We obtain estimates of the norm of Toeplitz operators on weighted Hardy and Besov spaces

AND BESOV SPACES

CARME CASCANTE, JOAN F `ABREGA, AND JOAQUIN M. ORTEGA

Abstract. We obtain estimates of the norm of Toeplitz operators on weighted Hardy and Besov spaces. As an application we give character- izations of some spaces of pointwise multipliers.

1. Introduction

The main goal of this work is the study of the norms of the Toeplitz operators in a large scale of spaces of holomorphic functions on B, which includes weighted Hardy and Besov spaces. In order to introduce the prob- lem and to state our results, we recall some results on Toeplitz operators in the classical setting of the Hardy space H^p.

Let B be the open unit ball in Cⁿ, ¯B its closure and S its boundary.

By dν and dσ we denote the normalized Lebesgue measures on B and S respectively. We denote by H(B) (resp. H( ¯B)) the space of holomorphic functions on B (resp. on ¯B). If p > 0, a holomorphic function f on B is in the Hardy space H^p(B) if and only if

kf k_H^p = sup

r<1

Z

S

|f (rζ)|^pdσ(ζ) < +∞.

If p ≥ 1, the operator which assigns to each f its boundary values f (ζ) = lim

r%1f (rζ), a.e ζ ∈ S defines an isometry from H^p(B) onto H^p(S),

2000 Mathematics Subject Classification. 47B35, 32A25, 32A37.

Key words and phrases. Toeplitz operators, weighted Hardy spaces, pointwise multipliers.

Authors partially supported by DGICYT Grant MTM2005-08984-C02-02, Grant MTM2006-26627-E and DURSI Grant 2005SGR 00611.

May 13, 2008.

1

the L^p-closure of the restriction to S of H( ¯B). A well known result about operators on the Hardy spaces H^p(S), 1 < p < ∞, states that for any ψ ∈ L^∞(dσ), the norm of the Toeplitz operator with symbol ψ, T_ψ : H^p(S) → H^p(B), defined by

Tψ(f )(z) = Z

S

ψ(ζ)f (ζ)

(1 − z ¯ζ)ⁿdσ(ζ), is equivalent to kψk_∞.

By composing the above mentioned isometry between H^p(B) and H^p(S) with T_ψ we can obtain maps from H^p(S) to itself, or from H^p(B) to itself.

All these operators will be denoted by T_ψ and we will simply write H^p to denote either H^p(S) or H^p(B).

We will consider the duality (H^p)⁰ = H^p0 with respect to the pairing

(1.1) hf, ¯gi_S = lim

r→1

Z

S

f_rg¯_rdσ,

where f_r(ζ) = f (rζ), in the sense that each Λ ∈ (H^p)⁰ is given by Λ(f ) = hf, ¯gi_S, for some g ∈ H^p0 and kΛk_(H^p₎⁰ ≈ kgk_Hp0. Then, we have that the following equivalence between the norm of the bilinear Toeplitz form Γ_ψ(f, ¯g) =R

Sψ f ¯gdσ = hT_ψ(f ), ¯gi_S and the norm of the operator T_ψ holds:

(1.2) kΓ_ψk

Bil(H^p×H^p0→C) ≈ kT_ψk_L(H^p→H^p) ≈ kψk∞.

There exists an extensive literature on Toeplitz operators in spaces of holomorphic functions. See for instance [11], [2] and the references therein.

We want to extend these results to other Banach spaces of holomorphic functions on B. Throughout the paper we consider Banach spaces of holo- morphic functions on B, X and Y satisfying H( ¯B) ⊂ X, Y . We denote by X_c (resp. Y_c), the space H( ¯B) normed by k · k_X (resp. k · k_Y). If X, Y ⊂ H¹(B), then the functions f in X or Y , have boundary values f (ζ), a.e. ζ ∈ S in L¹(dσ). As in the case of H^p spaces, we will identify the space X with its space of boundary values.

The Toeplitz form Γ_ψ is well-defined on X_c× Y_cfor any ψ ∈ L¹(dσ), and the associated Toeplitz operator T_ψ defines an operator from X_c to H(B).

Moreover, notice that if 1 ≤ p ≤ ∞, ψf ∈ L^p(dσ) and g ∈ H^p0(B), then Γ_ψ(f, ¯g) = hT_ψ(f ), ¯gi_S.

Therefore, if Y⁰ is the dual space of Y with respect to the pairing (1.1), the norm of the Toeplitz operator T_ψ : X → Y⁰ is equivalent to the norm of Γ_ψ : X × Y → C.

The norm of this form Γψ will be computed in terms of extensions of ψ to B, given by generalized Poisson-Szeg¨o operators. This computation will be used to obtain estimates of the norms of Toeplitz operators for different spaces of holomorphic functions on B. For m ≥ 0, ζ ∈ S and z ∈ B, we consider the integral operators

P_m(ψ)(z) = c_n,m Z

S

ψ(ζ)(1 − |z|²)^n+2m

|1 − z ¯ζ|^2n+2mdσ(ζ),

where c_n,m is a normalizing constant. If m = 0, we recover the Poisson- Szeg¨o kernel. These operators give the solution of some generalized Dirichlet problems (see Section 2 for more details).

The fact that for any z ∈ B, f_z(w) = (1 − w¯z)^−n−m∈ H( ¯B) gives that (1.3) |P_m(ψ)(z)| ≤ kΓ_ψk_Bil(Xc×Yc)ω_m(z),

where

ω_m(z) = ω_m,X,Y(z)

= c_n,m(1 − |z|²)^n+2mk(1 − w¯z)^−n−mk_Xk(1 − w¯z)^−n−mk_Y. (1.4)

Inequality (1.3) gives immediately two necessary conditions on ψ such that Γ_ψ : X_c× Y_c→ C is bounded.

The first condition is just that,

(1.5) sup

z∈B

|P_m(ψ)(z)|

ω_m(z) ≤ kΓ_ψk_Bil(X

c×Y_c→C).

The second condition is given in terms of ψ. If ϕ ≥ 0 is a continuous function on B and ζ ∈ S, let lim inf

r%1 ϕ(rζ) = sup

r

r<t<1inf ϕ(rζ).

Let

(1.6) ω˜_m(ζ) = lim inf

r%1 ω(rζ).

From (1.3), we obtain

(1.7) |ψ(ζ)| ≤ kΓ_ψk_Bil(X

c×Yc)ω˜_m(ζ), a. e. ζ ∈ S.

Now, if we assume that 0 < ˜ω_m(ζ) < ∞, a.e. ζ ∈ S, we obtain

(1.8) sup

ζ∈S

|ψ(ζ)|

˜

ω_m(ζ) ≤ kΓ_ψk_{Bil(Xc×Yc→C)}.

Since kΓ_ψk_{Bil(Xc×Yc→C)} ≤ kΓ_ψkBil(X×Y →C) it is clear that the correspond- ing conditions (1.5) and (1.8) are also necessary to ensure that Γ_ψ : X ×Y → C is bounded. Moreover, if X^c and Y_c are dense in X and Y respectively, and Γ_ψ is bounded on X_c× Y_c, then Γ_ψ extends to an unique operator on X × Y also denoted by Γ_ψ.

In order to state conditions on X and Y such that sup

z∈B

|Pm(ψ)(z)|

ω_m(z) ≈ kΓ_ψk_{Bil(Xc×Yc→C)}, or sup

z∈B

|P_m(ψ)(z)|

ωm(z) ≈ kΓ_ψkBil(X×Y →C), (1.9)

we consider the functions τ_c, τ : [0, 1) → R defined by τ_c(r) = sup

kf_r¯g_r(ω_m)_rk_L¹_(dσ)

kf k_Xkgk_Y ; f, g ∈ H( ¯B), f, g 6= 0

 , and

τ (r) = sup

kf_rg¯_r(ω_m)_rk_L¹_(dσ)

kf k_Xkgk_Y ; f ∈ X, g ∈ Y, f, g 6= 0

 , where f_r(ζ) = f (rζ), 0 ≤ r < 1, ζ ∈ S.

With these notations we can now state the following result.

Theorem 1.1. Let m ≥ 0 and let X and Y be Banach spaces of holomorphic functions on B, such that H( ¯B) ⊂ X, Y .

Then sup

z∈B

 |P_m(ψ)(z)|

ω_m(z)



≤ kΓ_ψk_Bil(X

c×Y_c→C) ≤ kτ_ck∞sup

z∈B

 |P_m(ψ)(z)|

ω_m(z)

 . Moreover, if X, Y ⊂ H¹(B),

kΓ_ψkBil(X×Y →C)≤ kτ k_∞sup

z∈B

 |P_m(ψ)(z)|

ωm(z)

 .

Observe that the functions τ and τ_c depend only on the spaces X and Y and not on the function ψ. Of course the interesting case for applications is when the functions τ and τc are bounded.

Notice that if X = H^pand Y = H^p0, choosing m = 0, we have kP₀(ψ)k∞= kψk∞, ω₀(z) ≈ 1 (by Proposition 1.4.10 in [14]) and kτ k∞= 1 (by H¨older’s Inequality). Therefore, (1.2) can be obtained from Theorem 1.1. We also remark that if p = 2, then ω₀ = 1 and in consequence the equivalences in (1.2) can be replaced by identities (see for instance [8]).

Theorem 1.1 can be used to improve some well known results on Toeplitz operators in classical spaces. For instance, if B_s^p is the holomorphic Besov space on B (defined in Section 2) and T Symb(X → Y ) denotes the set of symbols of all the continuous Toeplitz operators T_ψ from X to Y , then, this space coincides with L^∞(dσ) if X and Y satisfy one of the following conditions:

(i) B_n¹ ⊂ X ⊂ H^∞ and BM OA ⊂ Y ⊂ B₀^∞. (ii) B_n/p¹ 0 ⊂ X ⊂ H^p ⊂ Y ⊂ B₀^∞, with 1 < p < ∞.

This result includes the cases where X = A is the ball algebra (the space of holomorphic functions on B and continuous on ¯B), and generalizes the well known result T Symb(H^∞ → BM OA) = T Symb(H^∞→ B₀^∞) = L^∞in the unit disk [9].

In addition to the results on pointwise multipliers between some classical spaces which are a consequence of the fact that, if h ∈ H¹and f ∈ H( ¯B), the

multiplication operator Mh(f ) = hf corresponds to the Toeplitz operator T_h, we also prove that:

(i) If B_n¹ ⊂ X ⊂ H^∞∩ V M OA, then M(X → V M OA) coincides with the multiplicative algebra H^∞∩ V M OA.

(ii) Let b^∞₀ be the little Bloch space, that is the closure of H( ¯B) in B₀^∞. If B_n¹ ⊂ X ⊂ H^∞∩ b^∞₀ , then M(X → b^∞₀ ), coincides with the multiplicative algebra H^∞∩ b^∞₀ .

Here, M(X → Y ) denotes the space of pointwise multipliers from X to Y .

The next theorem gives conditions on X and Y such that

(1.10) sup

ζ∈S

|ψ(ζ)|

˜

ωm(ζ) ≈ kΓ_ψk_Bil(X

c×Y_c→C).

Theorem 1.2. Let X and Y be Banach spaces of holomorphic functions on B containing H( ¯B).

If 0 < ˜ω_m(ζ) < ∞ a.e. ζ ∈ S, and there exists a constant C_X,Y such that kf gk_L¹_(˜ωm)≤ C_X,Ykf k_Xkgk_Y, then

ψ

˜ ω_m

∞

≤ kΓ_ψk_Bil(X

c×Yc→C) ≤ kΓ_ψkBil(X×Y →C) ≤ C_X,Y

ψ

˜ ω_m

∞

. This result permit us to obtain results on Toeplitz operators in some class of holomorphic weighted Hardy spaces. In order to precise these results we consider weights in S, that for simplicity we will call admissible weights, satisfying:

(i) θ(ζ) > 0 a.e. ζ ∈ S.

(ii) θ, θ^−p0/p ∈ L¹(dσ),

For this class of weights, let H^p(θ) denotes the closure of the restriction of H( ¯B) on S in L^p(θ).

Observe that if θ^−p0/p ∈ L¹(dσ), then for each ϕ ∈ L^p(θ), Z

S

|ϕ(ζ)|dσ(ζ) ≤ kθ^−p0/pk^1/p_L1(dσ)⁰ kϕk_L^p_(θdσ). Therefore, H^p(θ) is a subspace of H¹.

With these conditions we have:

Theorem 1.3. Let 1 < p < ∞, ψ ∈ L¹(dσ) and θ₀, θ₁ a pair of admissible weights. Then,

kΓ_ψk

Bil(H^p(θ0)×H^p0(θ1))→C) ≈

ψ θ^1/p₀ θ₁^1/p0

∞

.

From this last result we can obtain estimates on the norm of the corre- sponding Toeplitz operator T_ψ on weighted Hardy spaces, once we have a description of the dual of H^p0(θ1) with respect to the pairing (1.1). This is the case when θ₁ is in the Muckenhoupt class A_p whose the definition and some properties of these weights will be stated in Section 4. Among them, we remark that we give a characterization of the fact that θ ∈ A_p in terms of the generalized extensions P_m(θ), which extend the one obtained by [12], where it was consider the case m = 0 and p = 2.

We obtain the following theorem:

Theorem 1.4. Let 1 < p < ∞, ψ ∈ L¹(dσ), θ₀ an admissible weight and θ₁ ∈ A_p⁰.

Then

kT_ψk_L(H^p_(θ0)→H^p_(θ1)) ≈

ψ θ^1/p₀ θ₁^−1/p

_∞

.

In particular, if θ = θ₀ = θ₁ ∈ A_p, then kT_ψk_L(H^p_(θ)→H^p_(θ)) ≈ kψk∞

The paper also contains some additional results on weighted Hardy spaces and Besov spaces, that may be interesting by themselves.

If θ ∈ Ap, then H( ¯B) is dense in H^p(θ), and in this case H^p(θ) consists of holomorphic functions f on B, such that

kf k^p_Hp(θ) =: sup

0≤r<1

Z

S

|f (rζ)|^pθ(ζ)dσ(ζ) < ∞.

Let (I + R)^k be the linear differential operator of order k defined by (I + R)^kz^α= (1 + |α|)^kz^α, where α = (α₁, · · · , α_n) and |α| =

n

X

j=1

|α_j|.

If s ∈ R, we denote by H_s^p(θ) the Hardy-Sobolev space of holomorphic funtions on B such that k(I + R)^sf k_H^p_(θ) < ∞.

It is well known (see [10] p. 334) that if 1 < p < ∞ and θ ∈ A_p, the dual of H_s^p(θ) with the pairing given by (1.1) is H_−s^p0 (θ^−p0/p).

In order to consider weighted Besov spaces related to H^p(θ), we introduce an averaging function Θ of the weight θ in S given by Θ(z) = _|I¹

z|

R

Izθdσ. If the weight θ is in A_p, then its averaging Θ is in the class B_p, introduced in [5] (see Section 4 for more details). The characterizations of H^p(θ) in terms of the admissible maximal function, shows that, when θ ∈ A_p we can define an equivalent norm in H^p(θ) in terms of the averaging function Θ given by

kf k^p_Hp(Θ) =: sup

0≤r<1

Z

S

|f (rζ)|^pΘ(rζ)dσ(ζ).

In order to define weighted Besov spaces, as in the Hardy spaces cases, for 1 ≤ p < ∞ we consider admisible weights on B, that is weights Ψ > 0 a.e. on B such that Ψ, Ψ^−p0/p∈ L¹(dν).

If 1 ≤ p < ∞, s ∈ R, a non-negative integer k > s and Ψ is an admissible weight, we denote by B_s,k^p (Ψ) the completion of the space H( ¯B) endowed with the norm

kf k^p_Bp

s,k(Ψ)=:

Z

B

|(I + R)^kf (z)|^p(1 − |z|²)^(k−s)p−1Ψ(z)dν(z).

If 1 < p < ∞ and Ψ ∈ Bp, then different values of k gives equivalent norms (see [7]), and therefore in this case the space B_s,k^p (Ψ) will be simply denoted by B_s^p(Ψ). Moreover, if Ψ ∈ B_p, then the dual of B_s^p(Ψ) with respect to the pairing (1.1) is B_−s^p (Ψ^−p0/p).

If θ ∈ A_p the function Θ is in B_p and consequently, if q < p, the weight Θ^q/p is in B₁₊^q

p(p−1). Since 1 + ^q_p(p − 1) > q, we have that the weight Θ^q/p might not be in B_q. If the weight θ is in a smaller class, for instance, if θ ∈ Ap0, where p0 ≤ 1 + p/q⁰, we have that the weight Θ^q/p ∈ Bq and in particular we have that the space B_s,k^q (Θ^q/p) is independent of k.

Theorem 1.5. Let 1 < p < ∞, 1 ≤ q0 < q1 ≤ min{p, 2} and θ ∈ Ap0, where p₀ = 1 + p/q⁰₀. For j = 0, 1, let s_j = n/q_j − n/p .

Then, B_s^q₀⁰(Θ^q0/p) ⊂ B_s^q1₁(Θ^q1/p) ⊂ H^p(θ).

With this result we can prove the following theorem.

Theorem 1.6. Let 1 < q₀ < p < q₁ < ∞, θ₀ ∈ A_p0, θ₁ ∈ A_p⁰

1 where p0 = 1 + p/q₀⁰, p⁰₁ = 1 + p⁰/q1 and Θ0 and Θ1 the corresponding averaging functions. Then Θ^q₀^0/p∈ B_q0, Θ^q

0 1/p⁰ 1 ∈ B_q⁰

1 and

kΓ_ψk

Bil(B^q0

n/q0−n/p(Θ^q0/p₀ )×B^q01

n/q01−n/p0(Θ^q0₁^1/p0)→C)

≈ kT_ψk

L(B^q0

n/q0−n/p(Θ^q0/p₀ )→B^q1

n/q1−n/p(Θ^−q1/p0₁ )) ≈

ψ θ^1/p₀ θ₁^1/p0

∞

.

More general results of this type can be found in Section 4.

The paper is organized as follows. In Section 2, we recall some well known properties of the unweighted holomorphic Hardy-Sobolev and Besov spaces, and other technical results that will be needed in the next sections.

In Section 3, we prove Theorems 1.1 and 1.2, and its application to the study of the Toepliz operators and pointwise multipliers in classical spaces and weighted Hardy spaces (Theorem 1.3). In Section 4, we start recalling the main properties of the weights A_p and B_p, and we extend a result of S.

Petermichl and B. Wick about characterizations of the weights in A_p of the sphere in terms of a generalized invariant harmonic extension to the unit ball. In the second part of this section we recall some well known properties of the weighted Hardy spaces with weights in A_p, and of the weighted Besov spaces with weights in B_p, and we prove Theorems 1.5 and 1.6.

Notations: Throughout the paper, the letter C may denote various non-negative numerical constants, possibly different in different places. The notation f (z) . g(z) means that there exists C > 0, which does not depends of z, f and g, such that f (z) ≤ Cg(z).

2. Preliminaries

In this section we state some notations and some well known results, that will be needed in the forthcoming sections.

2.1. Non-isotropic balls, tents and admissible regions. For ζ ∈ S and 0 < t < 2, let U_ζ,t be the non-isotropic ball on S defined by U_ζ,t = {η ∈ S; |1 − η ¯ζ| < t}, and let ˆU_ζ,t be the non-isotropic tent on B defined by ˆUζ,t = {z ∈ B; |1 − z ¯ζ| < t}.

It is well known that the Lebesgue measure on S of U_ζ,t, denoted by |U_ζ,t|, is of order of tⁿ, and the Lebesgue measure on B of ˆU_ζ,t also denoted by

| ˆU_ζ,t| satisfies | ˆU_ζ,t| ≈ tⁿ⁺¹.

To each z = rζ, 0 < r < 1, ζ ∈ S, we consider the associated non-isotropic ball I_z = U_ζ,(1−|z|²₎ and the tent ˆI_z = ˆU_ζ,(1−|z|²₎.

For ψ ∈ L¹(dσ), let M_H−L(ψ) denote the non-isotropic Hardy-Littlewood maximal, defined by

M_H−L(ψ)(ζ) = sup

t>0

1

|U_ζ,t| Z

Uζ,t

ψ(η)dσ(η).

If ζ ∈ S, let Γ_ζ be the admissible region Γ_ζ = {z ∈ B; |1 − z ¯ζ| < 1 − |z|²}.

Observe that if z ∈ Γ_ζ, |1 − z ¯ζ| ≈ 1 − |z|². The admissible maximal function of a function ψ ∈ L¹(dν) is M (ψ)(ζ) = sup_z∈Γζ|ψ(z)|.

If ψ is a non-negative measurable function on B, Fubini’s Theorem gives that

Z

B

ϕ(z)dν(z) = Z

S

Z

Γ_ζ

ϕ(z)dν(z)

|I_z| dσ(ζ).

2.2. Integral operators. Let P be the Cauchy integral operator given by P (f )(z) =

Z

S

f (ζ)

(1 − z ¯ζ)ⁿdσ(ζ), z ∈ B.

For m ≥ 0, ζ ∈ S and z ∈ B, let P_m be the non-isotropic kernel defined by

P_m(ζ, z) = c_n,m(1 − |z|²)^n+2m

|1 − z ¯ζ|^2n+2m, where c_n,m = Γ(n + m)² (n − 1)!Γ(n + 2m).

Observe that for m = 0, P0 is the Poisson-Szeg¨o kernel.

We also denote by P_m the corresponding integral operator on L¹(dσ) defined by

P_m(ψ)(z) = Z

S

ψ(ζ)P_m(ζ, z)dσ(ζ), z ∈ B.

Let ∆_m be the differential operator ∆_m defined by

∆_m = (1 − |z|²) ( _n

X

i,j=1

(δ_ij− z_iz_j) ∂_i∂_j+ mR + mR − m²Id )

.

This family of differential operators ∆_m generalizes the invariant Lapla- cian, which corresponds to m = 0. It is shown in [1] that the generalized Dirichlet problem

∆_mu = 0, u ∈ C(B), u = ϕ on S, ϕ ∈ C(S), has a unique solution given by

u(z) = Z

S

ϕ(ζ)P_m(ζ, z) dσ(ζ) = P_m(ϕ)(z).

It is also shown in [1] that if ϕ ∈ L¹(dσ), then lim

r→1P_m(ϕ)(rζ) = ϕ(ζ) a.e.

ζ ∈ S.

Proposition 1.4.10 in [14] gives in particular that if 1 ≤ p < ∞, m ≥ 0 and f ∈ H( ¯B), then

(2.11) |f (z)|^p ≈     Z

S

f (ζ) (1 − |z|²)^n+2m

(1 − z ¯ζ)ⁿ(1 − ζ ¯z)^n+2mdσ(ζ)    

p

. P^m(|f |^p)(z).

Finally, we also point out that kP_m(ψ)k∞ ≈ kψk∞, and that sup_r|Pm(ψ)(rζ)| . M^H−L(|ψ|)(ζ).

To conclude we state the following lemma, which follows easily from Proposition 1.4.10 in [14].

Lemma 2.1. If k, m > 0, then for z, w ∈ B, Z

S

dσ(ζ)

|1 − z ¯ζ|^n+k|1 − w ¯ζ|^n+m ≈ (1 − |z|²)^−k

|1 − w¯z|^n+m +(1 − |w|²)^−m

|1 − z ¯w|^n+k .

2.3. Unweighted spaces of holomorphic functions. Let ∂_jψ = _∂z^∂

jψ and ∂_jψ = _∂z^∂

jψ. If α = (α₁, . . . , α_n) and |α| =Pn

j=1|α_j|, we denote by ∂^α the differential operator of order |α| defined by ∂₁^α1· · · ∂_n^αn.

We denote by R the radial derivative

n

X

j=1

z_j∂_j. For s ∈ R, we consider the invertible linear operator (I + R)^s on H(B), defined on the monomials z^α = z^α₁¹· · · z^α_nⁿ by (I + R)^sz^α = (1 + |α|)^sz^α.

For 1 ≤ p ≤ ∞ and s ∈ R, the holomorphic Besov space Bs^p is the set of holomorphic functions f on B such that

k(1 − |z|²)^k−s((I + R)^kf )(z)k_L^p_((1−|z|²₎⁻¹_dν(z))< +∞,

for some non-negative integer k > s. It is well known that different values of k > s give equivalent norms in B_s^p. Moreover, if we replace in the last expression (I + R)^kf by ∇^kf =:P

|α|≤k|∂^αf | we also obtain an equivalent norm.

Notice that the space B₀^∞ is the Bloch space, and that if s > 0, then B_s^∞ coincides with the holomorphic Lipschitz space Lip_s.

It is well known that if 1 ≤ p < ∞ and s ∈ R, then H( ¯B) is dense in the ball algebra A, in B_s^p and also in H_s^p. This density fails to be true for the spaces H^∞, BM OA or B_s^∞. The closure of H( ¯B) in BM OA is denoted by V M OA, and the closure of H( ¯B) in the Bloch space B₀^∞ is the little Bloch space b^∞₀ .

The following two theorems summarize the inclusion and duality results on the above spaces. Their proofs can be found in [3] and [4].

Theorem 2.2. Let 1 ≤ q < p < ∞ and s, t ∈ R satisfying s−n/p = t−n/q.

Then,

(i) B_t^q ⊂ B_s^p and H_t^q⊂ H_s^p.

(ii) If q ≤ min{p, 2}, then B_t^q ⊂ H_s^p.

(iii) B_n¹ ⊂ A, B_n/p^p , H_n/p^p ⊂ V M OA ⊂ BM OA ⊂ B₀^∞.

Theorem 2.3. Let 1 < p < ∞ and s ∈ R. The following duality results with respect to the pairing (2.12) holds:

(i) (V M OA)⁰ = H¹, and ((B₀^∞)c)⁰ = B₀¹. (ii) (H¹)⁰ = BM OA, and (B¹₀)⁰ = B₀^∞. (iii) (B_s^p)⁰ = B_−s^p0 , and (H_s^p)⁰ = H_−s^p0 .

We conclude this section with some remarks about the pairing hf, gi_S. The fact that any function in H^p has its radial maximal function M_r(f )(ζ) = sup_0≤r<1|f (rζ)| in L^p(dσ) gives that

hf, gi_S = lim

r%1

Z

S

f_rg¯_rdσ = Z

S

f ¯gdσ,

and hence Z

S

|f ||g|dσ is finite.

However, if f ∈ B_s^p and g ∈ B_−s^p0 , then f g is not necessarily in L¹(dσ) and we cannot, in general, interchange the limit with the integral. In these cases it is convenient to rewrite the pairing (2.12) as follows. The formula (see Section 1.4 in [14])

Z

S

|ζ^α|²dσ = (n + |α|) · · · (n + k − 1 + |α|) n (k − 1)!

Z

B

|z^α|²(1 − |z|²)^k−1dν(z),

the fact that Rz^α = |α|z^α, and the homogeneous expansion of the holomor- phic functions f and g, give that

Z

S

f (rζ)g(rζ)dσ(ζ) = Z

B

[p(R)f ](rz)[q(R)g](rz)(1 − |z|²)^k−1dν(z),

where p(R) and q(R) are the differential operators associated to any of the one variable real polynomials p(t) and q(t) satisfying

p(t)q(t) = (n + t) · · · (n + k − 1 + t) n (k − 1)! .

In particular, if we denote by R_j^l the differential operator of order l defined by R^l_j = (jI + R) · · · ((j + l − 1)I + R), then for l ≤ k

Z

S

f_r(ζ)g_r(ζ)dσ

= 1

n (k − 1)!

Z

B

[R^l_nf ](rz)[R_n+l^k−lg(rz)](1 − |z|²)^k−1dν(z).

(2.12)

We point out that if j > 0, then the operators R^l_j : H → H are invertible.

The inverse will be denoted by R_j^−l.

Formula (2.12) can be used to prove that (B_s^p)⁰ = B_−s^p0 with the pairing (1.1). The corresponding results on duality for the Hardy-Sobolev spaces, i.e. (H_s^p)⁰ = H_−s^p0 , can be deduced using instead the formula

Z

S

f_rg¯_rdσ = Z

S

[(I + R)^−sf ]_r[(I + R)^sg]_rdσ.

3. Norms of Toeplitz operators

We will assume that X and Y are two Banach spaces of holomorphic func- tions on B, containing both of them H( ¯B). We start this section obtaining necessary conditions on a functions ψ ∈ L¹(dσ), such that the bilinear Toeplitz forms Γ_ψ : X_c× Y_c→ C be continuous.

Since

P_m(ψ)(z) = c_n,m(1 − |z|²)^n+2m Z

S

ψ(ζ) (1 − ζ ¯z)^−n−m(1 − z ¯ζ)^−n−mdσ(ζ), it is clear that for any z ∈ B,

|P_m(ψ)(z)|

≤ c_n,mkΓ_ψk_{Bil(Xc×Yc→C)}

(1 − |z|²)^n/p0+m (1 − w¯z)^n+m

X

(1 − |z|²)^n/p+m (1 − w¯z)^n+m

Y

= kΓ_ψk_{Bil(Xc×Yc→C)}ω_m(z).

(3.13)

Therefore, if Γ_ψ is bounded on X_c× Y_c, we have that sup

z∈B

 |P_m(ψ)(z)|

ω_m(z)



≤ kΓ_ψk_Bil(X

c×Y_c→C).

This condition is given in terms of the generalized Poisson-Sz¨ego extension of ψ. A necessary condition in terms of the boundary values of ψ can be obtained as follows. If z = rη and we take lim inf_r%1 in (3.13), we have (3.14) |ψ(η)| ≤ kΓ_ψk_{Bil(Xc×Yc→C)}lim inf

r%1 ω_m(rη), a.e. η ∈ S.

Thus, if 0 < ˜ω_m(η) = lim inf

r%1 ω_m(rη) < ∞ a. e. η ∈ S, we obtain as a necessary condition

ψ

˜ ω_m

∞

≤ kΓψk_{Bil(Xc×Yc→C)}.

In order to obtain conditions on X and Y that assures that

(3.15) sup

z∈B

 |P_m(ψ)(z)|

ω_m(z)



≈ kΓ_ψk_Bil(X

c×Yc→C), observe that if f, g ∈ H( ¯B) and ψ ∈ L¹(S), then

(3.16) Γ_ψ(f, ¯g) = lim

r%1

Z

S

P_m(ψ)(rζ)

ωm(rζ) f (rζ)¯g(rζ)ω_m(rζ)dσ(ζ).

Therefore, if τ_c(r) = sup

kf_r¯g_r(ω_m)_rk_L¹_(dσ) kf kXkgkY

; f, g ∈ H( ¯B), f, g 6= 0

 , is a bounded function on [0, 1) then (3.15) holds.

If X_c and Y_c are dense in X and Y respectively, then the bilinear form Γ_ψ can be extended to a bilinear continuous form on X × Y preserving the norm. As usual, we also denote this extension by Γψ. However, in some important examples these conditions of density are not satisfied (as it happens in the case of H^∞, BM OA, B₀^∞). In these cases we can modify the above argument to also obtain conditions on X and Y such that

(3.17) sup

z∈B

 |P_m(ψ)(z)|

ω_m(z)



≈ kΓ_ψkBil(X×Y →C), is satisfied.

Assume that f, g ∈ H¹, and ψ ∈ L¹(S), then 

   Z

S

ψ(ζ)f (ζ)¯g(ζ)dσ(ζ)    

≤ Z

S

lim

r%1|P_m(ψ)(rζ)||f (rζ)¯g(rζ)|dσ(ζ).

We also assume that the function τ (r) = sup

kf_rg¯_r(ω_m)_rk_L¹_(dσ)

kf k_Xkgk_Y ; f ∈ X, g ∈ Y, f, g 6= 0

 , is bounded on [0, 1).

Then, taking a sequence {r_j} % 1 and applying Fatou’s Lemma, we have 

   Z

S

ψ(ζ)f (ζ)¯g(ζ)dσ(ζ)    

≤ sup

z∈B

 |P_m(ψ)(z)|

ω_m(z)

 sup

0≤r<1

Z

S

|f_r(ζ)||g_r(ζ)|(ω_m)_r(ζ)dσ(ζ)

≤ kτ k∞sup

z∈B

 |P_m(ψ)(z)|

ωm(z)



kf k_Xkgk_Y, Summarizing these conditions, we have

Theorem 3.1. Let m ≥ 0 and let X and Y be Banach spaces of holomorphic functions on B containing H( ¯B).

Then, sup

z

 P_m(ψ)(z) ω_m(z)



≤ kΓ_ψk_Bil(X

c×Y_c→C) ≤ kτ_ck∞sup

z∈B

 |P_m(ψ)(z)|

ω_m(z)

 .

If in addition X, Y ⊂ H¹, then

kΓ_ψkBil(X×Y →C)≤ kτ k∞sup

z∈B

 |P_m(ψ)(z)|

ω_m(z)

 . Now let us state conditions on X and Y such that

(3.18)

ψ

˜ ω_m

_∞

≤ kΓ_ψk_{Bil(Xc×Yc→C)}. is satisfied.

Assuming that 0 < ˜ωm(ζ) < ∞ a. e. ζ ∈ S, we have Γ_ψ(f, ¯g) .

ψ

˜ ω_m

_∞

Z

S

|f ||g|˜ω_mdσ.

Therefore,

Theorem 3.2. Let X and Y be Banach spaces of holomorphic functions on B containing H( ¯B).

If 0 < ˜ω_m(ζ) < ∞, a.e. ζ ∈ S, and there exists a constant C_X,Y such that kf gk_L¹_(˜ωm)≤ C_X,Ykf k_Xkgk_Y, then

ψ

˜ ω_m

_∞

≤ kΓψk_{Bil(Xc×Yc→C)} ≤ kΓψkBil(X×Y →C) ≤ CX,Y

ψ

˜ ω_m

_∞

. Theorem 3.2 will be used to obtain results on Toeplitz operators in weighted Hardy spaces with weights satisfying some additional conditions on S.

3.1. Toeplitz operators in some classical spaces. Since kP_m(ψ)k∞ ≈ kψk_∞, we have as immediately consequence of Theorem 3.1 the following result.

Corollary 3.3. Let X and Y be Banach spaces of holomorphic functions on B satisfying ω_m ≈ 1 a.e. on S, and kτ k∞ < ∞. Then kΓ_ψkBil(X×Y →C) ≈ kψk∞.

Corollary 3.4. Let 1 ≤ p ≤ ∞. Assume that X, Y satisfy B_n/p¹ 0 ⊂ X ⊂ H^p and B_n/p¹ ⊂ Y ⊂ H^p0 respectively. Then, Γ_ψ : X × Y → C is bounded if and only if ψ is in L^∞.

Proof. If l > 0, Proposition 1.4.10 in [14] gives that k(1 − w¯z)^−n−lk_L¹_(dσ) ≈ (1 − |z|²)^−l. Therefore, the norms of the function f_z(w) = (1 − w¯z)^−n−2m both in B_n/p¹ 0 and in H^p, and consequently in X, are equivalent to (1 −

|z|²)^−n/p0−m. Analogously, the norms of the function f_z(w) in B_n/p¹ and H^p0 and consequently in Y are equivalent to (1 − |z|²)^−n/p−m. From these estimates, we obtain ω_m≈ 1 a.e. on S.

Since kf_rg¯_r(ω_r)k_L¹_(dσ) . kfr¯g_rk_L¹_(dσ) . kf kH^pkgk_Hp0 . kf kXkgk_Y, we have that the function τ is bounded. Consequently, the result follows from

Corollary 3.3. 

Examples of spaces X and Y satisfying the conditions in the above corol- lary are the Hardy-Sobolev spaces H_s^p₀⁰ with 1 ≤ p₀ < p and s₀ − n/p₀ =

−n/p, or more generaly the scale of Besov spaces B_s^p₀^0,q0 and the scale of Triebel-Lizorkin spaces F_s^p0,q0

0 satisfying 1 ≤ p₀ ≤ p, 1 ≤ q₀ ≤ 2, and s₀− n/p₀ = −n/p (see H. Triebel’s book [18] and the references therein for a more complete list of embeddings between these spaces).

Corollary 3.5. Let X and Y be Banach spaces of holomorphic functions on B. Assume that ω_m(rζ) → 0 as r % 1, a.e. ζ ∈ S. Then the only bounded bilinear Toeplitz forms Γ_ψ are the ones corresponding to the trivial case where ψ = 0 a.e. ζ ∈ S.

Proof. By Theorem 3.1, |P_m(ψ)(rζ)| ≤ c_n,mkΓ_ψk_{L(Xc×Yc→C)}ω_m(rζ), and consequently P_m(ψ)(rζ) → 0 as r % 1, a.e. ζ ∈ S. Therefore, ψ(ζ) = 0 a.e.

ζ ∈ S. 

Corollary 3.6. Let 1 ≤ p < q ≤ ∞. Assume that X, Y satisfy B_n/p¹ 0 ⊂ X ⊂ H^p and B_n/q¹ ⊂ Y ⊂ H^q0 respectively. Then, Γ_ψ : X_c× Y_c→ C is not bounded except for the trivial case where ψ(ζ) = 0 a.e. ζ ∈ S.

Proof. The arguments used in the proof of Corollary 3.4, give that ω_m(rζ) ≈

(1 − r²)^n/p−n/q → 0 as r % 1. 

Corollary 3.4 permit us to obtain easily some well known characteriza- tions on the symbols of Toeplitz operators. Among them we have that if 1 < p < +∞, then T Symb(H^p → H^p) = L^∞. It also can be obtained some improvements of the classical results that we sumarize in the following theorem.

Theorem 3.7. Let X and Y two Banach spaces of holomorphic functions satisfying one of the following conditions:

(i) B_n¹ ⊂ X ⊂ H^∞ and BM OA ⊂ Y ⊂ B₀^∞.

(ii) B_n/p¹ 0 ⊂ X ⊂ H^p ⊂ Y ⊂ B_−n/p^∞ , for some 1 < p < ∞.

Then T Symb(X → Y ) = L^∞.

Proof. Assume that (i) is satisfied. In this case, we only need to show that T Symb(B_n¹ → B₀^∞) ⊂ L^∞⊂ T Symb(H^∞ → BM OA).

The second embedding is a consequence of the fact that the Cauchy pro- jection maps L^∞ into BM OA.

In order to prove the first embedding, we need to show that kψk∞ . kT_ψk_L(B¹

n→B^∞₀ ). And that is a consequence of the duality (B₀¹)⁰ = B₀^∞, of the Corollary 3.4 with p = ∞ and kT_ψk_L(Bn¹→B₀^∞) ≈ kΓ_ψk_Bil(B1

n×B¹₀→C). If X and Y satisfy (ii), then the result is a consequence of the fact that

T Symb(B_n/p^{1 0} → B^∞_−n/p) ⊂ L^∞ = T Symb(H^p → H^p),

where the first embedding is a consequence of (B_n/p¹ )⁰ = B_−n/p^∞ and Corollary

3.4. 

Remark 3.8. In the unit disk the equalities

T Symb(H^∞→ BM OA) = T Symb(H^∞→ B^∞₀ ) = L^∞ were proved in [9].

Examples of spaces X satisfying the hypothesis of Theorem 3.7 are for in- stance the ball algebra A, the Hardy-Sobolev space H_n¹ and the multiplicative algebras B_n/p^p ∩ H^∞. As Y we can consider all the scale of Triebel-Lizorkin spaces F₀^∞,q, 2 ≤ q ≤ ∞. We recall that BM OA = F₀^∞,2 and B₀^∞= F₀^∞,∞. 3.1.1. Pointwise multipliers. Let X and Y be Banach spaces of holo- morphic functions on B, both of them containing H( ¯B). We denote by M(X → Y ) the space of pointwise multipliers from X to Y , that is the space of holomorphic functions h on B such that the map M_h(f ) = hf is continuous from X to Y .

It is clear that if h ∈ H¹(S) and f ∈ H( ¯B) then M_h(f ) = T_h(f ). There- fore, the results on Toeplitz operators lead easily to results on pointwise multipliers.

We then have:

Theorem 3.9. Let 1 ≤ p ≤ ∞ and let X and Y Banach spaces satisfying B_n/p¹ 0 ⊂ X ⊂ H^p ⊂ Y ⊂ B_−n/p^∞ .

Then M(X → Y ) = H^∞ Proof. Observe that

H^∞ = M(H^p → H^p) ⊂ M(X → Y ) ⊂ M(B_n/p¹ 0 → B_−n/p^∞ ) = H^∞, where the last identity is a consequence of Theorem 3.7. 

It is also possible to obtain other characterizations of spaces of pointwise multipliers.

Theorem 3.10. Let X and Y Banach spaces of holomorphic functions on B, and let 1 ≤ p < ∞.

(i) If B_n¹ ⊂ X ⊂ H^∞ ∩ V M OA ⊂ Y ⊂ V M OA, then M(X → Y ) coincides with the multiplicative algebra H^∞∩ V M OA.

(ii) If B_n¹ ⊂ X ⊂ H^∞∩ b^∞₀ ⊂ Y ⊂ b^∞₀ , then M(X → Y ), coincides with the multiplicative algebra H^∞∩ b^∞₀ .

(b^∞₀ denotes the little Bloch space, that is the closure of H( ¯B) in B^∞₀ .) Proof. We recall that a holomorphic function f ∈ BM OA is in V M OA, if

limt→0

1 tⁿ

Z

|{z∈B; |1−z ¯ζ|<t}

|Rf (z)|²(1 − |z|²)dν(z) = 0 a.e. ζ ∈ S.

Therefore, it is clear that H^∞ ∩ V M OA is a multiplicative algebra, and (H^∞∩ V M OA) · X ⊂ H^∞∩ V M OA ⊂ Y , which proves that H^∞ ∩ V M OA ⊂ M(X → Y ).

In order to prove the converse, observe that M(X → Y ) ⊂ Y ⊂ V M OA, and that M(X → Y ) ⊂ M(X → V M OA) ⊂ M(X → BM OA) = H^∞, where the last equality is a consequence of Theorem 3.9 with p = ∞.

The same arguments and the fact that a holomorphic function f ∈ B₀^∞ is in b^∞₀ if lim

r%1(1 − r²)|Rf (rζ)| = 0 a.e. ζ ∈ S gives (ii).  3.2. Toeplitz operators on weighted Hardy spaces. Let us conclude this section with an application of Theorem 3.2 to the study of Toeplitz operators on weighted Hardy spaces.

We will consider weighted Hardy spaces for the class of admissible weights consisting of measurable functions θ on S such that

(i) θ(ζ) > 0 a.e. ζ ∈ S, (ii) θ, θ^−p0/p ∈ L¹(dσ).

For this class of weights, H^p(θ) will denote the closure of H( ¯B)|S in L^p(θ).

Observe that since θ^−p0/p ∈ L¹(dσ), then for each ϕ ∈ L^p(θdσ), Z

S

|ϕ(ζ)|dσ(ζ) ≤ kθ^−p0/pk^1/p_L1(dσ)⁰ kϕk_L^p_(θ).

Therefore, L^p(θ) ⊂ L¹(dσ), and if {f_j} ⊂ H( ¯B) and lim_jf_j = ϕ in L^p(θ), then {fj} converges in L¹ to a function f ∈ H¹, whose boundary values coincide with ϕ a.e. on S. Therefore, H^p(θ) is a subspace of H¹.

If (n + m)p − 2n ≥ 0, and f_z is the test function defined by f_z(ζ) = (1 − |z|²)^n/p0+m

(1 − ζ ¯z)^n+m , we have that

kfzk_H^p_(θ) =

Z

S

(1 − |z|²)^(n/p0+m)p

|1 − ζ ¯z|^(n+m)p θ(ζ)dσ(ζ)

^1/p

=

 1

cn,(n+m)p−2n

P_(n+m)p−2n(θ)(z)

1/p

,

which has radial limit θ^1/p c^1/pn,(n+m)p−2n

a.e. on S.

We then have:

Theorem 3.11. Let be 1 < p < ∞, ψ ∈ L¹(dσ) and θ0, θ1 admissible weights. Then,

kΓψk_Bil(Hp(θ

0)×H^p0(θ1)→C) '

ψ θ₀^1/pθ^1/p₁ ⁰

_∞

.

Proof. Observe that the above computations give that ˜ωn(θ) ≈ θ^1/p₀ θ₁^1/p0 a.e.

on S, and that, by H¨older’s Inequality, Z

S

|f ||g|˜ωdσ . kf kH^p(θ0)kgk_Hp0(θ1).

Therefore H^p(θ₀) and H^p0(θ₁) satisfy the conditions in Theorem 3.2, which

proves the result. 

The estimates on the norm of the bilinear form Γ_ψ on H^p(θ₀) × H^p0(θ₁) give estimates on the norm of the corresponding Toeplitz operator Tψ once we can identify the dual of H^p0(θ₁) with respect to the pairing (1.1). This is the case when θ₁ is in the Muckenhoupt class A_p. The next section is devoted to give the properties we will need on this class of weights and the associated weighted Hardy spaces.

4. Weighted Hardy and Besov spaces and Toeplitz operators The main goal of this section is to prove a weighted version of the embed- ding B_t^q ⊂ H_s^p, 1 ≤ q ≤ min{p, 2}, t − n/q = s − n/p. The proof will rely on properties of weights in the Muckenhoupt class A_p on S and weights in the class Bp on B.

4.1. A_p and B_p weights. Given a non negative weight θ ∈ L¹(dσ) and W a measurable set in S, let θ(W ) =R

Wθdσ. For z = rζ, ζ ∈ S, 0 < r < 1, we consider the average function on B associated to θ defined by Θ(z) = θ(Iz)

|I_z| , where I_z = U_ζ,1−r² = {η ∈ S; |1 − η ¯ζ| < 1 − r²}.

The Muckenhoupt class A_p on S, 1 < p < ∞, consists of the non-negative weights θ ∈ L¹(dσ) satisfying

(4.19) A_p(θ) = sup

z∈B

(Θ(z))^1/p(Θ⁰(z))^1/p0 < ∞

where θ⁰ = θ^−p0/p and Θ⁰(z) = θ⁰(I_z)

|I_z| . Observe that θ ∈ A_p, if and only if, θ⁰ ∈ Ap⁰.

If p = 1, the class A₁ on S is the set of non-negative weights θ ∈ L¹(dσ) satisfying

(4.20) A₁(θ) = sup

ζ∈S

M_H−L(θ)(ζ) θ(ζ) < ∞.

An immediate consequence of H¨older’s Inequality is that if 1 < q < p, then A₁ ⊂ A_q ⊂ A_p, that A_p(θ) ≥ 1, and that if 0 ≤ λ ≤ 1, θ^λ is in A_q, with q = 1 + λ(p − 1) ≤ p (see [16] pag. 218). In particular, for any θ ∈ A_p and 0 < λ ≤ 1, the weight θ^λ is also in A_p.

Weights in the class A_pappear in the study of the boundedness of singular operators on weighted spaces. In particular, we have that if 1 < p < ∞ and θ ∈ A_p, then the Cauchy projection maps L^p(θ) in H^p(θ), and as a consequence of this fact, the dual of H^p(θ) can be identified with H^p0(θ⁰) with the pairing given by hf, ¯gi_S (see [10] p.334).

The following characterization of the class A_p (see [16] p.195), will be considered in the following sections.

θ ∈ A_p, if and only if for any measurable set U ⊂ S and ϕ ≥ 0 in L^p(dσ), (4.21)

 1

|U | Z

U

ϕ dσ

p

≤ A_p(θ) θ(U )

Z

U

ϕ^pθdσ.

From the above characterization we deduce that if V ⊂ U are measurable sets on S, and ϕ is the characteristic function of V , then

 |V |

|U |

p

≤ A_p(θ)θ(V ) θ(U ).

Therefore, for any θ ∈ A_p, the measure θdσ is a doubling measure, in the sense that there exists a constant DS such that

θ(U_ζ,2t) ≤ D_Sθ(U_ζ,t), for all ζ ∈ S, 0 < t < 2.