Growth Spaces on Circular Domains Taking Values in a Banach Lattice, Embeddings and Composition Operators

Abstract

We introduce the space of holomorphic growth spaces with values in a Banach lattice. We provide norm and essential norm estimates of the embedding operator, and we completely characterize the bounded and compact embeddings of the growth spaces using vector-valued Carleson measures. As an application, we prove a characterization of weighted composition operators.

1. Introduction

In this paper, we introduce vector-valued growth spaces taking values on a Banach space. The definition of these spaces makes use of the notion of the so-called Gamma order which is nothing but a linear transformation of a Banach space $\mathcal{E}$ into a conditionally complete vector lattice $\mathcal{Y}$. If $\mathsf{\Omega }$ is a domain in ${\mathbb{C}}^n$ and $\omega$ is a continuous function from $\mathsf{\Omega }$ into the positive cone ${\mathcal{Y}}^{+}$ in $\mathcal{Y}$, then $A^{\omega }\left(\mathcal{E}\right)$ is the class of all holomorphic maps f from $\mathsf{\Omega }$ into $\mathcal{E}$ for which

$$\begin{array}{c}\hfill \mathsf{\Gamma }\left(\right|f\left(z\right){|}_{\mathsf{\Gamma }})\le a\left(f\right)\omega \left(z\right),z\in \mathsf{\Omega },\end{array}$$

(1)

for some $a\left(f\right)\in {\mathcal{Y}}^{+}$. The reader is referred to Section 2.2 for the unexplained notation.

The usual approach to define the $L^p$-norm or “sup”-norm of a vector-valued function f is to take the corresponding norm of the scalar function $\parallel f\parallel$. Our definition differs in (1) in that we make use of the order structure, which is assumed to hold on the given Banach space via the transformation $\mathsf{\Gamma }$. A natural example is when the Banach space $\mathcal{E}$ has an unconditional Schauder basis (Section 2.3). It turns out that in this case there exists a natural conditionally complete vector lattice $\mathcal{Y}$ and a bijective linear transformation $\mathsf{\Gamma }$ from $\mathcal{E}$ onto $\mathcal{Y}$; hence, one can consider the order structure on $\mathcal{E}$ induced by $\mathsf{\Gamma }$.

The central focus of this paper is on the uniform embedding of the growth space $A^{\omega }\left(\mathcal{E}\right)$ into the $L^p$ space of a family $\{{\mu }_{\alpha }:\alpha \in \mathcal{J}\}$ of vector-valued measures; that is, there is the existence of a constant $C>0$ independent of f so that

$$\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\mathsf{\Omega }}{\left|f\left(z\right)\right|}_{\mathsf{\Gamma }}^{p}d{\mu }_{\alpha }\left(z\right)∥}_{\mathcal{E}}^{1/p}\le C{\parallel f\parallel }_{\omega }$$

for every $f\in A^{\omega }\left(\mathcal{E}\right)$. Now, we consider only a bounded, circular and strictly convex domain $\mathsf{\Omega }$ with at least a $C^2$ boundary in ${\mathbb{C}}^n$. When $\mathcal{J}$ consists of one element only, such a measure is called p-Carleson for the space $A^{\omega }\left(\mathcal{E}\right)$. The earliest examples of such measures appear in the seminal work of L. Carleson in [1,2], where p-Carleson measures for the Hardy space $H^p$ were characterized. They were used in connection with interpolation and corona problems. There is a vast list of different results about p-Carleson measures for (scalar-valued) Hardy, Bergman-type, Dirichlet-type and Besov spaces.

Scalar-valued growth spaces and their embedding properties were investigated in the work of E. Abakumov and E. Doubtsov ([3,4,5]). In [5], the author completely characterizes the q-Carleson measures for the scalar-valued $A^{\omega }$ and provides a characterization for the bounded or compact composition operators on these spaces when the weight $\omega$ is of the form

$$\begin{array}{c}\hfill \omega \left(t\right)={\displaystyle \frac{1}{{(1-t)}^{\beta }}}\end{array}$$

for some $\beta >0$, or

$$\begin{array}{c}\hfill \omega \left(t\right)=log{\displaystyle \frac{e}{1-t}}\end{array}$$

for $0\le t<1$. The paper [3] is mainly focused on the case when the domain $\mathsf{\Omega }$ is the open unit ball in ${\mathbb{C}}^n$. The authors prove a characterization of a bounded or compact composition and Volterra operators on growth spaces in the setting of the open unit ball. The general case when $\mathsf{\Omega }$ is a bounded circular domain is treated in [4]. The key point in all these works is a two-sided estimate for functions in the growth space. This estimate together with a characterization of composition and multiplication operators, Volterra-type operators and extended Cesàro operators are given in [4] for when the domain is bounded, circular and strictly convex.

Our definitions and results generalize the previous work on (scalar-valued) growth spaces in two different ways: We consider vector-valued holomorphic functions. Secondly, the problem is to characterize the embedding of $A^{\omega }\left(\mathcal{E}\right)$ into a family of $L^p$ spaces as explained above. Our compactness characterization is new even in the settings of scalar-valued growth spaces. We also provide an essential norm estimate for the embedding operator (see Section 4).

Let $H(\mathsf{\Omega },\mathcal{E})$ denote the class of all holomorphic maps from $\mathsf{\Omega }$ into $\mathcal{E}$. For a scalar holomorphic function $\phi$ on $\mathsf{\Omega }$ with $\phi \left(\mathsf{\Omega }\right)\subset \mathsf{\Omega }$ and a holomorphic map $\psi$ on $\mathsf{\Omega }$ into the sequence space ${\ell }^{\infty }$ of bounded sequences, the weighted composition operator $C_{\phi ,\psi }$ is the operator that maps $f\in H(\mathsf{\Omega },\mathcal{E})$ into the function $\psi \left(z\right)f\left(\phi \right(z\left)\right)$ for all $z\in \mathsf{\Omega }$. Hence, $C_{\phi ,\psi }$ can be thought of as the composition followed by a multiplication. When $\psi \left(z\right)=\left\{1\right\}$ is the constant sequence, we have $C_{\phi ,\psi }=C_{\phi }$: that is, the usual composition operator. Composition operators on function spaces are a vast subject. We refer the reader to [6,7] for the general theory.

The structure of the paper is as follows. In Section 2.1, we introduce the Gamma order and show that the $\mathsf{\Gamma }$ transform induces a lattice structure on $\mathcal{E}$. The growth space $A^{\omega }\left(\mathcal{E}\right)$ is defined in Section 2.2, and we prove that this is a Banach space. The specific case when $\mathcal{E}$ has a Schauder basis is treated in Section 2.3. In Section 3, we treat (vector-valued) Carleson measures. We provide essential norm estimates on the embedding operator, and we completely characterize bounded and compact Carleson embeddings of growth spaces in Section 3 and Section 4. As an application, we provide the necessary and sufficient criteria for the bounded or compact generalized composition operators (Section 5).

2. Preliminaries

2.1. Gamma Order

Let $\mathcal{E}$ be a complex Banach space, and let $\mathcal{H}$ be a complex separable Hilbert space. We denote the class of all bounded linear operators on $\mathcal{H}$ by $B\left(\mathcal{H}\right)$. The subspaces of self-adjoint and compact operators in $B\left(\mathcal{H}\right)$ are denoted by $S\left(\mathcal{H}\right)$ and $K\left(\mathcal{H}\right)$, respectively. For self-adjoint operators, $A\ge B$ means the operator $B-A$ is positive semi-definite. The pre-order defined on $S\left(\mathcal{H}\right)$ this way is called the usual order. Let $\mathcal{Y}$ be a real vector subspace (not necessarily closed) of $S\left(\mathcal{H}\right)$ with the following properties:

($𝒴$1)

$AB=BA\in \mathcal{Y}$ for every $A\in \mathcal{Y}$ and $B\in \mathcal{Y}$.

($𝒴$2)

$\mathcal{Y}$ together with the usual order is a conditionally complete vector lattice: that is, each non-empty upper-bounded subset has a least one upper bound in $\mathcal{Y}$ (equivalently, each non-empty lower-bounded subset has its greatest lower bound in $\mathcal{Y}$).

($𝒴$3)

$(\mathcal{Y},\parallel ·{\parallel }_{\mathcal{Y}})$ is a real Banach space.

($𝒴$4)

The norm ${\parallel ·\parallel }_{\mathcal{Y}}$ is a lattice norm. This means if A, $B\in \mathcal{Y}$ and $0\le \left|A\right|\le \left|B\right|$, then ${\parallel A\parallel }_{\mathcal{Y}}\le {\parallel B\parallel }_{\mathcal{Y}}$.

($𝒴$5)

The inclusion map from $\mathcal{Y}$ into $B\left(\mathcal{H}\right)$ is continuous.

($𝒴$6)

There exists a bijective, real linear, norm-continuous map $\mathsf{\Gamma }:\mathcal{E}\to \mathcal{Y}$.

Let us consider the relation ⪯ on $\mathcal{X}$ given by

$$x⪯y⟺\mathsf{\Gamma }\left(x\right)\le \mathsf{\Gamma }\left(y\right),x,y\in \mathcal{E}.$$

We refer to [8] for basic notions of vector lattices. We use ⋎ (∨) and ⋏ (∧) for the “sup” and “inf” operations in $\mathcal{E}$ (respectively in $\mathcal{Y}$). $\mathcal{E}$ inherits a similar structure as $\mathcal{Y}$, and we prove this first.

Theorem 1. 

If $\mathcal{E}$ and $\mathcal{Y}$ satisfy ($\mathcal{Y}2$) and ($\mathcal{Y}6$), then $(\mathcal{E},⪯)$ is a conditionally complete vector lattice.

Proof. 

Clearly, ⪯ is reflexive and transitive. It is symmetric, since $\mathsf{\Gamma }$ is injective. Hence, ⪯ is a partial order on $\mathcal{E}$. For a, b, $c\in \mathcal{E}$ and a scalar $\lambda \ge 0$ with $a⪯b$, we have $\mathsf{\Gamma }(a+c)=\mathsf{\Gamma }\left(a\right)+\mathsf{\Gamma }\left(c\right)\le \mathsf{\Gamma }\left(b\right)+\mathsf{\Gamma }\left(c\right)=\mathsf{\Gamma }(b+c)$: that is, $a+c⪯b+c$. Likewise, $\mathsf{\Gamma }\left(\lambda a\right)=\lambda \mathsf{\Gamma }\left(a\right)\le \lambda \mathsf{\Gamma }\left(b\right)=\mathsf{\Gamma }\left(\lambda b\right)$: that is, $\lambda a⪯\lambda b$. Let $\left\{x_{\alpha }\right\}$ be a family in $\mathcal{E}$ and let $b\in \mathcal{E}$ so that $x_{\alpha }⪯b$ for each $\alpha$. Set $A={\vee }_{\alpha }\mathsf{\Gamma }\left(x_{\alpha }\right)$. Since $\mathcal{Y}$ satisfies ($\mathcal{Y}2$), A belongs to $\mathcal{Y}$. Hence, $a={\mathsf{\Gamma }}^{-1}\left(A\right)$ is in $\mathcal{E}$. Then, $A=\mathsf{\Gamma }\left(a\right)\le \mathsf{\Gamma }\left(b\right)$: that is, $a⪯b$. By definition, a is an upper bound for $\left\{x_{\alpha }\right\}$. Hence, $a=⋎x_{\alpha }\in \mathcal{E}$. Similarly, one can show that the infimum of a lower-bounded family in $\mathcal{E}$ exists in $\mathcal{E}$. □

For $x\in \mathcal{E}$, let $x^{+}=x⋎0$, $x^{-}=(-x)⋎0$ and $\left|x\right|=x⋎(-x)$. We define the $\mathsf{\Gamma }$-modulus of x as

$$\begin{array}{c}\hfill {\left|x\right|}_{\mathsf{\Gamma }}=⋎\{e^{i\theta }x:0\le \theta <2\pi \}.\end{array}$$

(2)

Then $0⪯\left|x\right|⪯{\left|x\right|}_{\mathsf{\Gamma }}$.

Remark 1. 

1. For $x\in \mathcal{E}$, ${\mathsf{\Gamma }\left(\right|x|}_{\mathsf{\Gamma }})\ge \mathsf{\Gamma }(\left|x\right|)=|\mathsf{\Gamma }\left(x\right)|\in {\mathcal{Y}}^{+}$, where ${\mathcal{Y}}^{+}$ denotes the positive semi-definite elements in $\mathcal{Y}$.

2. $AB\in {\mathcal{Y}}^{+}$ whenever $A\in {\mathcal{Y}}^{+}$, $B\in {\mathcal{Y}}^{+}$.

2.2. Growth Spaces

Let $\mathsf{\Omega }\subset {\mathbb{C}}^d$ be a domain. Let $H\left(\mathsf{\Omega }\right)$ denote the space of holomorphic functions in $\mathsf{\Omega }$. Denote the class of holomorphic functions from $\mathsf{\Omega }$ to $\mathcal{E}$ by $H(\mathsf{\Omega },\mathcal{E})$. For the moment, let $\omega$ be a continuous mapping of $\mathsf{\Omega }$ into ${\mathcal{Y}}^{+}$. Such a function $\omega$ is called a weight. We denote by $A^{\omega }\left(\mathcal{E}\right)$ the class of all functions $f\in H(\mathsf{\Omega },\mathcal{E})$ for which

$$\begin{array}{c}\hfill \mathsf{\Gamma }\left(\right|f\left(z\right){|}_{\mathsf{\Gamma }})\le a\left(f\right)\omega \left(z\right)\end{array}$$

(3)

for every $z\in \mathsf{\Omega }$ for some $a\left(f\right)\in {\mathcal{Y}}^{+}$. Let $n\left(f\right)=\wedge a\left(f\right)$, where the infimum is taken over all possible $a\left(f\right)$ in ${\mathcal{Y}}^{+}$ satisfying (3). Set ${\parallel f\parallel }_{\omega }={\parallel n\left(f\right)\parallel }_{\mathcal{Y}}$.

Theorem 2. 

Suppose that $\mathcal{E}$ and $\mathcal{Y}$ satisfy all five properties ($\mathcal{Y}2$)–($\mathcal{Y}6$). Then the space $A^{\omega }\left(\mathcal{E}\right)$ is Banach with the norm ${\parallel ·\parallel }_{\omega }$.

Proof. 

Clearly, the norm is well-defined. Since $\mathsf{\Gamma }$ is injective, $n\left(f\right)=0$ implies that $f\left(z\right)$ is the constant zero. Let $\lambda \in \mathbb{C}$, with $\left|\lambda \right|=1$. Then ${\left|\lambda f\right|}_{\mathsf{\Gamma }}={\left|f\right|}_{\mathsf{\Gamma }}$, $n\left(\lambda f\right)=n\left(f\right)$, and ${\parallel \lambda f\parallel }_{\omega }={\parallel f\parallel }_{\omega }$. For a non-zero scalar $\lambda \in \mathbb{C}$, ${\mathsf{\Gamma }\left(\right|\lambda f|}_{\mathsf{\Gamma }}{)=|\lambda \left|\mathsf{\Gamma }\right(\left|f\right|}_{\mathsf{\Gamma }})$, which means that $n\left(\lambda f\right)=\left|\lambda \right|n\left(f\right)$. Hence, ${\parallel \lambda f\parallel }_{\omega }={\left|\lambda \right|\parallel f\parallel }_{\omega }$. For each $0\le \theta <2\pi$, f, $g\in A^{\omega }\left(\mathcal{E}\right)$ and $z\in \mathsf{\Omega }$, $\mathsf{\Gamma }\left(e^{i\theta }(f\left(z\right)+g\left(z\right))\right)=\mathsf{\Gamma }\left(e^{i\theta }f\left(z\right)\right)+\mathsf{\Gamma }\left(e^{i\theta }g\left(z\right)\right)\le (n\left(f\right)+n\left(g\right))\omega \left(z\right)$. Then $\mathsf{\Gamma }\left(\right|f\left(z\right)+g\left(z\right){|}_{\mathsf{\Gamma }})\le (n\left(f\right)+n\left(g\right))\omega \left(z\right)$, and this means that $n(f+g)\le n\left(f\right)+n\left(g\right)$. Since $n(·)$ has values in the positive semi-definite cone of $\mathcal{Y}$, we get that ${\parallel f+g\parallel }_{\omega }\le {\parallel f\parallel }_{\omega }+{\parallel g\parallel }_{\omega }$. Here we use the property ($\mathcal{Y}4$). Let $\left\{f_k\right\}$ be a Cauchy sequence in $A^{\omega }\left(\mathcal{E}\right)$. For each $z\in \mathsf{\Omega }$,

$$\begin{array}{c}\hfill 0\le \mathsf{\Gamma }\left(\right|f_k\left(z\right)-f_m\left(z\right){|}_{\mathsf{\Gamma }})\le n(f_k-f_m)\omega \left(z\right).\end{array}$$

Now a consequence of the open mapping theorem implies that ${\mathsf{\Gamma }}^{-1}$ is also bounded, and using ($\mathcal{Y}4$), for each compact $K\subset \mathsf{\Omega }$,

$$\begin{array}{ccc}\hfill 0& \le & C\underset{z\in K}{sup}\parallel f_k\left(z\right)-f_m{\left(z\right)\parallel }_{\mathcal{E}}\le \underset{z\in K}{sup}\parallel \mathsf{\Gamma }\left(\right|f_k\left(z\right)-f_m{\left(z\right)|}_{\mathsf{\Gamma }}{)\parallel }_{\mathcal{Y}}\hfill \\ & \le & \parallel f_k-f_m{\parallel }_{\omega }\underset{z\in K}{sup}{\parallel \omega \left(z\right)\parallel }_{B\left(\mathcal{H}\right)}.\hfill \end{array}$$

Then $f_k\left(z\right)$ converges uniformly on compact sets to a function $f\in H(\mathsf{\Omega },\mathcal{E})$. For $\epsilon >0$, there exists $N_{\epsilon }\ge 1$ so that k, $m\ge N_{\epsilon }$ implies

$$\begin{array}{c}\hfill 0\le \mathsf{\Gamma }\left(\right|f_k\left(z\right)-f_m\left(z\right){|}_{\mathsf{\Gamma }})\le \epsilon \omega \left(z\right)\end{array}$$

for every $z\in \mathsf{\Omega }$. Letting $k\to \infty$, we have

$$\begin{array}{c}\hfill \mathsf{\Gamma }\left(\right|f\left(z\right)-f_m\left(z\right){|}_{\mathsf{\Gamma }})\le \epsilon \omega \left(z\right),z\in \mathsf{\Omega },\end{array}$$

for each $m\ge N_{\epsilon }$. Fixing $m\ge N_{\epsilon }$,

$$\begin{array}{c}\hfill {\mathsf{\Gamma }\left(\right|f\left(z\right)|}_{\mathsf{\Gamma }})\le \mathsf{\Gamma }(|f\left(z\right)-f_m{\left(z\right)|}_{\mathsf{\Gamma }})+\mathsf{\Gamma }(|f_m\left(z\right){|}_{\mathsf{\Gamma }})\le (\epsilon +n\left(f_m\right))\omega \left(z\right),z\in \mathsf{\Omega }.\end{array}$$

Hence, $f\in A^{\omega }\left(\mathcal{E}\right)$, and $\left\{f_k\right\}$ converges to f. □

An estimate observed in the proof of Theorem 2 deserves separate attention.

Proposition 1. 

For a compact $K\subset \mathsf{\Omega }$,

$$\begin{array}{c}\hfill \underset{z\in K}{max}{\parallel f\left(z\right)\parallel }_{\mathcal{E}}\le \underset{z\in K}{max}{\parallel \omega \left(z\right)\parallel }_{B\left(\mathcal{H}\right)}{\parallel f\parallel }_{\omega }\end{array}$$

for every $f\in A^{\omega }\left(\mathcal{E}\right)$.

In particular, the point evaluation map taking f to $f\left(z\right)$, $z\in \mathsf{\Omega }$, is continuous from $A^{\omega }\left(\mathcal{E}\right)$ to $\mathcal{E}$.

2.3. Schauder Basis

Let $\mathcal{E}$ be a complex Banach space with an unconditional Schauder basis $\{e_j^{\prime }:j\in \mathbb{N}\}$: that is, for each $x\in \mathcal{E}$, there exist uniquely determined ${\zeta }_j\in \mathbb{C}$ so that

$$\begin{array}{c}\hfill x=\sum _{j=1}^{\infty }{\zeta }_j{e}_j^{\prime },\end{array}$$

and the sum is independent of permutations of the set $\mathbb{N}$ of positive integers.

For ${\zeta }_j={\alpha }_j+i{\beta }_j$, where ${\alpha }_j\in \mathbb{R}$ and ${\beta }_j\in \mathbb{R}$, and a vector $x\in \mathcal{E}$,

$$\begin{array}{c}\hfill x=\sum _{j=1}^{\infty }({\alpha }_j+i{\beta }_j)e_j^{\prime }=\sum _{j=1}^{\infty }[{\alpha }_j{e}_j^{\prime }+{\beta }_j\left(i{e}_j^{\prime }\right)]=\sum _{k=1}^{\infty }{c}_k{e}_k,\end{array}$$

where $c_k\in \mathbb{R}$ in the last equality; we have rearranged the basis vectors $e_j^{\prime }$ and $i{e}_j^{\prime }$ so that $e_k=e_j^{\prime }$ if $k=2j-1$ and $e_k=i{e}_j^{\prime }$ if $k=2j$. That is, $\{e_j:j\in \mathbb{N}\}$ is a Schauder basis for the real space ${\mathcal{E}}_{\mathbb{R}}$, which is nothing but the space $\mathcal{E}$ when thought of as a real Banach space. One can normalize the basis elements, and we will do so, meaning that we will assume that $\parallel e_j\parallel =1$ for each j.

Let ${\ell }^2$ be the sequence space of square summable complex sequences. Let $\mathcal{D}$ be the class of bounded, self-adjoint diagonal operators on ${\ell }^2$. If $E_j$ denotes the diagonal operator that has 1 in the j-th place and all other entries equal to 0, then

$$\begin{array}{c}\hfill \mathcal{D}=\left\{\sum _{j=1}^{\infty }{c}_j{E}_j:c_j\in \mathbb{R},\underset{j}{sup}\left|c_j\right|<\infty \right\},{\mathcal{D}}^{+}=\left\{\sum _{j=1}^{\infty }{c}_j{E}_j\in \mathcal{D}:c_j\ge 0\right\},\end{array}$$

and

$$\begin{array}{c}\hfill \mathcal{D}\cap K\left({\ell }^2\right)=\left\{\sum _{j=1}^{\infty }{c}_j{E}_j:\underset{j\to \infty }{lim}{c}_j=0\right\}.\end{array}$$

Let $\mathsf{\Gamma }:\mathcal{E}\to \mathcal{D}\cap K\left({\ell }^2\right)$ be the linear map given by

$$\begin{array}{c}\hfill \mathsf{\Gamma }\left(x\right)=\sum _{j=1}^{\infty }{c}_j{E}_j,x=\sum _{j=1}^{\infty }{c}_j{e}_j\in \mathcal{E}.\end{array}$$

Theorem 3. 

Let $\mathcal{E}$ be a real Banach space with a Schauder basis $\left\{e_j\right\}$. The map Γ from $\mathcal{E}$ into $B\left({\ell }^2\right)$ is well-defined, real linear, norm-continuous and injective. Moreover, $\mathsf{\Gamma }\left(\mathcal{E}\right)$ is a conditionally complete lattice with the usual order if $\left\{e_j\right\}$ is unconditional.

Proof. 

It is well-known that for a vector $x={\sum }_{j=1}^{\infty }{c}_j{e}_j\in \mathcal{E}$, $c_j\to 0$ as $j\to \infty$. In fact, if $P_j$ denotes the projection map from $\mathcal{E}$ onto ${\mathcal{E}}_j=span\{e_1,\dots ,e_j\}$, there exists a constant $C>0$ so that $\parallel P_j\parallel \le C$ for every j. This observation gives that

$$\begin{array}{c}\hfill |c_j|=\parallel c_j{e}_j\parallel =\parallel P_j\left(x\right)-P_{j+1}\left(x\right)\parallel \to 0\end{array}$$

as $j\to \infty$ and ${\parallel \mathsf{\Gamma }\left(x\right)\parallel }_{B\left({\ell }^2\right)}={sup}_j|c_j|\le 2C{\parallel x\parallel }_{\mathcal{E}}$. Hence, $\mathsf{\Gamma }$ is a well-defined, continuous, clearly linear and injective map from $\mathcal{E}$ into $\mathcal{D}\cap K\left({\ell }^2\right)$.

Let $\mathcal{S}$ be a lower-bounded subset of $\mathsf{\Gamma }\left(\mathcal{E}\right)$; say $T\ge A$ for every $T\in \mathcal{S}$, where $A\in \mathsf{\Gamma }\left(\mathcal{E}\right)$. Replacing $\mathcal{S}$ by $\mathcal{S}-A$, we may assume that $T\ge 0$ for every $T\in \mathcal{S}$. Let $L=\wedge \{T:T\in \mathcal{S}\}$ and $T={\sum }_{j=1}^{\infty }{c}_j{E}_j\in \mathcal{S}$. Since $L={\sum }_{j=1}^{\infty }{s}_j{E}_j\le T$, $0\le s_j\le c_j$ for every j. If $\left\{e_j\right\}$ is unconditional, $l={\sum }_{j=1}^{\infty }{s}_j{e}_j\in \mathcal{E}$. Then $L=\mathsf{\Gamma }\left(l\right)\in \mathsf{\Gamma }\left(\mathcal{E}\right)$, and L is the greatest lower bound of $\mathcal{S}$ in $\mathsf{\Gamma }\left(\mathcal{E}\right)$. This argument shows that $\mathsf{\Gamma }\left(\mathcal{E}\right)$ is a conditionally complete lattice with the usual order. □

If $\mathsf{\Gamma }\left(\mathcal{E}\right)$ is given the norm induced from $\mathcal{E}$: that is, ${\parallel \mathsf{\Gamma }\left(x\right)\parallel }_{\mathsf{\Gamma }\left(\mathcal{E}\right)}={\parallel x\parallel }_{\mathcal{E}}$, we will denote that by $\mathcal{Y}$. Then $\mathsf{\Gamma }$ is obviously an isometric isomorphism of $\mathcal{E}$ onto $\mathcal{Y}$, and $\mathsf{\Gamma }\left({\mathcal{E}}^{+}\right)={\mathcal{Y}}^{+}$. If $\mathcal{E}$ has an unconditional basis, then $\mathcal{Y}$ satisfies the properties $\left(\mathcal{Y}1\right)-\left(\mathcal{Y}3\right)$ and $\left(\mathcal{Y}5\right)-\left(\mathcal{Y}6\right)$. If the norm on $\mathcal{E}$ is a lattice norm—that is, if $\mathcal{E}$ is a Banach lattice—then so is $\mathcal{Y}$. That is, $\mathcal{Y}$ satisfies $\left(\mathcal{Y}4\right)$.

2.4. Carleson Measures

Let $\mathsf{\Omega }$ be a domain in ${\mathbb{C}}^n$. The class of Borel subsets of $\mathsf{\Omega }$ is denoted by $\Sigma$. If $\mathcal{E}$ is a Banach space, $U(\mathsf{\Omega },\mathcal{E})$ denotes the class of bounded, Borel-measurable functions from $\mathsf{\Omega }$ into $\mathcal{E}$. The class of Borel measures on $\mathsf{\Omega }$ with values in $\mathcal{E}$ with bounded variation is denoted by $M(\mathsf{\Omega },\mathcal{E})$. We will use the Riesz representation theorem for vector-valued functions in the form rephrased below (cf. [9], pp. 145–152).

Proposition 2. 

Let $\mathcal{E}$ and $\mathcal{F}$ be Banach spaces, and let $L:U(\mathsf{\Omega },\mathcal{E})\to \mathcal{F}$ be a dominated linear operation such that for every scalar-valued decreasing sequence $\left({\phi }_j\right)$ of bounded Borel functions converging pointwise to zero, $L\left({\phi }_j\right)\to 0$. Then there exists a unique measure $\mu \in M(\mathsf{\Omega },B(\mathcal{E},\mathcal{F}\left)\right)$ so that

$$\begin{array}{c}\hfill L\left(f\right)={\int }_{\mathsf{\Omega }}fd\mu \end{array}$$

for every $f\in U(\mathsf{\Omega },\mathcal{E})$.

Let $\mathcal{E}$ be a Banach space with a Schauder basis, and let $\mathsf{\Omega }$ be a topological space. We say that a Borel measure on $\mathsf{\Omega }$ taking values in $B\left(\mathcal{E}\right)$ is positive if for every $\mu$-integrable $\phi :\mathsf{\Omega }\to \mathcal{E}$ with values in ${\mathcal{E}}^{+}$, the integral ${\int }_{\mathsf{\Omega }}\phi d\mu$ belongs to ${\mathcal{E}}^{+}$. The class of positive Borel measures on $\mathsf{\Omega }$ with bounded variation and taking values in $B\left(\mathcal{E}\right)$ is denoted by $M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$.

If $\mu \in M(\mathsf{\Omega },B(\mathcal{E}\left)\right)$, let L be the continuous linear map on $U(\mathsf{\Omega },\mathcal{Y})=\mathsf{\Gamma }\left(U\right(\mathsf{\Omega },\mathcal{E}\left)\right)$ given by

$$L\left(\mathsf{\Gamma }\left(u\right)\right)=\mathsf{\Gamma }\left({\int }_{\mathsf{\Omega }}ud\mu \right)\in \mathcal{Y}$$

(4)

for any $u\in U(\mathsf{\Omega },\mathcal{E})$. As a consequence of Proposition 2, there exists a unique measure $\tilde{\mu }$ on $\mathsf{\Omega }$ with values in $B\left(\mathcal{Y}\right)$ so that

$$L\left(\mathsf{\Gamma }\left(u\right)\right)={\int }_{\mathsf{\Omega }}\mathsf{\Gamma }\left(u\right)d\tilde{\mu }.$$

If $\mu$ is positive, then so is $\tilde{\mu }$. We use this association of the measure $\mu$ to the measure $\tilde{\mu }$ throughout.

Let $\mathcal{E}$ be a Banach space with an unconditional basis. For $p\ge 1$, we set ${\left|x\right|}^p={\mathsf{\Gamma }}^{-1}{\left(\right(\mathsf{\Gamma }\left(\right|x\left|\right))}^p)$. Note that ${\left|x\right|}^p\in \mathcal{E}$ when $p\ge 1$. A measure $\mu \in M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$ is said to be a p-Carleson measure for $A^{\omega }\left(\mathcal{E}\right)$ if

$$\begin{array}{c}\hfill {\left[\mathsf{\Gamma }\left({\int }_{\mathsf{\Omega }}{\left|f\left(z\right)\right|}_{\mathsf{\Gamma }}^{p}d\mu \left(z\right)\right)\right]}^{1/p}\le a\left(\mu \right)n\left(f\right)\end{array}$$

for every $f\in A^{\omega }\left(\mathcal{E}\right)$, where $a\left(\mu \right)\in {\mathcal{Y}}^{+}$ is independent of f. In the seminal work of Carleson ([1,2]), Carleson measures are defined to be used for the solution of the corona problem. The concept has found many applications, and it is still an active area of research.

3. Carleson Embeddings

3.1. Scalar Weights

Let $\mathsf{\Omega }\subset {\mathbb{C}}^n$ be a bounded, circular and strictly convex domain with at least a $C^2$ boundary. We denote the normalized Lebesgue measure on $\mathsf{\Omega }$ by $dV$ so that $V\left(\mathsf{\Omega }\right)=1$. Given $z\in \mathsf{\Omega }$, let $r_{\mathsf{\Omega }}\left(z\right)=inf\{r>0:z/r\in \mathsf{\Omega }\}$. It is clear that $r_{\mathsf{\Omega }}\left(z\right)<1$ for $z\in \mathsf{\Omega }$. When $\mathsf{\Omega }=\mathbb{B}$, the open unit ball is in ${\mathbb{C}}^n$; then $r_{\mathsf{\Omega }}\left(z\right)=\left|z\right|$. A scalar weight is a function $\delta :[0,1)\to (0,\infty )$ so that $\delta$ is non-decreasing, continuous and unbounded. We extend $\delta$ to $\mathsf{\Omega }$ by $\delta \left(r_{\mathsf{\Omega }}\left(z\right)\right)$, $z\in \mathsf{\Omega }$. The growth space $A^{\delta }$ consists of those functions $f\in H\left(\mathsf{\Omega }\right)$ for which

$$\begin{array}{c}\hfill {\parallel f\parallel }_{\delta }=\underset{z\in \mathsf{\Omega }}{sup}{\displaystyle \frac{\left|f\right(z\left)\right|}{\delta \left(r_{\mathsf{\Omega }}\left(z\right)\right)}}<\infty .\end{array}$$

Given functions u, $v:S\to (0,\infty )$ on a set S, we write $u\left(a\right)\approx v\left(a\right)$ for every $a\in S$ if there exist constants $C_1>0$, $C_2>0$ so that $C_{1}v\left(a\right)\le u\left(a\right)\le C_{2}v\left(a\right)$ for every $a\in S$. In this case, u and v are said to be equivalent on S. Also, $u\lesssim v$ on S means that $u\left(a\right)\le C_{1}v\left(a\right)$ for any $a\in S$ for some absolute constant $C_1>0$, and $u\gtrsim v$ means that $v\lesssim u$.

A scalar-valued weight function $\delta :[0,1)\to (0,\infty )$ is said to be log-convex if $log\delta \left(t\right)$ is a convex function of $logt$ on the interval $(0,1)$. A weight function $\delta$ is log-convex if $log\delta \left(t\right)$ is convex; however, the converse is not necessarily true. The following nice characterization of the estimate for functions in $A^{\delta }$ is obtained by E. Abamukov and E. Doubtsov in [4] (see also [3,5]).

Theorem 4. 

Let $\delta :[0,1)\to (0,\infty )$ be a weight function. Let $\mathsf{\Omega }\subset {\mathbb{C}}^n$ be a bounded, circular and strictly convex domain with a $C^2$ boundary. Then the following are equivalent:

(i) 

There exist $f_m\in H\left(\mathsf{\Omega }\right),1\le m\le M\mathrm{such}\mathrm{that}{\sum }_{m=1}^M\left|f_m\left(z\right)\right|\approx \delta \left(r_{\mathsf{\Omega }}\left(z\right)\right),z\in \mathsf{\Omega }.$

(ii) 

$\delta \left(t\right)$ is equivalent to a log-convex weight function on $[0,1).$

Moreover, $M=M\left(\mathsf{\Omega }\right)$ is an integer that depends on Ω only.

In the rest of this paper, Ω will always denote a bounded, circular and strictly convex domain with a $C^2$ boundary unless otherwise stated.

The following result is basically Corollary 3 of [3] (see also [5]) when $\mathcal{J}$ is a singleton set; we include a proof for convenience.

Theorem 5. 

Let $p>0$, δ be a scalar weight that is equivalent to a log-convex function, and let $\{{\nu }_{\alpha }:\alpha \in \mathcal{J}\}$ be a collection of positive measures on Ω. Then

$$\underset{\alpha \in \mathcal{J}}{sup}{\int }_{\mathsf{\Omega }}{\delta }^p\left(r_{\mathsf{\Omega }}\left(z\right)\right)d{\nu }_{\alpha }\left(z\right)<\infty$$

if and only if

$$\underset{\alpha \in \mathcal{J}}{sup}{\int }_{\mathsf{\Omega }}{\left|f\left(z\right)\right|}^{p}d{\nu }_{\alpha }\left(z\right)\lesssim {\parallel f\parallel }_{\delta }^p$$

for all $f\in A^{\delta }$.

Proof. 

Suppose first that ${sup}_{\alpha \in \mathcal{J}}{\int }_{\mathsf{\Omega }}{\left|f\left(z\right)\right|}^{p}d{\nu }_{\alpha }\left(z\right)\lesssim {\parallel f\parallel }_{\delta }^p$ for all $f\in A^{\delta }$. There exist $f_m\in H\left(\mathsf{\Omega }\right)$, $1\le m\le M$ such that ${\sum }_{m=1}^M{\left|f_m\left(z\right)\right|}^p\approx {\delta }^p\left(r_{\mathsf{\Omega }}\left(z\right)\right)$ by [3]. Hence,

$$\begin{array}{ccc}\hfill \underset{\alpha \in \mathcal{J}}{sup}{\int }_{\mathsf{\Omega }}{\delta }^p\left(r_{\mathsf{\Omega }}\left(z\right)\right)d{\nu }_{\alpha }\left(z\right)& \approx & \underset{\alpha \in \mathcal{J}}{sup}\sum _{m=1}^M{\int }_{\mathsf{\Omega }}{\left|f_m\left(z\right)\right|}^{p}d{\nu }_{\alpha }\left(z\right)\hfill \\ & \lesssim & \sum _{m=1}^M{\parallel f_m\parallel }_{\delta }^p<\infty .\hfill \end{array}$$

Conversely, suppose that the first supremum in the statement is finite. If $f\in A^{\delta }$, then

$$\begin{array}{ccc}\hfill \underset{\alpha \in \mathcal{J}}{sup}{\int }_{\mathsf{\Omega }}{\left|f\left(z\right)\right|}^{p}d{\nu }_{\alpha }\left(z\right)& \le & {C\parallel f\parallel }_{\delta }^p,\hfill \end{array}$$

where

$$C=\underset{\alpha \in \mathcal{J}}{sup}{\int }_{\mathsf{\Omega }}{\delta }^p\left(r_{\mathsf{\Omega }}\left(z\right)\right)d{\nu }_{\alpha }\left(z\right).$$

□

3.2. Vector-Valued Weights

A weight $\omega :\mathsf{\Omega }\to {\mathcal{D}}^{+}$ is said to be log-convex if

$$\begin{array}{c}\hfill \omega \left(z\right)=\sum _{j=1}^{\infty }{\delta }_j\left(r_{\mathsf{\Omega }}\left(z\right)\right)(E_{2j-1}+E_{2j}),\end{array}$$

where each ${\delta }_j\ge 0$ is scalar-valued, log-convex and ${sup}_j{\delta }_j\left(r_{\mathsf{\Omega }}\left(z\right)\right)<\infty$ for every $z\in \mathsf{\Omega }$. We say that a weight $\omega$ is equivalent to a log-convex function $\tilde{\omega }$ if there exist positive constants $C_1$, $C_2$ so that $C_1\tilde{\omega }\le \omega \le C_2\tilde{\omega }$ on $\mathsf{\Omega }$.

Also, we say that a weight $w\left(z\right)$ is admissible if there exist $\tau >0$ and holomorphic functions ${\gamma }_{j,k}$ on $\mathsf{\Omega }$, $j\ge 1$, $k=1,\dots ,M=M\left(\mathsf{\Omega }\right)$ so that

$$\begin{array}{c}\hfill \tau {\delta }_j\left(r_{\mathsf{\Omega }}\left(z\right)\right)\le \sum _{k=1}^M\left|{\gamma }_{j,k}\left(z\right)\right|\le {\delta }_j\left(r_{\mathsf{\Omega }}\left(z\right)\right)\end{array}$$

for every $z\in \mathsf{\Omega }$. The identity map on ${\ell }^2$ is denoted by I. The following is the main result of this paper.

Theorem 6. 

Let $\mathcal{E}$ be a Banach lattice with an unconditional Schauder basis. Let $p\ge 1$, ω be a weight that is equivalent to a log-convex function on Ω, and let $\{{\mu }_{\alpha }:\alpha \in \mathcal{J}\}$ be a collection of measures in $M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$. Consider the statements below:

(i) 

$$\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\mathsf{\Omega }}{\omega }^p\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}<\infty$$

(ii) 

$$\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\mathsf{\Omega }}{\left|f\left(z\right)\right|}_{\mathsf{\Gamma }}^{p}d{\mu }_{\alpha }\left(z\right)∥}_{\mathcal{E}}^{1/p}\le C{\parallel f\parallel }_{\omega }$$

for all $f\in A^{\omega }$, where C is a constant that is independent of f.

Then $\left(i\right)$ implies $\left(ii\right)$. If w is admissible and takes values in $\mathsf{\Gamma }\left(\mathcal{E}\right)$, then $\left(ii\right)$ implies $\left(i\right)$.

Proof. 

If $f\in A^{\omega }$, then

$$\begin{array}{ccc}\hfill 0\le {\left[\mathsf{\Gamma }\left({\int }_{\mathsf{\Omega }}{\left|f\left(z\right)\right|}_{\mathsf{\Gamma }}^{p}d{\mu }_{\alpha }\left(z\right)\right)\right]}^{1/p}& \le & {\parallel f\parallel }_{A^{\omega }}{\left({\int }_{\mathsf{\Omega }}{\omega }^p\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)\right)}^{1/p}\hfill \end{array}$$

for each $\alpha \in \mathcal{J}$. Keeping in mind that $\mathsf{\Gamma }$ is an isometry from $\mathcal{E}$ onto $\mathcal{Y}$, we see that

$$\begin{array}{c}\hfill {∥{\int }_{\mathsf{\Omega }}{\left|f\left(z\right)\right|}_{\mathsf{\Gamma }}^{p}d{\mu }_{\alpha }\left(z\right)∥}_{\mathcal{E}}^{1/p}\le C{\parallel f\parallel }_{\omega }\end{array}$$

for each $\alpha \in \mathcal{J}$, where

$$\begin{array}{c}\hfill C=\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\mathsf{\Omega }}{\omega }^p\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}^{1/p}\end{array}$$

Hence, if C is finite, $\left(ii\right)$ holds.

For the converse, suppose $\left(ii\right)$ is true and that w is admissible and takes values in $\mathsf{\Gamma }\left(\mathcal{E}\right)$. For each $j\ge 1$, choose M functions ${\gamma }_{j,k}\in A^{{\delta }_j}$ (M is independent of j) so that

$$\begin{array}{c}\hfill \tau {\delta }_j\left(r_{\mathsf{\Omega }}\left(z\right)\right)\le \sum _{k=1}^M\left|{\gamma }_{j,k}\left(z\right)\right|\le {\delta }_j\left(r_{\mathsf{\Omega }}\left(z\right)\right)\end{array}$$

for every $z\in \mathsf{\Omega }$ for some constant $\tau$. Let ${\gamma }_{j,k}=u_{j,k}+i{v}_{j,k}$ and $f_k\left(z\right)={\sum }_{j=1}^{\infty }{\gamma }_{j,k}\left(z\right)e_j^{\prime }$. Then $f_k$ is a holomorphic function on $\mathsf{\Omega }$ and takes values in $\mathcal{E}$. In fact, a computation shows that

$$\begin{array}{ccc}\hfill \mathsf{\Gamma }\left(\right|f_k{|}_{\mathsf{\Gamma }}\left(z\right))& =& \underset{0\le \theta <2\pi }{\bigvee }\sum _{j=1}^{\infty }[(cos\theta u_{j,k}-sin\theta v_{j,k})E_{2j-1}+(sin\theta u_{j,k}+cos\theta v_{j,k})E_{2j}]\hfill \\ & =& \sum _{j=1}^{\infty }\underset{0\le \theta <2\pi }{\bigvee }(cos\theta u_{j,k}-sin\theta v_{j,k})E_{2j-1}+\sum _{j=1}^{\infty }\underset{0\le \theta <2\pi }{\bigvee }(sin\theta u_{j,k}+cos\theta v_{j,k})E_{2j}\hfill \\ & =& \sum _{j=1}^{\infty }\left|{\gamma }_{j,k}\left(z\right)\right|(E_{2j-1}+E_{2j})\hfill \\ & \le & \omega \left(z\right)\in \mathsf{\Gamma }\left(\mathcal{E}\right).\hfill \end{array}$$

Hence, $f_k\in A^{\omega }\left(\mathcal{E}\right)$ and $\parallel f_k{\parallel }_{\omega \left(\mathcal{E}\right)}\le 1$. By $\left(ii\right)$, we obtain for each $\alpha \in \mathcal{J}$ that

$$\begin{array}{ccc}\hfill \sum _{k=1}^M{\int }_{\mathsf{\Omega }}\mathsf{\Gamma }\left(\right|f_k{|}_{\mathsf{\Gamma }}^p\left(z\right))d\widetilde{{\mu }_{\alpha }}\left(z\right)& =& {\int }_{\mathsf{\Omega }}\sum _{j=1}^{\infty }\sum _{k=1}^M{\left|{\gamma }_{j,k}\left(z\right)\right|}^p(E_{2j-1}+E_{2j})d\widetilde{{\mu }_{\alpha }}\left(z\right)\hfill \\ & \ge & C_0{\int }_{\mathsf{\Omega }}\sum _{j=1}^{\infty }{\delta }_j^p\left(r_{\mathsf{\Omega }}\left(z\right)\right)(E_{2j-1}+E_{2j})d\widetilde{{\mu }_{\alpha }}\left(z\right)\hfill \\ & =& C_0{\int }_{\mathsf{\Omega }}{\omega }^p\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)\hfill \end{array}$$

for some constant $C_0>0$. Hence,

$$\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\mathsf{\Omega }}{\omega }^p\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}\le \sum _{k=1}^M{∥{\int }_{\mathsf{\Omega }}\mathsf{\Gamma }\left(\right|f_k{|}_{\mathsf{\Gamma }}^p\left(z\right))d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}\le CM/C_0<\infty$$

and $\left(i\right)$ is proved. □

An observation that is worth noting separately stems from the proof of the theorem above.

Corollary 1. 

Let $\mathcal{E}$ be a Banach space with an unconditional Schauder basis. Let ω be an admissible weight that is equivalent to a log-convex function on Ω and takes values in $\mathsf{\Gamma }\left(\mathcal{E}\right)$. Then there exist a constant $\tau >0$ and functions $f_k\in A^{\omega }\left(\mathcal{E}\right)$, $k=1,\dots ,M=M\left(\mathsf{\Omega }\right)$ so that

$$\tau \omega \left(z\right)\le \sum _{k=1}^M\mathsf{\Gamma }\left(\right|f_k{|}_{\mathsf{\Gamma }}\left(z\right))\le \omega \left(z\right)$$

for every $z\in \mathsf{\Omega }$.

4. Compact Embeddings

In this section, we prove an estimate for the essential norm of the Carleson embedding, and we characterize compact embeddings of $A^{\omega }\left(\mathcal{E}\right)$. Theorem 7 below is new even for the scalar-valued settings. For $q\ge 1$ and $\mu \in M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$, let $A(\mu ,q)$ denote the space of holomorphic functions $f\in H(\mathsf{\Omega },\mathcal{E})$ so that

$$\begin{array}{c}\hfill {\parallel f\parallel }_{A(\mu ,q)}={∥\mathsf{\Gamma }\left({\int }_{\mathsf{\Omega }}{\left|f\right|}_{\mathsf{\Gamma }}^{q}d\mu \right)∥}_{\mathcal{Y}}^{1/q}<\infty .\end{array}$$

We assume that $\{{\mu }_{\alpha }:\alpha \in \mathcal{J}\}$ is a family of measures belonging to $M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$ and that $X\subset {\cap }_{\alpha \in \mathcal{J}}A({\mu }_{\alpha },q)$ is a Banach space with the norm

$$\begin{array}{c}\hfill {\parallel f\parallel }_X=\underset{\alpha \in \mathcal{J}}{sup}{\parallel f\parallel }_{A({\mu }_{\alpha },q)}.\end{array}$$

(5)

We also assume the following for X:

(i)

The closed unit ball of X is compact when X is given the topology of uniform convergence on compact sets.

(ii)

Point evaluations are continuous on X.

The essential norm of a bounded linear operator T from $A^{\omega }\left(\mathcal{E}\right)$ to X is denoted by ${\parallel T\parallel }_{e,A^{\omega }\left(\mathcal{E}\right)\to X}$ or simply by ${\parallel T\parallel }_e$, and it is equal to the distance from T to the class of compact operators from $A^{\omega }\left(\mathcal{E}\right)$ to X. In other words,

$$\begin{array}{c}\hfill {\parallel T\parallel }_e=inf\{\parallel T-K\parallel |K:A^{\omega }\left(\mathcal{E}\right)\to X\mathrm{is}\mathrm{compact}\}.\end{array}$$

Theorem 7. 

Let $\mathcal{E}$ be a Banach lattice with an unconditional Schauder basis. Let ω be an admissible weight that is equivalent to a log-convex function on Ω and that takes values in $\mathsf{\Gamma }\left(\mathcal{E}\right)$. Let $q\ge 1$, and let $\{{\mu }_{\alpha }:\alpha \in \mathcal{J}\}\subset M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$ be a collection of measures on Ω. Suppose that the inclusion map ι is bounded from $A^{\omega }\left(\mathcal{E}\right)$ into X. Then

$$\begin{array}{c}\hfill {\parallel \iota \parallel }_e\approx \underset{r\to 1^{-}}{lim\; sup}\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\{z:r_{\mathsf{\Omega }}\left(z\right)>r\}}{\omega }^q\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}\end{array}$$

(6)

In particular, the following statements are equivalent:

(i) 

$\iota :A^{\omega }\left(\mathcal{E}\right)\to X$ is compact.

(ii) 

${lim}_{r\to 1^{-}}{sup}_{\alpha \in \mathcal{J}}{∥{\int }_{\{z:r_{\mathsf{\Omega }}\left(z\right)>r\}}{\omega }^q\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}=0$.

Proof. 

We may assume without loss of generality that $\mathsf{\Omega }\subset \mathbb{B}$. Then $\left|z\right|\gtrsim r_{\mathsf{\Omega }}\left(z\right)$ for $z\in \mathsf{\Omega }$. For functions $f_k$ as in Corollary 1 and an integer $m\ge 1$, we let $f_{k,m}\left(z\right)=z^m{f}_k\left(z\right)$. Then $f_{k,m}\in A^{\omega }\left(\mathcal{E}\right)$ so that

$$\begin{array}{c}\hfill {\tau \left|z\right|}^m\omega \left(z\right)\le \sum _{k=1}^M\mathsf{\Gamma }(|f_{k,m}{|}_{\mathsf{\Gamma }}{\left(z\right))\le |z|}^m\omega \left(z\right)\end{array}$$

for $z\in \mathsf{\Omega }$. Hence, $\left\{f_{k,m}\right\}$ is a bounded sequence of functions in $A^{\omega }\left(\mathcal{E}\right)$ that converge to zero uniformly on compact sets. It follows that (cf. [10,11]) if $K:A^{\omega }\left(\mathcal{E}\right)\to X$ is compact, then ${lim}_{m}K{f}_{k,m}=0$ in X for every $k=1,\dots ,M$. We estimate

$$\begin{array}{cc}\hfill {\parallel \iota \parallel }_e^q& =\underset{K\mathrm{compact}}{inf}{\parallel \iota -K\parallel }^q\hfill \\ \hfill & \gtrsim \underset{m\to \infty }{lim\; sup}\left(\sum _{k=0}^M{\parallel f_{k,m}-K{f}_{k,m}\parallel }_X^q\right)\hfill \\ \hfill & \ge \underset{m\to \infty }{lim\; sup}\sum _{k=0}^M(\parallel f_{k,m}{\parallel }_X-\parallel K{f}_{k,m}{{\parallel }_X)}^q\hfill \\ \hfill & \gtrsim \underset{m\to \infty }{lim\; sup}\underset{\alpha \in \mathcal{J}}{sup}\sum _{k=0}^M{∥{\int }_{\mathsf{\Omega }}\mathsf{\Gamma }\left(\right|f_{k,m}{{|}_{\mathsf{\Gamma }})}^{q}d\widetilde{{\mu }_{\alpha }}∥}_{\mathcal{Y}}\hfill \\ \hfill & \gtrsim \underset{m\to \infty }{lim\; sup}\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\mathsf{\Omega }}{\left|z\right|}^{qm}{\omega }^q\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}\hfill \\ \hfill & \ge \underset{m\to \infty }{lim\; sup}\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\{r_{\mathsf{\Omega }}\left(z\right)>1-{\displaystyle \frac{1}{m}}\}}{r}_{\mathsf{\Omega }}^{qm}\left(z\right){\omega }^q\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}\hfill \\ \hfill & \ge {\displaystyle \frac{e^q}{2}}\underset{m\to \infty }{lim\; sup}\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\{r_{\mathsf{\Omega }}\left(z\right)>1-{\displaystyle \frac{1}{m}}\}}{\omega }^q\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}.\hfill \end{array}$$

Hence, ${\parallel \iota \parallel }_e^2\gtrsim {lim\; sup}_{r\to 1^{-}}{sup}_{\alpha \in \mathcal{J}}{∥{\int }_{\{r_{\mathsf{\Omega }}\left(z\right)>r\}}{\omega }^q\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}$.

Since $\mathsf{\Omega }$ is strictly convex and circular, 0 is in $\mathsf{\Omega }$. For the converse, for each $m\ge 1$, let $C_{m}f\left(z\right)=f\left({\displaystyle \frac{mz}{m+1}}\right)$. Then $\iota C_m:A^{\omega }\left(\mathcal{E}\right)\to X$ is compact for each m. By the Banach–Steinhaus theorem, the sequence of operators $(\iota -\iota C_m)$ converges to zero uniformly on the compact subsets of $H(\mathsf{\Omega },\mathcal{E})$ when $H(\mathsf{\Omega },\mathcal{E})$ is endowed with the compact open topology (cf. [10], p. 56). Since the closed unit ball of $A^{\omega }\left(\mathcal{E}\right)$ is a compact subset of $H(\mathsf{\Omega },\mathcal{E})$, we conclude that

$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}\underset{{\parallel f\parallel }_{A^{\omega }\left(\mathcal{E}\right)}\le 1}{sup}\underset{\{r_{\mathsf{\Omega }}\left(z\right)\le r\}}{sup}{\parallel (f-C_{m}f)\left(z\right)\parallel }_{\mathcal{E}}=0.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill {\parallel \iota \parallel }_e^q& \le \underset{m\to \infty }{lim\; sup}{\parallel \iota -\iota C_m\parallel }^q\hfill \\ \hfill & =\underset{m\to \infty }{lim\; sup}\underset{{\parallel f\parallel }_{A^{\omega }\left(\mathcal{E}\right)}\le 1}{sup}\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\mathsf{\Omega }}{\left|f\left(z\right)-f\left({\displaystyle \frac{mz}{m+1}}\right)\right|}_{\mathsf{\Gamma }}^{q}d{\mu }_{\alpha }\left(z\right)∥}_{\mathcal{E}}\hfill \\ \hfill & \le 4^{q}C\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\{r_{\mathsf{\Omega }}\left(z\right)>r\}}{\omega }^q\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}.\hfill \end{array}$$

We obtain the reverse estimate ${\parallel \iota \parallel }_e^q\lesssim {lim\; sup}_{r\to 1^{-}}{sup}_{\alpha \in \mathcal{J}}{∥{\int }_{\{r_{\mathsf{\Omega }}\left(z\right)>r\}}{\omega }^q\left(z\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}$. □

5. Applications: Weighted Composition Operators

Let $\mathcal{E}$ be a Banach space with an unconditional basis $\left\{e_j^{\prime }\right\}$. For vectors

$$\begin{array}{c}\hfill a=\left\{{\alpha }_j\right\}\in {\ell }^{\infty },b=\sum _j{\beta }_j{e}_j^{\prime }\in \mathcal{E}\end{array}$$

one can define the product $ab=ba={\sum }_j{\alpha }_j{\beta }_j{e}_j^{\prime }$ to be the element of $\mathcal{E}$ obtained by multiplying coordinate-wise the coefficients of a and b. For $a=\left\{{\alpha }_j\right\}\in {\ell }^{\infty }$ and $q>0$, let ${\left|a\right|}^q=\left\{\right|{\alpha }_j{|}^q\}$.

Let $\phi \in H\left(\mathsf{\Omega }\right)$, with $\phi \left(\mathsf{\Omega }\right)\subset \mathsf{\Omega }$ and $\psi \in H(\mathsf{\Omega },{\ell }^{\infty })$. The weighted composition operator $C_{\phi ,\psi }$ is an operator that maps $f\in H(\mathsf{\Omega },\mathcal{E})$ into the function $\psi \left(z\right)f\left(\phi \right(z\left)\right)$ for all $z\in \mathsf{\Omega }$. Hence, $C_{\phi ,\psi }$ is the composition followed by a multiplication operator. When $\psi \left(z\right)=\left\{1\right\}$, the constant sequence $C_{\phi ,\psi }=C_{\phi }$ is the usual composition operator. Composition operators on function spaces are a vast subject. We refer the reader to [6,7] for the general theory.

For $q\ge 1$ and a positive measure $\mu \in M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$, we define a measure ${\nu }_{\mu ,\phi ,\psi ,q}$ in $M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$ by setting

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}ud{\nu }_{\mu ,\phi ,\psi ,q}={\int }_{\mathsf{\Omega }}u\left(\phi \left(z\right)\right){\left|\psi \left(z\right)\right|}^{q}d\mu \left(z\right)\end{array}$$

for any non-negative, continuous u on $\mathsf{\Omega }$ that takes values in $\mathcal{E}$. Now, if $\tilde{\nu }$ and $\tilde{\mu }$ denote the measures obtained by applying (4) to ${\nu }_{\mu ,\phi ,\psi ,q}$ and $\mu$, respectively, as a result of Proposition 2, then

$$\begin{array}{ccc}\hfill {\int }_{\mathsf{\Omega }}\mathsf{\Gamma }ud\tilde{\nu }& =& {\int }_{\mathsf{\Omega }}{\mathsf{\Gamma }\left((u\circ \phi )\right|\psi |}^q)d\tilde{\mu }\hfill \\ & =& {\int }_{\mathsf{\Omega }}{\left|\psi \right|}^q\mathsf{\Gamma }(u\circ \phi )d\tilde{\mu }\hfill \\ & =& \mathsf{\Gamma }\left({\int }_{\mathsf{\Omega }}u\left(\phi \left(z\right)\right){\left|\psi \left(z\right)\right|}^{q}d\mu \left(z\right)\right).\hfill \end{array}$$

We assume in the rest of this section that $\{{\mu }_{\alpha }:\alpha \in \mathcal{J}\}$ is a family of measures from the class $M^{+}(\mathsf{\Omega },B\left(\mathcal{E}\right))$. Hence,

$$\begin{array}{c}\hfill \parallel C_{\phi ,\psi }{f\parallel }_{A(\mu ,q)}^q={∥{\int }_{\mathsf{\Omega }}{\left|f\right|}_{\mathsf{\Gamma }}^{q}d{\nu }_{\mu ,\phi ,\psi ,q}∥}_{\mathcal{E}}.\end{array}$$

We assume that $A({\mu }_{\alpha },q)$ and $X={\cap }_{\alpha \in \mathcal{J}}A({\mu }_{\alpha },q)$ are Banach spaces of holomorphic functions on $\mathsf{\Omega }$ with the norm given by

$$\begin{array}{c}\hfill {\parallel f\parallel }_X=\underset{\alpha \in \mathcal{J}}{sup}{\parallel f\parallel }_{A({\mu }_{\alpha },q)},\end{array}$$

(7)

and X satisfies all the properties in Section 4. An application of Theorems 6 and 7 gives norm and essential norm estimates for composition operators on $A^{\omega }$.

Theorem 8. 

Suppose that $\mathcal{E}$ is a Banach lattice with an unconditional Schauder basis, and ω is an admissible weight that is equivalent to a log-convex function on Ω and takes values in $\mathsf{\Gamma }\left(\mathcal{E}\right)$. Let $q\ge 1$, $\phi \in H\left(\mathsf{\Omega }\right)$ with $\phi \left(\mathsf{\Omega }\right)\subset \mathsf{\Omega }$ and $\psi \in H(\mathsf{\Omega },{\ell }^{\infty })$. Then the following are equivalent:

(i) 

$C_{\phi ,\psi }$ is bounded from $A^{\omega }$ into X.

(ii) 

${sup}_{\alpha \in \mathcal{J}}{∥{\int }_{\mathsf{\Omega }}{\left|\psi \left(z\right)\right|}^q{\omega }^q\left(\phi \left(z\right)\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}<\infty$.

For the essential norm, we have

$$\begin{array}{c}\hfill \parallel C_{\phi ,\psi }{\parallel }_e\approx \underset{r\to 1^{-}}{lim\; sup}\underset{\alpha \in \mathcal{J}}{sup}{∥{\int }_{\mathsf{\Omega }}{\left|\psi \left(z\right)\right|}^q{\omega }^q\left(\phi \left(z\right)\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}.\end{array}$$

Therefore, the following are equivalent:

(i) 

$C_{\phi ,\psi }:A^{\omega }\to X$ is compact.

(ii) 

${lim\; sup}_{r\to 1^{-}}{sup}_{\alpha \in \mathcal{J}}{∥{\int }_{\mathsf{\Omega }}{\left|\psi \left(z\right)\right|}^q{\omega }^q\left(\phi \left(z\right)\right)d\widetilde{{\mu }_{\alpha }}\left(z\right)∥}_{\mathcal{Y}}=0$.

6. Discussion

This paper introduces the growth spaces: spaces of holomorphic functions with certain growth conditions and that take values in a Banach space $\mathcal{E}$. We believe that this is the first paper focusing on Banach-valued growth spaces in the literature. Our definition makes use of the abstract notion of the Gamma order, which is also introduced for the first time in this paper. A natural example fitting in this notion is when $\mathcal{E}$ has an unconditional basis, in which case, the Gamma order on $\mathcal{E}$ can be explicitly defined. The class of Banach spaces that have an unconditional basis contains most of the important spaces, such as the classical Lebesgue spaces. Any separable Hilbert space has an unconditional basis.

The growth space itself is also a Banach space. In the paper, bounded or compact embeddings of the growth spaces are completely characterized. Our results extend the results related to scalar-valued growth spaces. The compactness criteria are new even for the scalar-valued settings. We provide essential norm estimates for the embedding operator, which are also new information and are interesting on their own. As an application, a characterization of the bounded or compact generalized composition operators defined on these spaces is explicitly given.

For future research, versions of classical Banach spaces, such as Bergman, Hardy, etc., consisting of holomorphic Banach-valued functions can be defined analogously. We are planning to expand the work in this direction and the study of operators on such spaces.

The author is grateful for the comments of the reviewers, which improved the exposition of this paper.