Show that $\|P(f\circ\varphi_{\lambda})-\widetilde{f}(\lambda)\|_p=\|P(f\circ\varphi_{\lambda}-\overline{P(\overline{f}\circ\varphi_{\lambda}}))\|_p.$

Question

Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman reproducing kernel. Let $\varphi_{\lambda}$ be an analytic automorphism of $\Omega$ having the properties $:$

$(1)$ $\varphi_{\lambda} (\lambda) = 0$

$(2)$ $\varphi_{\lambda} \circ \varphi_{\lambda} = \text{Id}_{\Omega}.$

It is proved in Section $2.2$ in Rudin's Function Theory in the Unit Ball of $\mathbb C^n$ that the real Jacobian of $\varphi_{\lambda}$ has the following form $:$ $$J_{\mathbb R} (\varphi_{\lambda}) (z) = \frac {\left \lvert k_{\lambda} (z) \right \rvert^2} {k_{\lambda} (\lambda)}.$$

For a given $f \in L^{\infty} (\Omega)$ we define the Berezin transform $\widetilde f$ of $f$ in the following way $:$

$$\begin{align*} \widetilde {f} (\lambda) & = \left \langle f \frac {k_{\lambda}} {\|k_{\lambda}\|_2}, \frac {k_{\lambda}} {\|k_{\lambda}\|_2} \right \rangle \\ & = \frac {1} {k_{\lambda} (\lambda)} \int_{\Omega} f(z)\ \left \lvert k_{\lambda} (z) \right \rvert^2\ dV(z) \end{align*}$$

By change of variable formula we have $$\widetilde {f} (\lambda) = \int_{\Omega} f \circ \varphi_{\lambda}\ dV.$$

In Corollary $10$ of the paper Toeplitz and Hankel Operators on Bergman Spaces authored by Karel Stroethoff and Dechao Zheng it is claimed that if $P$ denotes the Bergman projection then

$$\left\|P\left(f\circ\varphi_{\lambda}\right)-\widetilde{f}(\lambda)\right\|_p=\left\|P\left(f\circ\varphi_{\lambda}-\overline{P\left(\overline{f}\circ\varphi_{\lambda}\right)}\right)\right\|_p.$$

But I am having hard time following this claim. Any help in this regard would be warmly appreciated.

Thanks for your time.

Answer 1

I think it goes like this.

\begin{align*} P \big(\overline{P(\overline{f} \circ \varphi_\lambda}) \big) (w) & = \int_\Omega \overline{P(\overline{f} \circ \varphi_\lambda}) (z) \overline{k_w(z)} dV(z) \\ & = \int_\Omega \int_\Omega f \circ\varphi_\lambda(\eta) k_z(\eta) dV(\eta) \overline{k_w(z)} dV(z) \\ & = \int_\Omega f \circ \varphi_\lambda(\eta) \overline{\int_\Omega k_\eta(z)k_w(z) dV(z)} dV(\eta) \\ & = \int_\Omega f \circ \varphi_\lambda (\eta) dV(\eta) = \tilde{f}(\lambda). \end{align*}

The second to last inequality comes from the fact that for the unit ball and the polydisc if $F$ is a holomorphic function and $dV$ is the normalized area measure then $\int_\Omega F(z) dV(z) = F(0) $.