Schatten class composition operators on the Hardy space

Abstract

Suppose $2< p<\infty$ and $\varphi$ is a holomorphic self-map of the open unit disk $\mathbb {D}$. We show the following assertions:

1. (1) If $\varphi$ has bounded valence and0.1

   \begin{equation} \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p/2}\frac{\mathrm{d} A(z)}{(1-|z|^2)^2}<\infty, \end{equation}
   then $C_{\varphi }$ is in the Schatten $p$-class of the Hardy space $H^2$.

2. (2) There exists a holomorphic self-map $\varphi$ (which is, of course, not of bounded valence) such that the inequality (0.1) holds and $C_{\varphi }: H^2\to H^2$ does not belong to the Schatten $p$-class.

1. Introduction and main results

1.1 Backgrounds and motivations

Let $\mathbb {D}=\{z\in \mathbb {C} :|z|<1\}$ be the unit disk of the complex plane $\mathbb {C}$. Let $H(\mathbb {D})$ be the space of holomorphic functions on $\mathbb {D}$ and let $\varphi$ be a holomorphic function on $\mathbb {D}$ with $\varphi (\mathbb {D})\subset \mathbb {D}$. For $f\in H(\mathbb {D})$, the composition operator $C_{\varphi }$ is a linear operator defined by $C_{\varphi }(f)=f\circ \varphi$.

Recall that a positive $T$ on a separable Hilbert space $H$ is in the trace class if

\[ \mathrm{tr}(T)=\sum_{n=0}^\infty\langle Te_n,e_n\rangle_H<{+}\infty \]

for some (or all) orthonormal basis $\{e_n\}$ of $H$. For any $0< p<\infty$, the Schatten $p$-class $\mathcal {S}_p(H)$ of $H$ consists of bounded linear operators $T:H\to H$ such that $(T^\ast T)^{p/2}$ belongs to the trace class. In particular, $\mathcal {S}_1(H)$ is the trace class of $H$, and $\mathcal {S}_2(H)$ is called the Hilbert–Schmidt class. It is easy to check that $T\in \mathcal {S}_p(H)$ if and only if $T^*\in \mathcal {S}_p(H)$. For more details about Schatten $p$-class operators, we refer the readers to Zhu [16].

The Hardy space $H^2$ is a Hilbert space of analytic functions $f$ on $\mathbb {D}$ such that

\[ \|f\|_{H^2}^2=\sup_{0< r<1}\int_0^{2\pi}|f(r e^{i\theta})|^2\frac{\mathrm{d}\theta}{2\pi} <\infty. \]

For $\alpha >-1$, the weighted Bergman space $A_\alpha ^2$ consists of holomorphic functions $f$ on $\mathbb {D}$ satisfying

\[ \|f\|_{A_\alpha^2}^2= \int_{\mathbb{D}}|f(z)|^2 {\mathrm{d} A_\alpha(z)} <\infty, \]

where $\mathrm {d} A_\alpha (z)=(\alpha +1)(1-|z|^2)^\alpha \mathrm {d} A(z)$ and $\mathrm {d} A(z)$ is the normalized area measure on $\mathbb {D}$. When $\alpha =0$, the space $A_0^2$ is usually denoted by $A^2$. Properties of composition operator on $A_\alpha ^2$ and $H^2$ has been widely investigated for decades, see e.g. [3, 8, 16]. In particular, conditions for $C_{\varphi }$ that belong to $\mathcal {S}_p(A_\alpha ^2)$ and $\mathcal {S}_p(H^2)$ are also characterized, see [1, 2, 4–7, 9, 10, 12, 14].

It is well known (see e.g. Zhu [15]) that $H^2$ can be viewed as the limit case of $A^2_{\alpha }$ as $\alpha \to -1^+$ in some sense. It is also known that for $0< p<\infty$, $C_{\varphi }\in \mathcal {S}_p(H^2)$ if and only if

\[ \int_{\mathbb{D}}\left(\frac{N_\varphi(z)}{\log\frac1{|z|}}\right)^{p/2}\mathrm{d} \lambda(z)<\infty, \]

where

\[ \mathrm{d}\lambda(z)=(1-|z|^2)^{{-}2}\mathrm{d} A(z) \]

is the Möbius invariant measure on $\mathbb {D}$, and

\[ N_\varphi(z)=\sum_{w\in \varphi^{{-}1}(z)}\log\frac1{|w|} \]

is the Nevanlinna counting function of $\varphi$. Similarly, $C_{\varphi }\in \mathcal {S}_p(A_\alpha ^2)$ if and only if

\[ \int_{\mathbb{D}}\left(\frac{N_{\varphi,\alpha+2}(z)}{(\log \frac1{|z|})^{\alpha+2}}\right)^{p/2}\mathrm{d} \lambda(z)<\infty, \]

where $N_{\varphi,\alpha +2}(z)$ is a generalized Nevanlinna counting function of $\varphi$ given by

\[ N_{\varphi,\alpha+2}(z)=\sum_{w\in \varphi^{{-}1}(z)}\left(\log\frac1{|w|}\right)^{\alpha+2}. \]

See Luecking-Zhu [5].

1.2 Main results

A holomorphic map $\varphi :\mathbb {D}\to \mathbb {D}$ is of bounded valence if there is a positive integer $N$ such that for each $z\in \mathbb {D}$, the set $\varphi ^{-1}(z)$ contains at most $N$ points. Zhu [14] shows that if $\alpha >-1$, $2\le p<\infty$ and $\varphi :\mathbb {D}\to \mathbb {D}$ is an analytic function of bounded valence, then $C_{\varphi }$ is in the Schatten class $\mathcal {S}_p$ of $A_\alpha ^2$ if and only if

\[ \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p(\alpha+2)/2} {\mathrm{d} \lambda(z)} <\infty. \]

Meanwhile, Zhu [16, Exercise 11.6.7] says that if $p>2$ and $C_{\varphi }\in \mathcal {S}_p(H^2)$, then

\[ \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p/2} {\mathrm{d} \lambda(z)} <\infty. \]

These observations hint us to give the following result.

Theorem 1.1 If $2< p<\infty,$ $\varphi$ has bounded valence and

(1.1)\begin{equation} \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p/2} \mathrm{d}\lambda(z)<\infty, \end{equation}

then $C_{\varphi }\in \mathcal {S}_p(H^2)$.

For $p>2$, Xia [10] constructs a holomorphic map $\varphi :\mathbb {D}\to \mathbb {D}$ such that

\[ \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^p \mathrm{d}\lambda(z)<\infty \]

and such that $C_{\varphi }: A^2\to A^2$ does not belong to the Schatten class $\mathcal {S}_p(A^2)$. Motivated by Xia [10], we prove the following theorem:

Theorem 1.2 For any $2< p<\infty,$ there exists a holomorphic function $\varphi :\mathbb {D}\to \mathbb {D}$ such that

(1.2)\begin{equation} \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p/2} \mathrm{d}\lambda(z)<\infty, \end{equation}

but $C_{\varphi }:H^2\to H^2$ does not belong to the Schatten class $\mathcal {S}_p(H^2)$.

The proof of theorem 1.1 is based on Wirths-Xiao [9] and Zhu [14]. The proof of theorem 1.2 is modified from Xia [10]. Although the idea of the proof of theorem 1.2 is coming from [10], there are several technical barriers we need to overcome. Thus, we need to adapt Xia's construction for our situation.

Notation. Throughout this paper, we only write $U\lesssim V$ (or $V\gtrsim U$) for $U\le c V$ for a positive constant $c$, and moreover $U\approx V$ for both $U\lesssim V$ and $V\lesssim U$.

2. Preliminaries

For $\alpha >-1$, the Dirichlet-type space is a space of holomorphic functions $f$ on $\mathbb {D}$ for which

\[ \|f\|_{ \alpha}^2=|f(0)|^2+\|f'\|^2_{ A_\alpha^2}<\infty. \]

It is easy to check that $A_\alpha ^2=\mathcal {D}_{\alpha +2}$ and $H^2=\mathcal {D}_1$ with equivalent norms.

The following lemma is contained in [9, Theorem 3.2].

Lemma 2.1 Let $\alpha >-1$ and $0< p<\infty$. Suppose $\varphi :\mathbb {D}\to \mathbb {D}$ is holomorphic. Then $C_{\varphi }\in \mathcal {S}_p(\mathcal {D}_\alpha )$ if and only if

(2.1)\begin{equation} \int_{\mathbb{D}}\left(\int_{\mathbb{D}}\left(\frac{(1-|w|^2)^\varepsilon}{|1- \bar w \varphi(z)|^{1+\varepsilon}}\right)^{2+\alpha}|\varphi'(z)|^2 (1-|z|^2)^\alpha\mathrm{d} A(z)\right)^{p/2} \mathrm{d}\lambda(w)<\infty \end{equation}

for some (any) $\varepsilon >\max \{1/(2+\alpha ),\, 2/(2p+p\alpha )\}$.

For fixed $\alpha >0$, $f,\,g\in \mathcal {D}_\alpha$ with

\[ f(z)=\sum_{n=0}^\infty a_nz^n\quad\hbox{ and }\quad g(z)=\sum_{n=0}^\infty b^n z^n, \]

let

\[ \langle f,g\rangle_{\mathcal{D}_\alpha}= \sum_{n=0}^\infty \frac{ n!\Gamma(\alpha)}{\Gamma(n+\alpha)} a_n\overline{b_n}. \]

Then the reproducing kernel of $\mathcal {D}_\alpha$ associated with the inner product $\langle \cdot,\,\cdot \rangle _{\mathcal {D}_\alpha }$ is given by

\[ K_{\alpha,w}(z)=K_{\alpha}(z,w)=\frac1{(1- \bar w z)^\alpha},\quad z,w\in\mathbb{D}. \]

This means that for each $f\in \mathcal {D}_\alpha$,

\[ f(w)=\langle f,K_{\alpha,w}\rangle_{\mathcal{D}_\alpha} \quad w\in\mathbb{D}. \]

Meanwhile, if we write

\[ J_{\alpha,w}(z)= J_{\alpha}(z,w)=\frac{\partial }{\partial\bar w}K_\alpha(z,w)=\frac{\alpha z}{(1-\bar w z)^{\alpha+1}}, \]

then

(2.2)\begin{equation} f'(w)=\langle f,J_{\alpha,w}\rangle_{\mathcal{D}_\alpha}. \end{equation}

Let

\[ \|f\|_{\mathcal{D}_\alpha}^2=\langle f,f\rangle_{\mathcal{D}_\alpha}. \]

Then

\[ \|K_{\alpha,w}\|_{\mathcal{D}_\alpha}^2=\frac1{(1-|w|^2)^\alpha} \]

and

(2.3)\begin{equation} \|J_{\alpha,w}\|_{\mathcal{D}_\alpha}^2=\langle J_{\alpha,w},J_{\alpha,w}\rangle_{\mathcal{D}_\alpha}=J_{\alpha,w}'(w) =\frac{\alpha(1+\alpha|w|^2)}{(1-|w|^2)^{\alpha+2}}\approx \frac1 {(1-|w|^2)^{\alpha+2}}. \end{equation}

Let

\[ k_{\alpha,w}(z)=\frac{K_{\alpha,w}(z)}{\left\|K_{\alpha,w} \right\|_{\mathcal{D}_\alpha}} \quad \hbox{ and }\quad j_{\alpha,w}(z)=\frac{J_{\alpha,w}(z)}{\left\|J_{\alpha,w} \right\|_{\mathcal{D}_\alpha}} . \]

The following lemma comes from [11, Lemma 10].

Lemma 2.2 Suppose $\alpha >0$ and $T: \mathcal {D}_\alpha \to \mathcal {D}_\alpha$ is a positive operator. Let

\[ \widehat{ T}^{ \alpha,t}(w)=\langle T j_{\alpha,w} , j_{\alpha,w} \rangle_{\mathcal{D}_\alpha},\quad w\in\mathbb{D}. \]

1. (1) Let $0< p\le 1$. If $\,\widehat { T}^{ \alpha,t}\in L^p (\mathbb {D},\,\mathrm {d} \lambda ),$ then $T$ is in $\mathcal {S}_p(\mathcal {D}_\alpha )$.

2. (2) Let $1\le p<\infty$. If $\, T$ is in $\mathcal {S}_p(\mathcal {D}_\alpha ),$ then $\widehat { T}^{ \alpha,t}\in L^p (\mathbb {D},\,\mathrm {d} \lambda )$.

Immediately, we have the following theorem.

Theorem 2.3 Suppose $\alpha >0$ and $\varphi :\mathbb {D}\to \mathbb {D}$ is a holomorphic function.

1. (1) If $0< p\le 2$ and

   (2.4)\begin{equation} \int_{\mathbb{D}} \left(\frac{(1-|z|^2)^{\alpha+2}|\varphi '(z)|^2}{(1-|\varphi(z)|^2)^{\alpha+2}} \right)^{p/2}\mathrm{d} \lambda(z)<\infty, \end{equation}
   then $C_{\varphi }$ is in $\mathcal {S}_p$ of $\,\mathcal {D}_\alpha$.

2. (2) If $2\le p<\infty$ and $C_{\varphi }$ is in $\mathcal {S}_p$ of $\,\mathcal {D}_\alpha,$ then (2.4) holds.

Proof. Write $S=C_{\varphi }C_{\varphi }^\ast$, then $S:\mathcal {D}_\alpha \to \mathcal {D}_\alpha$ is a positive operator. We have

\begin{align*} \widehat{ S}^{ \alpha,t}(w)=\langle S j_{\alpha,w} , j_{\alpha,w} \rangle_{\mathcal{D}_\alpha}& = \langle C_{\varphi}^\ast j_{\alpha,w} ,C_{\varphi}^\ast j_{\alpha,w} \rangle_{\mathcal{D}_\alpha}\\& = \frac{\langle C_{\varphi}^\ast J_{\alpha,w} ,C_{\varphi}^\ast J_{\alpha,w} \rangle_{\mathcal{D}_\alpha}}{\|J_{\alpha,w}\|^2 _{\mathcal{D}_\alpha}} = \frac{\| C_{\varphi}^\ast J_{\alpha,w} \|^2_{\mathcal{D}_\alpha}}{\|J_{\alpha,w}\|^2 _{\mathcal{D}_\alpha}}. \end{align*}

For each $f\in \mathcal {D}_\alpha$, (2.2) implies that

\begin{align*} \langle f,C_{\varphi}^\ast J_{\alpha,w}\rangle_{\mathcal{D}_\alpha}& = \langle C_{\varphi} f,J_{\alpha,w} \rangle_{\mathcal{D}_\alpha}=f'(\varphi(w))\varphi'(w)\\ & = \varphi'(w)\langle f,J_{\alpha,\varphi(w)} \rangle_{\mathcal{D}_\alpha}= \langle f,\overline{\varphi'(w)}J_{\alpha,\varphi(w)} \rangle_{\mathcal{D}_\alpha}. \end{align*}

Thus,

\[ C_{\varphi}^\ast J_{\alpha,w} =\overline{\varphi'(w)}J_{\alpha,\varphi(w)} . \]

Then (2.3) implies that

\[ \| C_{\varphi}^\ast J_{\alpha,w}\|_{\mathcal{D}_\alpha}^2\approx \frac{|\varphi'(w)|^2}{(1-|\varphi(w)|^2)^{2+\alpha}} . \]

This gives that

\[ \langle C_{\varphi} C_{\varphi}^\ast j_{\alpha,w} , j_{\alpha,w} \rangle_{\mathcal{D}_\alpha} = \frac{\langle C_{\varphi}^\ast J_{\alpha,w} ,C_{\varphi}^\ast J_{\alpha,w} \rangle_{\mathcal{D}_\alpha}}{\|J_{\alpha,w}\|^2 _{\mathcal{D}_\alpha}} \approx \frac{(1-|w|^2)^{2+\alpha}|\varphi'(w)|^2}{(1-|\varphi(w)|^2)^{2+\alpha}} . \]

An application of lemma 2.2 gives the desired assertions.

By letting $p=2$ in theorem 2.3, we have the following corollary.

Corollary 2.4 Suppose $\alpha >0$ and $\varphi :\mathbb {D}\to \mathbb {D}$ is a holomorphic function. Then $C_{\varphi }$ is in the Hilbert–Schmidt class of $\mathcal {D}_\alpha$ if and only if

\[ \int_{\mathbb{D}} \frac{(1-|z|^2)^{\alpha }|\varphi '(z)|^2}{(1-|\varphi(z)|^2)^{\alpha+2}} \, \mathrm{d} A(z)<\infty. \]

There are several well-known characterizations of the Hilbert–Schmidt compositions on $H^2$ and $A_\alpha ^2$, see e.g. [3, 13, 16]. Combine these characterizations with corollary 2.4, we have the following corollaries.

Corollary 2.5 Suppose $\varphi :\mathbb {D}\to \mathbb {D}$ is holomorphic. Then the following statements are equivalent:

1. (1) $C_{\varphi }\in \mathcal {S}_2(H^2)$.

2. (2) The following inequality holds:

   \[ \int_{\mathbb{D}} \frac{(1-|z|^2) |\varphi '(z)|^2}{(1-|\varphi(z)|^2)^{3}} \, \mathrm{d} A(z)<\infty. \]

3. (3) The following inequality holds:

   \[ \int_{\mathbb{D}} \frac{N_\varphi(z)}{\log\frac1{|z|}} \, \mathrm{d} \lambda(z)<\infty. \]

4. (4) The following inequality holds:

   \[ \int_{0}^{2\pi} \frac{\mathrm{d}\theta }{(1-|\varphi(\mathrm{e}^{\mathrm{i}\theta})|^2) } <\infty. \]

Corollary 2.6 Suppose $\alpha >-1$ and $\varphi :\mathbb {D}\to \mathbb {D}$ is holomorphic. Then the following statements are equivalent:

1. (1) $C_{\varphi }\in \mathcal {S}_2(A^2_\alpha )$.

2. (2) The following inequality holds:

   \[ \int_{\mathbb{D}} \frac{(1-|z|^2)^{\alpha+2} |\varphi '(z)|^2}{(1-|\varphi(z)|^2)^{\alpha+4}} \, \mathrm{d} A(z)<\infty. \]

3. (3) The following inequality holds:

   \[ \int_{\mathbb{D}} \frac{N_{\varphi,\alpha+2}(z)}{(\log\frac1{|z|})^{\alpha+2}} \, \mathrm{d} \lambda(z)<\infty. \]

4. (4) The following inequality holds:

   \[ \int_{\mathbb{D}} \frac{(1-|z|^2)^{\alpha }}{(1-|\varphi(z)|^2)^{2+\alpha} }\,\mathrm{d} A(z) <\infty. \]

3. Proof of theorem 1.1

Theorem 1.1 is just the case $\alpha =1$ of the following proposition.

Proposition 3.1 Suppose $\alpha >0,$ $2\le p<\infty$ and $p\alpha >2$. Let $\varphi :\mathbb {D}\to \mathbb {D}$ is a holomorphic function which has bounded valence and

(3.1)\begin{equation} \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p\alpha/2} \mathrm{d}\lambda(z)<\infty, \end{equation}

then $C_{\varphi }$ is in the Schatten class $\mathcal {S}_p$ of $\,\mathcal {D}_\alpha$.

The condition $p\alpha > 2$ in the above proposition is necessary. Indeed, if $0 < p\alpha \le 2$, then the involved integral is trivially divergent.

Proof. When $p=2$, the condition $p\alpha >2$ implies that $\alpha >1$. Notice that in this case $\mathcal {D}_\alpha =A_{\alpha -2}^2$. According to [14], the condition (3.1) implies that $C_{\varphi }\in \mathcal {S}_p(A_{\alpha -2}^2)$.

Now we suppose $2< p<\infty$. According to lemma 2.1, if we can check the inequality (2.1) for some $\varepsilon >\max \{1/(2+\alpha ),\, 2/(2p+p\alpha )\}$, then we have $C_{\varphi }\in \mathcal {S}_p(\mathcal {D}_\alpha )$. Write $q=p/2$, then $q>1$. Let

\[ F(w)=\int_{\mathbb{D}} \frac{(1-|w|^2)^{(2+\alpha)\varepsilon} }{|1- \bar w \varphi(z)|^{({2+\alpha})(1+\varepsilon)}} |\varphi'(z)|^2 (1-|z|^2)^\alpha\mathrm{d} A(z). \]

Then it is sufficient to check that $F\in L^q(\mathbb {D},\,\mathrm {d} \lambda )$.

Let

\[ H(w,z)=\frac{(1-|w|^2)^{(\alpha+2)\varepsilon}(1-|\varphi(z)|^2)^{\alpha }(1-|z|^2)^2 |\varphi'(z)|^2}{|1-\bar w\varphi(z)|^{(2+ \alpha)(1+\varepsilon)}} \]

and

\[ h( z)=\left(\frac{(1-|z|^2) }{(1-|\varphi(z)|^{2})}\right)^\alpha. \]

Then,

\[ F(w)=\int_{\mathbb{D}} H(w,z) h(z)\mathrm{d} \lambda(z). \]

Recall that $\varphi :\mathbb {D}\to \mathbb {D}$ is holomorphic. Schwarz's lemma implies that

(3.2)\begin{equation} \frac{ (1-|z|^2)^2 |\varphi'(z)|^2}{(1-|\varphi(z)|^2)^{2 }}\le1. \end{equation}

Then, for each $\varepsilon >1/(2+\alpha )$, Forelli–Rudin's estimate implies that

(3.3)\begin{align} \int_{\mathbb{D}} H(w,z)\mathrm{d} \lambda(w) & =(1-|\varphi(z)|^2)^{\alpha }(1-|z|^2)^2 |\varphi'(z)|^2\int_{\mathbb{D}} \frac{(1-|w|^2)^{(\alpha+2)\varepsilon-2}\mathrm{d} A(w)}{|1-\bar w\varphi(z)|^{(2+ \alpha)(1+\varepsilon)}}\nonumber\\ & \lesssim \frac{(1-|\varphi(z)|^2)^{\alpha }(1-|z|^2)^2 |\varphi'(z)|^2}{(1-|\varphi(z)|^2)^{2+\alpha }}\nonumber\\ & \le1. \end{align}

Meanwhile, recall that $\varphi$ is of bounded valence. Let $n_\varphi (z)$ be the number of points in $\varphi ^{-1}(z)$. Then,

\[ \sup_{z\in\mathbb{D}} n_\varphi(z)<\infty \]

and

(3.4)\begin{align} \int_{\mathbb{D}} H(w,z)\mathrm{d} \lambda(z)& =\int_{\mathbb{D}} \frac{(1-|w|^2)^{(\alpha+2)\varepsilon}(1-|\varphi(z)|^2)^{\alpha } |\varphi'(z)|^2}{|1-\bar w\varphi(z)|^{(2+ \alpha)(1+\varepsilon)}} \,\mathrm{d} A(z) \nonumber\\ & =(1-|w|^2)^{(\alpha+2)\varepsilon}\int_{\mathbb{D}} \frac{n_\varphi(z) (1-| z|^2)^{\alpha } }{|1-\bar w z|^{(2+ \alpha)(1+\varepsilon)}} \,\mathrm{d} A(z)\nonumber\\ & \lesssim 1. \end{align}

Put (3.3) and (3.4) together. Application of Schur's test tells us that the integral operator with kernel $H(w,\,z)$ is bounded on $L^q(\mathbb {D},\,\mathrm {d}\lambda )$. Recall that condition (3.1) implies that $h\in L^q(\mathbb {D},\,\mathrm {d} \lambda )$. This gives that $F\in L^q(\mathbb {D},\,\mathrm {d} \lambda )$ as desired.

4. Proof of theorem 1.2

4.1 Construction of $\varphi$

The construction is modified from Xia [10]. We adapt some parameters for our argument. For $n=1,\,2,\,\dots$, let

\[ T_n=\left(2^{-(n+1)},2^{{-}n}\right]\quad \hbox{ and }\quad S_n=\left(( 4/3) 2^{-(n+1)}, ( 5/3) 2^{-(n+1)}\right]. \]

That is, $S_n$ is the middle third of $T_n$. Let $t_n= ( 4/3) 2^{-(n+1)}$ be the left end-point of $S_n$.

For fixed $p\in (2,\,\infty )$, let $\varepsilon$ be a fixed rational number such that

\[ 0<\varepsilon<\frac2p<1. \]

We can choose a strictly increasing sequence $k(1)<\dots < k(n)<\dots$ of positive integers such that

\[ 2^{-(\frac2p+\varepsilon)k(n)} \cdot 2\cdot 2^{ \varepsilon k(n) }=2^{- \frac2p k(n)+1} \le (1/3) 2^{-(n+1)}=|S_n| \]

for all $n$ and such that every $\varepsilon k(n)$ is an integer.

For integers $n\ge 1$ and $1\le j\le 2^{ \varepsilon k(n)}$, recall that $t_n$ is the left end-point of $S_n$. Define the intervals

\[ J_{n,j}=(a_{n,j},c_{n,j})=\left(t_n+2^{-(\frac2p+\varepsilon)k(n)} \cdot2\cdot (j-1), t_n+2^{-(\frac2p+\varepsilon)k(n)} \cdot2\cdot j\right) \]

and

\[ I_{n,j}=(a_{n,j},b_{n,j})=\left(t_n+2^{-(\frac2p+\varepsilon)k(n)} \cdot2\cdot (j-1), t_n+2^{-(\frac2p+\varepsilon)k(n)} \cdot (2j-1)\right). \]

It is easy to check that $I_{n,j}$ is the left half of $J_{n,j}$, $J_{n,j}$'s are pairwise disjoint,

\[ \bigcup_{j=1}^{2^{ \varepsilon k(n)}}J_{n,j}\subset S_n, \]

and the length of the interval $I_{n,j}$ is denoted by $\rho _n$, that is

(4.1)\begin{equation} \rho_n=|I_{n,j}|=b_{n,j}- a_{n,j}=2^{-(\frac2p+\varepsilon)k(n)}. \end{equation}

We now define a measurable function $u$ on the unit circle $\mathbb {T}=\{w\in \mathbb {C}:|w|=1\}$ as follows:

\begin{align*} u(\mathrm{e}^{\mathrm{i} t}) & =2^{{-}k(n)} \quad \hbox{ if } t\in \bigcup_{j=1}^{2^{ \varepsilon k(n)}}I_{n,j},n\ge1 ,\\ u(\mathrm{e}^{\mathrm{i} t}) & =1 \quad \hbox{ if } t\in (-\pi,\pi]\setminus\left(\bigcup_{n=1}^\infty \bigcup_{j=1}^{2^{ \varepsilon k(n)}}I_{n,j}\right). \end{align*}

The harmonic extension of $u$ to $\mathbb {D}$ is also denoted by $u$. Let

\[ h(z)=\frac1{2\pi}\int_{-\pi}^\pi\frac{\mathrm{e}^{\mathrm{i} t}+z}{\mathrm{e}^{\mathrm{i} t}-z} \,u(\mathrm{e}^{\mathrm{i} t})\,\mathrm{d} t \]

and

(4.2)\begin{equation} \varphi(z)=\exp({-}h(z)) \end{equation}

for all $z\in \mathbb {D}$. Then, $\mathrm {Re} (h(z))=u(z)>0$ for each $z\in \mathbb {D}$, and thus,

\[ |\varphi(z)|=\mathrm{e}^{\mathrm{Re} (h(z))}=\mathrm{e}^{{-}u(z)}<1. \]

This implies $\varphi (\mathbb {D})\subset \mathbb {D}$. We will need the fact that $\varphi \in H^2$ with

(4.3)\begin{equation} \|\varphi\|_{H^2}^2=\sup_{0< r<1}\frac1{2\pi}\int_{-\pi}^\pi\left| \varphi(r\mathrm{e}^{ \mathrm{i}\theta})\right|^2\mathrm{d}\theta= \frac1{2\pi}\int_{-\pi}^\pi\left| \varphi( \mathrm{e}^{ \mathrm{i}\theta})\right|^2\mathrm{d}\theta. \end{equation}

4.2 Estimates

For $z\in \mathbb {D}$ and $\mathrm {e}^{\mathrm {i} t}\in \mathbb {T}$, let

\[ P(z,\mathrm{e}^{\mathrm{i} t})=\frac{1-|z|^2}{|\mathrm{e}^{\mathrm{i} t}-z|^2} \]

be the Poisson kernel. It is shown in [10, p. 2508] that if $1/2\le r<1$ and $|\theta -t|\le 5$, then there exist constants $0<\alpha <\beta <\infty$ such that

(4.4)\begin{equation} \frac{\alpha(1-r)}{(1-r)^2+(\theta-t)^2}\le \frac1{2\pi} P(r\mathrm{e}^{\mathrm{i}\theta},\mathrm{e}^{\mathrm{i} t})\le \frac{\beta(1-r)}{(1-r)^2+(\theta-t)^2} . \end{equation}

We have the following lemma modified from [10, Lemma 4].

Lemma 4.1 For any positive integer $n$ and $1\le j\le 2^{ \varepsilon k(n)}$, let $G_{n,j}$ be the Carleson box based on $I_{n,j}$, i.e.

(4.5)\begin{equation} G_{n,j}=\left\{r\mathrm{e}^{\mathrm{i}\theta}: \theta\in I_{n,j},0<1-r\le\rho_n\right\}\!. \end{equation}

Then there is a constant $C_1$ independent of $n,\,j$ such that

(4.6)\begin{equation} \int_{G_{n,j}}\left(\frac{1-|z|}{1-|\varphi(z)|}\right)^{p/2}\mathrm{d} \lambda(z)\le C_1 2^{-\frac{p\varepsilon}2 k(n) }. \end{equation}

Proof. Given such a pair of $n,\,j$, we write

\[ G_{n,j}=\bigcup_{\nu=0}^{k(n)} G_{n,j}^\nu, \]

where

\[ G_{n,j}^0=\left\{r\mathrm{e}^{\mathrm{i}\theta}: \theta\in I_{n,j}, 0<1-r\le\rho_n\cdot 2^{{-}k(n)}\right\}\!, \]

and

\[ G_{n,j}^\nu=\left\{r\mathrm{e}^{\mathrm{i}\theta}: \theta\in I_{n,j}, \rho_n\cdot 2^{{-}k(n)}\cdot 2^{\nu-1}<1-r\le\rho_n\cdot 2^{{-}k(n)}\cdot 2^{\nu }\right\}\!, \]

for $1\le \nu \le k(n)$.

It is shown in [10, p. 2509] that there is a constant $0< c<1$ independent of $n,\,j$ such that

\[ 1-|\varphi(z)|=1-\mathrm{e}^{{-}u(z)}\ge 1-\exp({-}c 2^{{-}k(n)+\nu}) \]

if $z\in G_{n,j}^\nu$ and $0\le \nu \le k(n)$. Let $\delta =\inf _{0< x\le 1}x^{-1}(1-\mathrm {e}^{-x})$. Then,

(4.7)\begin{align} \inf_{z\in G_{n,j}^\nu}\left(1-|\varphi(z)|\right)^{p/2}\ge (\delta c)^{p/2}\cdot 2^{{-}p/2k(n)}\cdot 2^{p/2\nu}, \quad 0\le\nu \le k(n). \end{align}

This implies that

(4.8)\begin{align} & \int_{G_{n,j}}\left(\frac{1-|z|}{1-|\varphi(z)|}\right)^{p/2}\mathrm{d} \lambda(z)\nonumber\\& \quad=\int_{G^0_{n,j}}\left(\frac{1-|z|}{1-|\varphi(z)|}\right)^{p/2}\mathrm{d} \lambda(z)+\sum_{\nu=1}^{k(n)} \int_{G^\nu_{n,j}}\left(\frac{1-|z|}{1-|\varphi(z)|}\right)^{p/2}\mathrm{d} \lambda(z)\nonumber\\ & \quad\le\frac{2^{p/2k(n)}}{(\delta c)^{p/2}}\int_{G^0_{n,j}}(1-|z|^2)^{p/2-2}\mathrm{d} A(z)\nonumber\\& \qquad+ \sum_{\nu=1}^{k(n)}\frac{2^{p/2k(n)}}{(\delta c)^{p/2} \cdot 2^{p/2\nu}} \int_{G^\nu_{n,j}} (1-|z| )^{p/2-2}\mathrm{d} A(z). \end{align}

Notice that $p/2-2>-1$. Straightforward computation shows that

(4.9)\begin{align} \int_{G^0_{n,j}}(1-|z|^2)^{p/2-2}\mathrm{d} A(z) & =\frac1\pi \int_{I_{n,j}} \mathrm{d} \theta\int_{1-\rho_n\cdot2^{{-}k(n)}}^1(1-r^2)^{p/2-2} r\, \mathrm{d} r\nonumber\\& \le C_2 \rho_n^{p/2}\cdot 2^{-(p/2-1)k(n)} \end{align}

for some $C_2>0$, and

(4.10)\begin{align} \int_{G^\nu_{n,j}}(1-|z| )^{p/2-2}\mathrm{d} A(z) & =\frac1\pi \int_{I_{n,j}} \mathrm{d} \theta\int_{1-\rho_n\cdot2^{{-}k(n)}\cdot 2^\nu}^{1-\rho_n\cdot 2^{{-}k(n)}\cdot 2^{\nu-1}}(1-r )^{p/2-2}r\, \mathrm{d} r\nonumber\\& \le C_3 \rho_n^{p/2}\cdot 2^{-(p/2-1)k(n)} \cdot 2^{(p/2-1)\nu} \end{align}

for some $C_3>0$. Put (4.8), (4.9) and (4.10) together, we have

\begin{align*} & \int_{G_{n,j}}\left(\frac{1-|z|}{1-|\varphi(z)|}\right)^{p/2}\mathrm{d} \lambda(z)\\& \quad \le\frac{C_2 \cdot 2^{ k(n)} \cdot \rho_n^{p/2}}{(\delta c)^{p/2}} + \sum_{\nu=1}^{k(n)}\frac{2^{p/2k(n)}\cdot C_3 \rho_n^{p/2}\cdot 2^{-(p/2-1)k(n)} \cdot 2^{(p/2-1)\nu}}{(\delta c)^{p/2} \cdot 2^{p/2\nu}}\\ & \quad= 2^{ k(n)} \cdot \rho_n^{p/2}\cdot\left( \frac{C_2 }{(\delta c)^{p/2}}+ \frac{C_3 }{(\delta c)^{p/2}} \sum_{\nu=1}^{k(n)} 2^{-\nu}\right)\!. \end{align*}

Recall the inequality (4.1), we get the desired inequality (4.6) by letting

\[ C_1= \frac{C_2 }{(\delta c)^{p/2}}+ \frac{C_3 }{(\delta c)^{p/2}} \sum_{\nu=1}^{\infty} 2^{-\nu}= \frac{C_2 +C_3 }{(\delta c)^{p/2}}. \]

The following lemma is quoted from [10, Lemma 7].

Lemma 4.2 There is a $C_4>0$ such that

\[ u(z)\ge C_4 \quad\hbox{ for every }\quad z\in\mathbb{D}\setminus\left( \bigcup_{n=1}^\infty \bigcup_{j=1}^{2^{ \varepsilon k(n)}} G_{n,j}\right)\!, \]

where $G_{n,j}$ is defined by (4.5).

4.3 Proof of theorem 1.2

Let $\varphi$ be the holomorphic self-map of $\mathbb {D}$ given by (4.2). It is sufficient to check the inequality (1.2) for this $\varphi$, and $C_{\varphi }\notin \mathcal {S}_p(H^2)$.

Let

\[ G=\bigcup_{n=1}^\infty \bigcup_{j=1}^{2^{ \varepsilon k(n)}} G_{n,j}, \]

where $G_{n,j}$ is given by (4.5). For $z\in \mathbb {D}\setminus G$, lemma 4.2 implies that

\[ |\varphi(z)|=\mathrm{e}^{-\mathrm{Re}(h(z))}=\mathrm{e}^{{-}u(z)}\le \mathrm{e}^{{-}C_4}. \]

Since $p/2-2>-1$, we have

(4.11)\begin{align} \int_{\mathbb{D}\setminus G}\left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p/2}\mathrm{d} \lambda(z)& \le \frac1{(1-\mathrm{e}^{{-}C_4})^{p/2}} \int_{\mathbb{D}\setminus G} (1-|z|^2)^{p/2-2}\mathrm{d} A(z) \nonumber\\ & \le \frac1{(1-\mathrm{e}^{{-}C_4})^{p/2}} \int_{\mathbb{D}} (1-|z|^2)^{p/2-2}\mathrm{d} A(z)<\infty. \end{align}

Meanwhile, lemma 4.1 implies that

(4.12)\begin{align} \int_G \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p/2}\mathrm{d} \lambda(z)& \approx \int_G \frac{(1-|z|)^{p/2-2} }{(1-|\varphi(z)|)^{p/2} }\, \mathrm{d} A(z)\nonumber\\& =\sum_{n=1}^\infty \sum_{j=1}^{2^{\varepsilon k(n)}}\int_{G_{n,j}} \frac{(1-|z|)^{p/2-2} }{(1-|\varphi(z)|)^{p/2} }\, \mathrm{d} A(z) \nonumber\\ & \le C_1 \sum_{n=1}^\infty 2^{\varepsilon k(n)}\cdot 2^{-\frac{p\varepsilon}2 k(n) } \le C_1 \sum_{n=1}^\infty 2^{-(p/2-1)\varepsilon k(n)}<\infty, \end{align}

where the last inequality is following from the fact that $p/2-1>0$. Now (1.2) follows from (4.11) and (4.12) easily.

It remains to check that $C_{\varphi }\notin \mathcal {S}_p(H^2)$, or equivalently, $\mathrm {tr}((C_{\varphi }^\ast C_{\varphi })^{\frac p2})=\infty$. Let $e_\ell (z)=z^\ell$, $\ell =0,\,1,\,2,\,\dots$. It is well known that $\{e_\ell :\ell \ge 0\}$ is an orthonormal basis for $H^2$. Since $p/2>1$, we have

\begin{align*} & \left\langle \left(C_{\varphi}^\ast C_{\varphi}\right)^{p/2} e_\ell,e_\ell\right \rangle_{H^2}\ge \left(\left\langle C_{\varphi}^\ast C_{\varphi} e_\ell, e_\ell\right\rangle_{H^2}\right)^{p/2}\\ & \quad =\left\| C_{\varphi} e_\ell\right\|_{H^2}^p= \left\| \varphi^l \right\|_{H^2}^p=\left(\frac1{2\pi}\int_{-\pi}^\pi \left|\varphi(\mathrm{e}^{ \mathrm{i}\theta})\right|^{2\ell}\mathrm{d}\theta\right)^{p/2}. \end{align*}

Write

\[ I_n=\bigcup_{j=1}^{2^{\varepsilon k(n)}}I_{n,j}. \]

Then,

\[ |I_n|=2^{\varepsilon k(n)}\rho_n= 2^{- \frac2p k(n)}, \]

and

\[ \left|\varphi(\mathrm{e}^{\mathrm{i}\theta})\right|= \exp({-u(\mathrm{e}^{\mathrm{i}\theta})}) =\exp({-}2^{{-}k(n)}) \]

for almost every $\theta \in I_n$. Thus,

\[ \int_{-\pi}^\pi \left|\varphi(\mathrm{e}^{ \mathrm{i}\theta})\right|^{2\ell}\mathrm{d}\theta \ge\sum_{n=1}^\infty \int_{I_{n }} \left|\varphi(\mathrm{e}^{ \mathrm{i}\theta})\right|^{2\ell}\mathrm{d} \theta= \sum_{n=1}^\infty \mathrm{e}^{{-}2\ell\cdot 2^{{-}k(n)}}\cdot 2^{- \frac2p k(n)}. \]

Notice that

\[ \left(\sum_n a_n\right)^s\ge \sum_n a_n^s \]

if $s\ge 1$ and $a_n\ge 0$. We get

\[ \left( \int_{-\pi}^\pi \left|\varphi(\mathrm{e}^{ \mathrm{i}\theta})\right|^{2\ell}\mathrm{d}\theta \right)^{p/2}\ge \left(\sum_{n=1}^\infty \mathrm{e}^{{-}2\ell\cdot 2^{{-}k(n)}}\cdot 2^{- \frac2p k(n)}\right)^{p/2}\ge \sum_{n=1}^\infty \mathrm{e}^{{-}p\ell\cdot 2^{{-}k(n)}}\cdot 2^{- k(n)}. \]

This gives that

\begin{align*} \mathrm{tr}\left((C_{\varphi}^\ast C_{\varphi})^{p/2}\right)& =\sum_{\ell=0}^\infty \left\langle (C_{\varphi}^\ast C_{\varphi})^{p/2} e_\ell,e_\ell\right\rangle_{H^2}\ge \sum_{\ell=0}^\infty \left(\frac1{2\pi}\int_{-\pi}^\pi \left|\varphi(\mathrm{e}^{ \mathrm{i}\theta})\right|^{2\ell}\mathrm{d}\theta\right)^{p/2}\\ & \ge\frac1{(2\pi)^{p/2}} \sum_{\ell=0}^\infty \sum_{n=1}^\infty \mathrm{e}^{{-}p\ell\cdot 2^{{-}k(n)}}\cdot 2^{- k(n)}\\ & =\frac1{(2\pi)^{p/2}}\sum_{n=1}^\infty \left( 2^{- k(n)} \sum_{\ell=0}^\infty \mathrm{e}^{{-}p\ell\cdot 2^{{-}k(n)}}\right)\\ & = \frac1{(2\pi)^{p/2}}\sum_{n=1}^\infty 2^{- k(n)} \cdot\frac1{1-\mathrm{e}^{{-}p \cdot 2^{{-}k(n)}}}. \end{align*}

Since

\[ \sup_{x>0}\frac{1-\mathrm{e}^{{-}x}}x\le1. \]

We have

\[ \frac1{1-\mathrm{e}^{{-}p \cdot 2^{{-}k(n)}}}\ge \frac1{p \cdot 2^{{-}k(n)}}. \]

Then,

\[ \sum_{n=1}^\infty 2^{- k(n)} \cdot\frac1{1-\mathrm{e}^{{-}p \cdot 2^{{-}k(n)}}}\ge \sum_{n=1}^\infty 2^{- k(n)} \cdot\frac1{p \cdot 2^{{-}k(n)}}=\sum_{n=1}^\infty\frac1p=\infty. \]

This implies that $C_{\varphi }\notin \mathcal {S}_p(H^2)$ and the proof is complete.