2. Notation and tools

Throughout this note, we will denote by $\mathcal {O}(\mathbb {D},\mathbb {D})\cap \mathcal {C}^1(\overline {\mathbb {D}})$ the holomorphic self maps of the unit disc that are of class $\mathcal {C}^1$ on the boundary. In the same notation, by replacing $\mathbb {D}$ with $\mathbb {D}^2$ , we will talk about the holomorphic self maps of the bi-disc that are of class $\mathcal {C}^1$ at the (topological) boundary of the bi-disc, the bi-torus $\mathbb {T}^2$ . Whenever we refer to a holomorphic function on the disc, we will simply write $f\in H(\mathbb {D}).$ Whenever the notation $a\asymp b$ appears, it means that there exist two positive constants $C_1,C_2$ such that $C_1a\leq b\leq C_2 a$ and whenever we encounter the notation $a\lesssim b,$ it means that there is a positive constant $C>0$ such that $a\leq Cb.$ We recall that for $p>0$ , the Dirichlet type space of the unit disc, denoted by $\mathcal {D}_p(\mathbb {D}),$ consists of the holomorphic functions f on the unit disc $\mathbb {D},$ such that

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

By $A_{\beta }^p(\mathbb {D}^2)$ , $\beta \ge -1,$ we denote the classical weighted Bergman space of the bi-disc $\mathbb {D}^2,$ comprising the holomorphic functions on the bi-disc such that

$$ \begin{align*}\int_{\mathbb{D}^2}|f(z,w)|^p\,dA_{\beta}(z)\,dA_{\beta}(w)<+\infty.\end{align*} $$

The measure $dA(z)={\pi }^{-1}\,dx\,dy$ is the normalised Lebesgue measure on the unit disc, while $dA_{\beta }(z)=c_{\beta }(1-|z|^2)^{\beta }\,dA(z)$ for $\beta \ge -1.$ The following recent result of Balooch and Wu [Reference Balooch and Wu1] will be of critical importance in our note.

Theorem 2.1. Let $\sigma , \tau \ge -1 $ and $\beta \in \mathbb {R},$ with $\tfrac 12\max (\sigma ,\tau )-1<\beta \leq \tfrac 12 (\sigma +\tau ).$ Let $f\in H(\mathbb {D}).$ Then

$$ \begin{align*}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(w)|^2}{|1-\overline{w}z|^{2(\beta+2)}}\,dA_{\sigma}(z)\,dA_{\tau}(w)\asymp ||f||^2_{\mathcal{D}_{\sigma+\tau-2\beta}(\mathbb{D})}.\end{align*} $$

We now define a lift-type operator, similarly to [Reference Li, Wulan, Zhao and Zhu7], but with an exponent of two positive parameters $p,\gamma>0$ .

Definition 2.2. Let $p,\gamma>0$ and $f\in H(\mathbb {D}).$ The lift-type operator is defined by

$$ \begin{align*}L^{p,\gamma}:H(\mathbb{D})\to H(\mathbb{D}^2), \quad L^{p,\gamma}(f):=\frac{f(z)-f(w)}{(1-\overline{w}z)^{p/ \gamma}},\quad z,w \in \mathbb{D}.\end{align*} $$

The first observation that one makes immediately is that the fraction that defines this operator appears in the characterisation of the Dirichlet-type spaces. The following proposition follows immediately.

Proof. The proof follows by a simple use of the asymptotic equality of Theorem 2.1 but we will write it down for convenience.

$$ \begin{align*} ||L^{2,2(\beta+2)}(f)||^2_{A^2_{\sigma}(\mathbb{D}^2)}&=\int_{\mathbb{D}}\int_{\mathbb{D}}|L^{2,2(\beta+2)}(f)(z,w)|^2\,dA_{\sigma}(z)dA_{\sigma}(w) \\ &=\int_{\mathbb{D}}\int_{\mathbb{D}}\bigg|\frac{f(z)-f(w)}{(1-\overline{w}z)^{{2(\beta+2)}/{2}}}\bigg|^2\,dA_{\sigma}(z)dA_{\sigma}(w) \\ &\lesssim ||f||^2_{\mathcal{D}_{2\sigma-2\beta}(\mathbb{D})}.\\[-35pt] \end{align*} $$

The next theorem is a characterisation of the symbols $\Psi \in \mathcal {O}(\mathbb {D}^2,\mathbb {D}^2)\cap \mathcal {C}^1(\overline {\mathbb {D}^2})$ that induce a bounded composition operator on the weighted Bergman space $A_{\beta }^2(\mathbb {D}^2)$ , $\beta>-1$ .

Using this result, we can deduce the following lemma.

Proof. It is quite obvious that $\Phi \in \mathcal {O}(\mathbb {D}^2,\mathbb {D}^2)\cap \mathcal {C}^1(\overline {\mathbb {D}^2}). $ We only have to show that the derivative $d_{\zeta }\Phi $ is invertible for every $\zeta \in \mathbb {T}^2$ such that $\Phi (\zeta )\in \mathbb {T}^2$ . Let ${\zeta =(\zeta _1,\zeta _2)}$ be such that $\Phi (\zeta _1,\zeta _2)=(\varphi (\zeta _1),\varphi (\zeta _2))\in \mathbb {T}^2.$ Of course, this only occurs for the points $\zeta _1,\zeta _2\in \mathbb {T}$ such that $\varphi (\zeta _1),\varphi (\zeta _2)\in \mathbb {T},\zeta _1\neq \zeta _2.$ We calculate the Jacobian of the derivative $d_{\zeta }\Phi $ for such points $\zeta \in \mathbb {T}^2$ . It is an immediate observation that

$$ \begin{align*}\frac{\partial \varphi(z_1)}{\partial z_2}=\frac{\partial \varphi(z_2)}{\partial z_1}=0.\end{align*} $$

So the Jacobian is

$$ \begin{align*}|d_\zeta\Phi|=\varphi'(\zeta_1)\varphi'(\zeta_2)\neq 0.\end{align*} $$

By the fact that $\varphi $ is of class $\mathcal {C}^1$ , and hence continuous on the boundary, one can calculate the value of $\varphi '$ at the boundary points $\zeta _1,\zeta _2$ . By the Julia–Caratheodory theorem (see [Reference Caratheodory3]), it follows that the value of the derivative of $\varphi $ is nonzero. Hence, by Theorem 2.4, $C_{\Phi }$ defines a bounded composition operator on the weighted Bergman space $A_{\beta }^2(\mathbb {D}^2)$ and the proof is complete.

3. Main result and proof

Here we state our main result and the proof of it. For what follows, we define

$$ \begin{align*}k^{\varphi}(z,w)=\frac{1-\varphi(z)\overline{\varphi(w)}}{1-z\overline{w}}, \quad z,w\in \mathbb{D},\end{align*} $$

for a function $\varphi \in \mathcal {O}(\mathbb {D},\mathbb {D})\cap \mathcal {C}^1(\overline {\mathbb {D}}),$ and denote by $||\cdot ||_{\infty }$ the supremum norm over the bi-disc $\mathbb {D}^2,$ that is, $||k||_{\infty }=\sup _{(z,w)\in \mathbb {D}^2}|k(z,w)|$ .

Proof. Let $\sigma>0$ and $\beta \in \mathbb {R}$ satisfy the conditions that we gave in our statement. For convenience, set $q=2(\beta +2).$ Then

$$ \begin{align*} ||C_{\varphi}(f)||^2_{\mathcal{D}_{2\sigma-2\beta}(\mathbb{D})}&\asymp \int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f\circ\varphi(z)-f\circ \varphi(w)|^2}{|1-\overline{w}z|^{q}}\,dA_{\sigma}(z)\,dA_{\sigma}(w) \\ &=\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f\circ\varphi(z)-f\circ \varphi(w)|^2}{|1-\overline{\varphi(w)}\varphi(z)|^{q}}\frac{|1-\overline{\varphi(w)}\varphi(z)|^{q}}{|1-\overline{w}z|^{q}}\,dA_{\sigma}(z)\,dA_{\sigma}(w) \\ &\leq \sup_{(z,w)\in \mathbb{D}^2}|k^{\varphi}(z,w)|^{q}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f\circ\varphi(z)-f\circ \varphi(w)|^2}{|1-\overline{\varphi(w)}\varphi(z)|^{q}}\,dA_{\sigma}(z)\,dA_{\sigma}(w). \end{align*} $$

At this point, we observe that the integral on the right-hand side can be expressed as the norm of a composition operator $C_{\Phi }$ on the weighted Bergman space $A_{\sigma }^2(\mathbb {D}^2),$ where $\Phi =\Phi (z_1,z_2)=(\varphi (z_1),\varphi (z_2)),z_1,z_2\in \mathbb {D}.$ To be precise,

$$ \begin{align*}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f\circ\varphi(z)-f\circ \varphi(w)|^2}{|1-\overline{\varphi(w)}\varphi(z)|^{2(\beta+2)}}\,dA_{\sigma}(z)\,dA_{\sigma}(w)=||C_{\Phi}(L^{2,2(\beta+2)}(f))||^2_{A^2_{\sigma}(\mathbb{D}^2)}.\end{align*} $$

By Lemma 2.5, $C_\Phi :A^2_{\sigma }(\mathbb {D}^2)\to A^2_{\sigma }(\mathbb {D}^2)$ is bounded. Hence, by Proposition 2.3,

$$ \begin{align*}||C_{\Phi}(L^{2,2(\beta+2)}(f))||^2_{A^2_{\sigma}(\mathbb{D}^2)}\lesssim||L^{2,2(\beta+2)}(f)||^2_{A^2_{\sigma}(\mathbb{D}^2)}\lesssim ||f||^2_{\mathcal{D}_{2\sigma-2\beta}(\mathbb{D})}.\end{align*} $$

Summarising,

$$ \begin{align*}||C_{\varphi}(f)||^2_{\mathcal{D}_{2\sigma-2\beta}(\mathbb{D})}\lesssim ||k^{\varphi}(z,w)||_{\infty}^{2(\beta+2)}||f||^2_{\mathcal{D}_{2\sigma-2\beta}(\mathbb{D})},\end{align*} $$

which, by our assumption that the supremum is bounded, gives the desired result.

4. Concluding remark

The significance of the main result of this note lies mainly in the fact that we managed to apply results for the bi-disc to the case of the unit disc. Another interesting aspect of the result is the appearance of the De Branges–Rovnyak kernel $k^\varphi $ that is induced by the symbol $\varphi .$ Assuming that $\varphi \in \mathcal {O}(\mathbb {D},\mathbb {D})\cap \mathcal {C}^1(\overline {\mathbb {D}}),$ then it is known that $\varphi \in H^{\infty }(\mathbb {D})$ and also $||\varphi ||_{H^{\infty }(\mathbb {D})}\leq 1$ which means that the function

$$ \begin{align*}k^{\varphi}(z,w)=\frac{1-\varphi(z)\overline{\varphi(w)}}{1-z\overline{w}},\quad z,w\in \mathbb{D}\end{align*} $$

is the De Branges–Rovnyak reproducing kernel associated with $\varphi .$ We can reformulate the main result of the note in the following manner.

For more details about the De Branges–Rovnyak spaces and kernel, see [Reference de Branges and Rovnyak4]. Also, in the work of Jury (see [Reference Jury5]), one can study the connections of the De Branges–Rovnyak kernel associated to a holomorphic self map of the unit disc, and the corresponding composition operators in the Hardy and Bergman spaces of the unit disc and also of the unit ball of $\mathbb {C}^n.$