Nonabelian Ramified Coverings and $L^p$-boundedness of Bergman Projections in ${\mathbb {C}}^2$

Abstract

In this work,we explore the theme of $L^p$-boundedness of Bergman projections of domains that can be covered, in the sense of ramified coverings, by “nice” domains (e.g., strictly pseudoconvex domains with real analytic boundary). In particular, we focus on two-dimensional normal ramified coverings whose covering group is a finite unitary reflection group. In an infinite family of examples, we are able to prove $L^p$-boundedness of the Bergman projection for every $p\in (1,\infty )$.

Appendix A

Theorem A.1

Let $B_2$ be the unit ball in ${\mathbb {C}}^2$ and let $\pi : B_2\dashrightarrow \pi (B_2)=:D$ be the normal ramified covering defined by

$$\begin{aligned} \pi (z_1,z_2)=(z_1^2,z_2) \end{aligned}$$

with associated covering group $G=\{r,{{\,\textrm{id}\,}}\}$, where $r(z_1,z_2)=(-z_1,z_2)$. Then, the integral operator with positive kernel $|K_{G,p}|$ is $L^p(B_2)$-bounded for every $p\in (1,\infty )$.

Proof

Up to an immaterial positive multiplicative constant, we have

$$\begin{aligned} K_{G,p}(z,w)&= |z_1|^{\frac{2}{p}-1}\Big (\frac{1}{(1-z_1\overline{w}_1-z_2{{\overline{w}}}_2)^3}-\frac{1}{(1+z_1{{\overline{w}}}_1-z_2\overline{w}_2 )^3}\Big )|w_1|^{1-\frac{2}{p}}, \end{aligned}$$

where we are writing $z=(z_1,z_2)$ and $w=(w_1,w_2)$.

Set ${\mathcal {G}}=\{(z,w)\in B_2\times B_2: |z_1 \overline{w}_1|\ge \frac{1}{2}|1-z_2{{\overline{w}}}_2|\}$. In this region

$$\begin{aligned} \frac{|z_1|}{|w_1|}\le 2\frac{|z_1|^2}{|1-z_2\overline{w_2}|}\le 2\frac{1-|z_2|^2}{(1-|z_2||w_2|)}\le 4 \end{aligned}$$

and an identical bound holds for $\frac{|w_1|}{|z_1|}$. Thus, on ${\mathcal {G}}$ the factor $(|z_1|/|w_1|)^{\frac{2}{p}-1}$ is bounded and

$$\begin{aligned} |K_{G,p}(z,w)|\le C(p) \Big (|K_{B_2}(z,w)|+|K_{B_2}(z,rw)| \Big ) \end{aligned}$$

(20)

where $K_{B_2}$ is the Bergman kernel of the unit ball.

On the complement $(B_2\times B_2)\backslash {\mathcal {G}}$,we take advantage of the power series expansion

$$\begin{aligned} \frac{1}{(1-\alpha )^3}=\sum _{k=0}^{+\infty }\frac{(k+1)(k+2)}{2}\alpha ^k \qquad (|\alpha |<1), \end{aligned}$$

which allows to write

$$\begin{aligned} \frac{1}{(1-z_1{{\overline{w}}}_1-z_2\overline{w}_2)^3}=\frac{1}{(1-z_2\overline{w}_2)^3}\sum _{k=0}^{\infty }\frac{(k+1)(k+2)}{2} \left( \frac{z_1{{\overline{w}}}_1}{1-z_2{{\overline{w}}}_2}\right) ^k \end{aligned}$$

and an analogous identity for the term $(1+z_1\overline{w}_1-z_2{{\overline{w}}}_2)^{-3}$. We get

$$\begin{aligned} K_{G,p}(z,w)&=|z_1|^{\frac{2}{p}-1}|w_1|^{1-\frac{2}{p}}\frac{1}{(1-z_2{{\overline{w}}}_2)^3}\sum _{k=0}^{\infty }\frac{(k+1)(k+2)}{2}\frac{(z_1{{\overline{w}}}_1)^{k}}{(1-z_2{{\overline{w}}}_2)^k}\big (1-(-1)^{k}\big )\\&=|z_1|^{\frac{2}{p}-1}|w_1|^{1-\frac{2}{p}}\frac{z_1\overline{w}_1}{(1-z_2\overline{w}_2)^4}\sum _{k=0}^{\infty }(2k+2)(2k+3)\frac{(z_1\overline{w}_1)^{2k}}{(1-z_2{{\overline{w}}}_2)^{2k}}. \end{aligned}$$

Since $C:=\sum \limits _{k=0}^{\infty }\frac{(2k+2)(2k+3)}{4^k}<+\infty$, we obtain

$$\begin{aligned} |K_{G,p}(z,w)|\le C \frac{|z_1|^{\frac{2}{p}}|w_1|^{2-\frac{2}{p}}}{|1-z_2{{\overline{w}}}_2|^4}\qquad \forall (z,w)\notin {\mathcal {G}}. \end{aligned}$$

(21)

Notice that both powers in the numerator are positive. Hence, we may use $|z_1|^2\le 1-|z_2|^2\le 2|1-z_2{{\overline{w}}}_2|$ and the similar estimate $|w_1|^2\le 2|1-z_2{{\overline{w}}}_2|$, to bound the RHS of (21) with a constant times $\frac{1}{|1-z_2{{\overline{w}}}_2|^3}$.

We finally observe that, on the complement of ${\mathcal {G}}$, $|1-z_1{{\overline{w}}}_1-z_2{{\overline{w}}}_2|\le \frac{3}{2}|1-z_2{{\overline{w}}}_2|$ which allows to conclude that on this set, we have

$$\begin{aligned} |K_{G,p}(z,w)|\le C\frac{1}{|1-z_1{{\overline{w}}}_1-z_2\overline{w}_2|^3}=C'|K_{B_2}(z,w)|. \end{aligned}$$

(22)

The conclusion follows from (20) and (22) and the well-known $L^p$-boundedness, for $p\in (1,\infty )$, of the integral operator with kernel $|K_{B_2}(z,w)|$ (see, e.g., Chapter 7 of [38]). $\square$