Some revisited results about composition operators on Hardy spaces

arXiv:1001.3328v1 [math.FA] 19 Jan 2010 Some revisited results about composition operators on Hardy spaces Pascal Lefèvre, Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza January 19, 2010 Abstract. We generalize, on one hand, some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces HΨ: construction of a “slow” Blaschke product giving a non-compact composition operator on HΨ; construction of a surjective symbol whose composition operator is compact on HΨand, moreover, is in all the Schatten classes Sp(H2),p > 0. On the other hand, we revisit the classical case of composition operators on H2, giving first a new, and simplier, characterization of closed range composition operators, and then showing directly the equivalence of the two characterizations of membership in the Schatten classes of Luecking and Luecking and Zhu. Mathematics Subject Classification. Primary: 47B33 – Secondary: 47B10 Key-words. Blaschke product – Carleson function – Carleson measure – composition operator – Hardy-Orlicz space – Nevanlinna counting function – Schatten classes 1 Introduction The study of composition operators on Hardy spaces is now a classical subject (see [18], [3] for example). In [8] (see also [7]), we considered a more general setting and studied composition operators on Hardy-Orlicz spaces; we gave there a characterization of their compactness in terms of the Carleson function of their symbol (and in terms of the Nevanlinna counting function in [11]). This work was continued in [10]: we compared the compactness on Hardy spaces versus the compactness on Hardy-Orlicz spaces. For instance, we showed that there is, for every 1≤p < ∞, an Orlicz function Ψsuch that Hp+ε⊆HΨ⊆Hpfor every ε > 0, and a composition operator Cϕsuch that Cϕis compact on Hp and Hp+ε, but which is not compact on HΨ. We carry on this study in the present work. In a first part (Section 3 and Section 4), we shall improve, and extend to the Hardy-Orlicz case, results known for Hardy spaces; in a second part (Section 5 and Section 6), we shall give new 1 lights on some results concerning Hardy spaces. More precisely, the content of this paper is as following. B. McCluer and J. Shapiro ([14], Theorem 3.10; see also [18], § 3.2) proved that, when their symbol ϕis finitely-valent, compactness of composition operators Cϕon the Hardy space H2can be characterized by the behaviour of the modulus of ϕnear the frontier of D: compactness is equivalent to 1−|z|= 01−|ϕ(z)|as |z| → 1, but that is not equivalent in general ([14], Example 3.8; see also [18], § 10.2). In [11], Theorem 5.3, we gave such a characterization for composition operators, with finitely-valent symbol, on Hardy-Orlicz spaces. In Section 3, we construct a “slow” Blaschke product (generalizing [18], § 10.2 and [8], Proposition 5.5) showing that this condition is not sufficient in general. In Section 4, we construct a compact composition operator Cϕ:HΨ→HΨ with surjective symbol ϕand such that Cϕ:H2→H2is in all the Schatten classes Sp(H2),p > 0. This generalizes and improves a result of B. McCluer and J. Shapiro ([14], Example 3.12; see also the survey [16], § 2). In Section 5, we give a characterization of composition operators Cϕ:Hp→ Hp,1≤p < ∞, with a closed range, simpler than the former ones (see [1] and [20]). Finally, based on the main result of [11], we show directly, in Section 6, the equivalence of Luecking’s and Luecking-Zhu’s criteria ([12], [13]) for the membership of Cϕ:H2→H2in the Schatten classes. Acknowledgement. Part of this work was made during the fourth-named author visited the University of Lille 1 and the University of Artois (Lens) in June 2009. This fourth-named author is partially supported by a Spanish research project MTM2006-05622. 2 Notation The open unit disk is denoted by D={z∈C;|z|<1}and its boundary, the unit circle, by T={z∈C;|z|= 1}. The normalized Lebesgue measure dt/2πon Tis denoted by m. The normalized area measure dx dy/π is denoted by A. The Hardy space H1is the space of analytic functions f:D→Csuch that supr<1R2π 0|f(reiθ)|dθ < ∞. Every f∈H1has almost everywhere boundary values on T, which are denoted by f∗. An Orlicz function is a convex nondecreasing function Ψ: [0,∞)→[0,∞) such that Ψ(0) = 0 and Ψ(∞) = ∞. If µis a positive measure on some measurable space S, the Orlicz space LΨ(µ)is the set of all (classes of) measurable functions f:S→Csuch that RSΨ(|f|/C)dµ < ∞for some C > 0; the norm kfkΨis defined as the infimum of the positive numbers Cfor which RSΨ(|f|/C)dµ ≤1. The Hardy-Orlicz space HΨis the linear subspace of f∈H1such that f∗∈LΨ(m)(see [8]). 2 Every analytic self-map ϕ:D→Ddefines a bounded composition operator Cϕ:f∈HΨ7→ f◦ϕ∈HΨ(see [8]). For every ξ∈Tand 0< h < 1, the Carleson window is the set W(ξ, h) = {z∈D;|z| ≥ 1−hand |arg(z¯ ξ| ≤ h}. The Carleson function ρϕof the analytic self-map ϕ:D→Dis defined, for 0< h < 1, by: ρϕ(h) = sup ξ∈T m{eiθ ∈T;ϕ∗(eiθ)∈W(ξ, h)}. Alternatively, ρϕ(h) = supξ∈Tmϕ[W(ξ, h)], where mϕis the pull-back measure of mby ϕ. We shall also use, instead of W(ξ, h), the set S(ξ, h) = {z∈ D;|z−ξ| ≤ h}, which has an equivalent size. The Nevanlinna counting function Nϕis defined, for w∈ϕ(D)\{ϕ(0)}, by Nϕ(w) = X ϕ(z)=w log 1 |z|, each term log 1 |z|being repeated according to the multiplicity of z, and Nϕ(w) = 0for the other w∈D. 3 Slow Blaschke products B. McCluer and J. Shapiro ([14], Theorem 3.10; see also [18], § 3.2) proved that, when ϕis finitely-valent (meaning that, for some s≥1, the equation ϕ(z) = whas at most ssolutions), the composition operators Cϕ:Hp→Hpis compact, 1≤p < ∞, if and only if ϕhas an angular derivative at no point of T; that means that: (3.1) lim |z|→1 1−|z| 1−|ϕ(z)|= 0 . In [11], Theorem 5.3, we generalized this result to Hardy-Orlicz spaces and proved that if ϕis finitely-valent, the composition operator Cϕ:HΨ→:HΨis compact if and only if: (3.2) lim |z|→1 Ψ−11 1−|ϕ(z)| Ψ−11 1−|z|= 0 . Without the assumption that ϕis finitely-valent, condition (3.2) is no longer sufficient to ensure the compactness of Cϕ:HΨ→HΨ. Indeed, we are going to construct a Blaschke product satisfying (3.2), but whose associated composition operator is of course not compact on HΨ, as this is the case for every inner function. A Blaschke product satisfying (3.1) is constructed in [18], § 10.2; that construction uses Frostman’s Theorem. Our construction, which is more general, is entirely elementary. 3 Theorem 3.1 Let δ: (0,1) →(0,1/2] be any function such that lim t→0δ(t) = 0. Then, there exists a Blaschke product Bsuch that: (3.3) 1−|B(z)| ≥ δ(1 −|z|),for all z∈D. Corollary 3.2 For every Orlicz function Ψthere exists a Blaschke product B which satisfies: lim |z|→1 Ψ−11 1−|B(z)| Ψ−11 1−|z|= 0 . though the composition operator CB:HΨ→HΨis not compact. Proof. CBis not compact since every compact composition operator should satisfy |ϕ∗|<1a.e. (see [8], Lemma 4.8). It suffices then to chose δ(t) = 1/ΨpΨ−1(1/t), which satisfies the hypothesis of Theorem 3.1. Moreover: Ψ−11/δ(t) Ψ−1(1/t)=1 pΨ−1(1/t)−→ t→00, and condition (3.3) gives the result.  Proof of Theorem 3.1. We shall essentially construct our Blaschke product Bas an infinite product of finite Blaschke products Y n Bn, where each finite Blaschke product Bnhas pnzeros equidistributed in the circumference of radius rn. That is, we will have, writing θk= 2πk/pnand zk=rneiθk, for k= 1,2,...,pn: (3.4) Bn(z) = pn Y k=1 |zk| zk zk−z 1−zkz= pn Y k=1 rn−e−iθkz 1−rne−iθkz· We shall need the following estimate for the finite Blaschke product in (3.4). Lemma 3.3 Let p∈N, and 0< r < 1. Consider the finite Blaschke product (3.5) G(z) = p Y k=1 r−e−iθkz 1−re−iθkz, where θk=2kπ p, for k= 1,2,...,p. (a)Then, for every z∈Dwith |z|=r, (3.6) |G(z)| ≤ 2rp 1 + r2p= 1 −(1 −rp)2 1 + r2p· 4 (b)If besides we have p h ≤1/2, where h= 1 −r, we also have, for every z∈Dwith |z|=r, (3.7) |G(z)| ≤ 1−(p h)2 2e · Let us continue the proof of the theorem. Define χ: (0,1) →(0,1] by: (3.8) χ(x) = sup t≤xmax{2δ(t),√t}. Then χis non-decreasing, limx→0χ(x) = 0 and limx→1χ(x) = 1. We can find a decreasing sequence (hn)n≥0of point hn∈(0,1), such that χ(hn)≤2−n. This sequence converges to 0; in fact, √hn≤χ(hn)≤2−n, by (3.8), and hence: (3.9) hn≤2−2n. We now define, for every n∈N, a positive integer pn, by: (3.10) pn= min{p∈N;p2h2 n 2e >2−n}. We have pn>1because h2 n/2e < h2 n≤2−4n. So, for every n, we have 4(pn−1)2≥p2 n, and then: (3.11) 4·2−n≥4(pn−1)2h2 n 2e ≥p2 nh2 n 2e · This yields, for n≥7, that (pnhn)2≤8e 2−n≤1/4. Therefore pnhn≤1/2, and we can use the estimate in part (b)of Lemma 3.3. Now, for n≥7, let Bnbe the finite Blaschke product defined by (3.4), where rn= 1 −hn. Using (b)in Lemma 3.3, the Maximum Modulus Principle and the definition of pnin (3.10), we have: (3.12) |Bn(z)| ≤ 1−p2 nh2 n 2e <1−2−n,for |z| ≤ rn. Consider then the Blaschke product Ddefined by: (3.13) D(z) = ∞ Y n=7 Bn(z). This product is convergent since, by (3.11), we have: Xpn(1 −rn) = Xpnhn≤X√8e 2−n<+∞. Finally, take N∈Nbig enough to have rN 6<1/2, and define: (3.14) B(z) = zND(z). 5 Thus Bis a Blaschke product, and, if |z| ≤ r6, we have, since δ(t)≤1/2: (3.15) |B(z)| ≤ |zN| ≤ rN 6<1/2≤1−δ(1 −|z|). If 1>|z|> r6, there exists k≥7such that rk≥ |z|> rk−1. Therefore, thanks to (3.12), (3.16) |B(z)| ≤ |D(z)| ≤ |Bk(z)| ≤ 1−2−k. On the other hand rk≥ |z|> rk−1implies hk≤1−|z|< hk−1, and so: (3.17) δ(1 −|z|)≤1 2χ(1 −|z|)≤1 2χ(hk−1)≤2−k. Combining (3.16) and (3.17) we get |B(z)| ≤ 1−δ(1−|z|), when 1>|z|> r6. From this and (3.15), Theorem 3.1 follows.  Proof of Lemma 3.3. It is obvious that, for all a, z ∈C, p Y k=1 (z−aeiθk) = zp−ap. Using this we have: (3.18) G(z) = p Y k=1 r−e−iθkz 1−re−iθkz= p Y k=1 z−reiθk rz −eiθk=zp−rp (rz)p−1· Now, if |z|=r, we can write zp=rpu, for some uwith |u|= 1. Then |G(z)|=|T(u)|, where Tis the Moebius transformation T(u) = rp(u−1) r2pu−1· This transformation Tmaps the unit circle ∂Donto a circumference C. As T maps the extended real line R∞to itself, and ∂Dis orthogonal to R∞at the intersection points 1and −1,Cis the circumference orthogonal to R∞crossing through the points T(1) = 0 and T(−1) = α. It is easy to see that |w| ≤ |α|, for every w∈C; consequently: |G(z)| ≤ sup u∈∂D|T(u)|=|T(−1)|=2rp 1 + r2p· This finishes the proof of the statement (a). To prove part (b), observe that, 1 + r2p≤2, and so, for |z|=r, (3.19) |G(z)| ≤ 1−(1 −rp)2 1 + r2p≤1−(1 −rp)2 2· Remember that r= 1 −h, so r≤e−h, and rp≤e−ph. Thus 1−rp≥1−e−ph. Now, if x∈[0,1/2], we have, by the Mean Value theorem: 1−e−x≥x √e· 6 Since p h ≤1/2, we can apply this last estimate to (3.19) to get, as promised, |G(z)| ≤ 1−(1 −e−ph)2 2≤1−p2h2 2e , and ending the proof of Lemma 3.3.  Remark. The key point in the proof of Theorem 3.1 is the inequality (3.6) in Lemma 3.3. This inequality may be viewed as a consequence of the strong triangle inequality (applied to a=zp,b=rpand c= 0): (3.20) d(a, b)≤d(a, c) + d(c, b) 1 + d(a, c)d(c, b) for the pseudo-hyperbolic distance d(u, v) = |u−v| |1−¯uv|on D. Let us recall a proof for the convenience of the reader: by conformal invariance, we may assume that c= 0; then: 1−[d(a, b)]2=(1 −|a|2)(1 −|b|2) |1−¯ab|2≥(1 −|a|2)(1 −|b|2) (1 + |a||b|)2= 1 −[d(|a|,−|b|)]2, so that: d(a, b)≤d(|a|,−|b|) = |a|+|b| 1 + |a||b|, proving (3.20), since d(a, 0) = |a|and d(0, b) = |b|. 4 A compact composition operator with a surjective symbol A well-known result of J. H. Schwartz ([17], Theorem 2.8) asserts that the composition operator Cϕ:H∞→H∞is compact if and only if kϕk∞<1. In particular, the compactness of Cϕ:H∞→H∞prevents the surjectivity of ϕ. It may be therefore to be expected that, the bigger Ψ, the more difficult it will be to obtain both the compactness of Cϕ:HΨ→HΨand the surjectivity of ϕ. Nevertheless, this is possible, as says the following theorem, and the case H∞ appears really as a singular case (corresponding to an “Orlicz function” which is discontinuous and can take the value infinity). Theorem 4.1 For every Orlicz function Ψ, there exists a symbol ϕ:D→D which is 4-valent and surjective and such that Cϕ:HΨ→HΨis compact. Moreover, ϕcan be taken so as Cϕ:H2→H2is in all the Schatten classes Sp(H2),p > 0. In the case of H2(Ψ(x) = x2), B. McCluer and J. Shapiro ([14], Example 3.12) gave an example based on the Riemann mapping theorem and on the fact that, for a finitely valent symbol ϕ, we have the equivalence: (4.1) Cϕ:H2→H2compact ⇐⇒ lim |z|< →1 1−|ϕ(z)| 1−|z|=∞. 7 A specific example is as follows. Take (4.2) R=z=x+iy ∈C;x > 0and 1 x< y < 1 x+ 4π, let g:D→Rbe a Riemann map and set ϕ= e−g. Then, ϕis 2-valent, ϕ(D) = D∗(where D∗=D\ {0}), and the validity of (4.1) is tested through the use of the Julia-Carathéodory theorem (see [16] for details). To get a fully surjective mapping ϕ1, just compose ϕwith the square of a Blaschke product: ϕ1(z) = B◦ϕ, with B(z) = z−α 1−αz 2, α ∈D∗=D\{0} (note that B(0) = B(2α/1 + |α|2). Since Cϕ1=Cϕ◦CB, we see that Cϕ1is compact as well and we are done. Here, we can no longer rely on the Julia-Carathéodory theorem. But we shall use the following necessary and sufficient condition, in terms of the maximal Carleson function ρϕ, which is valid for any symbol, finitely-valent or not (see [8], Theorem 4.18 – or [7], Théorème 4.2, where a different, but equivalent, formulation is given): (4.3) Cϕ:HΨ→HΨcompact ⇐⇒ lim h> →0 Ψ−1(1/h) Ψ−11/ρϕ(h)= 0 . For the sequel, we shall set: (4.4) ∆(h) = Ψ−1(1/h) Ψ−11/ρϕ(h)· Our strategy will be to elaborate on the previous example to produce a (nearly) surjective ϕsuch that ρϕ(h)is very small (depending on Ψ) for small h. The tool will be the notion of harmonic measure for certain open sets of the extended plane ˆ C=C∪{∞}, called hyperbolic (see [2], Definition 19.9.3); for example, every conformal image of Dis hyperbolic (see [2], Proposition 19.9.2 (d) and Theorem 19.9.7). If Gis a hyperbolic domain and a∈G, the harmonic measure of Gat ais the probability measure ωG(a, . )supported by ∂G (here, and throughout the rest of this section, boundaries and closures will be taken in ˆ C) such that: u(a) = Z∂G u(z)dωG(a, z) for each bounded and continuous function uon G, which is harmonic in G(see [2], Definition 21.1.3). The harmonic measure at aof a Borel set A⊆∂G will be denoted by ωG(a, A). Clearly, ωD(0, . ) = m, the Haar measure (i.e. normalized Lebesgue measure) of ∂D. 8 R. Nevanlinna (see [2], Proposition 21.1.6) showed that harmonic measures share a conformal invariance property. Namely, assume that Gis a simply connected domain, in which the Dirichlet problem can be solved (a Dirichlet domain), and τ:D→Gis a continuous function which maps conformally D onto G; then τmaps ∂Donto ∂G, and, if τ(0) = a: (4.5) ωG(a, A) = mτ−1(A) for every Borel set A⊆∂G. This explains why harmonic measures enter the matter when we consider composition operators Cϕ: such an operator induces a map HΨ→LΨ(mϕ), where mϕ=ϕ∗(m)appears as an image measure of m, as it happens for the harmonic measure of Gat ain (4.5). A useful alternative way of defining the harmonic measure, due to S. Kakutani, and completed by J. Doob (see [19], page 454, and [6], Appendix F, page 477) is the following: Let (Bt)t>0be the 2-dimensional Brownian motion starting at a∈G(i.e. B0=a), and τbe the stopping time defined by: (4.6) τ= inf{t > 0 ; Bt/∈G}; we have: (4.7) ωG(a, A) = Pa(Bτ∈A), i.e. the harmonic measure of Aat ais the probability that the Brownian motion starting at aexits from Gthrough the Borel set A⊆∂G. The following lemma will be basic for the construction of our example. We shall provide two proofs, the second one being more illuminating. Lemma 4.2 (Hole principle) Let G0and G1be two hyperbolic open sets and H⊆∂G0a Borel set such that G0⊆G1and ∂G0⊆∂G1∪H. Then, for every a∈G0, we have the following inequality: (4.8) ωG1(a, ∂G1\∂G0)≤ωG0(a, H). Proof 1. From [2], Corollary 21.1.14, with ∆ = ∂G0∩∂G1, one has ωG0(a, ∆) ≤ ωG1(a, ∆). But ∂G1\∆ = ∂G1\∂G0, and hence, since harmonic measures are probability measures, ωG1(a, ∂G1\∂G0) = ωG1(a, ∂G1\∆) = 1 −ωG1(a, ∆) ≤1−ωG0(a, ∆); we get the result since ∂G0=H∪∆, which implies 1≤ωG0(a, H) + ωG0(∆).  Proof 2. Let us define (4.9) τ0= inf{t > 0 ; Bt/∈G0}, τ1= inf{t > 0 ; Bt/∈G1} 9 Remark. To make the link with Cima-Thomson-Wogen’s criterion, we shall see that condition 5.2 implies that the restriction of µto the boundary T=∂D of the disk dominates the Lebesgue measure m. In fact, let Ibe an arc of T. If m(I) = h, we can write: I=\ n≥1 n [ j=1 W(ξn,j, h/2n), with disjoint windows W(ξn,1, h/2n),...,W(ξn,n, h/2n); hence: µ(I) = lim n→∞ n X j=1 µ[W(ξn,j, h/2n)] ≥c n X j=1 h 2n=c 2h . 6 Composition operators in Schatten classes In [12], D. Luecking characterized composition operators Cϕ:H2→H2 which are in the Schatten classes, by using, essentially, the mϕ-measure of Carleson windows. Five years later, D. Luecking and K. Zhu ([13]) characterized them by using the Nevanlinna counting function of ϕ. We shall see in this section how the result of [11] makes these two characterizations directly equivalent. It will be convenient here to work with modified Carleson windows, namely: Wn,j =z∈D; 1 −2−n≤ |z| ≤ 1and (2j−1)π 2n≤arg z < (2j+ 1)π 2n (j= 0,1,...,2n−1,n= 1,2,...). We shall say that Wn,j is the Carleson window centered at e2πij/2nwith size 2−n. Theorem 6.1 For p > 0the two following conditions are equivalent: a)Nϕ(z) log(1/|z|)∈Lp/2(λ), where dλ(z) = (1 − |z|)−2dA(z)and Ais the normalized area measure on D; b) ∞ X n=1 2n−1 X j=0 2nmϕ(Wn,j)p/2<∞. Condition b)in the last theorem yields that limn→∞ maxj2nmϕ(Wn,j) = 0, and it is not difficult to see that this implies that mϕ(∂D) = 0, or equivalently, that |ϕ∗|<1almost evereywhere on ∂D. In this situation we know ([9], Proposition 3.3) that b)in Theorem 6.1 is equivalent to Luecking’s condition in [12]. In fact the characterization of belonging to a Schatten class in [12] includes the requirement mϕ(∂D) = 0. Proof. We may, and do, assume that ϕ(0) = 0. 1) Assume first that condition b)is satisfied. Let: Rn,j =nz∈D; 1−2−n≤ |z|<1−2−n−1and (2j−1)π 2n≤arg z < (2j+ 1)π 2no 16 be the (disjoint) Luecking windows (0≤j≤2n−1,n≥0). One has Rn,j ⊆ Wn,j. By [11], Theorem 3.1, there are a constant C > 0and an integer Ksuch that Nϕ(z)≤C mϕ(f Wn,j), for every z∈Rn,j, where f Wn,j is the window centered at e2πij/2n, as Wn,j, but with size 2K−n. The windows Wn−K,j ,j= 0,1,...,2n−K−1, have the same size as the windows f Wn,j, but may have a different center; nevertheless, each f Wn,j can be covered with two windows Wn−K,l: for n > K,f Wn,j ⊆Wn−K,l ∪Wn−K,l+1, for some l= 1,2,...,2n−K (where l+ 1 is understood as 0if l= 2n−K−1), we get (we shall use .to mean ≤up to a constant): ZDNϕ(z)p/2 (1 −|z|)p 2+2 dA(z)≤X n,j ZRn,j (2n)p 2+2Nϕ(z)p/2dA(z) .X n,j ZRn,j (2n)p 2+2mϕ(f Wn,j)p/2dA(z) .X n,j (2n)p/2mϕ(f Wn,j)p/2 .X ν,l (2ν)p/2mϕ(Wν,l)p/2<∞, and a)holds. 2) Conversely, assume that a)is satisfied. We shall use the following inequality, whose proof will be postponed (for p≥2, (6.1) follows directly from [11], Theorem 4.2, and Hölder’s inequality): (6.1) [mϕ(Wn,j)]p/2.1 A(f Wn,j)Zf Wn,j [Nϕ(z)]p/2dA(z), where f Wn,j is a window with the same center as Wn,j but with a bigger proportional size; say of size 2−n+L. We get: X n,j [2nmϕ(Wn,j )]p/2.X n,j 2np/222nZf Wn,j [Nϕ(z)]p/2dA(z) =ZDX n 2n(2+ p 2)hX j 1If Wn,j (z)i[Nϕ(z)]p/2dA(z). Let k= 0,1,... such that 1−2−k+1 <|z| ≤ 1−2−k. One has z∈f Wn,j only if n≤k+L, and then, for each such n,zis at most in 2Lwindows f Wn,j. It follows that: X n 2n(2+ p 2)X j 1If Wn,j (z)≤2(k+L+1)(2+ p 2)×2L. 17 But |z| ≥ 1−2−k+1 implies 2(k+L+1)(2+ p 2)≤Cp/(1 −|z|)2+ p 2; hence: X n,j [2nmϕ(Wn,j )]p/2.ZD [Nϕ(z)]p/2 (1 −|z|)p 2+2 dA(z)<∞, and b)holds. It remains to show (6.1). By [11], Theorem 4.1, we can find a window Wwith the same center as Wn,j, but with greater size ch (h= 2−nis the size of the window Wn,j), such that: mϕ(Wn,j).sup w∈W Nϕ(w). There is hence some w0∈Wsuch that: mϕ(Wn,j).Nϕ(w0). Take R=|w0|+ch (one has R≥1since w0∈Wand Whas size ch) and set ϕ0(z) = ϕ(z)/R. One has Nϕ0(z) = Nϕ(Rz)for |z|<1/R and Nϕ0(z) = 0 if |z| ≥ 1/R. Let now ube the upper subharmonic regularization of Nϕ0([13], Lemma 1, and its proof page 1140): uis a subharmonic function on D\ {0}such that u≥Nϕ0and u=Nϕ0almost everywhere, with respect to dA. A result of C. Fefferman and E. M. Stein ([5], Lemma 2), generously attributed by them to Hardy and Littlewood, asserts that for any q > 0, there exists a constant C=C(q)such that (6.2) [u(a)]q≤C AD(a, r)ZD(a,r) [u(z)]qdA(z) for every nonnegative subharmonic function uon a domain Gand every disk D(a, r)⊆G(see also [13], Lemma 3). If ∆is the disk centered at w0/R and of radius 1−|w0|/R (which is contained in D\{0}since R > |w0|), one has, by (6.2): [Nϕ(w0)]p/2= [Nϕ0(w0/R)]p/2≤[u(w0/R)]p/2 ≤C A(∆) Z∆ [u(z)]p/2dA(z) =C A(∆) Z∆ [Nϕ0(z)]p/2dA(z) =C A(∆) Z∆∩D(0,1/R) [Nϕ(Rz)]p/2dA(z) =C A(˜ ∆) Z˜ ∆∩D [Nϕ(w)]p/2dA(w), where ˜ ∆ = D(w0, R −|w0|) = D(w0, ch). 18 Since the center w0of ˜ ∆is in D,˜ ∆∩Dcontains more than a quarter of ˜ ∆(at least for ch ≤1), and hence A(˜ ∆∩D)≥A(˜ ∆)/4 = c2h2/4π. Now, let ˜ Wn,j be the window with the same center as Wn,j and of size 2ch. Since 2ch ≥ch + (1 − |w0|),˜ Wn,j contains ˜ ∆∩Dand A(˜ Wn,j)≈h2≈A(˜ ∆) (≈ meaning that the ratio is between two absolute constants). We therefore get: [Nϕ(w0)]p/2.1 A(˜ Wn,j)Z˜ Wn,j [Nϕ(w)]p/2dA(w), proving (6.1).  References [1] J. Cima, J. Thomson and W. Wogen, On some properties of composition operators, Indiana Univ. Math. J. 24 (3) (1974), 215–220. [2] J. B. Conway, Functions of One Complex Variable II, Graduate Texts in Math. 159, Springer-Verlag (1995). [3] C. C. Cowen and B. D. McCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1995). [4] P. L. Duren, Theory of Hpspaces, Second edition, Dover Publications (2000). [5] C. Fefferman and E. M. Stein, Hpspaces of several variables, Acta Math. 129 (1972), 137–193. [6] J. B. Garnett and D. E. 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Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), 345–368. [13] D. H. Luecking and K. Zhu, Composition operators belonging to the Schatten ideals, Amer. J. Math. 114 (1992), 878–906. [14] B. McCluer and J. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38, no. 4 (1986), 878–906. [15] C. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften 299, Springer-Verlag (1992). [16] H. Queffélec, Carleson measures and composition operators, Proceed. 2008 OT Conference in Timisoara, to appear. [17] H. J. Schwartz, Composition operators on Hp, Thesis, University of Toledo (1969). [18] J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext, Tracts in Mathematics, Springer-Verlag, New York (1993). [19] D. W. Stroock, Probability Theory, An Analytic View, Cambridge University Press, Cambridge (1994). [20] N. Zorboska, Composition operators with closed range, Trans. Amer. Math. Soc. 334 (2) (1994), 791–801. 20 Pascal Lefèvre, Univ Lille Nord de France F-59 000 LILLE, FRANCE UArtois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, F-62 300 LENS, FRANCE pascal.lefevr[email protected]r Daniel Li, Univ Lille Nord de France F-59 000 LILLE, FRANCE UArtois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, Faculté des Sciences Jean Perrin, Rue Jean Souvraz, S.P. 18, F-62 300 LENS, FRANCE daniel.[email protected] Hervé Queffélec, Univ Lille Nord de France F-59 000 LILLE, FRANCE USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524, F-59 655 VILLENEUVE D’ASCQ Cedex, FRANCE [email protected]le1.fr Luis Rodríguez-Piazza, Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático, Apartado de Correos 1160, 41 080 SEVILLA, SPAIN [email protected] 21