Essential norms of weighted composition operators between Hardy spaces $H^p$ and $H^q$ for $1<= p,q<=infty$.

Essential norms of weighted composition operators between Hardy spaces H

and H

for

1 ≤ p, q ≤ ∞

R. Demazeux

Abstract

We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spacesH^pandH^qfor 1≤p, q≤ ∞.

In particular we give some estimates for the cases 1 =p≤q≤ ∞and 1≤q < p≤ ∞.

1 Introduction

Let D = {z ∈ C | |z| < 1} denote the open unit disk in the complex plane. Given two analytic functions u and ϕ defined on D such that ϕ(D) ⊂ D, one can define the weighted composition operator uCϕ that maps any analytic function f defined on D into the function uC_ϕ(f) = u.f ◦ϕ. In [10], de Leeuw showed that the isometries in the Hardy space H¹ are weighted composition operators, while Forelli [8] obtained this result for the Hardy space H^p when 1 < p <∞, p6= 2. Another example is the study of composition operators on the half- plane. A composition operator in a Hardy space of the half-plane is bounded if and only if a certain weighted composition operator is bounded on the Hardy space of the unit disk (see [13]

and [14]).

When u ≡ 1, we just have the composition operator C_ϕ. The continuity of these oper- ators on the Hardy space H^p is ensured by the Littlewood’s subordination principle, which says that Cϕ(f) belongs to H^p whenever f ∈ H^p (see [4], Corollary 2.24). As a conse- quence, the condition u ∈ H^∞ suffices for the boundedness of uC_ϕ on H^p. Considering the image of the constant functions, a necessary condition is that u belongs to H^p.Nevertheless a weighted composition operator needs not to be continuous onH^p, and it is easy to find examples whereuCϕ(H^p)*H^p (see Lemma 2.1 of [3] for instance).

In this note we deal with weighted composition operators between H^p and H^q for 1≤p, q≤ ∞. Boundedness and compactness are characterized in [3] for 1 ≤ p ≤ q < ∞ by means of Carleson measures, while essential norms of weighted composition operators are es- timated in [5] for 1< p≤q <∞by means of an integral operator. For the case 1≤q < p <∞, boundedness and compactness of uCϕ are studied in [5], and Gorkin and MacCluer in [9] gave an estimate of the essential norm of a composition operator acting betweenH^p andH^q.

The aim of this paper is to complete the different cases remaining in the estimation of the essential norm of a weighted composition operator. In section 2 and 3, we give an estimate of the essential norm of uCϕ acting between H^p and H^q when p = 1 and 1 ≤ q < ∞ and when 1≤p <∞ andq =∞.Sections 4 and 5 are devoted to the case where ∞ ≥p > q≥1.

LetDbe the closure of the unit diskDandT=∂Dits boundary. We denote by dm= dt/2π the normalised Haar measure onT.IfAis a Borel subset ofT, the notation m(A) as well as|A|

2010Mathematics Subject Classification: Primary 47B33; Secondary 30H10, 46E15.

Key words and phrases: weighted composition operator, essential norm, Carleson measure, Hardy space.

will design the Haar measure of A. For 1 ≤ p < ∞, the Hardy space H^p(D) is the space of analytic functions f :D→C satisfying the following condition

kfk_p = sup

0<r<1

Z

T

|f(rζ)|^p dm(ζ) 1/p

<∞.

Endowed with this norm, H^p(D) is a Banach space. The space H^∞(D) is consisting of every bounded analytic function on D,and its norm is given by the supremum norm onD.

We recall that any functionf ∈H^p(D) can be extended onT to a functionf^∗ by the following formula: f^∗(e^iθ) = limr%1f(re^iθ). The limit exists almost everywhere by Fatou’s theorem, and f^∗ ∈ L^p(T). Moreover, f 7→ f^∗ is an into isometry from H^p(D) to L^p(T) whose image, denoted byH^p(T) is the closure (weak-star closure forp=∞) of the set of polynomials inL^p(T).

So we can identifyH^p(D) andH^p(T), and we will use the notationH^p for both of these spaces.

More on Hardy spaces can be found in [11] for instance.

The essential norm of an operatorT :X→X, denoted kTk_e, is given by kTk_e = inf{kT −Kk | K is compact on X}.

Observe that kTk_e ≤ kTk, and kTk_e is the norm of T seen as an element of the Calkin alge- braB(X)/K(X) where K(X) is the space of all compact operators onX.

Notation: we will note a≈bwhenever there exists two positive universal constants candC such thatcb≤a≤Cb.In the sequel, u will be anon-zero analytic function on Dand ϕwill be a non-constant analytic function defined onDsatisfying ϕ(D)⊂D.

2 uC

∈ B(H

, H

) for 1 ≤ q < ∞

Let’s first start with a characterization of the boundedness ofuC_ϕ acting between H^p and H^q for 1≤p≤q <∞:

Theorem 2.1 (see [5], Theorem 4). Let u be an analytic function onD and ϕ a analytic self- map of D.Then the weighted composition operator uC_ϕ is bounded from H^p to H^q if and only if

sup

a∈D

Z

T

|u(ζ)|^q

1− |a|²

|1−¯aϕ(ζ)|² q/p

dm(ζ)<∞.

As a consequence uCϕ is a bounded operator as soon as uCϕ is uniformly bounded on the set{k_a^1/p| a∈D}wherekais the normalized kernel defined byka(z) = (1−|a|²)/(1−¯az)², a∈D. Note thatk^1/pa ∈H^p and kk_a^1/pk_p= 1.These kernels play a crucial role in the estimation of the essential norm of a weighted composition operator:

Theorem 2.2 (see [5], Theorem 5). Assume that the weighted composition operator uCϕ is bounded from H^p to H^q with 1< p≤q <∞.Then

kuC_ϕk_e≈lim sup

|a|→1⁻

Z

T

|u(ζ)|^q

1− |a|²

|1−¯aϕ(ζ)|² q/p

dm(ζ)

!¹

q

.

The aim of this section is to give the corresponding estimate for the case p = 1. We shall prove that the previous theorem is still valid for p= 1:

Theorem 2.3. Suppose that the weighted composition operator uC_ϕ is bounded fromH¹ to H^q for a certain 1≤q <∞. Then we have

kuC_ϕk_e≈lim sup

|a|→1⁻

Z

T

|u(ζ)|^q

1− |a|²

|1−aϕ(ζ)|¯ ² q

dm(ζ)

!¹_q .

Let’s start with the upper estimate:

Proposition 2.4. Let uCϕ ∈ B(H¹, H^q) with 1 ≤ q < ∞. Then there exists a positive con- stant C such that

kuC_ϕk_e≤Clim sup

|a|→1⁻

Z

T

|u(ζ)|^q

1− |a|²

|1−¯aϕ(ζ)|² q

dm(ζ)

!¹

q

.

The main tool of the proof is the use of Carleson measures. Assume thatµis a finite positive Borel measure on D and let 1 ≤ p, q < ∞. We say that µ is a (p, q)-Carleson measure if the embedding J_µ :f ∈ H^p 7→ f ∈ L^q(µ) is well defined. In this case, the closed graph theorem ensures that Jµ is continuous. In other words, µ is a (p, q)-Carleson measure if there exists a constantC1>0 such that for everyf ∈H^p,

Z

D

|f(z)|^q dµ(z)≤C₁kfk^q_p. (2.1) LetI be an arc in T. ByS(I) we denote the Carleson window given by

S(I) ={z∈D| 1− |I| ≤ |z|<1, z/|z| ∈I}.

Let’s denote byµ_D and µ_T the restrictions ofµtoDand Trespectively. The following result is a version of a theorem of Duren (see [7], p.163) for measures on D:

Theorem 2.5 (see [1], Theorem 2.5). Let 1 ≤p < q < ∞. A finite positive Borel measure µ onD is a(p, q)-Carleson measure if and only if µ_T= 0 and there exists a constantC2 >0 such that

µ_D(S(I))≤C2|I|^q/p for any arc I ⊂T. (2.2) Notice that the best constantsC₁ andC₂ in (2.1) and (2.2) are comparable, meaning that there is a positive constant β independent of the measure µsuch that (1/β)C2 ≤C1≤βC2.

The notion of Carleson measure was introduced by Carleson in [2] as a part of his work on the corona problem. He gave a characterization of measures µ on D such that H^p embeds continuously inL^p(µ).

Examples of such Carleson measures are provided by composition operators. Let ϕ:D→D be an analytic map and let 1 ≤p, q < ∞. The boundedness of the composition operator C_ϕ : f 7→f ◦ϕbetween H^p and H^q can be rephrased in terms of (p, q)-Carleson measures. Indeed, denote bymϕ the pullback measure of m byϕ, which is the image of the Haar measure m ofT under the mapϕ^∗, defined by

mϕ(B) =m

ϕ^∗−1(B)

for every Borel subset B ofD.Then kC_ϕ(f)k^q_q =

Z

T

|f◦ϕ|^q dm= Z

D

|f|^q dm_ϕ =kJ_mϕ(f)k^q_q

for all f ∈ H^p. Thus Cϕ maps H^p boundedly into H^q if and only if mϕ is a (p, q)-Carleson measure.

In the sequel we will note the disk of radiusr byrD={z∈D| |z|< r} for 0< r <1.We will need the following lemma concerning (p, q)-Carleson measures:

Lemma 2.6. Take 0< r <1 and let µbe a finite positive Borel measure on D.Note N_r^∗ := sup

|a|≥r

Z

D

|k_a(w)|^qp dµ(w).

If µ is a (p, q)-Carleson measure for1 ≤p≤q <∞ then so is µr :=µ|

D\rD. Moreover one can find an absolute constant M >0 satisfyingkµ_rk ≤M N_r^∗ where kµ_rk:= sup

I⊂T

µr S(I)

|I|^q/p ·

We omit the proof of Lemma 2.6 here, which is a slight modification of the proof of Lemma 1 and Lemma 2 in [6] using Theorem 2.5.

In the proof of the upper estimate of Theorem 2.2 in [5], the authors use a decomposition of the identity on H^p of the form I =KN +RN where KN is the partial sum operator defined by K_N(P∞

n=0a_nzⁿ) = PN

n=0a_nzⁿ, and they use the fact that (K_N) is a sequence of compact operators that is uniformly bounded inB(H^p) and thatR_N converges pointwise to zero on H^p. Nevertheless the sequence (KN) is not uniformly bounded inB(H¹).In fact, (KN) is uniformly bounded inB(H^p) if and only if the Riesz projectionP :L^p →H^p is bounded [15, Theorem 2], which occurs if and only if 1< p <∞. Therefore we need to use a different decomposition for the case p = 1. Since KN is the convolution operator by the Dirichlet kernel on H^p,we shall consider the Fej´er kernel F_N of order N. Let’s define K_N : H¹ → H¹ to be the convolution operator associated to F_N that maps f ∈ H¹ to K_Nf = F_N ∗f ∈ H¹ and R_N = I −K_N. Then kK_Nk ≤ 1, KN is compact and for every f ∈ H¹, kf −KNfk₁ → 0 following Fej´er’s theorem. Iff(z) =P

n≥0fˆ(n)zⁿ∈H¹,then K_Nf(z) =

N−1

X

n=0

1− n

N

fˆ(n)zⁿ. Lemma 2.7. Suppose uC_ϕ∈B(H¹, H^q). Then

kuC_ϕk_e ≤lim inf

N kuC_ϕRNk.

Proof.

kuC_ϕk_e=kuC_ϕK_N +uCϕR_Nk_e

=kuC_ϕRNk_e sinceKN is compact

≤ kuC_ϕRNk

and the result follows taking the lower limit.

We will need the following lemma for an estimation of the remainder R_N: Lemma 2.8. Let ε >0 and 0< r <1.Then ∃N₀ =N₀(r)∈N, ∀N ≥N₀,

|R_Nf(w)|^q < εkfk^q₁, for every|w|< r and for every f in H¹.

Proof. LetKw(z) = 1/(1−wz), w¯ ∈D, z ∈D. Kw is a bounded analytic function on D.

It is easy to see that for everyf ∈H¹,

hR_Nf, K_wi=hf, R_NK_wi where|w|< r, N ≥1 and

hf, gi= 1 2π

Z 2π 0

f(e^iθ)g(e^iθ) dθ

for f ∈ H¹ and g ∈ H^∞. Then we have |R_Nf(w)| = | hR_Nf, Kwi | = | hf, R_NKwi | ≤ kfk₁kR_NK_wk_∞. Take |w| < r and choose N₀ ∈ N so that for every N ≥ N₀ one has r^N ≤ε^1/q(1−r)/2 and 1/NPN−1

n=1 nrⁿ≤(1/2)ε^1/q.Since RNKw(z) =RN

^∞ X

n=0

¯ wⁿzⁿ

=

N−1

X

n=0

n

Nw¯ⁿzⁿ+

∞

X

n=N

¯ wⁿzⁿ, one has

kR_NKwk_∞< 1 N

N−1

X

n=0

nrⁿ+

∞

X

n=N

rⁿ≤ε^1/q.

Thus |R_Nf(w)|^q ≤εkfk^q₁ for everyf inH¹.

Proof of Proposition 2.4. Denote by µ the measure which is absolutely continuous with respect tomand whose density is |u|^q,and letµ_ϕ=µ◦ϕ⁻¹ be the pullback measure ofµbyϕ.

Fix 0< r <1. For every f ∈H¹,we have k(uC_ϕRN)fk^q_q =

Z

T

|u(ζ)|^q

(RNf)◦ϕ (ζ)

q dm(ζ)

= Z

T

(R_Nf)◦ϕ (ζ)

q dµ(ζ)

= Z

D

|R_Nf(w)|^q dµ_ϕ(w)

= Z

D\rD

|R_Nf(w)|^q dµ_ϕ(w) + Z

rD

|R_Nf(w)|^q dµ_ϕ(w)

=I₁(N, r, f) +I₂(N, r, f). (2.3)

Let us first show that lim

N sup

kfk₁=1

I2(N, r, f) = 0. For ε > 0, Lemma 2.8 gives us an inte- gerN₀(r) such that for everyN ≥N₀(r),

I2(N, r, f) = Z

rD

|R_Nf(w)|^q dµϕ(w)

≤εkfk^q₁µϕ(rD)

≤εkfk^q₁µ_ϕ(D)

≤εkfk^q₁kuk^q_q. So,r being fixed, we have lim

N sup

kfk1=1

I2(N, r, f) = 0.

Now we need an estimate of I₁(N, r, f).The continuity of uC_ϕ :H¹ →H^q ensures thatµ_ϕ is a (1, q)-Carleson measure, and therefore µϕ,r := µϕ|

D\rD

is also a (1, q)-Carleson measure by using Lemma 2.6 for p= 1.We deduce the following inequalities:

Z

D\rD

|R_Nf(w)|^q dµϕ,r(w)≤βkµ_ϕ,rkkR_Nfk^q₁

≤2^qβM N_r^∗kfk^q₁

using Lemma 2.6 and the fact that kR_Nk ≤ 1 +kK_Nk ≤ 2 for every N ∈ N. We take the supremum over B_H¹ and take the lower limit asN tend to infinity in (2.3) to obtain

lim inf

N→∞ kuC_ϕR_Nk^q≤2^qβM N_r^∗. Now asr going to 1 we have:

r→1limN_r^∗= lim sup

|a|→1⁻

Z

D

|k_a(w)|^q dµϕ(w)

= lim sup

|a|→1⁻

Z

T

|u(ζ)|^q

1− |a|²

|1−¯aϕ(ζ)|² q

dm(ζ)

and we obtain the estimate announced using Lemma2.7.

Now let’s turn to the lower estimate in Theorem 2.2. Let 1≤q <∞.ConsiderF_N the Fej´er kernel of order N, and define KN : H^q → H^q the convolution operator associated to FN and RN =I−KN. Then (KN)N is a sequence of uniformly bounded compact operators in B(H^q), and kR_Nfk_q →0 for allf ∈H^q.

Lemma 2.9. There exists 0 < C ≤2 such that whenever uCϕ is a bounded operator from H¹ to H^q, one has

1

Clim sup

N

kR_NuC_ϕk ≤ kuC_ϕk_e.

Proof. Take K ∈ B(H¹, H^q) a compact operator. Since (K_N) is uniformly bounded, one can find C >0 satisfying kR_Nk ≤1 +kK_Nk ≤C for all N >0,and we have:

kuC_ϕ+Kk ≥ 1

CkR_N(uCϕ+K)k

≥ 1

CkR_NuC_ϕk − 1

CkR_NKk.

Now use the fact that (R_N) goes pointwise to zero in H^q, and consequently (R_N) converges strongly to zero over the compact setK(B_H1) asN goes to infinity. It follows thatkR_NKk −→

N 0, and

kuC_ϕ+Kk ≥ 1

C lim sup

N

kR_NuCϕk

for every compact operator K:H¹ →H^q.

Proposition 2.10. Let uC_ϕ ∈B(H¹, H^q) with 1≤q <∞.Then kuC_ϕk_e≥ 1

Clim sup

|a|→1⁻

Z

T

|u(ζ)|^q

1− |a|²

|1−¯aϕ(ζ)|² q

dm(ζ)

!¹_q . Proof. Since k_ais a unit vector in H¹,

kR_NuC_ϕk=kuC_ϕ−K_NuC_ϕk ≥ kuC_ϕk_ak_q− kK_NuC_ϕk_ak_q. (2.4) First case: q >1

(ka) converges to zero for the topology of uniform convergence on compact sets in Das|a|goes to 1, so does uCϕ(ka). The topology of uniform convergence on compact sets in D and the weak topology agree onH^q, therefore it follows thatuC_ϕ(k_a) goes to zero for the weak topology inH^q as |a|goes to 1.Since K_N is a compact operator, it is completely continuous and carries weak-null sequences to norm-null sequences. So kK_N uCϕ(ka)

k_q→0 when |a| →1, and kR_NuCϕk ≥lim sup

|a|→1⁻

kuC_ϕ(ka)k_q.

Taking the upper limit whenN → ∞, we obtain the result using Lemma 2.9.

For the second case we will need the following computation lemma:

Lemma 2.11. Take a ∈ D and N ∈ N^∗. Denote by αp(a) the p-th Fourier coefficient of C_ϕ k_a/(1− |a|²)

, so that for everyz∈Dwe have k_a ϕ(z)

= (1− |a|²)

∞

X

p=0

α_p(a)z^p.

Then there exists a constant M = M(N) > 0 depending on N such that |α_p(a)| ≤ M for every p≤N and every a∈D.

Proof of Lemma 2.11. Write ϕ(z) = a0 +ψ(z) with a0 = ϕ(0) ∈ D and ψ(0) = 0. If we developk_a(z) as a Taylor series and replacez byϕ(z) we obtain:

ka ϕ(z)

= (1− |a|²)

∞

X

n=0

(n+ 1)(¯a)ⁿϕ(z)ⁿ.

Then

α_p(a) =

* _∞ X

n=0

(n+ 1)(¯a)ⁿϕ(z)ⁿ, z^p +

=

∞

X

n=0

(n+ 1)(¯a)ⁿ

n

X

j=0

n j

a^n−j₀

ψ(z)^j, z^p .

wherehf, gi=R

Tfg¯dm. Note that

ψ(z)^j, z^p

= 0 if j > psinceψ(0) = 0,and consequently

αp(a) =

∞

X

n=0

(n+ 1)(¯a)ⁿ

min(n,p)

X

j=0

n j

a^n−j₀

ψ(z)^j, z^p

=

p

X

j=0

∞

X

n=j

(n+ 1)(¯a)ⁿ n

j

a^n−j₀

ψ(z)^j, z^p

=

p

X

j=0

ψ(z)^j, z^p

∞

X

n=j

(n+ 1)(¯a)ⁿ n

j

a^n−j₀ .

In the case where a0 6= 0 we obtain α_p(a) =

p

X

j=0

ψ(z)^j, z^p a^−j₀

∞

X

n=j

(n+ 1) n

j

(¯aa₀)ⁿ

=

p

X

j=0

ψ(z)^j, z^p

a^−j₀ (j+ 1)(¯aa0)^j (1−aa¯ 0)^j+2

=

p

X

j=0

ψ(z)^j, z^p (j+ 1)(¯a)^j (1−¯aa0)^j+2 using the following equalities forx= ¯aa₀ ∈D:

∞

X

n=j

(n+ 1) n

j

xⁿ=





∞

X

n=j

n j

xⁿ⁺¹





0

=

x^j+1 (1−x)^j+1

0

= (j+ 1)x^j (1−x)^j+2

Note that the last expression obtained for α_p(a) is also valid for a₀ = 0. Thus, for 0≤p ≤N we have the following estimates:

|α_p(a)| ≤

p

X

j=0

|

ψ(z)^j, z^p

| j+ 1 (1− |a₀|)^j+2

≤

p

X

j=0

kψ^jk∞ N + 1 (1− |a₀|)^N+2

≤ (N+ 1)²

(1− |a₀|)^N+2 max

0≤j≤Nkψ^jk_∞

≤M,

whereM is a constant independent from a.

Second case: q= 1

In this case, it is no longer for the weak topology but for the weak-star topology of H¹ that

uCϕ(ka) tends to zero when |a| → 1. Nevertheless, it is still true that kK_NuCϕ(ka)k₁ → 0 as|a| →1.Indeed if f(z) =P

n≥0fˆ(n)zⁿ∈H¹,then K_Nf(z) =

N−1

X

n=0

1− n

N

fˆ(n)zⁿ.

We have the following development:

k_a ϕ(z)

= (1− |a|²)

∞

X

n=0

α_n(a)zⁿ. Denote by un then-th Fourier coefficient ofu, so that

uC_ϕ(k_a)(z) = (1− |a|²)

∞

X

n=0

n

X

p=0

α_p(a)un−p

zⁿ, ∀z∈D.

It follows that

kK_NuCϕ(ka)k₁ ≤(1− |a|²)

N−1

X

n=0

1− n

N

n

X

p=0

αp(a)un−p

kzⁿk₁.

Now using estimates from Lemma 2.11, one can find a constantM >0 independent fromasuch that|α_p(a)| ≤M for everya∈Dand 0≤p≤N−1.Use the fact thatkzⁿk₁ = 1 and|u_p| ≤ kuk₁ to deduce that there is a constant M⁰>0 independent fromasuch that

kK_NuC_ϕ(k_a)k₁ ≤M⁰(1− |a|²)kuk₁

for alla∈D.Thus K_NuCϕ(ka) converges to zero in H¹ when|a| →1,and take the upper limit in 2.4 whenatends to 1⁻ to obtain

kR_NuC_ϕk ≥lim sup

|a|→1

kuC_ϕ(k_a)k₁, ∀N ≥0.

We conclude with Lemma 2.9 and observe thatC= supkR_Nk ≤2 sincekR_Nk ≤1 +kK_Nk ≤2.

3 uC

∈ B(H

, H

) for 1 ≤ p < ∞

Let u be a bounded analytic function. Characterizations of boundedness and compactness of uC_ϕ as a linear map betweenH^p and H^∞have been studied in [3] for p≥1.Indeed,

uCϕ∈B(H^p, H^∞) if and only if sup

z∈D

|u(z)|^p

1− |ϕ(z)|² <∞ and

uCϕ is compact if and only if kϕk∞<1 or lim

|ϕ(z)|→1

|u(z)|^p

1− |ϕ(z)|² = 0.

In the case where kϕk_∞= 1 we will note

Mϕ(u) = lim sup

|ϕ(z)|→1

|u(z)|

(1− |ϕ(z)|²)^1p

·

As regarding Theorem 1.7 in [12], it seems reasonable to think that the essential norm of uC_ϕ is equivalent to the quantity M_ϕ(u).We first have a majorization:

Proposition 3.1. Suppose that uCϕ is a bounded operator fromH^p toH^∞and thatkϕk_∞= 1.

Then

kuC_ϕk_e≤2Mϕ(u).

Proof. Letεbe a real positive number, and pick r <1 satisfying sup

|ϕ(z)|≥r

|u(z)|

(1− |ϕ(z)|²)^1p

≤Mϕ(u) +ε.

We approximate uC_ϕ by uC_ϕK_N where K_N : H^p → H^p is the convolution operator by the Fej´er kernel of order N, where N is chosen so that |R_Nf(w)| < εkfk₁ for every f ∈ H¹ and every |w|< r (Lemma 2.8 for q = 1). We want to show that kuC_ϕ−uCϕKNk=kuC_ϕRNk ≤ max(2M_ϕ(u) + 2ε, εkuk_∞),which will prove our assertion. Iff is a unit vector inH^p, then

kuC_ϕRN(f)k∞= max sup

|ϕ(z)|≥r

|u(z)(R_Nf)◦ϕ(z)|, sup

|ϕ(z)|<r

|u(z)(R_Nf)◦ϕ(z)|

! .

We want to estimate the first term. Ifω∈D,we noteδ_ω the linear functional onH^p defined by δω(f) =f(ω).Then δω ∈(H^p)^∗ and

sup

|ϕ(z)|≥r

|u(z)(R_Nf)◦ϕ(z)| ≤ sup

|ϕ(z)|≥r

|u(z)|kδ_ϕ(z)k_(Hp)^∗kR_Nfk_p

≤2 sup

|ϕ(z)|≥r

|u(z)|

(1− |ϕ(z)|²)^1p

≤2 (Mϕ(u) +ε),

where we use the fact that kR_Nfk_p ≤2 and kδ_wk_(Hp)^∗ = 1/(1− |w|²)^1/p for everyw∈D. For the second term, since|ϕ(z)|< rwe have

|u(z)R_Nf(ϕ(z))| ≤ kuk∞|R_Nf(ϕ(z))| ≤εkuk∞kfk₁ ≤εkuk∞

which ends the proof.

On the other hand, we have the lower estimate:

Proposition 3.2. Suppose that uCϕ is a bounded operator fromH^p toH^∞and thatkϕk∞= 1.

Then 1

2Mϕ(u)≤ kuC_ϕk_e.

Proof. Assume that uC_ϕ is not compact, implying M_ϕ(u)>0.Let (z_n) be a sequence inD satisfying

limn |ϕ(z_n)|= 1 and lim

n

|u(z_n)|

(1− |ϕ(z_n)|²)^1p

=Mϕ(u).

Consider the sequence (f_n) defined by

f_n(z) =k_ϕ(zn)(z)^1/p= 1− |ϕ(z_n)|²¹_p

1−ϕ(z_n)z²_p

·

Each f_n is a unit vector ofH^p.LetK :H^p →H^∞be a compact operator.

First case: p >1

Since the sequence (f_n) converges to zero for the weak topology of H^p and K is completely continuous, the sequence (Kfn) converges to zero for the norm topology in H^∞. Use that

kuC_ϕ+Kk ≥ kuC_ϕ(fn)k_∞− kKf_nk_∞ and take the upper limit when n tends to infinity to obtain

kuC_ϕ+Kk ≥lim sup

n

kuC_ϕ(fn)k_∞

≥lim sup

n

|u(z_n)| |f_n(ϕ(zn))|

≥lim sup

n

|u(z_n)|

(1− |ϕ(z_n)|²)^1p

≥M_ϕ(u).

Second case: p= 1

Let ε > 0. Since the sequence (f_n) is no longer weakly convergent to zero in H¹, we cannot assert that (Kfn)n goes to zero in H^∞.Nevertheless, passing to subsequences, one can assume that (Kfn_k)_kconverges inH^∞,and hence is a Cauchy sequence. So we can find an integerN >0 such that for every kand m greater thanN we have kKf_nk−Kf_nmk< ε. We deduce that

kuC_ϕ+Kk ≥

(uC_ϕ+K)

f_nk −f_nm 2

_∞

≥ 1

2kuC_ϕ(f_nk−f_nm)k_∞− ε 2

≥ 1

2|u(z_nk)| |f_nk(ϕ(z_nk))−f_nm(ϕ(z_nk))| − ε 2

≥ |u(z_nk)|

2 (1− |ϕ(z_nk)|²) −|u(z_nk)| 1− |ϕ(z_nm)|² 2

1−ϕ(z_nm)ϕ(z_nk)

2 − ε 2

Now take the upper limit asmgoes to infinity (kbeing fixed) and remind that lim_m|ϕ(z_nm)|= 1 and |ϕ(z_nk)|<1 to obtain

kuC_ϕ+Kk ≥ |u(z_nk)|

2 (1− |ϕ(z_nk)|²) −ε 2 for everyk≥N.It remains to make ktend to infinity to have

kuC_ϕ+Kk ≥ 1

2M_ϕ(u)−ε 2.

Combining Proposition 3.1 and Proposition 3.2 we obtain the following estimate:

Theorem 3.3. Suppose that uC_ϕ is a bounded operator from H^p to H^∞ and that kϕk_∞ = 1.

Then kuC_ϕk_e≈Mϕ(u).More precisely, we have the following inequalities:

1

2M_ϕ(u)≤ kuC_ϕk_e≤2M_ϕ(u).

Note that ifp >1 one can replace the constant 1/2 by 1.

4 uC

∈ B(H

, H

) for ∞ > q ≥ 1

In this setting, boundedness of the weighted composition operator uCϕ is equivalent to saying thatu belongs toH^q, anduCϕ is compact if and only ifu= 0 or|E_ϕ|= 0 where Eϕ={ζ ∈T| ϕ^∗(ζ)∈T}is the extremal set of ϕ(see [3]). We give here some estimates of the essential norm of uCϕ that appear in [9] for the special case of composition operators:

Theorem 4.1. Let u∈H^q.Then kuC_ϕk_e≈ R

Eϕ|u(ζ)|^q dm(ζ)¹_q

. More precisely, 1

2 Z

Eϕ

|u(ζ)|^q dm(ζ)

!¹_q

≤ kuC_ϕk_e≤2 Z

Eϕ

|u(ζ)|^q dm(ζ)

!¹_q . We start with the upper estimate:

Proposition 4.2. Let u∈H^q.Then kuC_ϕk_e≤2

Z

Eϕ

|u(ζ)|^q dm(ζ) ¹_q

.

Proof. Take 0 < r < 1. Since krϕk_∞ ≤ r < 1, the set E_rϕ is empty and therefore the operatoruCrϕ is compact. ThuskuC_ϕk_e≤ kuC_ϕ−uCrϕk.But

kuC_ϕ−uCrϕk^q= sup

kfk∞≤1

Z

T

|u(ζ)|^q

f ϕ(ζ)

−f rϕ(ζ)

q dm(ζ). (4.1)

For ε > 0, note Eε = {ζ ∈ T | |ϕ^∗(ζ)| < 1 −ε}. We assume that uCϕ is not compact, henceE_ε 6=∅.We will use the pseudohyperbolic distanceρ defined forz andw in the unit disk byρ(z, w) =|z−w|/|1−wz|.¯ The Pick-Schwarz’s theorem ensures thatρ f(z), f(w)

≤ρ(z, w) for every function f ∈BH^∞.As a consequence the inequality|f(z)−f(w)| ≤2ρ(z, w) holds for everyw and zin D.

Ifζ is an element ofE_ε then

ρ ϕ(ζ), rϕ(ζ)

= (1−r)|ϕ(ζ)|

1−r|ϕ(ζ)|² ≤ 1−r 1−r(1−ε)². One can choose 0< r <1 satisfying sup_Eερ ϕ(ζ), rϕ(ζ)

< ε/2,and therefore f ϕ(ζ)

−f rϕ(ζ)

≤2 sup

Eε

ρ ϕ(ζ), rϕ(ζ)

≤ε

for all ζ ∈ E_ε and for every function f in the closed unit ball of H^∞. It follows from these estimates and (4.1) that

kuC_ϕ−uC_rϕk^q ≤ sup

kfk∞≤1

Z

Eε

|u(ζ)|^qε^q dm(ζ) + Z

T\Eε

2^q|u(ζ)|^q dm(ζ)

≤ε^qkuk^q_q+ 2^q Z

T\E_ε

|u(ζ)|^q dm(ζ).

Makeεtend to zero to deduce the upper estimate.

Let’s turn to the lower estimate:

Proposition 4.3. Suppose that u∈H^q. Then kuC_ϕk_e≥ 1

2 Z

Eϕ

|u(ζ)|^q dm(ζ)

!¹_q .

Proof. Take a compact operator K ∈B(H^∞, H^q).Since the sequence (zⁿ)n∈N is bounded in H^∞, there exists an increasing sequence of integers (n_k)_k≥0 such that (K(z^nk))_k≥0 con- verges in H^q. For any ε > 0 one can find N ∈ N such that for every k, m ≥ N we have kKz^nk−Kz^nmk_q< ε. If 0< r <1,we note g_r(z) = g(rz) for a functiong defined on D. Take k≥N.Then there exists 0< r <1 such that

k(uϕ^nk)_rk_q≥ kuϕ^nkk_q−ε.

For allm≥N we have

kuC_ϕ+Kk ≥

(uCϕ+K)

z^nk −z^nm 2

q

≥ 1

2ku(ϕ^nk −ϕ^nm)k_q−ε 2

≥ 1

2k(uϕ^nk)_r−(uϕ^nm)_rk_q−ε 2

≥ 1 2

k(uϕ^nk)_rk_q− k(uϕ^nm)_rk_q

−ε 2

≥ 1 2

kuϕ^nkk_q− k(uϕ^nm)_rk_q

−ε.

Let us makem tend to infinity, keeping in mind that 0< r <1 andkϕ_rk∞<1:

k(uϕ^nm)_rk_q ≤ kuk_qk(ϕ_r)^nmk_∞≤ kuk_qkϕ_rkⁿ_∞^m−→

m 0.

ThuskuC_ϕ+Kk ≥(1/2)kuϕ^nkk_q−εfor all k≥N.We conclude noticing that kuϕ^nkk_q=

Z

T

|u(ζ)ϕ(ζ)^nk|^q dm(ζ) ¹_q

−→

k

Z

Eϕ

|u(ζ)|^q dm(ζ)

!¹_q .

5 uC

∈ B(H

, H

) for ∞ > p > q ≥ 1

In [9], the authors give an estimate of the essential norm of a composition operator betweenH^p and H^q for 1 < q < p < ∞. The proof makes use of the Riesz projection from L^q onto H^q, which is a bounded operator for 1< q <∞.Since it is not bounded fromL¹ toH¹ (H¹ is not even complemented inL¹) there is no mean to use a similar argument. So we need a different approach to get some estimates for q = 1. A solution is to make use of Carleson measures.

First, we give a characterization of the boundedness of uC_ϕ in terms of a Carleson measure.

In the case where p > q, Carleson measures on Dare characterized in [1]. Denote by Γ(ζ) the Stolz domain generated by ζ ∈ T, i.e. the interior of the convex hull of the set {ζ} ∪(αD), where 0< α <1 is arbitrary but fixed.

Theorem 5.1 ( [1], Theorem 2.2). Letµ be a measure onD,1≤q < p <∞ ands=p/(p−q).

Then µ is a (p, q)-Carleson measure on D if and only if ζ 7→ R

Γ(ζ)

dµ(z)

1− |z|² belongs to L^s(T) and µ_T=Fdm for a function F ∈L^s(T).

This leads to a characterization of the continuity of a weighted composition operator be- tweenH^p and H^q:

Corollary 5.2. uC_ϕ : H^p → H^q is bounded if and only if G : ζ ∈ T 7→ G(ζ) = R

Γ(ζ) dµϕ(z) 1−|z|²

belongs toL^s(T) fors=p/(p−q)andµϕ_|

T =Fdmfor a certain F ∈L^s(T), wheredµ=|u|^qdm and µ_ϕ=µ◦ϕ⁻¹ is the pullback measure of µ by ϕ.

Proof. uCϕ is a bounded operator if and only if there exists c > 0 such that for anyf ∈H^p, R

T|u(ζ)|^q|f ◦ϕ(ζ)|^q dm(ζ)≤ckfk^q_p,which is equivalent (via a change of variables) toR

D|f(z)|^q dµϕ(z)≤ckfk^q_p for everyf ∈H^p. This exactly means thatµϕ is a (p, q)-Carleson measure. This is equivalent by Theorem 5.1 to the condition announced.

Iff ∈H^p,the Hardy-Littlewood maximal non tangential functionM fis defined byM f(ζ) = sup_z∈Γ(ζ)|f(z)|forζ ∈T.For 1< p <∞, M is a bounded operator from H^p toL^p and we will denote its norm by kMk_p. The following lemma is the analogue version of Lemma 2.6 for the case p > q.

Lemma 5.3. Letµbe a positive Borel measure onD.Assume thatµis a(p, q)-Carleson measure for 1 ≤ q < p∞. Let 0 < r <1 and µr := µ_|

D\rD. Then µr is a (p, q)-Carleson measure, and there exists a positive constant C such that for every f ∈H^p,

Z

D

|f(z)|^q dµr(z)≤(kFk_s+CkMk^q_pkG_rk_s)kfk^q_p where dµ_T=Fdm and Gr(ζ) =R

Γ(ζ) dµr(z)

1−|z|². In addition,kG_rk_s→0 as r →1.

Proof. Being a (p, q)-Carleson measure only depends on the ratiop/q (see [1], Lemme 2.1), so we have to show that µ_r is a (p/q,1)-Carleson measure.

It is clear that Gr ≤G∈L^s(T).Moreover dµr|

T = dµ_T =Fdm∈L^s(T).Corollary 5.2 ensures the fact that µr is a (p, q)-Carleson measure.

Letf be inH^p.Then Z

T

|f(ζ)|^q dµ_r(ζ) = Z

T

|f(ζ)|^q dµ(ζ) = Z

T

|f(ζ)|^qF(ζ) dm(ζ)

≤ Z

T

|f(ζ)|^p dm(ζ) ^q_p

kFk_s

≤ kfk^q_pkFk_s (5.1)

using H¨older’s inequality with conjugate exponentsp/q and s.

For z 6= 0, z ∈ D, note ˜I(z) = {ζ ∈ T | z ∈ Γ(ζ)}. In other words ζ ∈ I(z)˜ ⇔ z ∈ Γ(ζ).

Then m I(z)˜

≈1− |z|and Z

D

|f(z)|^q dµ_r(z)≈ Z

D

|f(z)|^q Z

I(z)˜

dm(ζ)

! dµr(z) 1− |z|²

= Z

T

Z

Γ(ζ)

|f(z)|^q dµr(z)

1− |z|² dm(ζ)

≤ Z

T

M f(ζ)^q Z

Γ(ζ)

dµr(z)

1− |z|² dm(ζ)

where M f(ζ) = sup_z∈Γ(ζ)|f(z)|is the Hardy-Littlewood maximal non tangential function. We apply H¨older’s inequality to obtain

Z

D

|f(z)|^q dµr(z)≤CkM fk^q_pkG_rk_s≤CkMk^q_pkG_rk_skfk^q_p. (5.2) whereC is a positive constant. Combining (5.1) and (5.2) it follows that

Z

D

|f(z)|^q dµ_r(z)≤(kFk_s+CkMk^q_pkG_rk_s)kfk^q_p.

It remains to show that G_r tends to zero in L^s(T) whenr tends to 1.We will make use of Lebesgue’s dominated convergence theorem. Clearly we have 0≤G_r ≤G∈L^s(T),so we need to show that Gr(ζ) → 0 as r → 1 for m−almost every ζ ∈ T. Let A={ζ ∈T| G(ζ)<∞}.

It is a set of full measure (m(A) = 1) since G ∈ L^s(T). Write Gr(ζ) = R

Γ(ζ)fr(z) dµ(z) withfr(z) = 1I

D\rD(z)(1− |z|²)⁻¹, z∈Γ(ζ).For every ζ∈A one has

|f_r(z)| ≤ 1

1− |z|² ∈L¹(Γ(ζ), µ) since ζ ∈A, fr(z)−→

r→10 for all z∈Γ(ζ)⊂D.

Lebesgue’s dominated convergence theorem in L¹(Γ(ζ), µ) ensures that G_r(ζ) = kf_rk_L1(Γ(ζ),µ)

tends to zero asr tends to 1 form−almost everyζ ∈T,which ends the proof.

Theorem 5.4. Let uCϕ be a bounded operator from H^p to H^q,with ∞> p > q≥1.Then kuC_ϕk_e≤2kC_ϕk^1/q_p/q

Z

Eϕ

|u(ζ)|^p−qpq dm(ζ)

!^p−q_pq , where kC_ϕk_p/q denotes the norm of C_ϕ acting on H^p/q.

Proof. We follow the same lines as in the proof of the upper estimate in Proposition 2.4: we have the decomposition I =K_N +R_N inB(H^p), where K_N is the convolution operator by the Fej´er kernel, and

kuC_ϕk_e ≤lim inf

N kuC_ϕRNk.

We also have, for every 0< r <1, k(uC_ϕR_N)fk^q_q =

Z

D\rD

|R_Nf(w)|^q dµ_ϕ(w) + Z

rD

|R_Nf(w)|^q dµ_ϕ(w)

=I₁(N, r, f) +I₂(N, r, f).

As in thep≤q case, we show that

limN sup

kfkp≤1

I₂(N, r, f) = 0.

The measure µ_ϕ being a (p, q)-Carleson measure, we use Lemma 5.3 to have the following inequality

I1(N, r, f)≤(kFk_s+CkMk^q_pkG_rk_s)kR_Nfk^q_p for everyf ∈H^p.As a consequence

kuC_ϕk_e≤lim inf

N sup

kfk_p≤1

I₁(N, r, f)

!¹_q

≤2(kFk_s+CkMk^q_pkG_rk_s)^1q

using the fact that sup_NkR_Nk ≤2.Now we maker tend to 1, keeping in mind thatkG_rk_s→0.

We obtain

kuC_ϕk_e ≤2kFk^1/q_s . It remains to see that we can choose F in such a way that

kFk_s≤ kC_ϕk_p/q Z

Eϕ

|u(ζ)|^p−qpq dm(ζ)

!1/s

.

Indeed, iff ∈C(T)∩H^p/q, we apply H¨older’s inequality with conjugates exponents p/q and s to have

Z

T

f dµ_ϕ,T

= Z

Eϕ

|u|^qf◦ϕdm

≤ Z

Eϕ

|u|^q|f◦ϕ|dm≤ kC_ϕ(f)k_p/q Z

Eϕ

|u|^sq dm

!1/s

,

meaning that µ_ϕ,T ∈ H^p/q∗

, which is isometrically isomorphic toL^s(T)/H₀^s, where H₀^s is the subspace of H^s consisting of functions vanishing at zero. If we denote by N(µ_ϕ,T) the norm of µ_ϕ,T viewed as an element of H^p/q∗

, then one can choose F ∈ L^s(T) satisfying kFk_s = N(µϕ,T) ≤ kC_ϕk_p/q

R

Eϕ|u|^pq/(p−q) dm 1/s

and µϕ,T = F dm (see for instance [11], p. 194).

Finally we have

kuC_ϕk_e≤2kC_ϕk^1/q_p/q Z

Eϕ

|u(ζ)|^p−qpq dm(ζ)

!^p−q_pq .

Although we haven’t be able to give a corresponding lower bound of this form for the essential norm ofuC_ϕ, we have the following result:

Proposition 5.5. Let 1≤q < p <∞, and assume thatuC_ϕ ∈B(H^p, H^q). Then kuC_ϕk_e≥

Z

Eϕ

|u(ζ)|^q dm(ζ)

!¹_q .

Proof. Take a compact operator K from H^p toH^q. Since it is completely continuous, and the sequence (zⁿ) converges weakly to zero in H^p,(K(zⁿ))_n converges to zero inH^q.Hence

kuC_ϕ+Kk ≥ k(uC_ϕ+K)zⁿk_q≥ kuC_ϕ(zⁿ)k_q− kK(zⁿ)k_q for everyn≥0.Taking the limit as ntends to infinity, we have

kuC_ϕk_e≥ Z

Eϕ

|u(ζ)|^q dm(ζ)

!¹

q

.

References

[1] O. Blasco and H. Jarchow. A note on Carleson measures for Hardy spaces. Acta Sci.

Math. (Szeged)71(2005), 371-389.

[2] L. Carleson Interpolations by bounded analytic functions and the corona problem. Ann.

of Math. (2)76 (1962), 547-559.

[3] M. D. Contreras and A. G. Hern´andez-D´ıaz Weighted Composition Operators be- tween Different Hardy Spaces. Integr. eq. oper. theory 46(2003), 165-188.

[4] C. C. Cowen and B. D. MacCluer. Composition operators on spaces of analytic functions. CRC Press, Boca Raton, 1995.

[5] ˘Z. C˘u˘ckovi´c and R. Zhao. Weighted composition operators between different weighted Bergman spaces and different Hardy spaces. Illinois J. Math.51no. 2 (2007), 479-498.

[6] ˘Z. C˘uckovi´˘ c and R. Zhao. Weighted composition operators on the Bergman spaces. J.

London Math. Soc. (2)51(2004), 499-511.

[7] P. Duren. Theory of H^p spaces. Academic Press, New York, 1970.

[8] F. Forelli. The isometries of H^p. Canad. J. Math. 16(1964), 721-728.

[9] P. Gorkin and B. D. MacCluer. Essential Norms of Composition Operators. Integr.

eq. oper. theory 48(2004), 27-40.

[10] K. Hoffman. Banach spaces of analytic functions. Dover Publications, Inc., 1988 [11] P. Koosis. Introduction to Hp spaces. Cambridge University Press, 1980.

[12] P. Lef`evre. Generalized Essential Norm of Weighted Composition Operators on some Uniform Algebras of Analytic Functions. Integr. eq. oper. theory 63(2009), 557-569.

[13] V. Matache. Composition operators on Hardy Spaces of a half-plane. Proc. Amer. Math.

Soc127 (1999), 1483-1491.

[14] S. D. Sharma and R. K. Singh. Composition operators on Hardy spaces of the upper half-plane. Bull. Allahabad Math. Soc.14 (1999), 129-145.

[15] K. Zhu. Duality of Bloch Spaces and Norm Convergence of Taylor Series. Michigan Math.

J.38(1991), no. 1, 89?101.

Romain Demazeux,Univ Lille Nord de France, FRANCE UArtois, Laboratoire de Math´ematiques de Lens EA 2462, F´ed´eration CNRS Nord-Pas-de-Calais FR 2956,

F-62 300 LENS, FRANCE

romain.demazeux@euler.univ-artois.fr