp,∞m (ℓ1n) , ℓp,∞n(ℓ1m) on the finitemeasure spaces (M, µ), (N, ν) from section 1 and consider the functionals|| a || = || a ||ℓp,∞m(ℓ1n) and || a ||⊤= || a⊤||ℓp,∞n(ℓ1m) .We have for p > 1 and E ⊂M , F ⊂N(2.3)|| 1E×F •a ||ℓ1 ≤p∗min {µ(E)1/p∗|| a ||, ν(F)1/p∗|| a ||⊤} .We introduce the norm ||| . |||p,t as follows||| a |||p,t = supE,Fµ(E)1/p∗∨t−1ν(F)1/p∗−1|| 1E×F •a ||ℓ1 .As in section 1 we want to compare the Kt– functional Kp,t( . ) for the interpolation coupleℓp,∞m (ℓ1n) , ℓp,∞n(ℓ1m) with the norm ||| .

|||p,t . In view of (2.3) we have a lower estimate ofKp,t(a)(2.4)||| a |||p,t ≤p∗Kp,t(a) .Now we establish the corresponding upper estimate.Proposition 2.1.

Let t > 0 , p > 1. For any matrix a with ||| a |||p,t ≤1 we can splitM ×N into a disjoint union M ×N = A ∪B , A ∩B = ̸⃝, such that(2.5)|| 1A •a || ≤1 ,|| 1B •a ||⊤≤1/t ,henceKp,t(a) ≤2 ||| a |||p,t .6

Proof. Similar to the proof of Proposition 1.1.

For m = 1 take k = n if ν(N) ≤µ1 tp∗,if not, suppose |a(1, 1)| ≥|a(1, 2)| ≥. .

. ≥|a(1, n)| and choose k , 0 ≤k < n , as large aspossible such that ν({1, .

. .

, k}) ≤µ1 tp∗. Inserting E = {1} and F = {1, .

. .

, k} we haveµ({1})1/p Xj≤k|a(1, j)| νj ≤1 ,and for E = {1} and F = {1, . .

. , l} , l > k ,ν({1, .

. .

, l})1/p|a(1, l)|µ1 ≤1/t .So we can put A = {1}×{1, . .

., k} , B = (M ×N)\A to obtain (2.5).For m>1 let us put k=n if ν(N)≤µ(M) tp∗, if not, choose the largest k , 0≤k

. , k}) ≤µ(M) tp∗.

Taking E = M and F = {1, . .

. , k} we have|| 1E×F •a ||ℓ1 ≤µ(M)1/p∗,consequently (we use the notations τm, σl of the preceding proof)(2.6)µ({1, .

. .

, m})1/pτm ≤1 ,and for E = M and F = {1, . .

. , l} , l > k ,(2.7)ν({1, .

. .

, l})1/p σl ≤1/t .The induction hypothesis on {1, . .

. , m−1}×{1, .

. ., k} together with (2.6) and (2.7) yieldsthe result.Remark 2.2.

The inequality (2.4) is meaningless for p = 1 , but the corresponding norm||| a |||1,t = supE,F (1∧t) || 1E×F •a ||ℓ1 = (1∧t) || a ||ℓ1(M×N)is proportional to the norm || a ||ℓ1(M×N) .Remark 2.3. For M = N = {1, .

. .

, m} , µi = νj = 1 , and for t ≥1 one can prove withonly slight changes the equivalence of ||| . |||p,t and|| a ||p,t = sup {µ(E)−1/p∗|| 1E×F •a ||ℓ1 | t−1ν(F)1/p∗≤µ(E)1/p∗} ,namely, with constants independent of m and t(1/p∗) || a ||p,t ≤(1/p∗) ||| a |||p,t ≤Kp,t(a) ≤3 || a ||p,t .We leave the details to the reader.

¿From the example M ={1} , N ={1, . .

. , n} , the νjand µ1 equal to 1 and a ≡1 , where ||| a |||p,1 = 1 , but Kp,1(a) = n1/p , we see that thereshould be a restriction M =N in order to have the occurring constants independent of n.7

Remark 2.4. If we replace || a || by || a ||ℓp,qm (ℓ1n) and || a ||⊤by || a⊤||ℓp,qn (ℓ1m) then|| 1E×F •a ||ℓ1 ≤C(p, q) min {µ(E)1/p∗|| a ||, ν(F)1/p∗|| a ||⊤}for the constant C(p, q) = (p∗/q∗)1/q∗.

Therefore the Kt–functional for the interpolationcouple ℓp,qm (ℓ1n) , ℓp,qn (ℓ1m) is greater than C(p, q)−1||| . |||p,t .

But there is no upper bound ofthis Kt– functional by ||| . |||p,t independent of m and n.Take e.g.

M = N = {1, . .

. , n} , µi = νj = 1 anda =12−1/p.

. .n−1/p00.

. .0.........00. .

.0then ||| a |||p,t ≤p∗, Kt(a) ∼(log n)1/q .If we now put for 1 ≤p, q ≤∞(2.8)|| a || = || a ||ℓp,∞m(ℓqn) ,|| a ||⊤= || a⊤||ℓp,∞n(ℓqm) ,define the Kt– functional Kp,q,t( . ) for the interpolation couple ℓp,∞m (ℓqn) , ℓp,∞n(ℓqm) and let,with the abbreviation α=1/q−1/p ,(2.9)||| a |||p,q,t = supE,Fµ(E)α∨t−1ν(F)α−1|| 1E×F •a ||ℓq ,then we derive from Proposition 2.1 by q-convexification (cf.

[LT]) the following theorem.Theorem 2.5. Let t > 0 , 1 ≤q < p ≤∞.

Denoting C(p, q) = (1−q/p)1/q we canestimate the Kt– functional Kp,q,t( . ) for the interpolation couple ℓp,∞m (ℓqn) , ℓp,∞n(ℓqm) by(2.10)C(p, q) ||| a |||p,q,t ≤Kp,q,t(a) ≤2 ||| a |||p,q,t .More precisely, for ||| a |||p,q,t ≤1 we find A , B with M ×N = A ∪B , A ∩B = ̸⃝, and|| 1A •a || ≤1 ,|| 1B •a ||⊤≤1/t .Remark 2.6.

An analogous result can be stated for measure spaces M = N with equalatoms µi = νj = 1 and t ≥1 for the norm|| a ||p,q,t = sup {µ(E)−α|| 1E×F •a ||ℓ1 | t−1ν(F)α≤µ(E)α} ,(α=1/q−1/p)(cf. Remark 2.3).8

Theorem 2.7. The statement (2.10) remains true for arbitrary discrete measure spaces.Proof.

Let M, N be arbitrary discrete measure spaces. It is very easy to check that foran element a = (a(i, j)) we have∥a∥ℓp,∞M (ℓqN) = sup ∥1A×B •a∥ℓp,∞M (ℓqN) ,where the sup runs over all finite subsets A ⊂M, B ⊂N.

Hence the result follows by asimple pointwise compactness argument left to the reader.9

§3. The Kt– functional for the couple Lp,∞(dµ ; Lq(dν)) , Lp,∞(dν ; Lq(dµ))Now we generalize the results from the previous sections to arbitrary measure spaces andtreat the generic case (M, µ)=(N, ν)=(R, λ) of non–atomic measure spaces, where λ isthe Lebesgue –measure.

For a=a(x, y) ∈Lqloc(R2) let us define the functionals|| a || = || a ||Lp,∞x(Lqy) ,|| a ||⊤= || a⊤||Lp,∞y(Lqx) .These functionals are used to define the Kt– functional Kp,q,t( . ) of the interpolation coupleLp,∞(dx ; Lq(dy)) , Lp,∞(dy ; Lq(dx)) in the obvious manner.

For t > 0, 1 ≤q < p ≤∞andα=1/q−1/p, let us introduce the norm||| a |||p,q,t = supE,Fλ(E)α∨t−1λ(F)α−1ZZE×F |a(x, y)|q dx dy1/q .Obviously (c.f. Theorem 2.5) for C(p, q) = (1 −p/q)1/qC(p, q) ||| a |||p,q,t ≤Kp,q,t(a) .We will now prove the counterpartKp,q,t(a) ≤2 ||| a |||p,q,t .Theorem 3.1.

For any t > 0 we have the inequality(3.1)C(p, q) ||| a |||p,q,t ≤Kp,q,t(a) ≤2 ||| a |||p,q,tbetween the norm ||| . |||p,q,t defined above and the Kt– functional of the interpolation coupleLp,∞(dx ; Lq(dy)) , Lp,∞(dy ; Lq(dx)) .Proof.

Let us call countably simple a function a : R2 →R of the form a = P aij 1Ai×Bjwhere (Ai) and (Bj) are countable measurable partitions of R. One can easily check thatthe subset of countably simple functions is dense in Lp,∞(dx ; Lq(dy))+Lp,∞(dy ; Lq(dx)) ,hence it suffices to check the inequality (3.1) for those functions. But in that case (3.1)follows from Theorem 2.7.

(This argument presupposes that the functional Kp,q,t is continuous with respect to thenorm ||| . |||p,q,t, but this can be checked using the equivalent renorming of weak Lp, condi-tional expectations and a weak compactness argument.

)Corollary 3.2. By combination of Theorems 2.7 and 3.1 we obtain the inequality (3.1)for all measure spaces.10

Remark 3.3. In the case of non–atomic measure spaces (M, µ), and (N, ν) of infinitetotal measures µ(M)=ν(N)=∞the difference between ||| a |||p,q,t and|| a ||p,q,t = sup {µ(E)−α|| 1E×F a ||Lq(M×N) | t−1ν(F)α ≤µ(E)α}(α=1/q−1/p)disappears completely.

Obviously||| a |||p,q,t = || a ||p,q,t .If the (non–atomic) measure spaces (M, µ) , (N, ν) have the same finite total measure, say1, the equivalence of these two norms with constants independent of t can be establishedfor t ≥1. The example of the function a ≡1 on [ 0, 1 ]×[ 0, 1 ] (equipped with the Lebesguemeasure) gives for 0

For non–atomic measure spaces (M, µ) , (N, ν) with the same finitetotal measure, say µ(M)=ν(N)=1, 1≤qµ(E)α and(3.2)t ν(F)−α ZZE×F |a(x, y)|q dx dy1/q > 1 .In order to construct ˜E and ˜F , t−1ν( ˜F)α ≤µ( ˜E)α with(3.3)µ( ˜E)−α ZZ˜EטF |a(x, y)|q dx dy1/q > tq/p−q ,we can suppose that t−1ν(F)α >µ(M)α and E =M in (3.2).

If not, ˜F =F and any ˜E ⊃Ewith µ( ˜E)=t−1/α ν(F) verify (3.3).By [BS], Theorem II, 2.7, p. 51, we find for all 0<λ≤1 an Fλ ⊂N , ν(Fλ)=λ ν(F) , suchthatt ν(Fλ)−α ZZM×Fλ|a(x, y)|q dx dy1/q > λ1/p .If we put ˜E =M and ˜F =Fλ forλ = µ(M)ν(F) t1/α ≥t1/α ,then (3.2) is obvious.11

§4. ApplicationsLet us denote by B(X, Y ) the space of bounded operators between two Banach spacesX, Y .

Let (M, µ), (N, ν) be arbitrary measure spaces. Let B0 = B(L∞(dν), L∞(dµ))and B1 = B(L1(dν), L1(dµ)).

Clearly we may view (B0 , B1) as a compatible couple in thesense of interpolation theory by identifying an element of B0 or B1 with a linear operatorfrom L∞(dν) ∩L1(dν) into L∞(dµ) + L1(dµ). For simplicity we will assume (althoughthis is inessential) that all the spaces we consider are over the field of real scalars.Let L, Λ be real Banach lattices and let u : L →Λ be a bounded operator.

Thenu is called regular if there is a positive operator v : L →Λ such that|u(x)| ≤v(|x|)for all x in L .We define∥u∥r = inf{∥v∥}where the infimum runs over all such dominating operators v. We denote by Br(L, Λ) thespace of all regular operators equipped with this norm. If Λ is Dedekind complete in thesense of [MN] then Br(L, Λ) is a Banach lattice and we have simply∥u∥r = ∥|u| ∥B(L,Λ) .This applies in particular when Λ is a Lorentz space, as in the situation we consider below.We should mention that any bounded operator between L∞– spaces (or L1– spaces) isautomatically regular, so that B0 = Br(L∞(dν), L∞(dµ)) and B1 = Br(L1(dν), L1(dµ)).Theorem 4.1.

For all u in B0 + B1, let||| u |||t = supnµ(E) ∨t−1ν(F)−1D|u|(1F), 1EEowhere the supremum runs over all measurable subsets E ⊂M, F ⊂N with positive andfinite measure. Then for all t > 012 Kt(u ; B0 , B1) ≤||| u |||t ≤Kt(u ; B0 , B1) .Proof.

Assume first that (M, µ) and (N, ν) are purely atomic measure spaces each withonly finitely many atoms. Then B0 and B1 can be identified (via their kernels) respectivelywith L∞(dµ ; L1(dν)) and L∞(dν ; L1(dµ)).

Therefore, in that case the nontrivial part ofTheorem 4.1 is but a reformulation of Proposition 1.1.The general case can be deduced from this using conditional expectations and a simpleweak compactness argument. We leave the details to the reader.12

Let (A0, A1) be any compatible couple of Banach spaces. We recall that (A0, A1)θ,qis defined for 0 < θ < 1, 1 ≤q ≤∞as the space of all x in A0 + A1 such that∥x∥θ,q = Z ∞0t−θKt(x ; A0, A1)q dtt1/q< ∞with the usual convention when q = ∞.

When equipped with the norm ∥. ∥θ,q, the space(A0, A1)θ,q is a Banach space.

We refer to [BL] for more informations.Theorem 4.2. Let 0 < θ < 1 and θ = 1/p .

We have(B0, B1)θ,∞= BrLp,1(dν), Lp,∞(dµ)with equivalent norms.Proof. Consider an operator u : Lp,1(dν) →Lp,∞(dµ) .

Let us define for θ = 1/p(4.1)[ u ]p = supν(F)−θµ(E)θ−1 Du(ϕ 1F) , ψ 1EEwhere the supremum runs over all ϕ (resp. ψ) in the unit ball of L∞(dν) (resp.

L∞(dµ))and over all measurable subsets E ⊂M, F ⊂N with finite positive measure. By a wellknown property of the Lorentz spaces Lp,1(dν) and Lp∗,1(dµ) (cf.

[SW, Theorem 3.13])there is a positive constant Cp depending only on p, 1 < p < ∞, such that we haveC−1p [ u ]p ≤∥u∥Lp,1(dν)→Lp,∞(dµ) ≤Cp [ u ]p ,for all u : Lp,1(dν) →Lp,∞(dµ) . If u is positive, (4.1) can be simplified.

In particular wehave(4.2)[ |u| ]p = supE⊂MF ⊂Nν(F)−θµ(E)θ−1 D|u|(1F) , 1EE Assume ν(F) = t µ(E) then ν(F)−θµ(E)θ−1 = t−θµ(E)−1, hence[ |u| ]p = supt>0 t−θ||| u |||t .Therefore by Theorem 4.1 ∥u∥(B0,B1)θ,∞is equivalent to [ |u| ]p or equivalently to the normof u in BrLp,1(dν), Lp,∞(dµ). This yields immediately the announced result.Remark 4.1.

We refer to [P1] for the analogue of Theorem 4.2 for the complex interpo-lation method.13

We conclude this paper with an application to Hp– spaces in the framework alreadyconsidered in [P3]. Let (A0, A1) be a compatible couple of Banach spaces and S0 ⊂A0,S1 ⊂A1 be closed subspaces.

Following the terminology in [P2], we will say that (S0, S1)is K– closed in (A0, A1) if there is a constant C such that for all x in S0 + S1 and all t > 0we haveKt(x ; S0, S1) ≤C Kt(x ; A0, A1) .Let (T, m) be the unit circle equipped with the normalized Lebesgue measure. LetB be a Banach space.

We will denote for 1 ≤p ≤∞by Hp(dm) (resp. Hp(dm ; B)) thesubspace of Lp(dm) (resp.

Lp(dm ; B)) of all the functions f such that ˆf(n) = 0, ∀n < 0.Let (M, µ) be any measure space. Consider the coupleX0 = L1dµ ; L∞(dm),X1 = L1dm ; L∞(dµ)and the subspacesY0 = L1dµ ; H∞(dm),Y1 = H1dm ; L∞(dµ).It is proved in [P3, Lemma 2] that (Y0, Y1) is K– closed in (X0, X1).By the simpleduality principle emphasized in [P2] (cf.

Proposition 1.11 and Remark 1.12 in [P2]) thisimplies that a similar property holds for the orthogonal subspaces Y ⊥0 , Y ⊥1 . More precisely,consider the subspacesS0 = L∞dµ ; H1(dm),S1 = H∞dm ; L1(dµ)of the spacesA0 = L∞dµ ; L1(dm),A1 = L∞dm ; L1(dµ).Then by this duality principle, (S0, S1) is K– closed in (A0, A1).

Therefore, our computa-tion of the Kt– functional for the couple (A0, A1) (cf. Theorem 0.1 above) is applicable tothe couple (S0, S1).

Taking for simplicity M = N equipped with the counting measure,we obtain:Theorem 4.3. There is a numerical constant C with the following property.

Let (fn)be a sequence in H1(dm) and let t > 0. Assume that for all subsets E ⊂N and allmeasurable subsets F ⊂T we have(4.3)Xn∈EZF |fn| dm ≤|E| ∨t−1m(F) .Then there is a decomposition fn = gn + hn with gn ∈H1(dm), hn ∈H∞(dm) such thatsupn∈N ∥gn∥H1(dm) ≤C andXn∈N|hn|L∞(dm) ≤C t−1 .14

§5. References[BF] R. C. Blei and J. J. F. Fournier.

Mixed–norm conditions and Lorentz norms. Proc.SLU–GTE Conference on Commutative Harmonic Analysis .

in: Contemp.Math. 91 ().

57–78[BL] J. Bergh and J. L¨ofstr¨om. Interpolation Spaces: An Introduction.

Grundlehren derMathematischen Wissenschaften 223. Springer–Verlag.

. [BS] C. Bennett and R. Sharpley.

Interpolation of Operators. Academic Press.

. [LT] J. Lindenstrauss and L. Tzafriri.

Classical Banach Spaces II. Springer Verlag.

. [MN] P. Meyer–Nieberg.

Banach Lattices. Springer Verlag.

Univerisitext. .

[P1] G. Pisier. Complex interpolation and regular operators between Banach lattices.Archiv der Math.

[ to appear ][P2] G. Pisier. Interpolation between Hp-spaces and non-commutative generalizations I.Pacific J.

Math. 155 ().

341–368. [P3] G. Pisier.

Interpolation between Hp-spaces and non-commutative generalizations II.Revista Mat. Iberoamericana () [ to appear ][SW] E. Stein and G. Weiss.Introduction to Fourier Analysis on Euclidean Spaces.Princeton University Press.

. [V] N. Varopoulos.

On an inequality of von Neumann and an application of the metrictheory of tensor products to operators theory. J. Funct.

Anal. 16 ().

83–100.Gilles PisierTexas A. & M. UniversityCollege StationTX 77843 – U. S. A.andUniversit´e Paris VIBoˆıte 1864, Place Jussieu75252 Paris Cedex 05 – FranceAlbrecht HeßMathematische Fakult¨atFriedrich–Schiller–Universit¨atUni–Hochhaus, 17.

OG07740 Jena – Germany15

---

출처: arXiv:9306.209 • 원문 보기