1 Introduction

Let $\mathbb {D}$ be the open unit disc in the complex plane $\mathbb {C}$ and let $\mathbb {T}=\partial \mathbb {D}$ be the unit circle. We denote by $H(\mathbb {D})$ the space of all analytic functions on $\mathbb {D}$ . In this paper, we study several analytic function spaces on $\mathbb {D}$ whose norm is defined in terms of a sup-condition. We begin with the definition of weighted Dirichlet spaces.

For $p>0$ and $t>-1$ , we let ${\mathcal D}^p_t$ denote the space of all functions $f\in H(\mathbb {D})$ such that

$$ \begin{align*} \|f\|^p_{{\mathcal D}^p_t}=\int_{\mathbb{D}}|f'(z)|^p(1-|z|^2)^t\,dA(z)<\infty, \end{align*} $$

where $dA$ is area measure normalized so that the unit disc has area $1$ . A norm of f in ${\mathcal D}^p_t$ is $|f(0)|+\|f\|_{{\mathcal D}^p_t}$ .

Two special cases of weighted Dirichlet spaces are worth mentioning. First, for $p>0$ and $t>-1$ , it follows from a Hardy–Littlewood type theorem that the space ${\mathcal D}^p_{p+t}$ coincides with the weighted Bergman space $A^p_t$ , consisting of all functions $f\in H(\mathbb {D})$ such that

$$ \begin{align*} \int_{\mathbb{D}}|f(z)|^p(1-|z|^2)^t\,dA(z)<\infty. \end{align*} $$

Second, by a classical Littewood–Paley theorem, the space ${\mathcal D}^2_1$ is nothing but the Hardy space $H^2$ . Recall that for any $p>0$ , the Hardy space $H^p$ consists of all functions ${f\in H(\mathbb {D})}$ such that

$$ \begin{align*} \|f\|_{H^p}^p=\sup_{0<r<1}\int_0^{2\pi}|f(re^{it})|^p\,dt<\infty. \end{align*} $$

Recall that for any $\alpha>0$ , the $\alpha $ -Bloch space $\mathcal {B}_\alpha $ (see [Reference ZhuZhu2]) consists of all functions $f\in H(\mathbb {D})$ such that

$$ \begin{align*} \|f\|_{\mathcal{B}_\alpha}=\sup_{z\in\mathbb{D}}(1-|z|^2)^\alpha|f'(z)|<\infty. \end{align*} $$

The special case $\alpha =1$ results in the classical Bloch space, which we denote by $\mathcal {B}$ . As usual, we let BMOA be the space of functions in $H^2$ whose boundary function is of bounded mean oscillation on $\mathbb {T}$ ; see [Reference Girela and AulaskariGi].

Denote by $\mathrm {Aut}(\mathbb {D})$ the set of all Möbius maps (bijective analytic maps) of the unit disc. It is a group under ordinary compositions and it is called the Möbius group or sometimes the automorphism group. Every $\sigma \in \mathrm {Aut}(\mathbb {D})$ can be written in the form

$$ \begin{align*} \sigma(z)=e^{i\theta}\sigma_a(z)=\sigma_b(e^{it}z), \end{align*} $$

where $\theta $ and t are real angles, while a and b are points in $\mathbb {D}$ . Here,

$$ \begin{align*} \sigma_a(z)=\frac{a-z}{1-\overline az},\quad z\in\mathbb{D}. \end{align*} $$

A subspace X of $H(\mathbb {D})$ endowed with a semi-norm $\|\ \|_X$ is said to be Möbius invariant if $f\circ \sigma \in X$ and $\|f\circ \sigma \|_X \lesssim \|f\|_X$ for all $f\in X$ and $\sigma \in \mathrm {Aut}(\mathbb {D})$ . A general construction of Möbius-invariant function spaces is as follows: let Y be any subspace of $H(\mathbb {D})$ endowed with a semi-norm $\|\ \|_Y$ and let $M[Y]$ consist of all functions $f\in Y$ such that

$$ \begin{align*} \|f\|_{M[Y]}=\sup_{\sigma\in\mathrm{Aut}(\mathbb{D})}\|f\circ\sigma-f(\sigma(0))\|_Y<\infty. \end{align*} $$

Then, $M[Y]$ is a Möbius-invariant function space with the above semi-norm and $M[Y]$ is also called the Möbius-invariant version of Y. For $p>0$ and $t>-1$ , it is known (see [Reference Aleman and SimbotinAS, Reference AxlerAx, Reference Girela and AulaskariGi]) that $M[A^p_t]=\mathcal {B}$ and $M[H^p]=\mathrm {BMOA}$ . We refer to [Reference Arazy, Fisher and PeetreAFP] or [Reference Wulan and ZhuWZ, Ch. 1] for the general theory on Möbius-invariant function spaces.

From [Reference Rubel and TimoneyRT], the Bloch space $\mathcal {B}$ is maximal among all Möbius-invariant spaces with an additional mild assumption. For every $\alpha>0$ , is $\mathcal {B}_\alpha $ maximal among some family of analytic function spaces? This question was posed in [Reference ZhuZhu2] and the answer is shown to be positive by Zhao in [Reference ZhaoZ].

Recall from [Reference ZhaoZ] that for $f\in H(\mathbb {D})$ , $\sigma \in \mathrm {Aut}(\mathbb {D})$ , and $\alpha>0$ , the weighted $\alpha $ -composition of f and $\sigma $ is defined by

$$ \begin{align*} f\circ_\alpha\sigma(z)=\int_0^zf'(\sigma(w))[\sigma'(w)]^\alpha\,dw +f(\sigma(0))[\sigma'(0)]^{\alpha-1}. \end{align*} $$

Note that each $\sigma \in \mathrm {Aut}(\mathbb {D})$ is univalent, so $\log \sigma '(z)$ , $[\sigma '(z)]^\alpha =\exp [\alpha \log \sigma '(z)]$ , and $[\sigma '(z)]^{\alpha -1}=\exp [(\alpha -1)\log \sigma '(z)]$ are all well-defined functions on $\mathbb {D}$ . A subspace X of $H(\mathbb {D})$ endowed with a semi-norm $\|\ \|_X$ is said to be $\alpha $ -Möbius invariant if $f\circ _\alpha \sigma \in X$ and $\|f\circ _\alpha \sigma \|_X \lesssim \|f\|_X$ for all $f\in X$ and $\sigma \in \mathrm {Aut}(\mathbb {D})$ . In 1996, Zhao [Reference ZhaoZ] proved that $\mathcal {B}_\alpha $ is maximal among $\alpha $ -Möbius invariant function spaces. In 2021, Bao and Zhu [Reference Bao and ZhuBZ] identified the minimal nontrivial $\alpha $ -Möbius invariant function space, and proved the existence and uniqueness of a nontrivial $\alpha $ -Möbius invariant semi-Hilbert space of analytic functions on $\mathbb {D}$ . We refer to [Reference Bao, Lou and ZhouBLZ, Reference Lou and ZhouLZ, Reference ZhouZhou] for more investigations on $\alpha $ -Möbius invariant function spaces.

More examples of $\alpha $ -Möbius invariant function spaces are $F(p,q,s)$ spaces. Recall that $F(p,q,s)$ , where $p>0$ , $q>-2$ , and $q+s>-1$ , consists of functions $f\in H(\mathbb {D})$ such that

$$ \begin{align*}\|f\|^p_{F(p,q,s)}=\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}|f'(z)|^p(1-|z|^2)^q (1-|\sigma_a(z)|^2)^s\,dA(z)<\infty.\end{align*} $$

The spaces $F(p,q,s)$ were introduced and studied in [Reference ZhaoZ]. In particular, $F(p,q,s)$ is a subset of $\mathcal {B}_{(q+2)/p}$ . For $s>1$ , we actually have $F(p,q,s)=\mathcal {B}_{(q+2)/p}$ . It is worth mentioning that $F(2, 0, s)$ is the well-studied Möbius-invariant space $\mathcal {Q}_s$ (see [Reference Aulaskari, Xiao and ZhaoAXZ, Reference XiaoX1, Reference XiaoX2]). It was proved in [Reference ZhaoZ] that $F(p,q,s)$ is a $(q+2)/p$ -Möbius invariant function space. Very recently, Zhou [Reference ZhouZhou] introduced and investigated a new class of $\alpha $ -Möbius invariant function spaces that contains $F(p, q, s)$ spaces as special cases.

We also recall Campanato spaces of analytic functions, which are closely associated with $F(p, q, s)$ spaces. For $p>0$ and $\lambda>0$ , the Campanato space ${\mathcal {H}}^{p,\lambda }$ consists of functions $f\in H^p$ such that

$$ \begin{align*} \|f\|^p_{{\mathcal{H}}^{p,\lambda}}=\sup_{I\subset\mathbb{T}}\frac{1}{|I|^\lambda} \int_I|f-f_I|^p\,dt<\infty, \end{align*} $$

where the supremum is taken over all arcs $I\subset \mathbb {T}$ , $|I|$ denotes the length of I, and $f_I$ is the mean of f over I with respect to $dt$ . We mention some special cases as follows.

• • If $0<\lambda <1$ , then ${\mathcal {H}}^{p,\lambda }$ is also called a Morrey space.

• • If $\lambda =1$ , then ${\mathcal {H}}^{p,\lambda }$ is BMOA.

• • If $\lambda \in (1, 1+p]$ , then ${\mathcal {H}}^{p,\lambda }$ is the analytic Lipschitz space $\Lambda _{(\lambda -1)/p}$ .

• • If $\lambda \in (1+p, \infty )$ , then ${\mathcal {H}}^{p,\lambda }$ contains constant functions only.

See [Reference Xiao and YuanXY, page 1003], for example. Thus, the really interesting case for us here is when $0<\lambda <1$ . Also, $\mathcal {H}^{2, \lambda }$ is equal to $F(2, 1-\lambda , \lambda )$ . We refer to [Reference Yuan and TongYT] for more investigations on ${\mathcal {H}}^{p,\lambda }$ and $F(p, q, s)$ spaces.

In 2021, Aleman and Mas [Reference Aleman and MasAM] introduced and studied another version of Möbius invariant function spaces. For $\beta>0$ , a subspace X of $H(\mathbb {D})$ endowed with a semi-norm $\|\ \|_X$ is said to be Möbius invariant of index $\beta $ if $W^{\beta }_\sigma f \in X$ and $\|W^{\beta }_\sigma f\|_X\lesssim \|f\|_X$ for all ${f\in X}$ and $\sigma \in \mathrm {Aut}(\mathbb {D})$ , where $W^{\beta }_\sigma f=f\circ \sigma (\sigma')^\beta $ . Roughly speaking, $\alpha $ -Möbius invariant spaces place many analytic function spaces defined by the derivative of functions in the same framework, such as $\alpha $ -Bloch spaces, ${\mathcal D}^p_{\alpha p-2}$ spaces, and ${F(p, \alpha p-2, s)}$ spaces. However, Möbius-invariant function spaces of index $\beta $ include many analytic function spaces defined without the derivative of functions. For example, as shown in [Reference Aleman and MasAM], the Hardy space $H^p$ is Möbius invariant of index $1/p$ and the Bergman space $A^p_t$ is Möbius invariant of index $(2+t)/p$ .

When a subspace X of $H(\mathbb {D})$ is defined by a derivative condition, it is natural to seek derivative-free characterizations of X. For instance, it is well known (see [Reference ZhuZhu1]) that an analytic function f on $\mathbb {D}$ belongs to the Bloch space $\mathcal {B}$ if and only if $|f(z)-f(w)|\lesssim \beta (z, w)$ for all z, $w\in \mathbb {D}$ , where $\beta (z, w)$ is the Bergman distance between z and w. See [Reference Li and StevićLS, Reference Li and WulanLW] for other derivative-free characterizations of the Bloch space. We refer to [Reference XiaoX2, Ch. 2] for a summary of derivative-free characterizations of $\mathcal {Q}_s$ spaces. Recently, Yang and Yuan [Reference Yang and YuanYY] gave some derivative-free characterizations of $F(p, q, s)$ spaces.

The notion of Carleson measures was introduced by Carleson [Reference CarlesonC1, Reference CarlesonC2] when he studied the interpolation problem and the corona problem for the space of bounded analytic functions on $\mathbb {D}$ . He characterized positive Borel measures $\mu $ on $\mathbb {D}$ such that $H^p\subseteq L^p(\mathbb {D}, d\mu )$ . Carleson measures are useful tools for studying analytic function spaces and related operators on them. In 2008, Xiao [Reference XiaoX3] studied a variation of Carleson measures for $\mathcal {Q}_s$ spaces (the embedding from $\mathcal {Q}_s$ to a certain tent-type space), which can be used to characterize the boundedness and compactness of Volterra-type operators on $\mathcal {Q}_s$ spaces. Pau and Zhao [Reference Pau and ZhaoPZ] further studied this variation of Carleson measures for $F(p, q, s)$ spaces.

In this paper, we continue to investigate $\alpha $ -Bloch spaces $\mathcal {B}_\alpha $ , Campanato spaces, and $F(p,q,s)$ spaces. In Section 3, we give some derivative-free characterizations of $\mathcal {B}_\alpha $ spaces. Some of them involve the weighted $\alpha $ -composition. In Section 4, we show that the characterizations of $\mathcal {B}_\alpha $ in Section 3 improve a conclusion in [Reference Pau and PeláezPP] and a result in [Reference Pau and ZhaoPZ]. In Section 5, we consider a variation of Carleson measures for Campanato spaces. In Section 6, we provide a derivative-free characterization and a variation of Carleson measures for $F(p, q, s)$ spaces.