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Absolutely summing Carleson embeddings on Hardy spaces
Pascal Lefèvre, Luis Rodríguez-Piazza
To cite this version:
Pascal Lefèvre, Luis Rodríguez-Piazza. Absolutely summing Carleson embeddings on Hardy spaces.
2017. �hal-01376926v2�
Absolutely summing Carleson embeddings on Hardy spaces
Pascal Lefèvre and Luis Rodríguez-Piazza∗
January 18, 2017
Abstract. We consider the Carleson embeddings of the classical Hardy spaces (on the disk) into a Lp(µ) space, where µ is a Carleson measure on the unit disk. This includes the case of composition operators. We characterize such operators which are r-summing on Hp, where p >1 and r ≥1. This completely extends the former results on the subject and solves a problem open since the early seventies.
Mathematics Subject Classification. Primary: 30H10; 47B10 – Secondary: 30H20; 47B33 Key-words. composition operator – absolutely summing operators – Hardy spaces – Carleson measures
1 Introduction
In this paper, we investigate Carleson embeddings on classical Hardy spaces Hp when p >1.
In the following, the unit disk of the complex plane is denotedD= z∈C
|z|<1 . Its boundary, the torus, is denoted T=
z ∈C
|z| = 1 =∂D. We denote byH(D)the class of holomorphic functions on the unit disk. At last the Hardy spaces are defined by
Hp=
f ∈ H(D) sup
r<1
Z
T
f rz
pdλ <∞
and
kfkp= sup
r<1
Z
T
|f(rz)|pdλ1/p
= sup
r<1
kfrkLp(T).
Hereλstands for the normalized Haar measure on the torus (it is the normalized arc length), and fr(z) =f(rz)withr∈(0,1)andz∈D.
Now, let us turn to our main subject. Given a positive Borel measureµon the closed unit disk D, we consider the formal identityJµ from the Hardy spaceHp into Lp(µ)(we keep the notation Jµ instead ofJp,µ in the sequel for sake of lightness):
Jµ: Hp −→ Lp(µ)
f 7−→ f
Thanks to a famous result of Carleson (see [C]), this is well defined and bounded if and only if µis a Carleson measure, i.e.
sup
ξ∈T
µ
W(ξ, h)
=O(h), whenh→0,
∗Partially supported by the project MTM2015-63699-P (Spanish MINECO and FEDER funds)
whereW(ξ, h)is the Carleson window
W(ξ, h) ={z∈C|1−h≤ |z| ≤1 and |arg(zξ)| ≤¯ h}.
Let us recall that Jµ is compact if and only ifµis a vanishing Carleson measure (see [Po], or [McC] for composition operators in higher dimension):
sup
ξ∈T
µ
W(ξ, h)
=o(h), whenh→0.
Moreover let us mention that there is no real restriction in assuming in the sequel of this paper thatµis actually a measure carried by the open unit disk: let us assumea priorithatµis carried by the closed unit disk and thatJµ is either order bounded orr-summing for somer≥1, then in both cases,Jµ is actually compact and a necessary condition is that µ(T) = 0. Hence from now on to the end of the paper, we are going to assume thatµis actually a positive Borel measure µ on the open unit diskD.
One motivation to get interested in the Carleson embedding is that it allows to treat the case of composition operators onHp.
Let us recall that, given a symbol, i.e. an analytic function ϕ : D → D, the composition operatorCϕ:Hp→Hp is well defined and automatically bounded (see the monographs [CmC] or [S1] for example). Moreover many operator properties ofCϕcan be expressed in terms of Carleson measures thanks to the transfert formula. Indeed, sinceϕ∈H∞, it admits boundary values almost everywhere: lim
r→1−ϕ(rξ) exists for almost every ξ ∈T. We shall write simply ϕ(ξ)in the sequel (although in the literature, it is often denotedϕ∗(ξ)). The pullback measure ofλassociated toϕ plays now a crucial role:
λϕ(E) =λ
ξ∈T|ϕ(ξ)∈E
for every Borel subsetsE ofD. The transfer formula gives
kf ◦ϕkHp=kfkLp(D,λϕ) for every f ∈Hp.
Hence many properties of the operator Cϕ are common with the ones of the operator Jλϕ, in particular compactness,r-summingness,...
The case of weighted composition operators can also be treated in the same manner.
We are going to characterize those operators Jµ, which arer-summing for somer≥1. Before standing the results, let us recall the definitions
Definition 1.1 Suppose1≤r <+∞and let T:X →Y be a (bounded) operator between Banach spaces. We say thatT is an r-summing operator if there existsC≥0 such that
Xn
j=1
kT xjkr1/r
≤C sup
x∗∈BX∗
Xn
j=1
|hx∗, xji|r1/r
=C sup
a∈B`r0
n
X
j=1
ajxj
for every finite sequence x1,x2, . . . , xn in X.
The r-summing norm of T, denoted byπr(T), is the least suitable constant C≥0.
The class of r-summing operators forms an operator ideal (for instance see [DJT] for more details).
We shall use several times the following well known fact about summing operators (see [DJT]):
a bounded operatorT :X →Y isr-summing if and only if there existsC >0such that, for every X-valued random variableF on any measure space(Ω, ν), we have
Z
Ω
kT◦Fkrdν≤C sup
ξ∈BX∗
Z
Ω
|ξ◦F|rdν (1.1)
Actually the best admissibleC is(πr(T))r.
Very few results are known on absolutely summing composition operators: there is a character- ization of r-summing composition operators onHp due to Shapiro and Taylor in [ST] only when r=p≥2. The same result (with an obviously adapted proof) is actually valid for general Carleson embeddings:
Theorem 1.2 [ST] Let p≥2.
Jµ isp-summing onHp if and only if Z
D
1
1− |z|dµ <∞. (1.2)
Moreover, for every p≥1, the condition (1.2) is sufficient to ensure that Jµ is a p-summing operator onHp. When1≤p≤2,Jµ is actually even absolutely summing sinceHp has cotype2.
A natural question then arises: is (1.2) the good condition (i.e. a necessary condition) when 1≤p≤2? This is false in general: Domenig proved in [Do] that, givenp∈[1,2) there exists an absolutely summing composition operator onHpwhich is not order bounded. He was able to give some sufficient condition for the construction of his example, but without any characterization.
Let us mention that, in this case, it is equivalent to be an order bounded Carleson embedding and to verify condition (1.2). Indeed, the following is known from the specialists (see for instance [LLQR1] in the case of composition operators).
Proposition 1.3 Let µbe a Carleson measure on the open unit disk Dandp≥1.
Jµ: Hp→Lp(µ)is order bounded if and only if (1.2) is satisfied.
Proof. By definition,Jµis order bounded if and only if there exists someh∈Lp(µ)such that for everyf in the unit ball ofHp, we have|f| ≤ha.e. onD. SinceHpis separable, it suffices to test this control on a dense countable subset of the unit ball deHp. HenceJµis order bounded if and only if
Z
D
sup
f∈Hp kfk≤1
|f(z)|pdµ <∞
which is equivalent to
Z
D
kδzkpdµ <∞,
whereδz is the evaluation at the pointz∈D, viewed as a functional onHp. It is well known that kδzk= 1
(1− |z|2)1/p and the result follows.
As far as we know, there is no other result on the characterization of Carleson embeddings (or merely summing composition operators). In particular, the following problem was fully open (except whenr=p≥2).
Problem:
Given p, r≥1andµ a Carleson measure on the unit disk, which condition onµcharacterizes the fact thatJµ is ar-summing operator ?
In this paper, we are going to solve completely this problem for p > 1 and r ≥ 1. Our characterizations involve Carleson windows when p≥2 and r > p0, or integral conditions when p∈(1,2)orr≤p0. Here, as usual,p0 is the conjugate exponent ofp≥1: 1
p+ 1 p0 = 1.
Concretely, let us describe the organization of the paper. This first section is devoted to the introduction of the notions and questions. In the second one, we state our main results and specify the special case of composition operators. We finish with a few examples enlightening our statements. The third one treats what we call the diagonal case: we restrict the domain to the space spanned by the monomialszN+1, . . . , z2N and the measure to the corona|z| ∈[1−1/N,1−1/2N).
It turns out that in this framework, the embedding acts as a diagonal operator on the classical`p space. The difficulty is then to glue the pieces: section 4 provides some tools to do so and section 5 makes explicits some consequences for our purpose. In section 6, we focus on the case p >2:
most of the cases follow the results obtained in the previous sections, but the caser≤p0 requires a specific approach. The casep <2 cannot be treated following the same ideas and we treat it in the section 7. At last section 8 is devoted to some examples and remarks. For instance, we focus on the (formal) identity from the Hardy space Hp to the Bergman space Bq, and on the other hand we compare the different classes ofr-summing composition operators.
Results of this paper were announced (without proof) in [LR].
We state here a first statement which is an elementary necessary condition when 1 ≤p≤2, and has a double interest: it first gives a useful (practical) test in some cases. On the other hand, this necessary condition is a step, which turns out to be mandatory for a step in the proof of our characterization. Without waiting for the right characterization, we can notice that this is surely not a general sufficient condition.
Proposition 1.4 Let 1≤p≤2.
1. Let 1 ≤ q ≤2. We assume that the formal identity f ∈ Hp 7−→ Lq(D, µ) is a r-summing operator for some r≥1. Then the measure dµ
(1− |z|)q2 is finite:
Z
D
1
(1− |z|)q2 dµ <∞. (1.3) In particular, if we assume thatJµ:Hp→Lp(µ)is anr-summing operator for some rthen
Z
D
1
(1− |z|)p/2 dµ <∞. (1.4) 2. When1≤p <2, the preceding condition 1.4 is not sufficient in general.
For instance, applying this result to the normalized (area) measure A on D, we can already point out that the formal identity fromHpto the Hilbert Bergman space is notr-summing for any p∈[1,2]and anyr≥1.
Proof. 1. Since Hp and Lq(µ) have cotype 2, the operator is actuallyr-summing for every r≥1. In particular, it is aq-summing operator. Thanks to the Pietsch domination theorem, there existC >0and a probability measureν on the unit ball of(Hp)∗ such that, for everyf ∈Hp, we have
Z
D
|f|qdµ≤C Z
B(Hp)∗
|α(f)|qdν(α).
We apply this inequality to
fω(z) =
N
X
n=0
rn(ω)zn
where N ≥1 and(rn)is a sequence of i.i.d. random variables (Rademacher). Then we can take the expectation with respect toω to obtain via Fubini and the Khinchin inequalities:
Z
D
XN
n=0
|z|2nq/2
dµ≤C0 Z
B(Hp)∗
E|α(fω)|qdν(α).
for some constantC0 depending only on qandC.
But, for anyα∈B(Hp)∗, we can writeα(f) =R
Tg z
f(z)dλ, whereg belongs to the unit ball ofLp0, so that (recall thatq≤2)
E|α(fω)|q ≤
E|α(fω)|2q/2
= E
N
X
n=0
rn(ω)bg(n)
2q/2
≤ kgkq2≤1,
asp0≥2and everyg∈BLp0 belongs toBL2. We get for arbitrary largeN:
Z
D
1− |z|2(N+1) 1− |z|2
q2
dµ≤C0. Taking the limit whenN →+∞, the conclusion follows.
2. We consider the examples contructed in Lemma 4.3.[LLQR3]. There exists some function ϕ : D → D analytic where
ϕ eit
= e−f(t) with f(t) ∼ |t| for t in the neighborhood of 0 and ϕ eit
<1 out of this neighborhood of0. Sincep/2<1, it is clear that condition 1.4 is fulfilled althoughCϕis not summing onHp: indeed it would be compact onHp(thanks to Sarason’s result in [Sa] forp= 1), equivalently, compact on H2 which is not.
2 The main results.
We state below our main results which characterize any absolutely summing Carleson embed- ding. As usual, the notationA≈B means that there exist two constantsc, c0 >0(depending on randponly) such thatA≤cB≤c0A.
In addition to the Carleson windows, these characterizations involve special domains. We first divide the open unit diskDinto dyadic annuli
Γn = z∈D
1− 1
2n ≤ |z|<1− 1
2n+1 , where n= 0,1,2,· · ·
Then eachΓn is divided into 2n similar piecesRn,j, 0≤j <2n, that we call Luecking boxes (or Luecking rectangles):
Rn,j= z∈D
1− 1
2n ≤ |z|<1− 1
2n+1 and arg(z)∈ 2πj/2n,2π(j+ 1)/2n . So clearly the family of theRn,j withn≥0 and0≤j <2n, forms a partition ofD.
We will also use the Stolz domain Σξ at ξ ∈ T which is the interior of the convex hull of D(0,1/2)∪ {ξ}.
Characterization of absolutely summing Carleson embeddings:
Letµbe a Carleson measure onD.
1) Let1< p≤2. The natural injectionJµ:Hp(D)→Lp(µ)is2-summing if and only if
Φ :ξ7−→
Z
Σξ
1
1− |z|21+p2dµ(z)
belongs toL2/p(T, dλ), whereΣξ is the Stolz domain at pointξ∈T. Moreover we have
π2(Jµ)≈ kΦk1/p2/p≈ kΨk1/p2/p, (2.1) whereΨ(ξ) =
Z
D
1
|1−ξz|1+p2 dµ(z), for everyξ∈T.
2) Letp≥2andr≥1.
• When1≤r≤p0 , we have
πr(Jµ)≈π1(Jµ)≈ kFkLp0
(T), whereF(ξ) =
"
X
n≥0
22n
µ W(ξ,2−n2p
#1/2
(2.2)
• Whenp0 < r≤p, we have
πr(Jµ)≈
"
X
n≥0
X
0≤j<2n
2nµ(Rn,j)r/p
#1/r
(2.3)
• Whenp≤r, we have
πr(Jµ)≈πp(Jµ)≈
"
X
n≥0
X
0≤j<2n
2nµ(Rn,j)
#1/p
≈ Z
D
1 1− |z| dµ
!1/p
(2.4)
Before giving the corollaries for composition operators, let us mention several remarks:
i) In the casep= 2, we recover the characterization of2-summing operators (i.e. Hilbert-Schmidt operators):
Z
T
Z
Σξ
1
(1− |z|)2dµ dλ(ξ) = Z
D
Z
{ξ|z∈Σξ}
1
(1− |z|)2dλ(ξ)dµ≈ Z
D
1 1− |z| dµ
thanks to Fubini’s theorem and since {ξ|z∈Σξ}is an interval of length ≈1− |z|.
ii) On the other hand, we recover the necessary condition given by Prop 1.4: when p≤ 2, we have 2/p≥1, and hence, by Fubini’s theorem,
Z
T
Z
Σξ
1
(1− |z|2)1+p2dµ
!2/p
dλ
!p/2
≥ Z
T
Z
Σξ
1
(1− |z|2)1+p2 dµdλ
= Z
D
Z
{ξ|z∈Σξ}
dλ
(1− |z|2)1+p2 dµ
≈ Z
D
1 (1− |z|2)p2dµ
iii) Our characterizations show that ther-summing character ofJµ depends only on the sequence of values
µ Rn,j n≥0;j<2n. More precisely, we point out that when two positive (finite) measuresµandνsatisfyµ Rn,j
≤ν Rn,j
for every Luecking rectangleRn,j, thenπr Jµ . πr Jν
for any r ≥ 1. This can be checked from our characterizations: it is obvious when
p≥2 andr > p0. In the other cases, just use the fact that the Luecking rectangles forms a partition of the unit disc. For instance, the functionΨin (2.1) is equivalent to
X
n≥0 j<2n
d ξ, Rn,j−(1+p/2)
µ Rn,j .
In particular, whenµ Rn,j
=ν Rn,j
for everynandj, thenJµ isr-summing if and only if Jν isr-summing.
iv) This leads to the natural question to wonder whether the characterizations depend on the order of enumeration of the values µ Rn,j
. More precisely, when p ≥ 2 and r > p0, the r-summing character ofJµ is clearly invariant by permutation of the values of
µ Rn,j j<2n
(for each fixed integern). It turns out that it is no more true in the other cases. We have the following examples
• Example 1. Letp >2. There exist two (finite) measuresµand ν onDsuch that – For every n ≥ 1, the sequence
µ Rn,j
j<2n is a permutation of the sequence ν Rn,j j<2n.
– Jµ:Hp→Lp(µ)is1-summing.
– Jν:Hp→Lp(ν)is notp0-summing.
Consider the centers zn,j of the Rn,j and two sequences: (mn)is defined as the integer part of 2n
n2 andαn= 2−npnγp withγfixed in the interval p10,p20 −12 . Now our measures are
µ= X
n>n0
αnµn where µn =αn mn
X
j=0
δzn,j
and
ν= X
n>n0
αnνn where νn=αn ln+mn
X
j=ln
δzn,j
where n0is large enough andln+1= 2 ln+mn .
Let us point out that the radial projection on Tof the rectanglesRn,j charged byν are pairwise disjoints arcs whereas for the measure µ, these projections are arcs tending to the point1, whenntends to infinity.
• Example 2. Letp∈(1,2). There exist two (finite) measuresµandν onDsuch that – For every n ≥ 1, the sequence
µ Rn,j j<2n is a permutation of the sequence ν Rn,j j<2n.
– Jµ:Hp→Lp(µ)is1-summing.
– Jν:Hp→Lp(ν)is notr-summing for anyr≥1.
The construction is similar to the previous one but we have to interchange the way we defineµandν in order to get now that the radial projection onTof the rectanglesRn,j charged byµare pairwise disjoints arcs whereas for the measureν, these projections are arcs tending to the point1. Also the parametersαn have to be adapted.
We leave the check of the details to the reader.
As an immediate corollary, we can characterize r-summing weighted composition operator for any r≥1and anyp >1. Indeed, dealing with the operator f 7→w(f◦ϕ), it suffices to consider the measure µ=νϕ wheredν =|w|pdλ. Nevertheless, we state precisely these particular results only for composition operators. One involves the Nevanlinna counting function:
Nϕ(w) =
P
ϕ(z)=w
log|z|1 ifw6=ϕ(0)andw∈ϕ(D)
0 else.
where the sum runs over the roots counted with multiplicity.
Characterization of absolutely summing composition operators:
Letϕ:D→Dbe an analytic map such that its boundary values satisfy|ϕ(w)|<1 for almost everyw∈T. IfCϕ is the composition operator fromHp toHp, we have
1) When1< p≤2, for anyr≥1:
πr(Cϕ)≈π1(Cϕ)≈
"
Z
T
Z
T
1
|1−ξϕ(w)|1+p2 dλ(w)
!2/p
dλ(ξ)
#1/2
2) Whenp≥2,
• for1≤r≤p0,
πr(Cϕ)≈π1(Cϕ)≈ kFkLp0(T) where F(ξ) =
"
X
n≥0
22n
λϕ W(ξ,2−n2p
#1/2
(2.5)
• forp0< r≤p,
πr(Cϕ) ≈
"
X
n≥0
X
0≤j<2n
2nλϕ(Rn,j)r/p
#1/r
≈
"
Z
D
Nϕ(z) 1− |z|2
!r/p
dA (1− |z|2)2
#1/r (2.6)
• forp≤r,
πr(Cϕ)≈ Z
T
1 1− |ϕ| dλ
!1/p
(2.7)
In the case p0 < r ≤p(see (2.6)), the equivalence with the integral quantity comes from the fact that the r-summing norm ofCϕ (onHp), thanks to Luecking’s characterization [Lu1], turns out to be equivalent (up to an exponent) to the 2rp-Schatten norm (on H2) . Then we use the characterization of the membership of Cϕ to the Schatten classes, given by Luecking and Zhu [LZ] and involving the Nevanlinna counting function. We could also use directly the results on equivalence between the measure of Carleson’s windows and the Nevanlinna counting function, see [LLQR2] and also [EK] for a recent new proof.
3 The diagonal case
In this section, we fix an integer N ≥1 and we characterize the absolutely summing norm in the case of a measure concentrated on theNthcorona. More precisely, we consider the restriction µN ofµto the coronaGN, defined byµN(A) =µ(A∩ GN), where
GN = z∈D
1− 1
N ≤ |z|<1− 1 2N .
We are then interested in the behavior of the operatorJµN. It turns out that ther-summing norm of JµN is equivalent to that of its restriction to the space spanned by the monomials zk, where N ≤k <2N.
Our characterization will rely on the values of the measure of the boxes RN,j =n
z∈D 1− 1
N ≤ |z|<1− 1
2N and arg(z)∈ 2πj/N,2π(j+ 1)/No .
Of courseGN =∪0≤j<NRN,j. Later we shall use the results of this section with the dyadic version N = 2n recovering the Luecking boxes since we haveRn,j=R2n,j andΓn=G2n.
The following proposition makes the link between the behavior of the restricted Carleson em- bedding and a suitable diagonal operator on classical`pN space. We define the multiplier operator Mβ from`pN to`pN byMβ(ej) =βjej whereβ ∈CN and{ej}0≤j<N is the canonical basis of`pN. Proposition 3.1 For every r≥1 andp >1, we have
πr JµN
≈πr Mβ
whereβj= N µ(RN,j)1/p
, for0≤j < N.
In the previous statement, the underlying constants depend only onp. Actually we are able to prove a more general statement. Before stating it, we shall give some other definitions.
Definition 3.2 We say that a normαof operator ideal is a monotone ideal norm if the following property is fulfilled: whenever we have three Banach spaces X, Y1 andY2 and two operatorsT : X →Y1 andS:X →Y2 such thatkT xk ≥ kSxk, for everyx∈X, we have α(T)≥α(S).
Allr-summing norms are monotone. Ifαis monotone, we have in particularα(T) =α(j◦T), for every T : X → Y and any isometry j : Y → Z. We will need the following result about monotone norms.
Lemma 3.3 Let I be an operator ideal with a monotone norm α. Let X, Y be Banach spaces and(Ω,Σ, µ)a measure space. Assume that we have defined operatorsS:X →Y andTω:X →Y, ω∈Ω, such that the mapω7→Tω is measurable (with values in I(X, Y)) and we have
kSxk ≤ Z
Ω
kTωxkdµ(ω), for every x∈X. (3.1) Thenα(S)≤
Z
Ω
α(Tω)dµ(ω).
Proof. We can assume that R
α(Tω)dµ(ω) < +∞. Let Z = L1(µ, Y) be the space of Y- valued Bochner integrable functions defined on (Ω,Σ, µ), and defineT:X →Z byT x(ω) =Tωx, ω∈Ω. Condition (3.1) means thatkSxk ≤ kT xk, for every x∈X. Sinceαis monotone we have α(S)≤α(T). We finish because it is not difficult to see that
α(T)≤ Z
Ω
α(Tω)dµ(ω).
In fact, this is clear ifω7→Tωis a step function and the general case follows by density.
We denote byHNp the subspace ofHpspanned by{1, . . . , zN−1}and byHgNp =zNHNp the space spanned by {zN, . . . , z2N−1}. Clearly HgNp and HNp are isometrically isomorphic via the mapping f ∈HNp 7→zNf.
Theorem 3.4 Letp >1andαbe a monotone norm on an operator idealI. LetN be an integer, withN ≥1 and consider the following operators:
(a) JµN:f ∈Hp7−→f ∈Lp(D, µN).
(b) Tp,N:f ∈HNp 7−→f ∈Lp(D, µN).
(c) Tep,N:f ∈HgNp 7−→f ∈Lp(D, µN).
(d) Mβ: (aj)0≤j<N ∈`pN 7−→(βjaj)0≤j<N ∈`p, where βj= N µ(RN,j)1/p
. We have
α JµN
≈α Tep,N
≈α Tp,N
≈α Mβ where the underlying constants are independent ofN andα.
We shall need some lemmas. The following one is an obvious extension of a classical result (see 7.10[Z], p.30). We shall writeθj= exp2ijπ
N
for anyj∈ {0, . . . , N−1}.
Lemma 3.5 Let 1< p <∞. Then we have, for everyf ∈HNp,
kfkp≈ 1 N
N−1
X
j=0
f(θj)
p
!1/p .
Note that the estimations are uniform onN and depend only onp.
Lemma 3.6 Let 1< p <∞.
Let us define Q(z) = 1 +· · ·+zN−1 andQj(z) =N−1/p0Q θjz . Then, there exist ap, bp>0such that for everyu0, . . . , uN−1∈C:
ap N−1
X
j=0
|uj|p
!1/p
≤
N−1
X
j=0
ujQj
Hp
≤bp N−1
X
j=0
|uj|p
!1/p
Proof. Point out that Qj(θl) =N1/pδj,l (the Kronecker symbol) for allj, l∈ {0, . . . , N −1}.
Using Lemma 3.5, we get
N−1
X
j=0
ujQj
p
Hp
≈ 1 N
N−1
X
l=0
N−1
X
j=0
ujQj(θl)
p
=
N−1
X
l=0
ul
p.
Proof of Theorem 3.4.
It is obvious by restriction thatα JµN
≥α Tp,N . The equivalenceα Tep,N
≈α Tp,N
follows from the fact that there is an isometric isomorphism betweenHgNp and HNp (described above) and the fact that for anyf ∈Hp, theLp(D, µN)-norm of f andzNf are equivalent since|z|N belongs to 14,√1e
whenz∈ GN (andN >1).
