Cyclic Polynomials in Dirichlet-Type Spaces in the Unit Ball of ${\mathbb {C}}^2$

Abstract

We characterize polynomials that are cyclic in Dirichlet-type spaces in the unit ball in ${\mathbb {C}}^2$.

1 Introduction

Three classical Hilbert spaces of holomorphic functions in the unit ball of ${\mathbb {C}}^n$ are the Hardy, Bergman and Drury–Arveson spaces. All of them are special cases of a space family that depends on a real parameter, called Dirichlet-type spaces. A general introduction to this theory can be found in [16].

Our purpose is to characterize the polynomials that are cyclic for the shift operators on these spaces in two variables. An analogous problem for the bidisk was solved in [1] and shortly after extended in [7]. Sola [15] studied this problem in the unit ball. His paper is a main motivation for our research. Sola asked, in particular, for a characterization of cyclic polynomials analogous to that achieved for the bidisc. Note that it looks like a harder problem in the ball because of the absence of determinantal representations. The main aim of the paper is to give an answer to this question.

To attack the problem we shall study the zeros in the sphere of a polynomial non-vanishing in the ball. Using some tools coming from semi-analytic geometry we shall show that this zero set is either finite or contains an analytic curve. The first possibility is in principle simpler to deal with and can be overcome with tools analogous to those used in [1]. If, in turn, the zero set contains an analytic curve, we shall make use of the necessity capacity argument from [15] as well as a radial dilation argument. In particular, we shall show a result interesting in its own right: a polynomial p is cyclic if and only if $p/p_r\rightarrow 1$, where $p_r(z,w)=p(rz, rw)$ is a radial dilation of p. This is connected with the problem of approximating 1/f in a space of analytic functions (see [14] for the study of this subject). In the one dimensional case the above-mentioned radial dilation observation was proven in [7]. One-variable Dirichlet-type spaces are discussed in the textbook [5].

1.1 Dirichlet-Type Spaces in the Unit Ball

Denote the unit ball by

$$\begin{aligned} {{\mathbb {B}}_2}=\{(z,w)\in {{\mathbb {C}}}^2:|z|^2+|w|^2<1\}, \end{aligned}$$

and its boundary, the unit sphere by

$$\begin{aligned} {\mathbb {S}}_2=\{(\zeta ,\eta )\in {{\mathbb {C}}}^2:|\zeta |^2+|\eta |^2=1\}. \end{aligned}$$

Let $f:{{\mathbb {B}}_2}\rightarrow {\mathbb {C}}$ be a holomorphic function with power series expansion

$$\begin{aligned} f(z,w)=\sum _{k=0}^{\infty }\sum _{l=0}^{\infty }a_{k,l}z^{k}w^{l}. \end{aligned}$$

We say that f belongs to the $Dirichlet-type$ space $D_{\alpha }({{\mathbb {B}}_2}),$ where $\alpha \in {\mathbb {R}}$ is a fixed parameter, if

$$\begin{aligned} || f||_{\alpha }^{2}=\sum _{k=0}^{\infty }\sum _{l=0}^{\infty }(2+k+l)^{\alpha }\frac{k!l!}{(1+k+l)!}|a_{k,l}|^{2}<\infty . \end{aligned}$$

(1)

Note that these spaces are Hilbert spaces. The case when $\alpha =0$ corresponds to the Hardy space and $\alpha =-1$ to the Bergman one. When $\alpha =1$, $D_1({\mathbb {B}}_2)$ coincides with the Drury-Arveson space. The Dirichlet space corresponds to the parameter $\alpha =2.$ A general introduction to function theory in the ball can be found in [13] and [16]. Some crucial facts about Dirichlet-type spaces and cyclic vectors in the unit ball can also be found in [15].

The following results from function theory are well known. If X is a normed space and $C\subset X$ a convex set, then C is closed in norm if and only if it is weakly closed. Moreover, since $D_\alpha ({{\mathbb {B}}_2})$ is a reflexible Banach space, given a sequence $\{f_n\}\subset D_\alpha ({{\mathbb {B}}_2}),$ then $f_n \rightarrow 0$ weakly if and only if $f_n\rightarrow 0$ pointwise and $\sup _n\{||f_n||_\alpha \}<\infty .$

In Dirichlet-type spaces the integral representation of the norm is achieved in a limited range of parameters:

Lemma 1

(see [10, 11]) If $\alpha \in (-1,1)$, then $||f||_\alpha$ is equivalent to $|f(0)| + |f|_\alpha$, where

$$\begin{aligned} |f|^2_\alpha :=\int _{{{\mathbb {B}}_2}} \frac{||\nabla (f)||^2 - |R(f)|^2}{ (1-|z|^2 - |w|^2)^\alpha } dA(z,w). \end{aligned}$$

Above, $\nabla (f)(z,w)=(\partial _zf(z,w),\partial _wf(z,w))$ denotes the holomorphic gradient of a holomorphic function f and

$$\begin{aligned} R(f)(z,w)=z\partial _zf(z,w)+w\partial _wf(z,w) \end{aligned}$$

is its radial derivative. Moreover, dA(z, w) denotes the normalized area measure.

Lemma 1 allows us to deal with Dirichlet norms $D_\alpha$ using analytic methods whenever $\alpha \in (-1,1).$

Polynomials are dense in the spaces $D_{\alpha }({{\mathbb {B}}_2}),$ $\alpha \in {{\mathbb {R}}},$ and $z\cdot f,w\cdot f\in D_\alpha ({{\mathbb {B}}_2})$ whenever $f\in D_\alpha ({{\mathbb {B}}_2}).$ Also, if $\alpha >2$ the spaces $D_\alpha ({{\mathbb {B}}_2})$ are algebras, see [15], and, by definition, $D_\alpha ({{\mathbb {B}}_2})\subset D_\beta ({{\mathbb {B}}_2}),$ when $\alpha \ge \beta .$

A crucial relation among these spaces is the following:

Lemma 2

Let f be a holomorphic function in ${{\mathbb {B}}_2}.$ Then

$$\begin{aligned} f\in D_\alpha ({{\mathbb {B}}_2})\quad \text {if and only if} \quad 2f+R(f)\in D_{\alpha -2}({{\mathbb {B}}_2}). \end{aligned}$$

This elementary observation allows us to use Lemma 1 for a wide range of parameters $\alpha$.

A multiplier of $D_\alpha ({{\mathbb {B}}_2})$ is a holomorphic function $\phi :{{\mathbb {B}}_2}\rightarrow {{\mathbb {C}}}$ that satisfies $\phi \cdot f\in D_\alpha ({{\mathbb {B}}_2})$ for all $f\in D_\alpha ({{\mathbb {B}}_2}).$ The set of all multipliers will be denoted by $M(D_\alpha ({{\mathbb {B}}_2})).$ As mentioned above, polynomials, as well as holomorphic functions in a neighbourhood of the closed unit ball, are multipliers in every space $D_\alpha ({{\mathbb {B}}_2})$.

1.2 Shift Operators and Cyclic Vectors

Consider two bounded linear operators $S_1,S_2:D_{\alpha }({{\mathbb {B}}_2})\rightarrow D_{\alpha }({{\mathbb {B}}_2})$ defined by $S_i:f\mapsto z_if.$ We say that $f\in D_\alpha ({{\mathbb {B}}_2})$ is a cyclic vector if the closed invariant subspace, i.e.

$$\begin{aligned}{}[f]:=\textrm{clos}\, \textrm{span}\{z_1^k z_2^lf:k=0,1,...,l=0,1,...\} \end{aligned}$$

coincides with $D_\alpha ({{\mathbb {B}}_2})$ (the closure is taken with respect to the $D_\alpha ({{\mathbb {B}}_2})$ norm). In addition, we have the following equivalent definition of cyclicity: f is cyclic if and only if there exist a sequence of polynomials $\{p_n\}$ such that $p_nf\rightarrow 1$ in norm. In other words, f is cyclic if and only if $1\in [f].$

Examples of cyclic functions in various Dirichlet-type spaces were provided in [15]. It is well known (see e.g. [15]) that the cyclicity of a function $f\in D_\alpha ({{\mathbb {B}}_2})$ is intimately connected with its zero set

$$\begin{aligned} {\mathcal {Z}}(f) =\{(z,w)\in {{\mathbb {C}}}^2:f(z,w)=0\}. \end{aligned}$$

Since $D_\alpha ({{\mathbb {B}}_2})$ enjoys the bounded point evaluation property a function that is cyclic cannot vanish inside the unit ball. Any non-zero constant function is cyclic in each space $D_\alpha ({{\mathbb {B}}_2}).$ Moreover, if $\alpha >2,$ then $f\in D_\alpha ({{\mathbb {B}}_2})$ is cyclic precisely when f has no zeros in the closed unit ball. Points lying in the set ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2,$ where p is a given polynomial, will be called boundary zeros.

An important result that helps us to restrict the cyclicity problem of polynomials to irreducible ones is the following: if $f\in D_\alpha ({{\mathbb {B}}_2})$ and $\phi \in M(D_\alpha ({{\mathbb {B}}_2})),$ then $\phi f$ is cyclic if and only if both f and $\phi$ are cyclic.

1.3 Main Result

Our aim is to characterize the polynomials that are cyclic in Dirichlet-type spaces in the setting of the ball. As we shall see the situation in the ball mirrors the bidisk case, meaning that the cyclicity of a function is inextricably linked to the nature of the boundary zeros. Our goal is to separate the problem into two parts: polynomials with finitely many boundary zeros and polynomials with infinitely many boundary zeros.

The main result is as follows:

Theorem 3

Let $p\in {{\mathbb {C}}}[z,w]$ be an irreducible polynomial non-vanishing in the unit ball.

• (1) If $\alpha \le 3/2,$ then p is cyclic in $D_\alpha ({{\mathbb {B}}_2}).$

• (2) If $3/2<\alpha \le 2,$ then p is cyclic in $D_\alpha ({{\mathbb {B}}_2})$ if and only if ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2$ is empty or finite.

• (3) If $\alpha >2,$ then p is cyclic in $D_\alpha ({{\mathbb {B}}_2})$ if and only if ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2=\emptyset .$

2 The Zero Set of a Polynomial Non-vanishing in the Ball

We begin by studying the boundary zeros of a polynomial non-vanishing in the ball.

Lemma 4

${\mathcal {Z}}(p)\cap {\mathbb {S}}_2$ is either a finite set or there is a non-constant analytic curve contained in it.

It is convenient to look at ${\mathcal {Z}}(p)$ $\cap$ ${\mathbb {S}}_2$ as at a semi-algebraic set. We are interested in the case where ${\mathcal {Z}}(p)$ $\cap$ ${\mathbb {S}}_2$ contains at least one accumulation point.

Recall that a set $A\subset {{\mathbb {R}}}^N$ is said to be semi-analytic (resp. semi-algebraic), if for any $x\in {{\mathbb {R}}}^N,$ there are a neighbourhood $U=U(x)$ and a finite number of real analytic functions (resp. polynomials) $f_i,$ $g_{ij}$ in U such that

$$\begin{aligned} A\cap U=\bigcup \limits _{j=1}^{p}\bigcap \limits _{i=1}^{q}\{x\in U:f_i(x)=0, g_{ij}(x)>0\}. \end{aligned}$$

According to this definition, ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2$ is a semi-algebraic (and thus semi-analytic), as it is an intersection of the sphere and two semi-algebraic sets $\{\text {Re} (p)=0\}$ and $\{\text {Im} (p) =0\}$. Hence Lemma 4 is a consequence of the following semi-analytic version of the Bruhat–Cartan–Wallace Curve Selecting Lemma:

Lemma 5

(see [4]) Let A be a semi-analytic set and suppose that $a\in \overline{A\setminus \{a\}}.$ Then there exists an analytic function $\gamma :(0,1)\rightarrow A$ yielding a semi-analytic curve and such that $\lim _{t\rightarrow 0^+}\gamma (t)=a.$

3 Polynomials with Finitely Many Boundary Zeros

The case when a polynomial does not vanish on the closed ball is obvious to deal with. The simplest non-trivial case occurs when ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2$ is finite. This case is relatively simple and can be overcome with tools that were used for Dirichlet-type spaces over the bidisc:

Theorem 6

Let $p\in {{\mathbb {C}}}[z,w]$ be a polynomial non-vanishing in the unit ball with finitely many zeros in ${\mathbb {S}}_2.$ Then p is cyclic in $D_\alpha ({{\mathbb {B}}_2})$ precisely when $\alpha \le 2.$

The idea is to compare a polynomial with a product of cyclic polynomials. The function $(z,w)\mapsto |p(z,w)|^2$ is real analytic on ${{\mathbb {C}}}^2,$ and hence, one can apply Łojasiewicz’s inequality to it on the compact set ${\mathbb {S}}_2$, see [8]. Moreover, there are a constant $C>0$ and a natural number q such that

$$\begin{aligned} |p(\zeta ,\eta )|\ge C\cdot \textrm{dist}((\zeta , \eta ), {\mathcal {Z}}(p)\cap {\mathbb {S}}_2)^q, \end{aligned}$$

for all $(\zeta ,\eta )\in {\mathbb {S}}_2.$ The distance above is considered with respect to the Euclidean norm.

Proof of Theorem 6

Let p be a polynomial, non-vanishing in the ball with finitely many boundary zeros. Let ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2=\{(\zeta _1,\eta _1),...,(\zeta _n,\eta _n)\}.$ Take polynomials $s_1,...,s_n$, which are cyclic in $D_\alpha ({{\mathbb {B}}_2})$ precisely when $\alpha \le 2,$ and such that ${\mathcal {Z}}(s_i)\cap {\mathbb {S}}_2=\{(\zeta _i,\eta _i)\}.$ The polynomials $s_i$ can be trivially constructed as compositions of $\pi (z,w):=1-z$ with unitary matrices ${\mathcal {U}}_i$ that satisfy ${\mathcal {U}}_i(\zeta _i,\eta _i)^{T}=(1,0)$, i.e. $s_i = \pi \circ {\mathcal {U}}_i$. Since the distance is invariant under unitary transformations, we find that $\textrm{dist}((\zeta ,\eta ),(\zeta _i,\eta _i))\ge |\pi \circ {\mathcal {U}}_i(\zeta ,\eta )|=|s_i(\zeta ,\eta )|.$ Clearly all $s_j$ are trivially bounded from above on ${\mathbb {S}}_2$ by 2, whence $\textrm{dist}((\zeta ,\eta ),{\mathcal {Z}}(p)\cap {\mathbb {S}}_2)\ge C_1\prod _{i=1}^{n}|s_i(\zeta ,\eta )|,$ for some constant $C_1>0.$

Summing up, by the above-mentioned Łojasiewicz’s inequality there exist a constant $C_2>0$ and $q\in {\mathbb {N}}$ such that

$$\begin{aligned} |p(\zeta ,\eta )|\ge C_2 \prod _{i=1}^{n}|s_i(\zeta ,\eta )|^q, \end{aligned}$$

for all $(\zeta ,\eta )\in {\mathbb {S}}_2.$ Then the rational function Q defined by

$$\begin{aligned} Q(z,w)=\frac{\prod _{i=1}^{n}s_i(z,w)^q}{p(z,w)}, \end{aligned}$$

is bounded on $\overline{{\mathbb {B}}}_2.$ Increasing $q\in {\mathbb {N}}$ we make this function as smooth as we like in ${\mathbb {S}}_2.$ In particular, the function $2Q+R(Q)$ lies in the Hardy space for q big enough. Thus, making use of the Lemma 2 we conclude that Q lives in $D_2({{\mathbb {B}}_2})$.

The function pQ is cyclic in $D_\alpha ({{\mathbb {B}}_2})$, since it is a product of cyclic polynomials, and $pQ\in D_\alpha ({{\mathbb {B}}_2}),$ for $\alpha \le 2.$ The assertion follows as p is a multiplier. $\square$

4 Cyclicity Via Radial Dilations: Cyclicity for Infinitely Many Boundary Zeros

The aim of this section is to prove the following:

Theorem 7

Let $p\in {{\mathbb {C}}}[z,w]$ be a polynomial non-vanishing in the unit ball. Then p is cyclic in $D_\alpha ({{\mathbb {B}}_2})$ for any $\alpha \le 3/2.$

To prove Theorem 7 we shall use radial dilation of a function $f:{{\mathbb {B}}_2}\rightarrow {{\mathbb {C}}}$. It is defined for $r\in (0,1)$ by $f_r(z,w)=f(rz,rw).$ To prove Theorem 7 it is enough to prove the following:

Lemma 8

If $p\in {{\mathbb {C}}}[z,w]$ does not vanish on ${{\mathbb {B}}_2}$ and $\alpha \le 3/2$, then $||p/p_r||_\alpha <\infty$ as $r\rightarrow 1^-$.

Indeed, if Lemma 8 holds, then $\phi _r\cdot p\rightarrow 1$ weakly, where $\phi _r:=1/p_r.$ Since $\phi _r$ extends holomorphically past the closed unit ball, $\phi _r$ are multipliers, and hence, $\phi _r\cdot p\in [p].$ Finally, 1 is weak limit of $\phi _r\cdot p$ and [p] is weakly closed. It is clear that $1\in [p],$ and hence, p is cyclic.

Moreover, it is enough to prove that $||p/p_r||_\alpha <\infty ,$ as $r\rightarrow 1^{-},$ for $\alpha =3/2.$ Then the case $\alpha <3/2$ follows since the inclusion $D_{3/2}({{\mathbb {B}}_2})\hookrightarrow D_\alpha ({{\mathbb {B}}_2})$ is a compact linear map and weak convergence in $D_{3/2}({{\mathbb {B}}_2})$ gives weak convergence in $D_\alpha ({{\mathbb {B}}_2}).$

Remark 9

As was mentioned above, it is enough to show that $||p/p_r||_{3/2}< \infty ,$ as $r\rightarrow 1^{-}.$ Lemma 2 shows that this is equivalent to $||2p/p_r+R(p/p_r)||_{-1/2}<\infty ,$ as $r\rightarrow 1^{-}.$ The advantage of this approach is that now one can use an equivalent integral norm. In what follows we shall show that $|f_r|_{-1/2}<\infty ,$ as $r\rightarrow 1^{-},$ where $f_r$ is one of the following functions: $p/p_r,$ $z\partial _z(p/p_r)$ or $w\partial _w(p/p_r).$

A standard compact-type argument allows us to estimate the integral norm locally in the sense that it is enough to show that for every point $P\in {\mathbb {S}}_2$ there exists a neighborhood $U=U(P)$ such that

$$\begin{aligned} \int _{{{\mathbb {B}}_2}\cap U} \left( ||\nabla (f_r)||^2 - |R(f_r)|^2 \right) \sqrt{1- |z|^2 - |w|^2} dA(z,w)<\infty , \end{aligned}$$

(2)

as $r\rightarrow 1^-$, where $f_r\in \{p/p_r,z\partial _z(p/p_r),w\partial _w(p/p_r)\}$.

It is convenient to expand in (2) the term $||\nabla (f_r)||^2 - |R(f_r)|^2$ as follows:

$$\begin{aligned} ||\nabla (f_r)||^2 - |R(f_r)|^2 = ||\nabla (f_r)||^2(1-|z|^2- |w|^2) + |{\bar{z}} \partial _wf_r - {\bar{w}} \partial _zf_r|^2. \end{aligned}$$

(3)

Of course, the only problematic points P are those that lie in ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2.$

An idea is to expand p in its Weierstrass form. To do it properly we need to rotate p. The following result is clear:

Remark 10

Let $\alpha \in (-1,1).$ Let ${\mathcal {U}}$ be a unitary matrix. Then $(f\circ$ ${\mathcal {U}})_r=f_r\circ$ ${\mathcal {U}}$ and $|f\circ$ ${\mathcal {U}}|_\alpha = |f|_\alpha$.

Let us take a point $P\in {\mathcal {Z}}(p)\cap {\mathbb {S}}_2.$ Composing with a unitary matrix we may assume that $P=(0,1).$ Near $P=(0,1)$ we can expand

$$\begin{aligned} p(z,w)= \alpha (z)(1-h_1(z)w)\cdots (1-h_n(z)w), \quad (z,w)\in {{\mathbb {D}}}(\epsilon )\times {{\mathbb {D}}}(1,\epsilon ), \end{aligned}$$

where $\alpha$ is a non-vanishing single-valued holomorphic function in ${{\mathbb {D}}}(\epsilon )$ for some $\epsilon >0$. Here ${{\mathbb {D}}}(\epsilon ):=\{z\in {{\mathbb {C}}}:|z|<\epsilon \}$ and ${{\mathbb {D}}}(1,\epsilon ):=\{z\in {{\mathbb {C}}}:|z-1|<\epsilon \}$.

The functions $h_j$ are branches of algebraic functions. Sometimes it is convenient to look at them as single-valued holomorphic functions that are well defined on dense simply connected subdomains of ${\mathbb {D}}(\epsilon )\setminus \{0\}$, called slight domains. Note that this does not affect values of area integrals. See [7] for more on the properties of these functions.

In particular, we may define the following function:

$$\begin{aligned} H(z,w):=(w-h_1(z))\cdots (w-h_n(z)), \quad (z,w)\in {{\mathbb {D}}}(\epsilon )\times {{\mathbb {C}}}. \end{aligned}$$

The function H is a monic polynomial of degree n with respect to w,  whose coefficients are holomorphic functions in ${{\mathbb {D}}}(\epsilon ).$ Recall the following consequence of Puiseux’s theorem.

Corollary 11

(see [9], Corollary, p.171) If H(z, w) is a polynomial with respect to $w\in {{\mathbb {C}}}$ which is monic of degree m and whose coefficients are holomorphic in a neighbourhood of zero in ${{\mathbb {C}}},$ then there exist an integer exponent $k>0$ and holomorphic functions $\Phi _1,...,\Phi _m$ in the disk $\Delta =\{z\in {{\mathbb {C}}}:|z|<\delta \}$ such that

$$\begin{aligned} H(z^k,w)=(w-\Phi _1(z))\cdots (w-\Phi _m(z)) \quad \text {in}\quad \Delta \times {{\mathbb {C}}}. \end{aligned}$$

Take $j=1,...,n.$ By Puiseux’s theorem there is a non-negative integer $k=k(j)$ and a holomorphic function $\Phi$ such that $\Phi (z^{1/k})=h_j(z)$ for properly chosen branch of $z^{1/k}$, as $z\rightarrow 0.$ The polynomial does not vanish in the ball and therefore $|h_j(z)|^2\le (1-|z|^2)^{-1}.$ It thus follows from the assumption that

$$\begin{aligned} |\Phi (z)|^2\le \frac{1}{1 - |z|^{2k}}, \end{aligned}$$

(4)

as $z\rightarrow 0$. Whence,

$$\begin{aligned} h_j(z)=1+\gamma z^2+o(z^2), \quad z\rightarrow 0 \end{aligned}$$

(5)

where $|\gamma |\le 1/2.$

With these tools in hands we are ready to start the proof of the main result of this section. For $r\in (0,1)$ set

$$\begin{aligned} q_r(z,w)=\frac{p(z,w)}{p(rz,rw)}=\frac{\alpha (z)}{\alpha (rz)} \prod _{j=1}^{n}\frac{1-h_j(z)w}{1-rh_j(rz)w}. \end{aligned}$$

Let us express partial derivatives of $q_r$ in a useful way. Let

$$\begin{aligned} H_j(z,w):=\frac{1-h_j(z)w}{1-rh_j(rz)w}. \end{aligned}$$

Remark 12

Simple computations give

$$\begin{aligned} \partial _zq_r(z,w)= & {} A_1(z,w)+\sum _{j}B_j(z,w)\partial _zH_j(z,w),\nonumber \\ \partial _wq_r(z,w)= & {} \sum _{j}B_j(z,w)\partial _wH_j(z,w),\nonumber \\ \partial _{zz}q_r(z,w)= & {} A_2(z,w)\nonumber \\{} & {} +\sum _{j}\Big (2\Gamma _j(z,w)\partial _zH_j(z,w)+B_j(z,w)\partial _{zz}H_j(z,w)\Big )\nonumber \\{} & {} +\sum _{i\ne j}\Delta _{i,j}(z,w)\partial _zH_j(z,w)\partial _zH_i(z,w), \end{aligned}$$

(6)

$$\begin{aligned} \partial _{wz}q_r(z,w)= & {} \sum _{j}\Big (\Gamma _j(z,w)\partial _wH_j(z,w)+B_j(z,w)\partial _{wz}H_j(z,w)\Big )\nonumber \\{} & {} +\sum _{i\ne j}\Delta _{i,j}(z,w)\partial _zH_j(z,w)\partial _wH_i(z,w), \end{aligned}$$

(7)

$$\begin{aligned} \partial _{ww}q_r(z,w)= & {} \sum _{j}B_j(z,w)\partial _{ww}H_j(z,w)\nonumber \\{} & {} +\sum _{i\ne j}\Delta _{i,j}(z,w)\partial _wH_j(z,w)\partial _wH_i(z,w), \end{aligned}$$

(8)

$$\begin{aligned} \partial _{zw}q_r(z,w)= & {} \sum _{j}\Big (\Gamma _j(z,w) \partial _wH_j(z,w)+B_j(z,w)\partial _{zw}H_j(z,w)\Big )\nonumber \\{} & {} +\sum _{i\ne j}\Delta _{i,j}(z,w)\partial _wH_j(z,w)\partial _zH_i(z,w), \end{aligned}$$

(9)

Functions $A_1,A_2,B_j,\Gamma _j,\Delta _{i,j}$ above are products of the terms $H_j(z,w)$ and $\alpha (z)/\alpha (rz),$ as well as derivatives of $\alpha (z)/\alpha (rz).$

To simplify the notation all different positive constants that appear in inequalities below and that are uniform with respect to $r\rightarrow 1^{-}$ will be denoted by $C>0.$

Remark 13

As pointed out before, $\alpha$ does not vanish in ${{\mathbb {D}}}(\epsilon )$, and hence, $A_1$ and $A_2$ are bounded in a neighbourhood of zero, as are the terms $H_j(z,w)$:

$$\begin{aligned} \left| 1- H_j(z,w)\right| \le C \frac{|h_j (z) - rh_j(rz)|}{|1 - rh_j(rz) w|}\le C \frac{1-r}{|1 - rh_j(rz) w|}\le C. \end{aligned}$$

Consequently, all the functions $A_1,A_2,B_j,\Gamma _j,\Delta _{i,j}$ are bounded.

In the sequel the term ${\bar{z}} h_j(rz)-r|w|^2h_j'(rz)$ will appear. To estimate it we need the following elementary lemma, due to Z. Błocki:

Lemma 14

([3], Lemma 3.1) Let $\Omega$ be a domain in ${{\mathbb {R}}}^n$ and $\psi \in \mathcal C^{1,1}(\Omega )$ be non-negative. Then $\sqrt{\psi }\in \mathcal C^{0,1}(\Omega )$.

If $\psi$ is smooth and positive, the above lemma says that $|\partial _{x_j}\psi |^2 \le C |\psi |$, where C is a locally uniform constant.

We have the following preparatory result:

Lemma 15

There exists $C>0$ such that

$$\begin{aligned} |{\bar{z}} h_j(z) - |w|^2 h'_j(z)|\le C |1 - h_j(z)w|^{1/2}, \end{aligned}$$

for $(z,w)\in {{\mathbb {B}}_2}$ and z close to 0.

Proof

Let $\Phi$ holomorphic in a neighborhood of 0 be such that $h_j(z) = \Phi (z^{1/k})$, for some integer $k\ge 1$, and define $\psi (z)=1-(1-|z|^{2k})|\Phi (z)|^2$. Put

$$\begin{aligned} \varphi (z)= \frac{\psi (z)}{|z|^{2(k-1)}}. \end{aligned}$$

Note that $\varphi \ge 0$ by (4) and one can check that $\varphi$ is ${\mathcal {C}}^{1,1}$ smooth. It follows from Lemma 14 that $|\partial _{z} \varphi | \le C \sqrt{\varphi }$. Simple calculations lead to

$$\begin{aligned} |z \partial _z \psi (z) - (k-1) \psi (z)|\le C |z|^k \sqrt{\psi (z)}, \end{aligned}$$

and therefore

$$\begin{aligned} |z\partial _z \psi (z)|\le C(|z|^k \sqrt{\psi (z)} + \psi (z)). \end{aligned}$$

(10)

Since $\psi (z)=O(|z|^{2k})$, one has $\psi (z) \le C |z|^k \sqrt{\psi (z)}$. Thus, (10) implies that

$$\begin{aligned} |\partial _z \psi (z)| \le C |z|^{k-1} \sqrt{\psi (z)}. \end{aligned}$$

Computing the derivative of $\psi$ we find that

$$\begin{aligned} |k z^{k-1} {\bar{z}}^k \Phi (z) - (1-|z|^{2k}) \Phi '(z)|^2 \le C |z|^{k-1} (1 - (1-|z|^{2k}) |\Phi (z)|^2). \end{aligned}$$

From this we immediately get that

$$\begin{aligned} |{\bar{z}} h_j(z) - (1-|z|^2)h'_j(z)|^2\le C (1-(1-|z|^2)|h_j(z)|^2), \end{aligned}$$

as $z\rightarrow 0$

On the other hand $|1-h_j(z)w|\ge 1 - |h_j(z)||w|\ge C(1-|h_j(z)|^2 |w|^2)\ge C( 1- (1-|z|^2) |h_j(z)|^2),$ so the assertion follows. $\square$

Remark 16

Lemma 15 applied to points (rz, rw) gives that

$$\begin{aligned} |{\bar{z}} h_j(rz) - r|w|^2 h'_j(rz)|\le C |1 - rh_j(rz)w|^{1/2} \end{aligned}$$

for $(z,w)\in {\mathbb {B}}_2$, z close to 0 and $r\rightarrow 1^-$. This inequality will play a crucial role in the sequel.

Lemma 17

The term $||\nabla (q_r)||$ is bounded from above by a constant multiple of the term

$$\begin{aligned} \sum _{j}\frac{1-r}{|1-rh_j(rz)w|^3}. \end{aligned}$$

Moreover, $||\nabla (z \partial _z q_r)||$ and $||\nabla (w\partial _wq_r)||$ are estimated (up to a positive constant) by

$$\begin{aligned} \sum _{j}\frac{1-r}{|1-rh_j(rz)w|^3} +\sum _{i\ne j}\frac{(1-r)^2}{|1-rh_j(rz)w|^2|1-rh_i(rz)w|^2}. \end{aligned}$$

Proof

Let us start with the first estimate. According to Remark 12 it is enough to carry out computation for terms $\partial _z H_j(z,w)$ and $\partial _w H_j(z,w)$. We have

$$\begin{aligned} \partial _{z}H_j(z,w)=\frac{\sigma (z,w)}{(1-rh_j(rz)w)^2}, \end{aligned}$$

(11)

where

$$\begin{aligned} \begin{aligned} \sigma (z,w)&=rw(rh_j'(rz)-h_j'(z)) +wh_j'(z)(r-1)\\&\quad \ +\,rw^2h_j'(z)(h_j(rz)-h_j(z))+rw^2h_j(z)(h_j'(z)-rh_j'(rz)). \end{aligned} \end{aligned}$$

(12)

It is clear from (5) that

$$\begin{aligned} |\sigma (z,w)|\le C(1-r). \end{aligned}$$

In particular,

$$\begin{aligned} |\partial _{z}H_j(z,w)|\le C(1-r)|1-rh_j(rz)w|^{-3} . \end{aligned}$$

(13)

Similarly,

$$\begin{aligned} \partial _{w}H_j(z,w)=\frac{rh_j(rz)-h_j(z)}{(1-rh_j(rz)w)^2}. \end{aligned}$$

and, by (5),

$$\begin{aligned} |\partial _wH_j(z,w)|\le C(1-r)|1-rh_j(rz)w|^{-3} . \end{aligned}$$

(14)

To prove the second part of the assertion we need to estimate $z \partial _{zz} q_r$, $\partial _{wz}q_r,$ $\partial _{zw} q_r$, and $\partial _{ww} q_r$. Let us start with $z\partial _{zz}q_r.$ According to (6) we have

$$\begin{aligned} |z\partial _{zz}q_r(z,w)|\le & {} C\sum _{j}\Big (|\partial _zH_j(z,w)|+|z\partial _{zz}H_j(z,w)|\Big )\\{} & {} +C\sum _{i\ne j}|\partial _{z}H_j(z,w)||\partial _z H_i(z,w)|. \end{aligned}$$

Thus, what we need to do is to estimate $z\partial _{zz}H_j(z,w).$ From (11) we get

$$\begin{aligned} z\partial _{zz}H_j(z,w)=\frac{z\partial _z\sigma (z,w)}{(1-rh_j(rz)w)^2}+\frac{2r^2wzh_j'(rz)\sigma (z,w)}{(1-rh_j(rz)w)^3}, \end{aligned}$$

(15)

where

$$\begin{aligned} \begin{aligned} z\partial _z\sigma (z,w)&=rw(r^2zh_j''(rz)-zh_j''(z))+zwh_j''(z)(r-1)\\&\quad +rzw^2h_j''(z)(h_j(rz)-h_j(z))+rzw^2h_j'(z)(rh_j'(rz)-h_j'(z))\\&\quad +rzw^2h_j'(z)(h_j'(z)-rh_j'(rz))+rw^2h_j(z)(zh_j''(z)-r^2zh_j''(rz)). \end{aligned} \end{aligned}$$

(16)

It is then clear that

$$\begin{aligned} |z\partial _{zz}H_j(z,w)|\le & {} C\frac{1-r}{|1-rh_j(rz)w|^2}+C\frac{1-r}{|1-rh_j(rz)w|^{3}}\\\le & {} C \frac{1-r}{|1-rh_j(rz)w|^{3}}. \end{aligned}$$

Thus,

$$\begin{aligned} |z\partial _{zz}q_r(z,w)|&\le C\sum _{j}\frac{1-r}{|1-rh_j(rz)w|^{3}}\\&\quad +C\sum _{i\ne j}\frac{(1-r)^2}{|1-rh_j(rz)w|^2|1-rh_i(rz)w|^{2}} \end{aligned}$$

Let us look at $z\partial _{wz}q_r$. According to (7) we have

$$\begin{aligned} |z\partial _{wz}q_r(z,w)|\le & {} C\sum _{j}\Big (|\partial _wH_j(z,w)|+|\partial _{wz}H_j(z,w)|\Big )\\{} & {} +C\sum _{i\ne j}|\partial _zH_j(z,w)||\partial _w H_i(z,w)|. \end{aligned}$$

The only term that has not been estimated yet is $\partial _{wz}H_j(z,w).$ From (11) we get

$$\begin{aligned} \partial _{wz}H_j(z,w)=\frac{\partial _w\sigma (z,w)}{(1-rh_j(rz)w)^2}+\frac{2rh_j(rz)\sigma (z,w)}{(1-rh_j(rz)w)^3}. \end{aligned}$$

(17)

Whence,

$$\begin{aligned} |\partial _{wz}H_j(z,w)|\le C(1-r)|1-rh_j(rz)w|^{-3}. \end{aligned}$$

Finally, one can estimates $\partial _{zw}q_r$ and $\partial _{ww}q_r$ in the same way as presented above.

$\square$

The next lemma requires more subtle estimations.

Lemma 18

Let $f_r$ denote one of the functions $q_r,z\partial _zq_r,w\partial _wq_r.$ Then $|{\bar{z}} \partial _wf_r - {\bar{w}} \partial _zf_r|$ is bounded from above by a constant multiple of

$$\begin{aligned} \sum _{j}\frac{1-r}{|1-rh_j(rz)w|^{5/2}} +\sum _{i\ne j}\frac{(1-r)^2}{|1-rh_j(rz)w|^2|1-rh_i(rz)w|^{3/2}}. \end{aligned}$$

Proof

Let us take $f_r=q_r.$ Since $|{\bar{z}} \partial _wq_r - {\bar{w}} \partial _zq_r|$ can be estimated by $||\nabla (q_r)||,$ the assertion for this term follows from Lemma 17.

Consider the case $f_r=z\partial _zq_r.$ Since

$$\begin{aligned} |{\bar{z}} \partial _w(z\partial _zq_r) - {\bar{w}} \partial _z(z\partial _zq_r)|\le C ||\nabla (q_r)||+C\left| |z|^2\partial _{wz}q_r-\bar{w}z\partial _{zz}q_r\right| , \end{aligned}$$

it suffices to estimate $\left| |z|^2\partial _{wz}q_r-\bar{w}z\partial _{zz}q_r\right| .$ According to (6) and (7) we get

$$\begin{aligned} \left| |z|^2\partial _{wz}q_r-{\bar{w}}z\partial _{zz}q_r\right|\le & {} C\sum _{j}\Big (|\partial _zH_j(z,w)|+|\partial _wH_j(z,w)|\Big )\\{} & {} +C\sum _{j}\left| |z|^2\partial _{wz}H_j(z,w)-{\bar{w}} z\partial _{zz}H_j(z,w)\right| \\{} & {} +C\sum _{i\ne j}|\partial _zH_j(z,w)|\left| |z|^2\partial _{w}H_i(z,w)-{\bar{w}} z\partial _{z}H_i(z,w)\right| . \end{aligned}$$

Note that the terms $\partial _zH_j(z,w),$ $\partial _wH_j(z,w)$ have already been estimated in (13), (14).

By (15) and (17), to deal with $|z|^2\partial _{wz}H_j(z,w)-{\bar{w}} z\partial _{zz}H_j(z,w)$ it suffices to estimate

$$\begin{aligned} \frac{|z|^2\partial _w\sigma (z,w)-{\bar{w}} z\partial _z\sigma (z,w)}{(1-rh_j(rz)w)^2} \text { and } \frac{2r\sigma (z,w) z({\bar{z}} h_j(rz)-r|w|^2h_j'(rz))}{(1-rh_j(rz)w)^3}. \end{aligned}$$

By (12), (16) and (5) one has $|\sigma (z,w)|,$ $|\partial _z\sigma (z,w)|,$ $|\partial _w\sigma (z,w)|\le C(1-r).$ Thus, $\left| |z|^2\partial _{wz}H_j(z,w)-{\bar{w}} z\partial _{zz}H_j(z,w)\right|$ is bounded by a constant times

$$\begin{aligned} \frac{1-r}{|1-rh_j(rz)w|^2}+\frac{(1-r)|{\bar{z}} h_j(rz)-r|w|^2h_j'(rz)|}{|1-rh_j(rz)w|^3}. \end{aligned}$$

Applying Remark 16 to the second term above we get

$$\begin{aligned} \left| |z|^2\partial _{wz}H_j(z,w)-{\bar{w}} z\partial _{zz}H_j(z,w)\right|\le & {} \frac{C(1-r)}{|1-rh_j(rz)w|^2} +\frac{C(1-r)}{|1-rh_j(rz)w|^{5/2}}. \end{aligned}$$

Let us focus on $|z|^2\partial _{w}H_i(z,w)-{\bar{w}} z\partial _{z}H_i(z,w).$ Carrying out some computations we get that

$$\begin{aligned} |z|^2\partial _{w}H_i(z,w)-{\bar{w}} z\partial _{z}H_i(z,w)= & {} \frac{rz|w|^2(h_i'(z)-rh_i'(rz))(1-wh_i(z))}{(1-rh_i(rz)w)^2}\\{} & {} +\frac{(1-r)z(|w|^2h_i'(z)-{\bar{z}}h_i(z))}{(1-rh_i(rz)w)^2}\\{} & {} +\frac{r|z|^2(h_i(rz)-h_i(z))(1-wh_i(z))}{(1-rh_i(rz)w)^2}\\{} & {} +\frac{rzw(h_i(rz)-h_i(z))({\bar{z}}h_i(z)-|w|^2h_i'(z))}{(1-rh_i(rz)w)^2}. \end{aligned}$$

Note that $1-wh_i(z)=1-rh_i(rz)w+w(rh_i(rz)-h_i(z)).$ It follows from Lemma 15 and (5) that

$$\begin{aligned} \left| |z|^2\partial _{w}H_i(z,w)-{\bar{w}} z\partial _{z}H_i(z,w)\right| \le C\frac{1-r}{|1-rh_i(rz)w|^{3/2}}. \end{aligned}$$

Finally, take $f_r=w\partial _wq_r.$ Then $|{\bar{z}} \partial _w(w\partial _wq_r) - {\bar{w}} \partial _z(w\partial _wq_r)|$ is bounded by a constant times $||\nabla (q_r)||+\left| |w|^2\partial _{zw}q_r-\bar{z}w\partial _{ww}q_r\right| .$ According to (8) and (9) we get

$$\begin{aligned} \left| |w|^2\partial _{zw}q_r-{\bar{z}}w\partial _{ww}q_r \right|\le & {} C\sum _{j}|\partial _wH_j(z,w)|\\{} & {} +C\sum _{j}\left| |w|^2\partial _{zw}H_j(z,w) -{\bar{z}} w\partial _{ww}H_j(z,w)\right| \\{} & {} +C\sum _{i\ne j}|\partial _wH_j(z,w)|\left| |w|^2\partial _{z}H_i(z,w)-{\bar{z}} w\partial _{w}H_i(z,w)\right| . \end{aligned}$$

Let us expand

$$\begin{aligned} |w|^2\partial _{z}H_i(z,w)-{\bar{z}} w\partial _{w}H_i(z,w)= & {} \frac{rw|w|^2(rh_i'(rz)-h_i'(z))(1-wh_i(z))}{(1-rh_i(rz)w)^2}\\{} & {} +\frac{(1-r)w({\bar{z}}h_i(rz)-|w|^2h_i'(z))}{(1-rh_i(rz)w)^2}\\{} & {} +\frac{w^2(h_i(rz)-h_i(z))(r|w|^2h_i'(z)-{\bar{z}}h_i(z))}{(1-rh_i(rz)w)^2}\\{} & {} +\frac{{\bar{z}} w(h_i(rz)-h_i(z))(wh_i(z)-1)}{(1-rh_i(rz)w)^2}. \end{aligned}$$

In particular, the following estimation holds:

$$\begin{aligned} \left| |w|^2\partial _{z}H_i(z,w)-{\bar{z}} w\partial _{w}H_i(z,w)\right| \le C\frac{1-r}{|1-rh_i(rz)w|^{3/2}}. \end{aligned}$$

Similarly, one can show that

$$\begin{aligned} \Big ||w|^2\partial _{zw}H_j(z,w)-{\bar{z}} w\partial _{ww}H_j(z,w)\Big |\le C\frac{1-r}{|1-rh_j(rz)w|^{5/2}}. \end{aligned}$$

$\square$

Our goal is to prove (2). Thanks to Lemmas 17, 18 and (3) it is suffices to show that

$$\begin{aligned} \int _{{{\mathbb {B}}_2}\cap U_j}\frac{(1-r)^2 (1- |z|^2 - |w|^2)^{\alpha }}{|1 - rh_j(rz)w |^{\beta }} dA(z,w)<\infty , \text { as }r\rightarrow 1^-,\end{aligned}$$

(18)

where $(\alpha ,\beta )=(3/2,6),$ or $(\alpha , \beta )=(1/2,5)$, and

$$\begin{aligned} \int _{{{\mathbb {B}}_2}\cap U_{i,j}}\frac{(1-r)^4 (1- |z|^2 - |w|^2)^{\alpha '}}{|1 - rh_j(rz)w |^{\beta '}|1 - rh_i(rz)w|^{\gamma '}} dA(z,w)<\infty , \text { as } r\rightarrow 1^{-}, \end{aligned}$$

where $(\alpha ',\beta ',\gamma ')=(3/2,4,4),$ or $(\alpha ', \beta ', \gamma ')=(1/2,4,3).$ Applying Hölder’s inequality to the second integral above we see that it is enough to prove (18).

The following integral estimate of Forelli and Rudin is crucial for our computations:

Remark 19

(see [6], Theorem 1.7) Let $a\in (-1,\infty )$ and $b \in (0,\infty ).$ Then

$$\begin{aligned} \int _{{{\mathbb {D}}}}\frac{(1-|w|)^{a}}{|1-{\overline{z}} w|^{2+a+b}}dA(w)\asymp (1-|z|^2)^{-b}, \end{aligned}$$

(19)

as $|z|\rightarrow 1^{-}.$

Lemma 20

Let $\alpha \in (-1,\infty )$ and $\beta -2-\alpha \in (0,\infty ).$ Set $U_j=\Omega _j\times {{\mathbb {D}}}(1,\epsilon ),$ where $\Omega _j$ is the slight domain on which $h_j$ is defined. Then

$$\begin{aligned} \int _{{{\mathbb {B}}_2}\cap U_j}\frac{(1- |z|^2 - |w|^2)^\alpha }{|1 - rh_j(rz)w |^\beta } dA(z,w) \le C\int _{\Omega _j} \frac{1}{ (1- r^2 |h_j(rz)|^2 (1-|z|^2))^b} dA(z), \end{aligned}$$

where $b=\beta -\alpha -2.$

Proof

Let us estimate

$$\begin{aligned}&\int _{{{\mathbb {B}}_2}\cap U_j}\frac{(1- |z|^2 - |w|^2)^\alpha }{|1 - rh_j(rz)w |^\beta } dA(z,w)\\&\quad \le \int _{\Omega _j}(1-|z|^2)^{\alpha } \int _{\{|w|<\sqrt{1-|z|^2}\}} \frac{\left( 1 - |(1-|z|^2)^{-1/2}w|^2\right) ^\alpha }{ |1 - rh_j(rz) w|^\beta } dA(w)dA(z)\\&\quad \le \int _{\Omega _j}(1-|z|^2)^{\alpha +2} \int _{{{\mathbb {D}}}} \frac{(1 - |w|^2)^\alpha }{ |1 - rh_j(rz) \sqrt{1-|z|^2}w|^\beta } dA(w)dA(z)\\&\quad \le C \int _{\Omega _j} \frac{1}{ (1- r^2 |h_j(rz)|^2 (1-|z|^2))^b} dA(z). \end{aligned}$$

In the computations carried out above the change-of-variables $w\mapsto w(1-|z|^2)^{-\frac{1}{2}}$ and Remark 19 have been used. $\square$

Set $h_j=h$ and $\Omega _j=\Omega .$ According to Lemma 20, to get (18), it suffices to show that the integral

$$\begin{aligned} \int _{\Omega }\frac{(1-r)^2}{(1-r^2(1-|z|^2)|h(rz)|^2)^{5/2}}dA(z) \end{aligned}$$

is bounded as $r\rightarrow 1^{-}.$

We showed in (5) that $h(z)=1+\gamma z^2+o(z^2),$ where $|\gamma |\le 1/2.$ If $|\gamma |<1/2$, then $|h(z)|^2\le 1+|z|^2,$ and we end up with the integral

$$\begin{aligned} \int _{0}^{\epsilon }\frac{(1-r)^2x}{(1-r^2+r^2x^4)^{5/2}}\textrm{d}x, \end{aligned}$$

which is bounded as $r\rightarrow 1^{-}.$

The case $|\gamma |=1/2$ needs some more attention. Without loss of generality we may then assume that $\gamma =1/2,$ and hence it suffices to consider

$$\begin{aligned} \int _{\Omega }\frac{(1-r)^{2}}{(1-r^2+r^2(|z|^2-r^2Re(z^2)+o(z^2)))^{5/2}}dA(z), \end{aligned}$$

as $r\rightarrow 1^-.$ Note that $z\mapsto o(z^2)$ is a real-valued function, smooth on $\Omega$. Setting $z=x+iy,$ it is enough to estimate the integral

$$\begin{aligned} \iint _{\Omega } \frac{(1-r)^{2}}{(1-r^2+r^2((1-r^2)x^2+(1+r^2)y^2+o(z^2)))^{5/2}}\textrm{d}x\textrm{d}y. \end{aligned}$$

Fix $r\in (0,1)$ and define u in ${{\mathbb {D}}}$ by

$$\begin{aligned} u(x,y)=\frac{(1-r)^2}{(1-r^2+r^2((1-r^2)x^2+(1+r^2)y^2))^{5/2}}\ge 0. \end{aligned}$$

Next, define $G\in C^\infty (\Omega )$ by

$$\begin{aligned} G(x,y)=\Big (x, y\sqrt{1+\frac{o(z^2)}{1+r^2}}\Big ). \end{aligned}$$

Note that the Jacobian of G at (x, y) is close to 1 as (x, y) is close to zero. This allows us to make a proper change of variables. Therefore, it is enough to estimate $\iint _{{{\mathbb {D}}}}u(x,y)\textrm{d}x\textrm{d}y$ which, in turn, boils down to

$$\begin{aligned} \int _{0}^{1}\int _{0}^{2\pi }\frac{(1-r)^{2}t}{\Big (1 -r^2+r^2t^2(1-r^2\cos (\theta ))\Big )^{5/2}}\textrm{d}t\textrm{d}\theta . \end{aligned}$$

It is elementary to see that the last integral is bounded as $r\rightarrow 1^-$, so the desired result follows.

5 Non-cyclicity for Infinitely Many Boundary Zeros

All that is left to prove that any polynomial non-vanishing in the ball that vanishes along a curve in the sphere fails to be cyclic in certain $D_{\alpha }({{\mathbb {B}}_2}).$ We have the following:

Theorem 21

Let $p\in {{\mathbb {C}}}[z,w]$ be a polynomial non-vanishing in the unit ball with infinitely many zeros in ${\mathbb {S}}_2.$ Then p is not cyclic in $D_\alpha ({{\mathbb {B}}_2})$ whenever $\alpha >3/2.$

This result is a consequence of capacitary conditions and the nature of the boundary zeros of a polynomial non-vanishing in the ball. We consider Riesz $\alpha$-capacity for a fixed $\alpha \in (0,2]$ with respect to the anisotropic distance in ${\mathbb {S}}_2$ given by

$$\begin{aligned} d(\zeta ,\eta )=|1-\langle \zeta ,\eta \rangle |^{1/2} \end{aligned}$$

and the positive kernel $K_\alpha :(0,\infty )\rightarrow [0,\infty )$ given by

$$\begin{aligned} K_\alpha (t)= {\left\{ \begin{array}{ll} t^{\alpha -2}, &{} \alpha \in (0,2) \\ \log (e/t), &{} \alpha =2 \end{array}\right. } \end{aligned}$$

If $\mu$ is a Borel probability measure supported on some Borel set $E\subset {\mathbb {S}}_2,$ then the Riesz $\alpha$-energy of $\mu$ is given by

$$\begin{aligned} I_\alpha [\mu ]=\iint _{{\mathbb {S}}_2}K_\alpha (|1-\langle \zeta ,\eta \rangle |)\textrm{d}\mu (\zeta )\textrm{d}\mu (\eta ) \end{aligned}$$

and the Riesz $\alpha$-capacity of E by

$$\begin{aligned} \text {cap}_\alpha (E)=1/\inf \{I_\alpha [\mu ]:\mu \in {\mathcal {P}}(E)\}, \end{aligned}$$

where ${\mathcal {P}}(E)$ is the set of Borel probability measures supported on E.

The following theorem is crucial to identify non-cyclicity in Dirichlet-type spaces:

Theorem 22

([15], Remark 4.5.) Let $\alpha \in {{\mathbb {R}}}$ be fixed. Suppose $f\in D_\alpha ({{\mathbb {B}}_2})$ has $\textrm{cap}_\alpha ({\mathcal {Z}}(f^*))>0.$ Then f is not cyclic in $D_\alpha ({{\mathbb {B}}_2}).$

Note that ${\mathcal {Z}}(f^*)$ is the zero set in the sphere of the radial limits of f. In particular, ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2$ coincides with ${\mathcal {Z}}(p^*)$ when p is a polynomial.

Proof of Theorem 21

Let p be a polynomial non-vanishing in the ball that has infinitely many boundary zeros. According to Lemma 4, the set of the boundary zeros contains at least one non-constant analytic curve. The model polynomial $1-2zw$ vanishes on an analytic curve ${\mathcal {Z}}(1-2zw) \cap {\mathbb {S}}_2 = \{(e^{i\theta }/\sqrt{2}, e^{-i\theta }/\sqrt{2}): \theta \in {\mathbb {R}}\}$ and satisfies $\text {cap}_\alpha ({\mathcal {Z}}(1-2zw)\cap {\mathbb {S}}_2)>0,$ when $\alpha >3/2,$ see [15].

Let $\gamma :(-\epsilon , \epsilon )\rightarrow {\mathcal {Z}}(p)\cap {\mathbb {S}}_2$ and $\omega :(-\epsilon , \epsilon )\rightarrow {\mathcal {Z}}(1-2zw)\cap {\mathbb {S}}_2$ be analytic functions yielding the above-mentioned curves. Of course, $\omega$ can be written with an explicit formula $\omega (\theta ) = (e^{i\theta } /\sqrt{2}, e^{-i\theta }/\sqrt{2})$. Note that $d(\omega (\theta ), \omega (\theta '))\asymp |\theta -\theta '|$, where d denotes the anisotropic distance in ${\mathbb {S}}_2.$ The same distance estimate is true for $\gamma$, namely $d(\gamma (\theta ), \gamma (\theta '))\asymp |\theta -\theta '|$. To see it write $\gamma =(\gamma _1,\gamma _2)$. We can make, for the sake of simplicity, some additional assumptions: $\theta '=0$, $\gamma (0)= (0,1)$ (compose with a unitary matrix) and $\gamma _1'(0)\ne 1$. It follows from (5) that $1 - \gamma _2(t) = O(\gamma _1^2(t)).$ Let us write $\gamma _1(t) = a_1 t + O(t^2),$ $\gamma _2(t) = 1+b_2 t^2 + O(t^3)$. Since $1 = |\gamma _1(t)^2| + |\gamma _2(t)|^2$, $t\in (-\epsilon , \epsilon )$, we get that $b_2\ne 0$. Then $d(\gamma (0), \gamma (\theta )) = |1-\gamma _2(\theta )|^{1/2} \asymp |\theta |$, as claimed.

In particular, $T:=\omega \circ \gamma ^{-1}$ transforms the analytic curve contained in ${\mathcal {Z}}(p)\cap {\mathbb {S}}_2$ into the one contained in ${\mathcal {Z}}(1-2zw)\cap {\mathbb {S}}_2$ so that $K_\alpha (|1- \langle T(\zeta ), T(\eta )\rangle |) \asymp K_\alpha (1 - \langle \zeta , \eta \rangle )$. Now we can proceed as in [12, Theorem 5.3.1]. Precisely, it follows from [12, Theorem A.4.4] that for a probability measure supported in $\omega ((-\epsilon ,\epsilon ))$ the measure $\nu =\mu T^{-1}$ is supported in $\gamma ((-\epsilon , \epsilon ))$. In particular, $I_\alpha [\mu ]\asymp I_\alpha [\nu ]$ for $\alpha >3/2$. From this we deduce that $\gamma ((-\epsilon ,\epsilon ))>0$ whenever $\omega ((-\epsilon ,\epsilon ))>0.$ Consequently, $\textrm{cap}_\alpha ({\mathcal {Z}}(p)\cap {\mathbb {S}}_2)>0$ for $\alpha >3/2.$

The assertion thus follows from Theorem 22. $\square$

Remark 23

Many of ideas presented above may be extended to Dirichlet-type spaces of ${\mathbb {B}}_n$ with $n\ge 3$. One can expect that the solution of this general problem depends on $\dim _{\mathbb R}({\mathcal {Z}}(p)\cap {\mathbb {S}}_n).$

Let us mention here that some partial results for the polydisc ${\mathbb {D}}^n$ with $n\ge 3$ were obtained in [2]. A crucial difference between the polydisc and the ball lies in their geometry. They are not biholomorphic to each other, even their group of automorphisms (both transitive) are not isomorphic. What played a key role in studying the cyclicity of a polynomial in Dirichlet-type spaces over the bidisc (see [1, 7]) was the size of its zero set on the Shilov (distinguished) boundary. Recall that the Shilov boundary of a given domain is the smallest subset of its topological boundary where an analog of the maximum modulus principle holds. Let us mention that it shows yet another important difference between these domains: the distinguished boundary of the ball coincides with its topological boundary, which is not the case for the polydisc – its distinguished boundary is the torus.

It is worth mentioning that Theorem 3 has a simpler formulation on ${\mathbb {B}}_2$ comparing to its counterpart on ${\mathbb {D}}^2$. This is because in the bidisc one needs to treat $(1-z)$ and $(1-w)$ separately: they are cyclic even thought their zero sets ${\mathcal {Z}}(p)\cap \partial {\mathbb {D}}^2$ are quite large.