Composition operators on the normal weight Dirichlet type space in the unit disc

Abstract

Let p>0 and \nu be a normal function on [0,1). In this paper, several equivalent characterizations are given for which composition operators are bounded or compact on the normal weight Dirichlet type space {\mathcal{D}}_{\nu }^p \left(D\right) in the unit disc.

Keywords

Composition operator / normal weight Dirichlet type space / boundedness / compactness / unit disc

1 Introduction

Let \mathbb{C} denote the complex plane, and denote the unit disc in \mathbb{C}. The class of analytic functions on D is denoted by H(D). Suppose that \mathrm{d}v denotes the normalized Lebesgue measure on D.

Definition 1.1　A positive continuous function on [0,1) is called normal if there exist constants and b\ge a>0 such that \frac{\nu (\rho )}{(1-{\rho }^2{)}^b} is increasing and \frac{\nu (\rho )}{(1-{\rho }^2{)}^a} is decreasing on [{\rho }_0, 1). For example, \nu (\rho )=(1 -{\rho }^2{)}^{\alpha }{\left(\log \frac{e}{1-{\rho }^2}\right)}^{\beta } {\left(\log \log \frac{e^2}{1-{\rho }^2}\right)}^{\gamma } (\alpha >0,\beta \text{and}\gamma \text{real}),

n=1,2,\dots ,b>a>0, are such normal functions.

For the convenience of proof, let {\rho }_0= 0 in this paper.

Definition 1.2　Let p>0 and \nu be a normal function on [0,1). If h\in H(D ) and

then we say that h belongs to the normal weight Dirichlet type space {\mathcal{D}}_{\nu }^p(D).

In particular, {\mathcal{D}}_{\nu }^p(D) is the Dirichlet type space {\mathcal{D}}_{\alpha }^p (D) when \nu (\rho )=(1-{\rho }^2{)}^{\alpha +1}( \alpha >-1 ). Further, {\mathcal{D}}_{\nu }^p (D) is the Dirichlet space when \nu (\rho )= 1-{\rho }^2 and p=2.

Definition 1.3　Let \phi :D\to D be an analytic mapping. Then \phi induces a linear operator C_{\phi }h=h\circ \phi on H(D). We say that C_{\phi } is a composition operator on H(D).

Composition operators can act on the function space of various support sets to itself or between different function spaces. There have been a lot of references in this field, and many of them involve in composition operator problems on the Dirichlet space or the weighted Dirichlet space (for example, [1-14,16-18,20]). In each case, the main purpose is to find out the relationship between the properties of C_{\phi } and the induced symbol \phi. The results closely related to this paper are as follows:

Theorem 1.1 [4, Theorem 3.5]　 Let \mu be a positive function on the unit interval with such that, for each q>1, there is a constant k=k(q) satisfying \mu (s)\le k\nu (t) whenever s\le qt. For p\ge 1, suppose Y is the Banach space of all analytic functions on the disc for which the norm given by

If \phi is an automorphism of the unit disc, then C_{\phi } is a bounded operator on Y.

Theorem 1.2 [4, p142]　 For \alpha >-1, let \phi :D\to D be an analytic mapping. Define the measure {\mu }_{\alpha } on D by \mathrm{d}{\mu }_{\alpha } =|{\phi }^{\prime }(\omega ){|}^2(1 -|\omega {|}^2{)}^{\alpha }\mathrm{d}v(\omega ). Then

(1) C_{\phi } is bounded on D_{\alpha }^2(D) if and only if {\mu }_{\alpha }{\phi }^{-1}S(\xi ,h)=O(h^{\alpha +2}),

(2) C_{\phi } is compact on D_{\alpha }^2(D) if and only if \underset{\xi \in \mathrm{\partial }D}{sup}{\mu }_{\alpha }{\phi }^{-1}S(\xi ,h)=o(h^{\alpha +2}) (h\to 0^{+}).

Theorem 1.3 [10, Theorem 2.5 and Theorem 3.4]　For \alpha >0, let \phi \in D_{\alpha }^2(D) such that \phi (D) \subseteq D and . Then

(1) C_{\phi } is bounded on D_{\alpha }^2(D) if and only if

\underset{\xi \in \mathrm{\partial }D}{sup}{\int }_{E(\xi ,r)}\sum _{\phi (\omega )=w}(1-|\omega {|}^2{)}^{\alpha } \mathrm{d}v(w )=O(r^{\alpha }),

(2) C_{\phi } is compact on D_{\alpha }^2(D) if and only if

\underset{\xi \in \mathrm{\partial }D}{sup}{\int }_{E(\xi ,r)}\sum _{\phi (\omega )=w}(1-|\omega {|}^2{)}^{\alpha } \mathrm{d}v(w )=o(r^{\alpha })(r\to 0^{+}).

On the basis of the above results, we give some equivalent characterizations for which composition operators are bounded or compact on {\mathcal{D}}_{\nu }^p (D) for the general normal weight on [0,1) and all p>0 in this paper.

In the following, if there exist constants l>0 and m>0 such that lQ\le P\le mQ, then we say that P and Q are equivalent, written as “P\approx Q”. If there exists a constant l>0 such that P\le lQ (P\ge lQ), then we write “P\lesssim Q” (“P\gtrsim Q”).

2 Some lemmas

For \omega \in D, let {\phi }_{\omega } be the involutive automorphism of D with {\phi }_{\omega } (0)=\omega and {\phi }_{\omega } (\omega )=0. For \rho >0 and \omega \in D, we define the Bergman disc on D, where

\beta (w,\omega )=\frac{1}{2}\log \frac{1+|{\phi }_{\omega }(w)|}{1-|{\phi }_{\omega }(w)|}.

For \theta \in [0,2\pi ] and t>0, let . We know that S({\mathrm{e}}^{\mathrm{i}\theta }, t)=D when t\ge 2. Therefore, we only need to consider the smaller t.

Lemma 2.1 [15]　 Let ϵ>0 and \delta >- 1. Then

{\int }_D\frac{(1-|\omega {|}^2{)}^{\delta }}{|1-w\overline{\omega }{|}^{2+\delta +ϵ}}\mathrm{d}v(\omega )\approx \frac{1}{(1-|w{|}^2{)}^{ϵ}}forallw\in D.

This result comes from Proposition 1.4.10 in [15].

Lemma 2.2 [21]　 There exists a positive integer N such that for any we can find a sequence \{a_k\} \subset D with the following properties: D=\bigcup _{k=1}^{\mathrm{\infty }}D(a_k,r), and each point z\in D belongs to at most N of the sets D(a_k,4r).

These results come from Theorem 2.23 in [21].

Lemma 2.3 [19]　 Let r>0 and \nu be a normal function on [0,1). Then

(1) \nu (|w|)\approx \nu (|\omega |) when w\in D and \omega \in D(w, r),

(2) \frac{\nu (|w|)}{\nu (|\omega |)}\le {\left(\frac{1-|w{|}^2}{1-|\omega {|}^2}\right)}^a+{\left(\frac{1-|w{|}^2}{1-|\omega {|}^2}\right)}^b for all w,\omega \in D.

These results come from Lemma 2.2 in [19].

Lemma 2.4　 Let p>0 and \nu be a normal function on [0,1). Suppose that \phi is an analytic self mapping on D, and \phi \in D_{\nu }^p (D). If g is a nonnegative measurable function on D, then

{\int }_{D}g(\omega ) \mathrm{d}{\mu }_{p,\nu ,\phi }(\omega )={\int }_{D}g [\phi (\omega )]\frac{|{\phi }^{\prime }(\omega ){|}^p\nu (|\omega |)}{1-|\omega {|}^2} \mathrm{d}v(\omega ),

where

{\mu }_{p,\nu ,\phi }(A)={\int }_{{\phi }^{-1}(A)}\frac{|{\phi }^{\prime }(\omega ){|}^p\nu (|\omega |)}{1-|\omega {|}^2}\mathrm{d}v(\omega ),

A is any Borel measurable set on D.

Proof　The condition \phi \in D_{\nu }^p(D) guarantees that \frac{|{\phi }^{\prime }(\omega ){|}^p\nu (|\omega |)}{1-|\omega {|}^2}\mathrm{d}v(\omega ) is a finite measure on D. This is a general variable substitution formula, which appears in different forms on different occasions. Detailed proof is omitted.

3 Main results and proofs

Theorem 3.1　 Let p>0 and \nu be a normal function on [0,1). Suppose that \phi is an analytic self mapping on D, and \phi \in D_{\nu }^p (D). Given r>0 and a finite Borel measure \mathrm{d}{\mu }_{p,\nu ,\phi }(z)= \mathrm{d}{m}_{p,\nu ,\phi }{\phi }^{-1}(z), where

\mathrm{d}{m}_{p,\nu ,\phi }(z)=\frac{|{\phi }^{\prime }(z){|}^p\nu (|z|)}{1-|z{|}^2}\mathrm{d}v(z ) (z\in D).

Then the following three conditions are equivalent:

(1) {\mu }_{p,\nu ,\phi }[S({\mathrm{e}}^{\mathrm{i}\theta },t)]\lesssim t \nu (1-t) for all \theta \in [0,2 \pi ] and ;

(2) {\mu }_{p,\nu ,\phi }[D(w,r )]\lesssim (1- | w{|}^2)\nu (|w|) for all w\in D;

(3) C_{\phi } is a bounded operator on {\mathcal{D}}_{\nu }^p (D).

Proof　 (1)\Rightarrow (2).

Let {\mu }_{p,\nu ,\phi }[S({\mathrm{e}}^{\mathrm{i}\theta },t)]\lesssim t \nu (1-t) for all \theta \in [0,2 \pi ] and .

For any 0\ne w\in D, let w/|w|={\mathrm{e}}^{\mathrm{i}\theta }. When z\in D(w ,r), it follows from Proposition 4.5 in [22] that

This shows that D(w,r) \subset S[{\mathrm{e}}^{\mathrm{i}\theta },2(1- | w{|}^2)/(1-\tanh r)].

Otherwise, recall the definition of normal function and

When |w{|}^2>(3+\tanh r )/4, we have . Therefore, we may obtain that

\begin{array}{rl}{\mu }_{p,\nu ,\phi }[D(w,r)]& \le {\mu }_{p,\nu ,\phi }\left[S\left({\mathrm{e}}^{\mathrm{i}\theta },\frac{2(1-|w{|}^2)}{1-\tanh r}\right)\right]\\ & \lesssim \frac{2(1-|w{|}^2)}{1-\tanh r}\nu \left[1-\frac{2(1-|w{|}^2)}{1-\tanh r}\right]\\ & \lesssim (1-|w{|}^2)\nu (|w|).\end{array}

When |w{|}^2\le (3+\tanh r )/4, we have (1-|w{|}^2)\nu (|w|)\ge \frac{1-\tanh r}{4}\nu \left(\sqrt{\frac{3+\tanh r}{4}}\right)

\Rightarrow {\mu }_{p,\nu ,\phi }[D(w,r )]\le {\mu }_{p,\nu ,\phi }(D)\lesssim 1\lesssim (1- | w{|}^2)\nu (|w|).

This shows that {\mu }_{p,\nu ,\phi }[D(w ,r)]\lesssim (1-|w{|}^2)\nu (|w|) for all w\in D.

(2) \Rightarrow (3).

Let {\mu }_{p,\nu ,\phi }[D(w,r )]\lesssim (1- | w{|}^2)\nu (|w|) for all w\in D.

For any f\in {\mathcal{D}}_{\nu }^p (D), by Lemma 2.20 and Lemma 2.24 in [21], Lemmas 2.2−2.4, we have

\begin{array}{rl}||C_{\phi }f|{|}_{{\mathcal{D}}_{\nu }^p}^p& ={\int }_D|f^{\prime }[\phi (z)]{|}^p|{\phi }^{\prime }(z){|}^p\frac{\nu (|z|)}{1-|z{|}^2}\mathrm{d}v(z)\\ & ={\int }_D|f^{\prime }(z){|}^p\mathrm{d}{\mu }_{p,\nu ,\phi }(z)\\ & \le \sum _{k=1}^{\mathrm{\infty }}{\int }_{D(a_k,r)}|f^{\prime }(z){|}^p\mathrm{d}{\mu }_{p,\nu ,\phi }(z)\\ & \le \sum _{k=1}^{\mathrm{\infty }}{\mu }_{p,\nu ,\phi }[D(a_k,r)]\underset{z\in D(a_k,r)}{sup}|f^{\prime }(z){|}^p\\ & \lesssim \sum _{k=1}^{\mathrm{\infty }}\frac{\nu (|a_k|)}{1-|a_k{|}^2}\underset{z\in D(a_k,r)}{sup}{\int }_{D(z,r)}|f^{\prime }(w){|}^p\mathrm{d}v(w)\\ & \le \sum _{k=1}^{\mathrm{\infty }}\frac{\nu (|a_k|)}{1-|a_k{|}^2}{\int }_{D(a_k,2r)}|f^{\prime }(w){|}^p\mathrm{d}v(w)\\ & \lesssim \sum _{k=1}^{\mathrm{\infty }}{\int }_{D(a_k,4r)}\frac{|f^{\prime }(w){|}^p\nu (|w|)\mathrm{d}v(w)}{1-|w{|}^2}\\ & \le N{\int }_D\frac{|f^{\prime }(w){|}^p\nu (|w|)\mathrm{d}v(w)}{1-|w{|}^2}\\ & =N||f|{|}_{{\mathcal{D}}_{\nu }^p}^p.\end{array}

This means that C_{\phi } is bounded on {\mathcal{D}}_{\nu }^p (D).

(3) \Rightarrow (1).

Let C_{\phi } be bounded on {\mathcal{D}}_{\nu }^p (D). For any \theta \in [0,2 \pi ] and , we take

f_{t,\theta }(z) =\frac{t^{\frac{b}{p}+1}}{{\nu }^{\frac{1}{p}}(1-t)[1-(1-t)z{e}^{-i\theta }{]}^{\frac{b+1}{p}}} (z\in D).

When z\in S( {\mathrm{e}}^{\mathrm{i}\theta },t), we have . By the boubdedness of C_{\phi } on {\mathcal{D}}_{\nu }^p (D) and Lemmas 2.1 and 2.2, we have

\begin{array}{rl}\frac{(b+1{)}^p{\mu }_{p,\nu ,\phi }[S({\mathrm{e}}^{\mathrm{i}\theta },t)]}{p^p{2}^{b+1+2p}t\nu (1-t)}& \le {\int }_{S(e^{i\theta },t)}|f_{t,\theta }^{\prime }(z){|}^p\mathrm{d}{\mu }_{p,\nu ,\phi }(z)\\ & \le {\int }_D|f_{t,\theta }^{\prime }(z){|}^p\mathrm{d}{\mu }_{p,\nu ,\phi }(z)\\ & \lesssim {\int }_D\frac{t^{b+p}}{\nu (1-t)|1-z(1-t){\mathrm{e}}^{-\mathrm{i}\theta }{|}^{1+b+p}}\frac{\nu (|z|)}{1-|z{|}^2}\mathrm{d}v(z)\\ & \le {\int }_D\frac{t^{b+p-a}(1-|z{|}^2{)}^{a-1}}{|1-z(1-t){\mathrm{e}}^{-\mathrm{i}\theta }{|}^{1+b+p}}\mathrm{d}v(z)\\ & +{\int }_D\frac{t^p(1-|z{|}^2{)}^{b-1}}{|1-z(1-t){\mathrm{e}}^{-\mathrm{i}\theta }{|}^{1+b+p}}\mathrm{d}v(z)\\ & \lesssim 1.\end{array}

This shows that {\mu }_{p,\nu ,\phi }[S({\mathrm{e}}^{\mathrm{i}\theta },t)] \lesssim t\nu (1-t) for all \theta \in [0,2 \pi ] and .

This proof is complete.□

Theorem 3.2　 Let p>0 and \nu be a normal function on [0,1 ). Suppose that \phi is an analytic self mapping on D, and \phi \in D_{\nu }^p(D). Give r>0 and a finite Borel measure \mathrm{d}{\mu }_{p,\nu ,\phi }(z) =\mathrm{d}{m}_{p,\nu ,\phi }{\phi }^{-1}(z), where

\mathrm{d}{m}_{p,\nu ,\phi }(z)=\frac{|{\phi }^{\prime }(z){|}^p\nu (|z|)}{1-|z{|}^2}\mathrm{d}v(z ) (z\in D).

Then the following three conditions are equivalent:

(1) \underset{\theta \in [0,2\pi ]}{sup}{\mu }_{p,\nu ,\phi }[S({\mathrm{e}}^{\mathrm{i}\theta },t)]=o [t\nu (1-t )] (t\to 0^{+});

(2) {\mu }_{p,\nu ,\phi }[D(w,r )]=o[(1-|w{|}^2)\nu (|w|)] (|w|\to 1^{-});

(3) C_{\phi } is a compact operator on {\mathcal{D}}_{\nu }^p (D).

Proof　 (1)\Rightarrow (2).

Let \underset{\theta \in [0,2\pi ]}{sup}{\mu }_{p,\nu ,\phi }[S({\mathrm{e}}^{\mathrm{i}\theta },t)]=o [t\nu (1-t )] (t\to 0^{+}). For any w\in D, let w/|w|={\mathrm{e}}^{\mathrm{i}\theta } when |w{|}^2>(3+\tanh r )/4. It follows Theorem 3.1 that

\frac{\mu [D(w,r)]}{(1-|w{|}^2)\nu (|w|)}\le \mu \left[S\left({\mathrm{e}}^{\mathrm{i}\theta },\frac{2(1-|w{|}^2)}{1-\tanh r}\right)\right]/(1-|w{|}^2)\nu (|w|).

When |w|\to 1^{-}, we have \frac{2(1-|w{|}^2)}{1-\tanh r}\to 0. Therefore, we have

\underset{|w|\to 1^{-}}{lim}\frac{\mu [D(w,r)]}{(1-|w{|}^2)\nu (|w|)}=0.

(2) \Rightarrow (3).

If \mu [D(w,r )]=o[(1-|w{|}^2)\nu (|w|)] (|w|\to 1^{-} ), then for any \epsilon >0, there exists a such that

(3.1)

Suppose that \{f_j\} is any analytic function sequence which converges to 0 uniformly on any compact subset of D and ||f_j|{|}_{{\mathcal{D}}_{\nu }^p}\le 1 for all j\in \{1 ,2,\dots \}. We take the sequence \{a_k\} in Lemma 2.2, and we may let |a_k|\to 1^{-}(k\to \mathrm{\infty }). Therefore, there is a positive integer K_0 such that |a_k|>{\delta }_0 when k>K_0. Let E=\bigcup _{k=1}^{K_0}D(a_k,r). By (3.1), Lemmas 2.2−2.4, Lemma 2.20 and Lemma 2.24 in [21], we have

\begin{array}{rl}||C_{\phi }{f}_j|{|}_{{\mathcal{D}}_{\nu }^p}^p& ={\int }_D|f_j(z){|}^p\mathrm{d}{\mu }_{p,\nu ,\phi }(z)\\ & \le \sum _{k=1}^{\mathrm{\infty }}{\int }_{D(a_k,r)}|f_j(z){|}^p\mathrm{d}{\mu }_{p,\nu ,\phi }(z)\\ & \le \sum _{k=1}^{K_0}{\mu }_{p,\nu ,\phi }(D)\underset{z\in \overline{E}}{sup}|f_j(z){|}^p+\sum _{k=K_0+1}^{\mathrm{\infty }}{\mu }_{p,\nu ,\phi }[D(a_k,r)]\underset{z\in D(a_k,r)}{sup}|f_j(z){|}^p\\ & \lesssim \sum _{k=1}^{K_0}{\mu }_{p,\nu ,\phi }(D)\underset{z\in \overline{E}}{sup}|f_j(z){|}^p\\ & +\epsilon \sum _{k=K_0+1}^{\mathrm{\infty }}\frac{\nu (|a_k|)}{1-|a_k{|}^2}\underset{z\in D(a_k,r)}{sup}{\int }_{D(z,r)}|f_j(w){|}^p\mathrm{d}v(w)\\ & \lesssim \sum _{k=1}^{K_0}{\mu }_{p,\nu ,\phi }(D)\underset{z\in \overline{E}}{sup}|f_j(z){|}^p+\epsilon \sum _{k=K_0+1}^{\mathrm{\infty }}{\int }_{D(a_k,4r)}\frac{|f_j(w){|}^p\nu (|w|)\mathrm{d}v(w)}{1-|w{|}^2}\\ & \le \sum _{k=1}^{K_0}{\mu }_{p,\nu ,\phi }(D)\underset{z\in \overline{E}}{sup}|f_j(z){|}^p+N\epsilon {\int }_D\frac{|f_j(w){|}^p\nu (|w|)\mathrm{d}v(w)}{1-|w{|}^2}\\ & \le \sum _{k=1}^{K_0}{\mu }_{p,\nu ,\phi }(D)\underset{z\in \overline{E}}{sup}|f_j(z){|}^p+N\epsilon \to N\epsilon (j\to \mathrm{\infty }).\end{array}

By the arbitrariness of \epsilon, we have \underset{j\to \mathrm{\infty }}{lim}||C_{\phi }{f}_j|{|}_{{\mathcal{D}}_{\nu }^p}=0. This shows that C_{\phi } is a compact operator on {\mathcal{D}}_{\nu }^p (D).

(3) \Rightarrow (1).

Let C_{\phi } be compact on {\mathcal{D}}_{\nu }^p (D). Let \{t_j\}\subset (0,1 /2 ) be any sequence tending to zero. For any \theta \in [0,2 \pi ], we take a function sequence

f_{j,\theta }(z) =\frac{t_j^{\frac{b}{p}+1}}{{\nu }^{\frac{1}{p}}(1-t_j)[1-(1-t_j)z{\mathrm{e}}^{-\mathrm{i}\theta }{]}^{\frac{1+b}{p}}}(z\in D ).

Then \{f_{j,\theta }\} is an analytic function sequence on D, which converges to 0 uniformly on any compact subset of D and ||f_{j,\theta } | {|}_{{\mathcal{D}}_{\nu }^p}\lesssim 1 for all j=1,2,\dots and \theta \in [0,2 \pi ].

By Lemma 2.4 and the compactness of C_{\phi } on {\mathcal{D}}_{\nu }^p (D), we have

\begin{array}{rl}\underset{\theta \in [0,2\pi ]}{sup}\frac{(b+1{)}^p\mu [S({\mathrm{e}}^{\mathrm{i}\theta },t_j)]}{p^p{2}^{1+b+2p}{t}_j\nu (1-t_j)}& \le \underset{\theta \in [0,2\pi ]}{sup}{\int }_{S({\mathrm{e}}^{\mathrm{i}\theta },t_j)}|f_{j,\theta }^{\prime }(z){|}^p\mathrm{d}{\mu }_{p,\nu ,\phi }(z)\\ & \le \underset{\theta \in [0,2\pi ]}{sup}{\int }_D|f_{j,\theta }^{\prime }(z){|}^p\mathrm{d}{\mu }_{p,\nu ,\phi }(z)\\ & =\underset{\theta \in [0,2\pi ]}{sup}||f_{j,\theta }|{|}_{{\mathcal{D}}_{\nu }^p}^p\to 0(j\to \mathrm{\infty }).\end{array}

This shows that \underset{\theta \in [0,2\pi ]}{sup}\mu [S({\mathrm{e}}^{\mathrm{i}\theta },t)]=o[t\nu (1-t)] ( t\to 0^{+}).

This proof is complete.□

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