Interpolation Between Hp Spaces and Non-Commutative Generalizations II*
Interpolation Between Hp Spaces and Non-Commutative Generalizations II*
arXiv:math/9212204v1 [math.FA] 4 Dec 1992Interpolation Between Hp Spaces and Non-Commutative Generalizations II*by Gilles PisierAbstractWe continue an investigation started in a preceding paper. We discuss the classicalresults of Carleson connecting Carleson measures with the ¯∂-equation in a slightly moreabstract framework than usual.We also consider a more recent result of Peter Joneswhich shows the existence of a solution of the ¯∂-equation, which satisfies simultaneouslya good L∞estimate and a good L1 estimate.
This appears as a special case of our mainresult which can be stated as follows: Let (Ω, A, µ) be any measure space.Considera bounded operator u : H1 →L1(µ). Assume that on one hand u admits an extensionu1 : L1 →L1(µ) bounded with norm C1, and on the other hand that u admits an extensionu∞: L∞→L∞(µ) bounded with norm C∞.
Then u admits an extension eu which isbounded simultaneously from L1 into L1(µ) and from L∞into L∞(µ) and satisfies∥˜u: L∞→L∞(µ)∥≤CC∞∥˜u: L1 →L1(µ)∥≤CC1where C is a numerical constant.IntroductionWe will denote by D the open unit disc of the complex plane, by T the unit circle andby m the normalized Lebesgue measure on T. Let 0 < p ≤∞. We will denote simply byLp the space Lp(T, m) and by Hp the classical Hardy space of analytic functions on D. Itis well known that Hp can be identified with a closed subspace of Lp, namely the closurein Lp (for p = ∞we must take the weak*-closure) of the linear span of the functions{eint|n ≥0}.
More generally, when B is a Banach space, we denote by Lp(B) the usualspace of Bochner-p-integrable B-valued functions on (T, m), so that when p < ∞, Lp ⊗Bis dense in Lp(B). We denote by Hp(B) (and simply Hp if B is one dimensional) theHardy space of B-valued analytic functions f such that supr<1(R∥f(rz)∥pdm(z))1/p < ∞.
* Supported in part by N.S.F. grant DMS 90035501
We denote∥f∥Hp(B) = supr<1(Z∥f(rz)∥pdm(z))1/p.We refer to [G] and [GR] for more information on Hp-spaces and to [BS] and [BL] for moreon real and complex interpolation.We recall that a finite positive measure µ on D is called a Carleson measure if thereis a constant C such that for any r > 0 and any real number θ, we haveµ({z ∈D, |z| > 1 −r, | arg(z) −θ| < r}) ≤Cr.We will denote by ∥µ∥C the smallest constant C for which this holds. Carleson (see [G])proved that, for each 0 < p < ∞, this norm ∥µ∥C is equivalent to the smallest constant C′such that(0.1)∀f ∈HpZ|f|pdµ ≤C′∥f∥pHp.Moreover, he proved that, for any p > 1 there is a constant Ap such that any harmonicfunction v on D admitting radial limits in Lp(T, m) satisfies(0.2)ZD|v|pdµ ≤Ap∥µ∥CZT|v|pdm.We observe in passing that a simple inner outer factorisation shows that if (0.1) holds forsome p > 0 then it also holds for all p > 0 with the same constant.It was observed a few years ago (by J.Bourgain [B], and also, I believe, by J.Garcia-Cuerva) that Carleson’s result extends to the Banach space valued case.
More precisely,there is a numerical constant K such that, for any Banach space B, we have(0.3)∀p > 0 ∀f ∈Hp(B)Z∥f∥pdµ ≤K∥µ∥C∥f∥pHp(B).Since any separable Banach space is isometric to a subspace of ℓ∞, this reduces to thefollowing fact. For any sequence {fn, n ≥1} in Hp, we have(0.4)Zsupn |fn|pdµ ≤K∥µ∥CZsupn |fn|pdm.2
This can also be deduced from the scalar case using a simple factorisation argument.Indeed, let F be the outer function such that |F| = supn |fn| on the circle. Note that bythe maximum principle we have |F| ≥supn |fn| inside D, hence (0.1) impliesZsupn|fn|pdµ ≤Z|F|pdµ ≤C′Z|F|pdm =Zsupn|fn|pdm.This establishes (0.4) (and hence also (0.3)).We wish to make a connection between Carleson measures and the following resultdue to Mireille L´evy [L]:Theorem 0.1.
Let S be any subspace of L1 and let u : S →L1(µ) be an operator. LetC be a fixed constant.
Then the following are equivalent(i) For any sequence {fn, n ≥1} in S, we haveZsupn |u(fn)|dµ ≤CZsupn |fn|dm. (ii) The operator u admits an extension eu : L1 →L1(µ) such that ∥eu∥≤C.Proof: This theorem is a consequence of the Hahn Banach theorem in the same styleas in the proof of theorem 1 below.
We merely sketch the proof of (i)⇒(ii). Assume (i).Let V ⊂L∞(µ) be the linear span of the simple functions (i.e.
a function in V is a linearcombination of disjointly supported indicators). Consider the space S ⊗V equipped withthe norm induced by the space L1(m; L∞(µ)).
Let w =nP1ϕi ⊗fi with ϕi ∈Vfi ∈S .We will write⟨u, w⟩=X⟨ϕi, ufi⟩.Then (i) equivalently means that for all such w|⟨u, w⟩| ≤C∥w∥L1(m;L∞(µ)).By the Hahn Banach theorem, the linear form w →⟨u, w⟩admits an extension of norm ≤Con the whole of L1(m; L∞(µ)). This yields an extension of u from L1 to L∞(µ)∗= L1(µ)∗∗,with norm ≤C.
Finally composing with the classical norm one projection from L1(µ)∗∗to L1(µ), we obtain (ii).In particular, we obtain as a consequence the following (known) fact which we wishto emphasize for later use.3
Proposition 0.2. Let µ be a Carleson measure on D, then there is a bounded operatorT : L1 →L1(µ) such that T(eint) = zn for all n ≥0, or equivalently such that T inducesthe identity on H1.Proof: We simply apply L´evy’s theorem to H1 viewed as a subspace of L1, and to theoperator u : H1 →L1(µ) defined by u(f) = f. By (0.1) we have ∥u∥≤K∥µ∥C, butmoreover by (0.4) and L´evy’s theorem there is an operator T : L1 →L1(µ) extending uand with ∥T∥≤K∥µ∥C.
This proves the proposition.Allthough we have not seen this proposition stated explicitly, it is undoubtedly knownto specialists (see the remarks below on the operator T ∗). Of course, for p > 1 there isno problem, since in that case the inequality (0.2) shows that the operator of harmonicextension (given by the Poisson integral) is bounded from Lp into Lp(µ) and of course itinduces the identity on Hp.
However this same operator is well known to be unboundedif p = 1. The adjoint of the operator T appearing in Proposition 0.2 solves the ¯∂-barequation in the sense that for any ϕ in L∞(µ) the function G = T ∗(ϕ) satisfies ∥G∥L∞(m) ≤∥T∥∥ϕ∥∞together with∀f ∈H1ZfGdm =Zfϕdµ,and by well known ideas of H¨ormander [H] this means equivalently that Gdm is the bound-ary value (in the sense of [H]) of a distribution g on ¯D such that ¯∂g = ϕ.µ.
In conclusion,we have¯∂g = ϕ.µand∥G∥L∞(m) ≤K∥µ∥C∥ϕ∥∞.This is precisely the basic L∞-estimate for the ¯∂-equation proved by Carleson to solvethe corona problem, (cf. [G], theorem 8.1.1, p.320).
More recently, P.Jones [J] proved arefinement of this result by producing an explicit kernel which plays the role of the operatorT ∗in the above. He proved that one can produce a solution g of the equation ¯∂g = ϕ.µwhich depends linearly on ϕ with a boundary value G satisfying simultaneously∥G∥L∞(m) ≤K∥µ∥C∥ϕ∥∞and∥G∥L1(m) ≤KZ|ϕ|dµ,where K is a numerical constant.
(Jones [J] mentions that A.Uchiyama found a differentproof of this. A similar proof, using weights, was later found by S.Semmes.) Taking into4
account the previous remarks, our theorem 1 below gives at the same time a different proofand a generalization of this theorem of Jones.Our previous paper [P] contains simple direct proofs of several consequences of Jones’result for interpolation spaces between Hp-spaces. We will use similar ideas in this paper.Let us recall here the definition of the Kt functional which is fundamental for the realinterpolation method.
Let A0, A1 be a compatible couple of Banach (or quasi-Banach)spaces. For all x ∈A0 + A1 and for all t > 0, we letKt(x, A0, A1) = inf∥x0∥A0 + t∥x1∥A1 | x = x0 + x1, x0 ∈A0, x1 ∈A1).Let S0 ⊂A0, S1 ⊂A1 be closed subspaces.
As in [P], we will say that the couple (S0, S1)is K-closed (relative to (A0, A1)) if there is a constant C such that∀t > 0∀x ∈S0 + S1Kt(x, S0, S1) ≤CKt(x, A0, A1).Main resultsTheorem 1. Let (Ω, A, µ) be an arbitrary measure space.
Let u: H∞→L∞(µ) be abounded operator with norm ∥u∥= C∞. Assume that u is also bounded as an operatorfrom H1 into L1(µ), moreover assume that there is a constant C1 such that for all finitesequences x1, .
. ., xn in H1 we haveZsupi|u(xi)|dµ ≤C1Zsup |xi|dm.Then there is an operator ˜u: L∞→L∞(µ) which is also bounded from L1 into L1(µ) suchthat∥˜u: L∞→L∞(µ)∥≤CC∞∥˜u: L1 →L1(µ)∥≤CC1where C is a numerical constant.Proof:Let w be arbitrary in L∞(µ) ⊗H∞.We introduce on L∞(µ) ⊗L∞(m) thefollowing two norms ∀w ∈L∞(µ) ⊗L∞(m)∥w∥0 =Z∥w(ω, ·)∥L∞(dm)dµ(ω)∥w∥1 =Z∥w(·, t)∥L∞(dµ)dm(t).5
Let A0 and A1 be the completions of L∞(µ) ⊗L∞(m) for these two norms. (Note that A0and A1 are nothing but respectively L1(dµ; L∞(dm)) and L1(dm; L∞(dµ)). ) Let S0 andS1 be the closures of L∞(µ) ⊗H∞in A0 and A1 respectively.The completion of the proof is an easy aplication (via the Hahn Banach theorem) ofthe following result which is proved further below:Lemma 2.
(S0, S1) is K-closed.Indeed, assuming the lemma proved for the moment, fix t > 0, and consider w inL∞(µ) ⊗H∞, we have (for some numerical constant C)∀t > 0Kt(w, S0, S1) ≤CKt(w, A0, A1).Recall that we denote by V ⊂L∞(µ) the dense subspace of functions taking only finitelymany values. Let w =nP1ϕi ⊗fi with ϕi ∈Vfi ∈H∞.
We will write, for every operatoru : H1 →L1(µ),⟨u, w⟩=X⟨ϕi, ufi⟩.ClearlyXϕi ⊗u(fi)L1µ(L∞m ) ≤C∞∥w∥0(1)andXϕi ⊗u(fi)L1m(L∞µ ) ≤C1∥w∥1(2)Moreover, by completion, we can extend (1) (resp. (2)) to the case when w is in S0 (resp.S1).
Hence, if w = w0 + w1 with w0 ∈S0, w1 ∈S1 we have by (1) and (2)|⟨u, w⟩| =X⟨ϕi, u(fi)⟩ ≤C∞∥w0∥0 + C1∥w1∥1≤C∞Ks(w, S0, S1)≤CC∞Ks(w, A0, A1)where s = C1(C∞)−1. By Hahn-Banach, there is a linear form ξ on A0 + A1 such that6
ξ(w) = ⟨u, w⟩∀w ∈S0 + S1(3)and|ξ(w)| ≤CC∞Ks(w, A0, A1)∀w ∈A0 + A1.Clearly this implies∀w ∈A0|ξ(w)| ≤CC∞∥w∥0(4)∀w ∈A1|ξ(w)| ≤CC∞s∥w∥1≤CC1∥w∥1. (5)Now (4) implies ∀ϕ ∈L∞(µ)∀f ∈L∞(dm)(6)|⟨ξ, ϕ ⊗f⟩| ≤CC∞∥ϕ∥1∥f∥∞.Define ˜u: L∞→L1(µ)∗= L∞(µ) as ⟨˜u(f), ϕ⟩= ⟨ξ, ϕ ⊗f⟩then (6) implies ∥˜u(f)∥L∞(µ) ≤CC∞∥f∥∞, while (5) implies|⟨ξ, ϕ ⊗f⟩| ≤CC1∥ϕ∥∞∥f∥1,hence ∥˜u(f)∥1 ≤CC1∥f∥1.
Finally (3) implies that ∀f ∈H∞∀ϕ ∈L∞(µ)⟨˜u(f), ϕ⟩= ⟨ϕ, u(f)⟩so that ˜u|H∞= u.q.e.d.Proof of Lemma 2: We start by reducing this lemma to the case when Ωis a finite setor equivalently, in case the σ-algebra A is generated by finitely many atoms, with a fixedconstant independent of the number of atoms. Indeed, let V be the union of all spacesL∞(Ω, B, µ) over all the subalgebras B ⊂A which are generated by finitely many atoms.7
Assume the lemma known in that case with a fixed constant C independent of the numberof atoms. It follows that for any w in H∞⊗V we have∀t > 0Kt(w, S0, S1) ≤CKt(w, A0, A1).Since H∞⊗V is dense in S0 + S1, this is enough to imply Lemma 2.Now, if (Ω, B, µ) is finitely atomic as above we argue exactly as in section 1 in [P]using the simple (so-called) “square/dual/square” argument, as formalized in Lemma 3.2in [P].
We want to treat by the same argument the pairH1(L∞(µ)) ⊂L1(L∞(µ))L1(µ; H∞) ⊂L1(µ; L∞).Taking square roots, the problem reduces to prove the following couple if K-closed:H2(L∞(µ)) ⊂L2(L∞(µ))L2(µ; H∞) ⊂L2(µ; L∞)provided we can check that(7)H2(L∞(µ)) · L2(µ; H∞) ⊂(H1(L∞(µ)), L1(µ; H∞)) 12 ∞We will check this auxiliary fact below. By duality and by Proposition 0.1 in [P] , we canreduce to checking the K-closedness for the coupleH2(L1(µ)) ⊂L2(L1(µ))L2(µ; H1) ⊂L2(µ; L1).Taking square roots one more time this reduces to prove that the following couple is K-closed( H4(L2(µ)) ⊂L4(L2(µ))L4(µ; H2) ⊂L4(µ; L2)provided we have8
(8)H4(L2(µ)) · L4(µ; H2) ⊂(H2(L1(µ)), L2(µ; H1)) 12 ∞.But this last couple is trivially K-closed (with a fixed constant independent of (Ω, B, µ))because, by Marcel Riesz’ theorem, there is a simultaneously bounded projectionL4(L2(µ)) →H4(L2(µ))L4(µ; L2) →L4(µ; H2).It remains to check the inclusions (7) and (8). We first check (7).
By Jones’ theorem (seethe beginning of section 3 and Remark 1.12 in [P])(9)H2(L∞(µ)) = (H1(L∞(µ)), H∞(L∞(µ))) 12 2also by an entirely classical result (cf. [BL] p.109)(10)L2(µ; H∞) = (L∞(µ; H∞), L1(µ; H∞)) 12 2.By the bilinear interpolation theorem (cf.
[BL] p.76) the two obvious inclusionsH1(L∞(µ)) · L∞(µ; H∞) ⊂H1(L∞(µ))H∞(L∞(µ)) · L1(µ; H∞) ⊂L1(µ; H∞),(note that H∞(L∞(µ)) = L∞(µ; H∞)), imply that(H1(L∞(µ)), H∞(L∞(µ))) 12 2·(L∞(µ; H∞), L1(µ; H∞)) 12 2 ⊂(H1(L∞(µ)), L1(µ; H∞)) 12 ∞.Therefore, by (9) and (10), this proves (7). We now check (8).
We will first prove ananalogous result but with the inverses of all indices translated by 1/r. More precisely, let2 < r < ∞, let p, r′ be defined by the relations 1/2 = 1/r + 1/p and 1 = 1/r + 1/r′.
Wewill first check(11)H2p(L2r′(µ)) · L2p(µ; H2r′) ⊂(Hp(Lr′(µ)), Lp(µ; Hr′)) 12 ∞.9
Indeed, we have(12) H2p(L2r′(µ))·L2p(µ; H2r′) ⊂L2p(L2r′(µ))·L2p(µ; L2r′) ⊂(Lp(Lr′(µ)), Lp(µ; Lr′)) 12 .The last inclusion follows from a classical result on the complex interpolation of Banachlattices, (cf. [C] p.125).
But now, since all indices appearing are between 1 and infinity,the orthogonal projection from L2 onto H2 defines an operator bounded simultaneouslyfrom Lp(Lr′(µ)) into Hp(Lr′(µ)) and from Lp(µ; Lr′) into Lp(µ; Hr′), hence also boundedfrom (Lp(Lr′(µ)), Lp(µ; Lr′)) 12 into (Hp(Lr′(µ)), Lp(µ; Hr′)) 12 . Since the latter space isincluded into (Hp(Lr′(µ)), Lp(µ; Hr′)) 12 ,∞, (cf.
[BL] p.102) we obtain the announced result(11).Then, we use the easy fact that any element g in the unit ball of H4(L2(µ))(resp. hin the unit ball of L4(µ; H2)) can be written as g = Gg1 (resp.
h = Hh1) with G and Hin the unit ball of H2r(L2r(µ)) = L2r(µ; H2r) and with g1 (resp. h1) in the unit ball ofH2p(L2r′(µ)) (resp.
L2p(µ; H2r′)). Then, by (11), there is a constant C such that∥g1h1∥(Hp(Lr′(µ)),Lp(µ;Hr′ )) 12 ∞≤C.Now, the product M = GH is in the unit ball of Hr(Lr(µ)) = Lr(µ; Hr), therefore theoperator of multiplication by M is of norm 1 both from Hp(Lr′(µ)) into H2(L1(µ)) andfrom Lp(µ; Hr′) into L2(µ; H1).By interpolation, multiplication by M also has norm1 from (Hp(Lr′(µ)), Lp(µ; Hr′)) 12 ∞into (H2(L1(µ)), L2(µ; H1)) 12 ∞.
Hence, we concludethat gh = Mg1h1 has norm at most C in the space (H2(L1(µ)), L2(µ; H1)) 12 ∞.Thisconcludes the proof of (8).References[BL] J.Bergh and J.L¨ofstr¨om, Interpolation spaces, An introduction, Springer Verlag 1976. [BS] C.Bennett and R.Sharpley, Interpolation of operators.Academic Press,1988.
[B] J.Bourgain. On the similarity problem for polynomially bounded operators on Hilbertspace, Israel J.
Math. 54 (1986) 227-241.
[C] A.Calder´on, Intermediate spaces and interpolation, Studia Math. 24 (1964) 113-190.
[G] J.Garnett, Bounded Analytic Functions. Academic Press 1981.10
[GR] J.Garcia-Cuerva and J.L.Rubio de Francia. Weighted norm inequalities and relatedtopics.
North Holland, 1985. [H] L.H¨ormander, Generators for some rings of analytic functions, Bull.Amer.Math.Soc.73 (1967) 943-949.
[J] P.Jones, L∞estimates for the ¯∂-problem in a half plane. Acta Math.
150 (1983)137-152. [L] M.L´evy.
Prolongement d’un op´erateur d’un sous-espace de L1(µ) dans L1(ν). S´emi-naire d’Analyse Fonctionnelle 1979-1980.
Expos´e 5. Ecole Polytechnique.Palaiseau.
[P] G.Pisier, Interpolation between Hp spaces and non-commutative generalizations I.Pacific J. Math.
(1992) To appear.Texas A. and M. UniversityCollege Station, TX 77843, U. S. A.andUniversit´e Paris 6 Equipe d’Analyse, Boˆıte 186, 4 Place Jussieu, 75230Paris Cedex 05, France11
출처: arXiv:9212.204 • 원문 보기