# The SHiP experiment and the RPC technology

G. De Lellis,11footnotetext: on behalf of the SHiP Collaboration

###### Abstract

The discovery of the Higgs boson has fully confirmed the Standard Model of
particles and fields. Nevertheless, there are still fundamental phenomena,
like the existence of dark matter and the baryon asymmetry of the Universe,
which deserve an explanation that could come from the discovery of new
particles. The SHiP experiment at CERN meant to search for very weakly coupled
particles in the few GeV mass domain has been recently proposed. The existence
of such particles, foreseen in Beyond Standard Models, is largely unexplored.
A beam dump facility using high intensity 400 GeV protons is a copious source
of such unknown particles in the GeV mass range. The beam dump is also a
copious source of neutrinos and in particular it is an ideal source of tau
neutrinos, the less known particle in the Standard Model. We report the
physics potential of such an experiment and describe the use of the RPC
technology therein. An anchillary measurement of the charm cross-section will
be carried out in July 2018 and RPC are used as a muon detector. We also
describe the design and construction of these new chambers.

##  1 The SHiP experiment

The discovery of the Higgs boson is certainly a big triumph of the Standard
Model. In particular, given its mass, it could well be that the Standard Model
is an effective theory working all the way up to the Planck scale.
Nevertheless, there are several phenomena deserving an explanation that the
Standard Model is unable to provide: the existence of dark matter and its
nature, the baryonic asymmetry of the Universe and neutrino masses. It is
therefore clear the new physics is there and presumably several new particles
have still to be discovered.

Searches for new physics with accelerators are being carried out at the LHC,
especially suited to look for high mass particles with ordinary couplings to
matter. Complementary searches for very weakly coupled and therefore long-
lived particles require a beam dump facility. Such a facility is made of a
high density proton target, followed by a hadron stopper and a muon shield.
Apart from residual muons, the only remaining particles are electron, muon and
tau neutrinos on top of hidden, long-lived particles produced either in proton
interactions or in secondary particle decays.

A new experiment, Search for Hidden Particles (SHiP), has been proposed [1],
designed to operate at a beam dump facility to be built at CERN and to search
for weakly coupled particles in the few GeV mass range. The physics case for
such an experiment is widely discussed in Ref. [2]. In five years, the
facility will integrate $2\times 10^{20}$ 400 GeV protons, produced by the SPS
accelerator complex, impinging on a 12 interaction length ($\lambda_{int}$)
target made of Molybdenum and Tungsten, followed by a 30 $\lambda_{int}$ iron
hadron absorber. Hidden particles in the GeV mass range would be produced
mostly by the decay of charmed hadrons produced in proton interactions.
$D_{s}$ mesons, copiously produced among charmed hadrons, are a source of tau
neutrinos through their fully leptonic decay. Therefore, the SHiP facility is
ideal also to study the physics of tau neutrinos, the less known particle in
the Standard Model.

Figure 1: The beam dump facility and the SHiP detector.

Figure 1 shows the SHiP facility to be placed in the North Area: downstream of
the target, the hadron absorber filters out all hadrons, therefore only muons
and neutrinos are left. An active muon shield is designed with two sections
with opposite polarity to maximize the muon flux reduction: it reduces the
muon flux from $~{}10^{10}$ down to $~{}10^{5}$ muons per spill. $4\times
10^{13}$ protons are extracted in each spill, designed to be 1s long to reduce
the detector occupancy [3]. A first successful test of the SPS cycle with a 1s
long spill was performed in April 2015.

The neutrino detector is located downstream of the muon shield, followed by
the decay vessel and the detector for hidden particles. The Collaboration will
prepare a document for the European Strategy by the end of 2018 and a
Comprehensive Design Report by 2019, in the framework of the Physics Beyond
Colliders working group, launched in 2016 at CERN. The construction and
installation of the detector will start in 2021 and last until the end of the
third LHC long shutdown such that the data taking is assumed to start in 2026.

The neutrino detector is made of a magnetised region, followed by a muon
identification system, as shown in Figure 2. The magnetised region will host
both the neutrino target and a particle spectrometer. The neutrino target is
based on the emulsion cloud chamber technology employed by the OPERA
experiment [4], with a compact emulsion spectrometer, made of a sequence of
very low density material and emulsion films to measure the charge and
momentum of hadrons in magnetic field. This detector is suitable for the
measurement of momenta up to 12 GeV$/c$. Indeed, this feature would allow to
discriminate between tau neutrinos and anti-neutrinos also in the hadronic
decay channels of the tau lepton. The emulsion target is complemented by high
resolution tracking chambers to provide the time stamp to the event, connect
muon tracks from the target to the muon system and measure the charge and
momentum for particles with momenta above 10 GeV$/c$. The muon system is based
on 23 iron slabs, 5 cm thick each, alternated with 24 RPCs providing the
tracking within the slabs. The muon system will also act as upstream veto
tagger for background processes to the hidden particle search, which motivates
the high sampling choice. Nevertheless, the muon system configuration is still
under optimisation.

Figure 2: The neutrino detector upstream of the decay vessel in different
views.

The emulsion target will also act as the target of dark matter as well as of
any very weakly in- teracting particle produced at the accelerator, when its
mass is in the GeV range. The ongoing optimisation of this detector concerns
the target material, the sampling frequency of the emulsion cloud chamber and
the timing performances of the target tracker that would enable the separation
between neutrinos and heavy particles based on the time of flight measurement.

The detector for hidden particles is located in the downstream part of a 60 m
long evacuated decay vessel, with a conical shape and an elliptical transverse
section at the very downstream end of $5\times 10$ m2, the longer axis being
vertical. The hidden particles are supposed to decay within the vessel. The
requirement to have less than 1 neutrino interaction in the vessel over five
years sets the pressure to about $10^{-3}$ mbar. A magnetic spectrometer is
located downstream of the vessel: it is made of straw tubes with a material
budget of 0.5% $X_{0}$ per station, achieving a position resolution of 120
$\mu$m per straw, with 8 hits per station on average. This gives a momentum
resolution of about 1%. The vessel would be sorrounded by a liquid
scintillator layer to tag particles coming from outside. Downstream of the
spectrometer, an hadronic and electromagnetic calorimeter and a muon filter
are used to identify particles. A timing detector complements the apparatus to
reject vertices from chance coincidences.

##  2 Search for hidden particles and physics with the neutrino detector

Extensions of the Standard Model in the low mass region foresee the existence
of particles as singlets with respect to the Standard Model gauge group. These
particles couple to different singlet composite operators (so-called Portals)
of the Standard Model. The SHiP detector has the potentiality to discover very
weakly interacting and long lived particles in a wide unexplored range of
their masses and couplings, within these Portals. As an example, we report in
the left plot of Figure 3 the sensitivity to heavy neutral leptons, when only
the muon coupling $U_{\mu}$ is considered [5]. For an overview of the
sensitivity to different portals and corresponding particles, we refer to [1,
2].

Figure 3: Left: SHiP sensitivity to heavy neutral leptons [5]. Right:
Improvement of the accuracy on $s^{+}$ with SHiP (red) compared to the present
status (blue) in the $0.02<x<0.35$ range.

The observation of tau neutrinos was confirmed by the DONUT experiment only in
2008 when 9 candidates events were reported [6]. The OPERA experiment [4] has
detected ten tau neutrinos [7, 8, 9, 10, 11, 12], leading to the discovery of
tau neutrino appearance from muon neutrino oscillations [11, 12]. The only
leptonic decay observed by OPERA [9] shows negative charge as expected from a
$\nu_{\tau}$ interaction. Therefore, so far there is no direct evidence for
tau anti-neutrinos. The SHiP facility is a $\nu_{\tau}$ factory, with
$6.6\times 10^{15}$ tau neutrinos produced in primary proton collisions,
equally divided in neutrinos and anti-neutrinos. Given the neutrino target
mass of about 10 tons, one expects more than 10000 interactions of tau
neutrinos and anti-neutrinos.

Charmed hadrons are produced in neutrino and anti-neutrino charged-current
interactions at the level of about 5% [13]. Experiments based on calorimetric
technology identify charmed hadrons only in their muonic decay channel, when
two opposite sign muons are produced in the final state. A cut of 5 GeV is
applied to muons in order to suppress the background due to punch-through
pions. The nuclear emulsion technology, instead, identifies topologically the
charmed hadron by detecting its decay vertex. Energy cuts are therefore much
looser, thus providing a better sensitivity to the charm quark mass. Moroever,
a large statistical gain is provided by the use of hadronic decay modes [13].
Indeed, SHiP will integrate about $10^{5}$ charm candidates, more than one
order of magnitude larger than the present statistics, with a large ($\sim
30$%) contribution from anti-neutrinos. Charm production in neutrino
scattering is extremely sensitive to the strange quark content of the nucleon,
especially with anti-neutrinos where the $s$-quark is dominant. SHiP will
improve significantly the uncertainty on the strange quark distribution in the
nucleon as shown in the right plot of Figure 3 in terms of
$s^{+}=s(x)+\bar{s}(x)$ in the $0.02<x<0.35$ range.

##  3 RPC technology in SHiP

The RPC technology is proposed to be used for two different applications in
SHiP: one is a tracking detector for the muon system of the neutrino detector,
also acting as a veto for the background of hidden particle decays. Prototypes
of these chambers are being constructed for the measurement of the charm
cross-section, where RPC will instrument the muon system. The other
application of the RPC technology in SHiP is the timing detector in the
downstream apparatus for the detection of hidden particles. We describe both
applications here.

Figure 4: Left: Setup of the charm measurement experiment including the
downstream muon filter based on the RPC technology. Right: structure of the 32
mm thick RPC chamber.

###  3.1 Muon system for the charm cross-section measurement

The prediction of the tau neutrino yield is affected by a large uncertainty:
indeed, simulation studies of proton interactions in heavy and thick targets
show that the charmed hadron yield is increased by a factor of 2.3 from the
cascade production [14]. Charmed hadrons are produced either directly from
interactions of the primary protons or from subsequent interactions of the
particles produced in the hadronic cascade showers, including the protons
after a primary elastic collision. The only available measurement of the
associated charm production per nucleon $\sigma_{c\bar{c}}=18.1\pm 1.7$
$\mu$barn [15] was indeed obtained with a thin target where the secondary
production is negligible.

The SHiP Collaboration has proposed the SHiP-charm project [16], aiming at
measuring the associated charm production by employing the SPS 400 GeV/c
proton beam. This proposal includes a study of the cascade effect to be
carried out using ECC techniques, i.e. slabs consisting of a replica of the
SHiP experiment target [1] interleaved with emulsion films. The detector is
hybrid, combining the emulsion technique with electronic detectors to provide
the charge and momentum measurement of charmed hadron decay daughters and the
muon identification. This setup shown in the left part of Figure 4 allows a
full kinematical reconstruction of the event. An optimisation run was approved
at CERN for July 2018 while the full measurement is planned after the long
shutdown LS2 of the CERN accelerator complex, with $5\times 10^{7}$ protons on
target and a charm yield of about 2500 fully reconstructed interactions.

The RPC chambers were designed to operate in avalanche mode, with a time
resolution of about 1 ns. Two orthogonal sets of strips are used for 2D
measurements with an expected position resolution of about 3 mm in both
directions. Their structure is shown in the right part of Figure 4. The 2D
measurement is necessary to cope with the large occupancy of the detector:
indeed, in each event there are on average several tens of particles entering
the first two RPC detectors of the muon system, only one or two being a muon
particle. The bakelite electrodes were produced at Puricelli s.r.l. in Italy.
Each RPC is using 118 horizontal and 184 vertical strips with a pitch of
10.625 mm produced at KODEL in Korea, with a total size of $2100\times 1350$
mm2 and an active area of $1900\times 1200$ mm2. Each RPC is equipped with 20
front-end cards, FEERIC developed by the ALICE Collaboration, 12 connected to
vertical and 8 to horizontal strips. 5 readout boards are used for each
chamber. In total 5 chambers were built and are being tested at CERN before
their installation at the H4 beamline at CERN in July 2018.

###  3.2 Timing detector

One of the SHIP timing detector prototypes is based on timing Resistive Plate
Chamber (tRPC) technology. The prototype uses a novel concept, i.e. the RPC
sensitive volume. With this approach, the gas volume and the High Voltage (HV)
insulation are confined inside a permanent sealed plastic box, decoupling it
from the pick up electrodes located on the top and on the bottom of the
sensitive volume. The main advantages of this sensitive volume are:
versatility, the same volume can be coupled to different readout electrodes;
ease of construction and low cost on the $1\div 2$ m2 scale with a complete
tightness of the plastic box allowing an operation with low gas flux.

The sensitive volume of the SHIP tRPC prototype houses a multi-gap RPC
structure‘[17] with six gas gaps defined by seven 1 mm thick float glass
electrodes of about $1550\times 1250$ mm2 separated by 0.3 mm nylon mono-
filaments. The HV electrodes are made up of a resistive layer applied to the
surface of the outermost glasses with airbrush techniques. The structure is
permanently sealed inside a PMMA gas tight box with a 1 mm lid thickness
equipped with feed-throughs for gas and HV connections.

The RPC chamber is composed of two identical sensitive modules, read out by a
pick-up electrode, located between the modules, made from FR4 Printed Circuit
Board with a thickness of 1.5 mm and equipped with $1600\times 30$ mm2 copper
strips. The set is enclosed in an aluminium case to guarantee the
electromagnetic insulation from the environment and enough mechanical
rigidity. The chamber was operated with pure C2H2F4 and open gas flow. Both
sides of each strip are directly connected to low-jitter high-gain/bandwidth
Front-End Electronics [18] and its digital output connected to a FPGA based
multi-hit TDC [19]. The time of the each strip is calculated as the average of
the times in each side.

Three fast plastic scintillators (BC420, $2\times 3\times 8$ cm3) readout on
both sides by Hamamatsu H6533 photomultipliers are used to trigger on cosmic
muons and to provide a time reference, in order to evaluate the response of
the prototype. The three scintillators are aligned with one of the strips, two
above and one below the chamber. The left plot of Figure 5 shows the time
distribution of the difference between one of the reference scintillators and
the prototype after the walk correction. The time precision of the chamber
after subtracting the contribution of the scintillator ($\sigma\sim 107$ ps)
shows a $\sigma\sim 105$ ps . The right plot of Figure 5 shows the time
accuracy and the chamber efficiency as a function of the HV/gap: a plateau in
the efficiency is reached above 2550 V/gap as well as an accuracy of about 105
ps. At the plateau, the dark counting rate of the detector is about 2.2
kHz/m2. These measurements were carried out at the University of Coimbra in
Portugal and further tests will be performed with the final gas mixture 90%
C2H2F4 and 10% SF6. Nevertheless, the performance already measured makes the
tRPC a good candidate to instrument the 50 m2 SHIP timing detector.

Figure 5:  Left: Distribution of the time difference between one of the
reference scintillators and the prototype chamber after the walk correction.
Right: Time accuracy and efficiency as a function of the HV/gap.

## References

  * [1] M. Anelli et al., _A facility to Search for Hidden Particles (SHiP) at the CERN SPS_ , arXiv:1504.04956.
  * [2] S. Alekhin et al., _A facility to search for hidden particles at the CERN SPS: the SHiP physics case_ , Rep. Prog. Phys. 79 (2016) 124201.
  * [3] A. Akmete et al., _The active muon shield in the SHiP experiment_ , JINST 12 (2017) no.05, P05011.
  * [4] N. Agafonova et al., _The OPERA experiment in the CERN to Gran Sasso neutrino beam_ , JINST 4 (2009) P04018.
  * [5] K. Bondarenko et al., _Phenomenology of GeV-scale Heavy Neutral Leptons_ , arXiv:1805.08567v2.
  * [6] K. Kodama et al., _Final tau-neutrino results from the DONuT experiment_ , Phys. Rev. D78 (2008) 052002.
  * [7] N. Agafonova et al., _Observation of a first $\nu_{\tau}$ candidate in the OPERA experiment in the CNGS beam_, Phys. Lett. B691 (2010) 138.
  * [8] N. Agafonova et al., _New results on $\nu_{\mu}\rightarrow\nu_{\tau}$ appearance with the OPERA experiment in the CNGS beam_, JHEP 1311 (2013) 036.
  * [9] N. Agafonova et al., _Evidence for $\nu_{\mu}\rightarrow\nu_{\tau}$ appearance in the CNGS neutrino beam with the OPERA experiment_, Phys. Rev. D89 (2014) 051102.
  * [10] N. Agafonova et al., _Observation of tau neutrino appearance in the CNGS beam with the OPERA experiment_ , PTEP 2014 (2014) 101C01.
  * [11] N. Agafonova et al., _Discovery of $\tau$ neutrino appearance in the CNGS Neutrino Beam with the OPERA experiment_, Phys. Rev. Lett. 115 (2015) 121802.
  * [12] N. Agafonova et al., _Final results of the OPERA experiment on $\nu_{\tau}$ appearance in the CNGS beam_, Phys. Rev. Lett. 120 (2018) 211801.
  * [13] G. De Lellis et al., _Charm physics with neutrinos_ , Physics Reports 399 (2004) 227.
  * [14] H. Dijkstra and T. Ruf, http://cds.cern.ch/record/2115534/files/SHiP-NOTE-2015-009.pdf.
  * [15] C. Lourenco, H.K. Wohri, _Heavy flavour hadro-production from fixed-target to collider energies_ , Physics Reports 433 (2006) 127.
  * [16] A. Akmete et al., _Measurement of associated charm production induced by 400 GeV/c protons_ , CERN-SPSC-2017-033, SPSC-EOI-017 (2017).
  * [17] E. Cerron Zeballos et al., _A New type of resistive plate chamber: The Multigap RPC_ , NIMA 374 (1996) 132 - 135.
  * [18] D. Belver et al., _Performance of the Low-Jitter High-Gain/Bandwidth Front-End Electronics of the HADES tRPC Wall_ , IEEE Transactions on Nuclear Science 57 (2010) 2848-2856.
  * [19] A Neiser et al., _TRB3: a 264 channel high precision TDC platform and its applications_ , JINST 8 (2013) C12043.


# A note on cyclic vectors in Dirichlet-type spaces in the unit ball of
${\mathbb{C}}^{n}$

Dimitrios Vavitsas<EMAIL_ADDRESS>Institute of
Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian
University, Łojasiewicza 6, 30-348 Kraków, Poland

###### Abstract.

We characterize model polynomials that are cyclic in Dirichlet-type spaces in
the unit ball of $\mathbb{C}^{n},$ and we give a sufficient capacity condition
in order to identify non-cyclic vectors.

###### Key words and phrases:

Dirichlet-type spaces, cyclic vectors, anisotropic capacities

###### 1991 Mathematics Subject Classification:

31C25, 32A37, 47A15

Partially supported by NCN grant SONATA BIS no. 2017/26/E/ST1/00723 of the
National Science Centre, Poland

##  1\. Introduction

Studying Dirichlet-type spaces in the unit ball of ${\mathbb{C}}^{n}$ we can
draw conclusions for classical Hilbert spaces of holomorphic functions such as
the Hardy, Bergman and Dirichlet spaces. General introduction to this theory
can be found in [18], [22].

The purpose of this note is to characterize model polynomials and to study
special families of functions that are cyclic for the shift operators on these
spaces. Moreover, we give a sufficient capacity condition in order to identify
non-cyclic functions. Norm comparisons, sharp decay of norms for special
subspaces, capacity conditions studied in [3], [4], [6], [21] are the main
motivation for this work. The cyclicity of a function $f$ in a space of
holomorphic functions is connected also with the problem of approximating
$1/f$, see [19], [20] for the study of this subject.

Full characterization of polynomials in more than two variables looks like a
hard problem either in the unit ball or the polydisc. The cyclicity problem of
polynomials for the bidisk was solved in [5] and shortly after extended in
[13]. The corresponding problem in the setting of the unit ball of
${\mathbb{C}}^{2}$ was solved in [14].

###  1.1. Dirichlet-type spaces in the unit ball

Denote the unit ball by

${\mathbb{B}_{n}}=\\{z\in{\mathbb{C}}^{n}:||z||<1\\},$

and its boundary, the unit sphere by

$\mathbb{S}_{n}=\\{z\in{\mathbb{C}}^{n}:||z||=1\\},$

where $||z||=\sqrt{|z_{1}|^{2}+...+|z_{n}|^{2}}$ is the associated norm of the
usual _Euclidean inner product_ $\langle
z,w\rangle=z_{1}\bar{w}_{1}+...+z_{n}\bar{w}_{n}.$ Denote the _class of
holomorphic functions_ in ${\mathbb{B}_{n}}$ by
$\textrm{Hol}({\mathbb{B}_{n}}).$ Any function
$f\in\textrm{Hol}({\mathbb{B}_{n}})$ has a power series expansion

(1)
$f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}=\sum_{k_{1}=0}^{\infty}...\sum_{k_{n}=0}^{\infty}a_{k_{1},...,k_{n}}z_{1}^{k_{1}}\cdots
z_{n}^{k_{n}},\quad z\in{\mathbb{B}_{n}},$

where $k=(k_{1},...,k_{n})$ is a n-tuple index of non-negative integers,
$k!=k_{1}!\cdots k_{n}!$ and $z^{k}=z_{1}^{k_{1}}\cdots z_{n}^{k_{n}}.$ The
power series in (1) exist, converges normal in ${\mathbb{B}_{n}}$ and it is
unique since the unit ball is a connected Reinhardt domain containing the
origin, i.e. $(z_{1},...,z_{n})\in{\mathbb{B}_{n}}$ implies
$(e^{i\theta_{1}}z_{1},...,e^{i\theta_{n}}z_{n})\in{\mathbb{B}_{n}}$ for
arbitrary real $\theta_{1},...,\theta_{n},$ (see [11]).

To simplify the notation we may write (1) as follows:

(2)
$f(z)=\sum_{m=0}^{\infty}\sum_{|k|=m}^{\infty}a_{k}z^{k}=\sum_{|k|=0}^{\infty}a_{k}z^{k},\quad
z\in{\mathbb{B}_{n}},$

where $|k|=k_{1}+...+k_{n}.$

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

(3)
$||f||^{2}_{\alpha}:=\sum_{|k|=0}^{\infty}(n+|k|)^{\alpha}\frac{(n-1)!k!}{(n-1+|k|)!}|a_{k}|^{2}<\infty.$

General introduction to the theory of Dirichlet-type spaces in the unit ball
of ${\mathbb{C}}^{n}$ can be found in [1], [2], [15], [16], [19], [21], [22].
One variable Dirichlet-type spaces are discussed in the textbook [12]. The
weights in the norm in (3) are chosen in such a way that
$D_{0}({\mathbb{B}_{n}})$ and $D_{-1}({\mathbb{B}_{n}})$ coincide with the
Hardy and Bergman spaces of the ball, respectively. The Dirichlet space having
Möbius invariant norm corresponds to the parameter choice $\alpha=n.$

By the definition, $D_{\alpha}({\mathbb{B}_{n}})\subset
D_{\beta}({\mathbb{B}_{n}}),$ when $\alpha\geq\beta.$ Polynomials are dense in
the spaces $D_{\alpha}({\mathbb{B}_{n}}),$ $\alpha\in{\mathbb{R}},$ and
$z_{i}\cdot f\in D_{\alpha}({\mathbb{B}_{n}}),$ $i=1,...,n$ whenever $f\in
D_{\alpha}({\mathbb{B}_{n}}).$

A _multiplier_ in $D_{\alpha}({\mathbb{B}_{n}})$ is a holomorphic function
$\phi:{\mathbb{B}_{n}}\rightarrow{\mathbb{C}}$ that satisfies $\phi\cdot f\in
D_{\alpha}({\mathbb{B}_{n}})$ for all $f\in D_{\alpha}({\mathbb{B}_{n}}).$
Polynomials, as well as holomorphic functions in a neighbourhood of the closed
unit ball, are multipliers in every space $D_{\alpha}({\mathbb{B}_{n}})$.

###  1.2. Shift operators and cyclic vectors

Consider the bounded linear operators
$S_{1},...,S_{n}:D_{\alpha}({\mathbb{B}_{n}})\rightarrow
D_{\alpha}({\mathbb{B}_{n}})$ defined by $S_{i}:f\mapsto z_{i}\cdot f.$ We say
that $f\in D_{\alpha}({\mathbb{B}_{n}})$ is a _cyclic vector_ if the closed
invariant subspace, i.e.

$[f]:=\mathrm{clos}\,\mathrm{span}\\{z_{1}^{k_{1}}\cdots
z_{n}^{k_{n}}f:k_{1},...,k_{n}=0,1,...\\}$

coincides with $D_{\alpha}({\mathbb{B}_{n}})$ (the closure is taken with
respect to the $D_{\alpha}({\mathbb{B}_{n}})$ norm). An equivalent definition
is that $f$ is cyclic if and only if $1\in[f].$

Since $D_{\alpha}({\mathbb{B}_{n}})$ enjoys the _bounded point evaluation
property_ a function that is cyclic cannot vanish inside the unit ball. Thus,
we focus on functions non-vanishing in the domain. Also, non-zero constant
functions are cyclic in every space $D_{\alpha}({\mathbb{B}_{n}}).$ More
information regarding cyclic vectors in Dirichlet-type spaces over the disk,
the polydisc and the unit ball can be found in [3], [4], [5], [6], [8], [12],
[13], [14], [20], [21].

Just as in the settings of the bidisk and the unit ball of two variables, the
cyclicity of a function $f\in D_{\alpha}({\mathbb{B}_{n}})$ is inextricably
linked with its zero set

$\mathcal{Z}(f)=\\{z\in{\mathbb{C}}^{n}:f(z)=0\\}.$

The zeros of a function lying on the sphere are called the _boundary zeros_.

###  1.3. Plan of the paper

Section 2 studies Dirichlet-type spaces. In particular, we give a crucial
relation among them. Using fractional radial derivatives and the Cauchy
formula of functions lying in the ball algebra $A({\mathbb{B}_{n}})$ which
contains functions that are continuous on the closed unit ball and holomorphic
in its interior, we give an equivalent norm of Dirichlet-type spaces for a
wide range of parameters $\alpha.$

Section 3 studies diagonal subspaces. In particular, we extend result from
[21]. It makes sense to define functions $f\in\mathrm{Hol}({\mathbb{B}_{n}})$
using functions $\tilde{f}\in\mathrm{Hol}({\mathbb{D}}(\mu))$ for a proper
$\mu>0.$ Geometrically speaking, we are looking at a disk embedded in the ball
but not in a coordinate plane. Thus, we may switch the problem of cyclicity
from the ball to spaces of holomorphic functions of one variable that are well
known. Then we use optimal approximants in order to identify cyclicity.

Moreover, we prove cyclicity for model polynomials for proper parameters. In
the setting of the unit ball of two variables, see [21], the model polynomials
are the following: $1-z_{1}$ which vanishes in the closed unit ball on a
singleton, i.e. $\mathcal{Z}(1-z_{1})\cap\mathbb{S}_{2}=\\{(1,0)\\},$ and
$1-2z_{1}z_{2}$ which vanishes along an analytic curve, i.e.
$\mathcal{Z}(1-2z_{1}z_{2})\cap\mathbb{S}_{2}=\\{(e^{i\theta}/\sqrt{2},e^{-i\theta}/\sqrt{2}):\theta\in{\mathbb{R}}\\}.$
In our case, the corresponding candidates are the following:

$p(z)=1-m^{m/2}z_{1}\cdots z_{m},\quad 1\leq m\leq n.$

They vanish in the closed unit ball along the following analytic sets:

$\mathcal{Z}(p)\cap\mathbb{S}_{n}=\\{1/\sqrt{m}(e^{i\theta_{1}},..,e^{i\theta_{m-1}},e^{-i(\theta_{1}+...+\theta_{m-1})},0,..,0):\theta_{i}\in\mathbb{R}\\}.$

These polynomials are also studied with respect to the Drury-Arveson space in
[19].

In two variables, $1-z_{1}$ is cyclic in $D_{\alpha}(\mathbb{B}_{2})$
precisely when $\alpha\leq 2,$ and $1-2z_{1}z_{2}$ is cyclic in
$D_{\alpha}(\mathbb{B}_{2})$ precisely when $\alpha\leq 3/2.$ Here, there are
more than two fixed parameters. The characterization of cyclicity of these two
polynomials was crucial in [14].

Section 4 studies the radial dilation of a polynomial. Using the equivalent
norm of Section 2, we identify cyclicity for the model polynomials via the
powerful radial dilation method. In particular, we show that if
$p/p_{r}\rightarrow 1$ weakly, where $p_{r}(z)=p(rz)$ is a radial dilation of
$p,$ then $p$ is cyclic, (see [13] for the bidisk settings and [14] for the
unit ball in two variables). This method is quite interesting since it can be
applied to an arbitrary polynomial. Note that in [13], [14] the radial
dilation method is one of the main tools of solving cyclicity problem for
polynomials. The main result of this section verifies the arguments made about
polynomials in Section 3.

Section 5 studies non-cyclic vectors. We use the notion of Riesz
$\alpha$-capacity in order to identify non-cyclic functions. Moreover, we
study Cauchy transforms of Borel measures supported on zero sets of the radial
limits of a given function $f\in D_{\alpha}({\mathbb{B}_{n}})$ and we give
asymptotic expansions of their norms. Then employing a standard scheme due to
Brown and Shields, see [8], we prove the main result. Note that this
sufficient capacity condition for non-cyclicity in Dirichlet-type spaces in
the unit ball of two variables was proved by A. Sola in [21].

## Standard tools

Let us give some standard tools which will be useful in the sequel.

The binomial series:

$\frac{1}{(1-x)^{\alpha}}=\sum_{k=0}^{\infty}\frac{\Gamma(k+\alpha)}{\Gamma(\alpha)k!}x^{k},$

where $|x|<1$ is a complex number and $\alpha$ is a non-negative real number.
The asymptotic behaviour of the $\Gamma$-function is the following:
$\Gamma(k+\alpha)\asymp(k-1)!k^{\alpha},$ where the symbol $\asymp$ denotes
that the ratio of the two quantities either tends to a constant as $k$ tends
to infinity or it is rather two sides bound by constants.

The multinomial formula:

$(x_{1}+...+x_{n})^{k}=\sum_{|j|=k}\frac{k!}{j!}x_{1}^{j_{1}}\cdots
x_{n}^{j_{n}},$

where $j=(j_{1},...,j_{n})$ is a $n$-tuple index of non-negative integers and
$x_{i}$ are complex numbers.

The Stirling formula that describes the asymptotic behaviour of the gamma
function:

$k!\asymp k^{1/2}k^{k}/e^{k}.$

Denote the normalized area measure on ${\mathbb{C}}^{n}={\mathbb{R}}^{2n}$ by
$du(z)$ and the normalized rotation-invariant positive Borel measure on
$\mathbb{S}_{n}$ by $d\sigma(\zeta),$ (see [18], [22]). The measures $du(z)$
and $d\sigma(\zeta)$ are related by the formula

$\int_{{\mathbb{C}}^{n}}f(z)du(z)=2n\int_{0}^{\infty}\int_{\mathbb{S}_{n}}\epsilon^{2n-1}f(\epsilon\zeta)d\sigma(\zeta)d\epsilon.$

The holomorphic monomials are orthogonal to each other in $L^{2}(\sigma),$
that is, if $k$ and $l$ are multi-indices such that $k\neq l,$ then

$\int_{\mathbb{S}_{n}}\zeta^{k}\bar{\zeta}^{l}d\sigma(\zeta)=0.$

Moreover,

$\int_{\mathbb{S}_{n}}|\zeta^{k}|^{2}d\sigma(\zeta)=\frac{(n-1)!k!}{(n-1+|k|)!}\quad\text{and}\quad\int_{{\mathbb{B}_{n}}}|z^{k}|^{2}du(z)=\frac{n!k!}{(n+|k|)!}.$

##  2\. Relation among Dirichlet-type spaces and equivalent norms

We study the structure of Dirichlet-type spaces. Note that

$R(f)(z)=z_{1}\partial_{z_{1}}f(z)+...+z_{n}\partial_{z_{n}}f(z)$

is the _radial derivative_ of a function $f.$ The radial derivative plays a
key role in the function theory of the unit ball. A crucial relation among
these spaces is the following.

######  Proposition 1.

Let $f\in\mathrm{Hol}({\mathbb{B}_{n}})$ and $\alpha\in{\mathbb{R}}$ be fixed.
Then

$f\in D_{\alpha}({\mathbb{B}_{n}})\quad\text{if and only if}\quad
n^{q}f+R^{q}(f)+q\sum_{i=1}^{q-1}n^{i}R^{q-i}(f)\in
D_{\upsilon}({\mathbb{B}_{n}}),$

where $\alpha=2q+\upsilon,$ $q\in\mathbb{N}$ and $R^{q}$ is the $q$-image of
the operator $R.$

###### Proof.

Indeed, it is enough to check that

$||nf+R(f)||^{2}_{\alpha-2}=\sum_{|k|=0}^{\infty}(n+|k|)^{\alpha-2}\frac{(n-1)!k!}{(n-1+|k|)!}(n+|k|)^{2}|a_{k}|^{2}=||f||_{\alpha}^{2}.$

∎

We continue by giving an equivalent characterization of Dirichlet-type norms.
In Dirichlet-type spaces in the unit ball, one of the integral representations
of the norm is achieved in a limited range of parameters.

######  Lemma 2 (see[16]).

If $\alpha\in(-1,1)$, then $||f||^{2}_{\alpha}$ is equivalent to

$|f|^{2}_{\alpha}:=\int_{{\mathbb{B}_{n}}}\frac{||\nabla(f)(z)||^{2}-|R(f)(z)|^{2}}{(1-||z||^{2})^{\alpha}}du(z).$

Above, $\nabla(f)(z)=(\partial_{z_{1}}f(z),...,\partial_{z_{n}}f(z))$ denotes
the _holomorphic gradient_ of a holomorphic function $f.$ Note that
Proposition 1 allows us to use Lemma 2 whenever $\upsilon\in(-1,1).$ . Let
$\gamma,t\in{\mathbb{R}}$ be such that neither $n+\gamma$ nor $n+\gamma+t$ is
a negative integer. If $f=\sum_{|k|=0}^{\infty}a_{k}z^{k}$ is the homogeneous
expansion of a function $f\in\textrm{Hol}({\mathbb{B}_{n}}),$ then we may
define an invertible continuous linear operator with respect to the topology
of uniform convergence on compact subsets of ${\mathbb{B}_{n}},$ denoted by
$R^{\gamma,t}:\textrm{Hol}({\mathbb{B}_{n}})\rightarrow\textrm{Hol}({\mathbb{B}_{n}})$
and having expression

$R^{\gamma,t}f(z)=\sum_{|k|=0}^{\infty}C(\gamma,t,k)a_{k}z^{k},\quad
z\in{\mathbb{B}_{n}},$

where

(4)
$C(\gamma,t,k)=\frac{\Gamma(n+1+\gamma)\Gamma(n+1+|k|+\gamma+t)}{\Gamma(n+1+\gamma+t)\Gamma(n+1+|k|+\gamma)}\asymp|k|^{t}.$

See [22] for more information regarding these fractional radial derivatives.

######  Lemma 3.

Let $t\in{\mathbb{R}}$ be such that $n-1+t\geq 0.$ If $f\in
A({\mathbb{B}_{n}}),$ then

$R^{-1,t}f(z)=\int_{\mathbb{S}_{n}}\frac{f(\zeta)}{(1-\langle
z,\zeta\rangle)^{n+t}}d\sigma(\zeta),\quad z\in{\mathbb{B}_{n}}.$

###### Proof.

The continuous linear operator $R^{\gamma,t},$ see [22], satisfies

$R^{\gamma,t}\Big{(}\frac{1}{(1-\langle
z,w\rangle)^{n+1+\gamma}}\Big{)}=\frac{1}{(1-\langle
z,w\rangle)^{n+1+\gamma+t}}$

for all $w\in{\mathbb{B}_{n}}.$ Next, define $f_{\epsilon}$ for
$\epsilon\in(0,1)$ by

$f_{\epsilon}(z)=\int_{\mathbb{S}_{n}}\frac{f(\zeta)}{(1-\langle
z,\epsilon\zeta\rangle)^{n}}d\sigma(\zeta),\quad z\in{\mathbb{B}_{n}}.$

The Cauchy formula holds for $f\in A({\mathbb{B}_{n}})$ and hence
$f=\lim_{\epsilon\rightarrow 1^{-}}f_{\epsilon}.$ It follows that

$\displaystyle R^{-1,t}f(z)$
$\displaystyle=R^{-1,t}\Big{(}\lim_{\epsilon\rightarrow
1^{-}}\int_{\mathbb{S}_{n}}\frac{f(\zeta)}{(1-\langle
z,\epsilon\zeta\rangle)^{n}}d\sigma(\zeta)\Big{)}$
$\displaystyle=\lim_{\epsilon\rightarrow
1^{-}}R^{-1,t}\Big{(}\int_{\mathbb{S}_{n}}\frac{f(\zeta)}{(1-\langle
z,\epsilon\zeta\rangle)^{n}}d\sigma(\zeta)\Big{)}$
$\displaystyle=\lim_{\epsilon\rightarrow
1^{-}}\int_{\mathbb{S}_{n}}f(\zeta)R^{-1,t}\Big{(}\frac{1}{(1-\langle
z,\epsilon\zeta\rangle)^{n}}\Big{)}d\sigma(\zeta)$
$\displaystyle=\lim_{\epsilon\rightarrow
1^{-}}\int_{\mathbb{S}_{n}}\frac{f(\zeta)}{(1-\langle
z,\epsilon\zeta\rangle)^{n+t}}d\sigma(\zeta)$
$\displaystyle=\int_{\mathbb{S}_{n}}\frac{f(\zeta)}{(1-\langle
z,\zeta\rangle)^{n+t}}d\sigma(\zeta)$

and the assertion follows. ∎

######  Theorem 4.

Let $\alpha\in{\mathbb{R}}$ be such that $n-1+\alpha/2\geq 0$ and $f\in
A({\mathbb{B}_{n}}).$ Then $f\in D_{\alpha}({\mathbb{B}_{n}})$ if and only if

$\int_{{\mathbb{B}_{n}}}(1-||z||^{2})\Big{|}\int_{\mathbb{S}_{n}}\frac{f(\zeta)\bar{\zeta}_{p}}{(1-\langle
z,\zeta\rangle)^{n+\alpha/2+1}}d\sigma(\zeta)\Big{|}^{2}du(z)<\infty$

and

$\int_{{\mathbb{B}_{n}}}\Big{|}\int_{\mathbb{S}_{n}}\frac{(\overline{z_{p}\zeta_{q}-z_{q}\zeta_{p}})f(\zeta)}{(1-\langle
z,\zeta\rangle)^{n+\alpha/2+1}}d\sigma(\zeta)\Big{|}^{2}du(z)<\infty,$

where $p,q=1,...,n.$

###### Proof.

Choose $t$ so that $\alpha=2t.$ Note that $n,t$ are fixed and hence

$||f||^{2}_{\alpha}\asymp\sum_{|k|=0}^{\infty}\frac{(n-1)!k!}{(n-1+|k|)!}||k|^{t}a_{k}|^{2}.$

Thus, (4) implies that $||R^{-1,t}f||_{0}\asymp||f||_{\alpha}.$ One can apply
then the integral representation of Dirichlet-type norms to
$R^{-1,t}f\in\textrm{Hol}({\mathbb{B}_{n}}),$ i.e. $||R^{-1,t}f||_{0}$ is
equivalent to $|R^{-1,t}f|_{0}.$ According to Lemma 3 we get that

$\partial_{z_{p}}(R^{-1,t}f)(z)=\int_{\mathbb{S}_{n}}\frac{f(\zeta)\bar{\zeta}_{p}}{(1-\langle
z,\zeta\rangle)^{n+t+1}}d\sigma(\zeta),\quad z\in{\mathbb{B}_{n}},$

where $p=1,...,n.$ Expand the term $||\nabla(f)||^{2}-|R(f)|^{2}$ as follows:

$||\nabla(f)||^{2}-|R(f)|^{2}=(1-||z||^{2})||\nabla(f)||^{2}+\sum_{p,q}|\bar{z}_{p}\partial_{z_{q}}f-\bar{z}_{q}\partial_{z_{p}}f|^{2}.$

The assertion follows by Lemma 2. ∎

##  3\. Diagonal subspaces

In [3], a method of construction of optimal approximants via determinants in
Dirichlet-type spaces in the unit disk is provided. Similarly, we may define
optimal approximants in several variables, (see [19]).

Fix $N\in\mathbb{N}.$ We define the space of polynomials
$p\in{\mathbb{C}}[z_{1},...,z_{n}]$ with degree at most $nN$ as follows:

$P_{N}^{n}:=\\{p(z)=\sum_{k_{1}=0}^{N}...\sum_{k_{n}=0}^{N}a_{k_{1},...,k_{n}}z_{1}^{k_{1}}\cdots
z_{n}^{k_{n}}\\}.$

######  Remark 5.

Let $(X,||\cdot||)$ be a normed space and fix $x\in X,$ $C\subset X.$ The
distance between $x$ and the set $C$ is the following:

$\mathrm{dist}_{X}(x,C):=\inf\\{||x-c||:c\in C\\}.$

It is well known that if $X$ is a Hilbert space and $C\subset X$ a convex
closed subset, then for any $x\in X,$ there exists a unique $y\in C$ such that
$||x-y||=\mathrm{dist}_{X}(x,C).$ Let $f\in D_{\alpha}({\mathbb{B}_{n}})$ be
non-zero constant. We deduce that for any $N\in\mathbb{N},$ there exists
exactly one $p_{N}\in P_{N}^{n}$ satisfying

$||p_{N}f-1||_{\alpha}=\mathrm{dist}_{D_{\alpha}({\mathbb{B}_{n}})}(1,f\cdot
P_{N}^{n}).$

Let $f\in D_{\alpha}({\mathbb{B}_{n}}).$ We say that a polynomial $p_{N}\in
P^{n}_{N}$ is an optimal approximant of order $N$ to $1/f$ if $p_{N}$
minimizes $||pf-1||_{\alpha}$ among all polynomials $p\in P_{N}^{n}.$ We call
$||p_{N}f-1||_{\alpha}$ the optimal norm of order $N$ associated with $f.$

Let $M=(M_{1},...,M_{n})$ be a multi-index, where $M_{i}$ are non-negative
integers, and $m\in\\{1,...,n\\}.$ Setting

$\mu(m):=\frac{(M_{1}+...+M_{m})^{M_{1}+...+M_{m}}}{M_{1}^{M_{1}}\cdots
M_{m}^{M_{m}}},$

we see that

(5) $\mu(m)^{1/2}|z_{1}|^{M_{1}}\cdots|z_{m}|^{M_{m}}\leq 1,\quad
z\in{\mathbb{B}_{n}}.$

Using (5) we may construct polynomials that vanish in the closed unit ball
along analytic subsets of the unit sphere.

######  Remark 6.

Let $\tilde{f}\in\mathrm{Hol}({\mathbb{D}}(\mu(m)^{-1/4})),$ where

${\mathbb{D}}(\mu)=\\{z\in{\mathbb{C}}:|z|<\mu\\},\quad\mu>0.$

According to (5) we define the following function:

$f(z)=f(z_{1},...,z_{n})=\tilde{f}(\mu(m)^{1/4}z_{1}^{M_{1}}\cdots
z_{m}^{M_{m}}),\quad z\in{\mathbb{B}_{n}}.$

Then $f\in\mathrm{Hol}({\mathbb{B}_{n}})$ and it depends on $m$ variables.
Note that we may change the variables $z_{1},...,z_{m}$ by any other $m$
variables. For convenience, we choose the $m$ first variables. The power $1/4$
will be convenient in the sequel.

Thus, the question that arises out is if we may define closed subspaces of
$D_{\alpha}({\mathbb{B}_{n}})$ passing through one variable functions. We
shall see that these subspaces are called diagonal subspaces due to the nature
of the power series expansion of their elements.

Instead of the classical one variable Dirichlet-type spaces of the unit disk,
we may consider spaces $d_{\beta},$ $\beta\in{\mathbb{R}},$ consisting of
holomorphic functions $\tilde{f}\in\mathrm{Hol}({\mathbb{D}}(\mu^{-1/4})).$
Moreover, such functions with power series expansion
$\tilde{f}(z)=\sum_{l=0}^{\infty}a_{l}z^{l}$ are said to belong to $d_{\beta}$
if

$||\tilde{f}||^{2}_{d_{\beta}}:=\sum_{l=0}^{\infty}\mu^{-l/2}(l+1)^{\beta}|a_{l}|^{2}<\infty.$

There is a natural identification between the function theories of
$D_{\beta}({\mathbb{D}})$: one variable Dirichlet-type spaces of the unit
disk, and $d_{\beta},$ and one verifies that the results in [3] are valid for
$d_{\beta}.$

We are ready to define diagonal closed subspaces. Set

$\beta(\alpha):=\alpha-n+\frac{m+1}{2}.$

Let $\alpha,$ $M,$ $m$ be as above. The diagonal closed subspace of
$D_{\alpha}({\mathbb{B}_{n}})$ is the following:

$J_{\alpha,M,m}:=\\{f\in D_{\alpha}({\mathbb{B}_{n}}):\exists\tilde{f}\in
d_{\beta(\alpha)},f(z)=\tilde{f}(\mu(m)^{1/4}z_{1}^{M_{1}}\cdots
z_{m}^{M_{m}})\\}.$

The existence of a holomorphic function $\tilde{f}$ is unique by identity
principle and hence there is no any amiss in the definition. Any function
$f\in J_{\alpha,M,m}$ has an expansion of the form

$f(z)=\sum_{l=0}^{\infty}a_{l}(z_{1}^{M_{1}}\cdots z_{m}^{M_{m}})^{l}.$

The relation of norms between one variable and diagonal subspaces follows.

######  Proposition 7.

If $f\in J_{\alpha,M,m},$ then
$||f||_{\alpha}\asymp||\tilde{f}||_{d_{\beta(\alpha)}}.$

###### Proof.

If $f\in J_{\alpha,M,m},$ then

$||f||^{2}_{\alpha}\asymp\sum_{l=0}^{\infty}(l+1)^{\alpha}\frac{(M_{1}l)!\cdots(M_{m}l)!}{(n-1+(M_{1}+\cdots
M_{m})l)!}|a_{l}|^{2}.$

By Stirling’s formula, we obtain

$||f||^{2}_{\alpha}\asymp\sum_{l=0}^{\infty}(l+1)^{\alpha-n+m/2+1/2}\mu(m)^{-l}|a_{l}|^{2}.$

On the other hand, define the function
$f^{\prime}(z)=\sum_{l=0}^{\infty}\mu(m)^{-l/4}a_{l}z^{l}.$ Then
$f^{\prime}(\mu(m)^{1/4}z_{1}^{M_{1}}\cdots z_{m}^{M_{m}})=f(z_{1},...,z_{n})$
and

$||f^{\prime}||^{2}_{d_{\beta(\alpha)}}\asymp\sum_{l=0}(l+1)^{\alpha-n+m/2+1/2}\mu(m)^{-l}|a_{l}|^{2}.$

The assertion follows since $f^{\prime}$ coincides with $\tilde{f}.$ ∎

The corresponding Lemma 3.4 of [4] in our case is the following.

######  Lemma 8.

Let $f\in J_{\alpha,M,m},$ where $\alpha,M,m$ be as above. Let $r_{N}\in
P_{N}^{n}$ with expansion

$r_{N}(z)=\sum_{k_{1}=0}^{N}\cdots\sum_{k_{n}=0}^{N}a_{k_{1},...,k_{n}}z_{1}^{k_{1}}...z_{n}^{k_{n}},$

and consider its projection onto $J_{\alpha,M,m}$

$\pi(r_{N})(z)=\sum_{\\{l:M_{1}l,...,M_{m}l\leq
N\\}}c_{M_{1}l,...,M_{m}l,0,...,0}z_{1}^{M_{1}l}\cdots z_{m}^{M_{m}l}.$

Then

$||r_{N}f-1||_{\alpha}\geq||\pi(r_{N})f-1||_{\alpha}.$

Moreover, just as in Proposition 7, there is a relation of optimal
approximants between one variable and diagonal subspaces.

######  Proposition 9.

If $f\in J_{\alpha,M,m},$ then

$\mathrm{dist}_{D_{\alpha}({\mathbb{B}_{n}})}(1,f\cdot
P_{N}^{n})\asymp\mathrm{dist}_{d_{\beta}(\alpha)}(1,\tilde{f}\cdot
P^{1}_{N}).$

###### Proof.

Let $r_{N},$ $\pi(r_{N})$ be as in Lemma 8. Then $\pi(r_{N})f-1\in
J_{\alpha,M,m}.$ It follows that

$\displaystyle||r_{N}f-1||_{\alpha}\geq||\pi(r_{N})f-1||_{\alpha}\asymp||\tilde{\pi}(r_{N})\tilde{f}-1||_{d_{\beta(\alpha)}}\geq\mathrm{dist}_{d_{\beta(\alpha)}}(1,\tilde{f}\cdot
P^{1}_{N}),$

since $\tilde{\pi}(r_{N})\in P^{1}_{N}.$ On the other hand, let

$\mathrm{dist}_{d_{\beta(\alpha)}}(1,\tilde{f}\cdot
P^{1}_{N})=||q_{N}\tilde{f}-1||_{d_{\beta(\alpha)}},\quad
q_{N}(z)=\sum_{l=0}^{N}a_{l}z^{l}.$

Then, the polynomial

$q^{\prime}_{N}(z_{1},...,z_{n})=\sum_{l=0}^{N}\mu(m)^{-l/4}a_{l}z_{1}^{M_{1}l}\cdots
z_{m}^{M_{m}l}$

satisfies $q_{N}^{\prime}\in J_{\alpha,M,m}\cap P_{N}^{n}$ and
$q_{N}^{\prime}f-1\in J_{\alpha,M,m}.$ Thus,

$\displaystyle||q_{N}\tilde{f}-1||_{d_{\beta(\alpha)}}=||\tilde{q}_{N}^{\prime}\tilde{f}-1||_{d_{\beta(\alpha)}}\asymp||q_{N}^{\prime}f-1||_{\alpha}\geq\mathrm{dist}_{D_{\alpha}({\mathbb{B}_{n}})}(1,f\cdot
P_{N}^{n})$

and the assertion follows. ∎

Define the function $\phi_{\beta}:[0,\infty)\rightarrow[0,\infty)$ by

$\phi_{\beta}(t)=\begin{dcases}t^{1-\beta},&\beta<1\\\
\log^{+}(t),&\beta=1\end{dcases},$

where $\log^{+}(t):=\max\\{\log t,0\\}.$ We have the following.

######  Theorem 10.

Let $\alpha\in{\mathbb{R}}$ be such that $\beta(\alpha)\leq 1.$ Let $f\in
J_{\alpha,M,m}$ be as above and suppose the corresponding $\tilde{f}$ has no
zeros inside its domain, has at least one zero on the boundary, and admit an
analytic continuation to a strictly bigger domain. Then $f$ is cyclic in
$D_{\alpha}({\mathbb{B}_{n}})$ whenever $\alpha\leq\frac{2n-m+1}{2}$ and

$\mathrm{dist}^{2}_{D_{\alpha}({\mathbb{B}_{n}})}(1,f\cdot
P_{N}^{n})\asymp\phi_{\beta(\alpha)}(N+1)^{-1}.$

###### Proof.

It is an immediate consequence of the identification between
$D_{\beta}({\mathbb{D}})$ and $d_{\beta}$ and previous lemmas and
propositions. ∎

If we focus on polynomials, then the following is true.

######  Theorem 11.

Consider the polynomial $p(z)=1-m^{m/2}z_{1}\cdots z_{m},$ where $1\leq m\leq
n.$ Then $p$ is cyclic in $D_{\alpha}({\mathbb{B}_{n}})$ whenever
$\alpha\leq\frac{2n+1-m}{2}.$

Note that the Theorem11 is not a characterization. We shall study the case
$\alpha>\frac{2n+1-m}{2}.$

##  4\. Cyclicity for model polynomials via radial dilation

The _radial dilation_ of a function
$f:{\mathbb{B}_{n}}\rightarrow{\mathbb{C}}$ is defined for $r\in(0,1)$ by
$f_{r}(z)=f(rz).$ To prove Theorem 11, it is enough to prove the following
lemma.

######  Lemma 12.

Consider the polynomial $p(z)=1-m^{m/2}z_{1}\cdots z_{m},$ where $1\leq m\leq
n.$ Then $||p/p_{r}||_{\alpha}<\infty$ as $r\rightarrow 1^{-}$ whenever
$\alpha\leq\frac{2n+1-m}{2}.$

We follow the arguments of [14], [13]. Indeed, if Lemma 12 holds, then
$\phi_{r}\cdot p\rightarrow 1$ weakly, where $\phi_{r}:=1/p_{r}.$ This is a
consequence of a crucial property of Dirichlet-type spaces: if
$\\{f_{n}\\}\subset D_{\alpha}({\mathbb{B}_{n}}),$ then $f_{n}\rightarrow 0$
weakly if and only if $f_{n}\rightarrow 0$ pointwise and
$\sup_{n}\\{||f_{n}||_{\alpha}\\}<\infty.$ 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 closed and convex or, equivalently, 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 $\alpha_{0}=\frac{2n+1-m}{2}.$ Then the case
$\alpha<\alpha_{0}$ follows since the inclusion
$D_{\alpha_{0}}({\mathbb{B}_{n}})\hookrightarrow D_{\alpha}({\mathbb{B}_{n}})$
is a compact linear map and weak convergence in
$D_{\alpha_{0}}({\mathbb{B}_{n}})$ gives weak convergence in
$D_{\alpha}({\mathbb{B}_{n}}).$

###### Proof of Lemma 12.

By Theorem 4 it is enough to show the following:

$I_{p}:=\int_{{\mathbb{B}_{n}}}(1-||z||^{2})\Big{|}\int_{\mathbb{S}_{n}}\frac{(1-\lambda\zeta_{1}\cdots\zeta_{m})\bar{\zeta}_{p}}{(1-r^{m}\lambda\zeta_{1}\cdots\zeta_{m})(1-\langle
z,\zeta\rangle)^{\beta}}d\sigma(\zeta)\Big{|}^{2}du(z)$

and

$I_{p,q}:=\int_{{\mathbb{B}_{n}}}\Big{|}\int_{\mathbb{S}_{n}}\frac{(\overline{z_{p}\zeta_{q}-z_{q}\zeta_{p}})(1-\lambda\zeta_{1}\cdots\zeta_{m})}{(1-r^{m}\lambda\zeta_{1}\cdots\zeta_{m})(1-\langle
z,\zeta\rangle)^{\beta}}d\sigma(\zeta)\Big{|}^{2}du(z)$

are finite, as $r\rightarrow 1^{-},$ where $\beta=n+t+1,$
$t=\frac{2n+1-m}{4},$ and $\lambda=m^{m/2}.$

Denote

$S_{p}:=\int_{\mathbb{S}_{n}}\frac{(1-\lambda\zeta_{1}\cdots\zeta_{m})\bar{\zeta}_{p}}{(1-r^{m}\lambda\zeta_{1}\cdots\zeta_{m})(1-\langle
z,\zeta\rangle)^{\beta}}d\sigma(\zeta),$

where the last integral is equal to

$\frac{1}{2\pi}\int_{\mathbb{S}_{n}}\int_{0}^{2\pi}\frac{(1-\lambda
e^{im\theta}\zeta_{1}\cdots\zeta_{m})e^{-i\theta}\bar{\zeta}_{p}}{(1-r^{m}\lambda
e^{im\theta}\zeta_{1}\cdots\zeta_{m})(1-e^{-i\theta}\langle
z,\zeta\rangle)^{\beta}}d\theta d\sigma(\zeta).$

Let $z,\zeta$ be fixed. Then

$\int_{0}^{2\pi}\frac{e^{-i\theta}}{(1-e^{-i\theta}\langle
z,\zeta\rangle)^{\beta}}d\theta=0.$

Thus, replacing $p(e^{i\theta}\zeta)/p(re^{i\theta}\zeta)$ by
$p(e^{i\theta}\zeta)/p(re^{i\theta}\zeta)-1$ we obtain

$S_{p}=\frac{\lambda(r^{m}-1)}{2\pi}\int_{\mathbb{S}_{n}}\int_{0}^{2\pi}\frac{\bar{\zeta}_{p}\zeta_{1}\cdots\zeta_{m}e^{i(m-1)\theta}}{(1-r^{m}\lambda
e^{im\theta}\zeta_{1}\cdots\zeta_{m})(1-e^{-i\theta}\langle
z,\zeta\rangle)^{\beta}}d\theta d\sigma(\zeta).$

Next, expand the binomials

$\displaystyle\int_{0}^{2\pi}$
$\displaystyle\frac{e^{i(m-1)\theta}}{(1-r^{m}\lambda
e^{im\theta}\zeta_{1}\cdots\zeta_{m})(1-e^{-i\theta}\langle
z,\zeta\rangle)^{\beta}}d\theta$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{\Gamma(k+\beta)}{\Gamma(\beta)k!}(r^{m}\lambda\zeta_{1}\cdots\zeta_{m})^{l}\langle
z,\zeta\rangle^{k}\int_{0}^{2\pi}e^{i(m(l+1)-k-1)\theta}d\theta$
$\displaystyle=2\pi\sum_{k=0}^{\infty}\frac{\Gamma(m(k+1)-1+\beta)}{\Gamma(\beta)(m(k+1)-1)!}(r^{m}\lambda\zeta_{1}\cdots\zeta_{m})^{k}\langle
z,\zeta\rangle^{m(k+1)-1}$
$\displaystyle=2\pi\sum_{k=0}^{\infty}\sum_{|j|=m(k+1)-1}\frac{\Gamma(m(k+1)-1+\beta)}{\Gamma(\beta)j!}(r^{m}\lambda\zeta_{1}\cdots\zeta_{m})^{k}z^{j}\bar{\zeta}^{j}.$

Therefore,

$S_{p}=\lambda(r^{m}-1)\sum_{k=0}^{\infty}\sum_{|j|=m(k+1)-1}\frac{\Gamma(m(k+1)-1+\beta)}{\Gamma(\beta)j!}(r^{m}\lambda)^{k}z^{j}c(k),$

where
$c(k)=\int_{\mathbb{S}_{n}}\zeta^{\alpha(k)}\bar{\zeta}^{b(k)}d\sigma(\zeta),$
$\alpha(k)=(k+1,..,k+1\text{(m-comp.)},0,..,0)$ and
$b(k)=(j_{1},...,j_{p-1},j_{p}+1,j_{p+1},...,j_{n}).$ Whence, $1\leq p\leq m.$
Since the holomorphic monomials are orthogonal to each other in
$L^{2}(\sigma)$ we get that

$|S_{p}|\asymp(1-r^{m})\Big{|}z^{\prime}_{p}\sum_{k=0}^{\infty}(k+1)^{\beta-n}(r^{m}\lambda
z_{1}\cdots z_{m})^{k}\Big{|},$

where $z^{\prime}_{p}=z_{1}\cdots z_{p-1}z_{p+1}\cdots z_{m}.$ Hence we obtain

$I_{p}\asymp(1-r^{m})^{2}\sum_{k=0}^{\infty}(k+1)^{2(\beta-n)}(r^{m}\lambda)^{2k}\int_{{\mathbb{B}_{n}}}(1-||z||^{2})|z^{\prime}_{p}|^{2}|z_{1}\cdots
z_{m}|^{2k}du(z),$

where has been used again the orthogonality of the holomorphic monomials in
$L^{2}(\sigma).$ To handle the integral above we use polar coordinates

$\displaystyle\int_{{\mathbb{B}_{n}}}$
$\displaystyle(1-||z||^{2})|z^{\prime}_{p}|^{2}|z_{1}\cdots z_{m}|^{2k}du(z)$
$\displaystyle\asymp\int_{0}^{1}\int_{\mathbb{S}_{n}}\epsilon^{2n-1}(1-\epsilon^{2})\epsilon^{2km+2m-2}|\zeta_{p}^{\prime}|^{2}|\zeta_{1}\cdots\zeta_{m}|^{2k}d\sigma(\zeta)d\epsilon$
$\displaystyle\asymp\frac{[(k+1)!]^{m-1}k!}{(n+m(k+1)-2)!}\cdot\frac{1}{(k+1)^{2}}.$

If we recall that $\beta=n+t+1,$ $t=\frac{2n+1-m}{4}$ and
$\lambda^{2k}=m^{mk},$ then applying the Stirling formula more than one time
we see that

$I_{p}\asymp(1-r^{m})^{2}\sum_{k=0}^{\infty}(k+1)r^{2mk}.$

This proves the assertion made about $I_{p}.$

It remains to estimate the following term:

$I_{p,q}=\int_{{\mathbb{B}_{n}}}\Big{|}\int_{\mathbb{S}_{n}}\frac{(\overline{z_{p}\zeta_{q}-z_{q}\zeta_{p}})(1-\lambda\zeta_{1}\cdots\zeta_{m})}{(1-r^{m}\lambda\zeta_{1}\cdots\zeta_{m})(1-\langle
z,\zeta\rangle)^{\beta}}d\sigma(\zeta)\Big{|}^{2}du(z).$

We shall show that $I_{p,q}\asymp I_{p}.$ Denote again the inner integral by
$S_{p,q}$ which is convenient to expand it as
$S_{p,q}=\bar{z}_{p}S_{q}-\bar{z}_{q}S_{p}.$ Recall that
$z_{p}^{\prime}=z_{1}\cdots z_{p-1}z_{p+1}\cdots z_{m}.$ Similar calculations
to the one above lead to

$|S_{p,q}|\asymp(1-r^{m})|\bar{z}_{p}z_{q}^{\prime}-\bar{z}_{q}z_{p}^{\prime}|\Big{|}\sum_{k=0}^{\infty}(k+1)^{\beta-n}(r^{m}\lambda
z_{1}\cdots z_{m})^{k}\Big{|}.$

Moreover, the orthogonality of the holomorphic monomials in $L^{2}(\sigma)$
gives the following estimation:

$I_{p,q}\asymp(1-r^{m})^{2}\sum_{k=0}^{\infty}(k+1)^{2\beta-2n}(r^{m}\lambda)^{2k}\int_{{\mathbb{B}_{n}}}|\bar{z}_{p}z_{q}^{\prime}-\bar{z}_{q}z_{p}^{\prime}|^{2}|z_{1}\cdots
z_{m}|^{2k}du(z).$

It is easy to see that
$|\bar{z}_{p}z_{q}^{\prime}-\bar{z}_{q}z_{p}^{\prime}|^{2}=|z_{p}|^{2}|z^{\prime}_{q}|^{2}+|z_{q}|^{2}|z^{\prime}_{p}|^{2}-2|z_{1}\cdots
z_{m}|^{2}.$ Let us estimate the integral

$\int_{{\mathbb{B}_{n}}}(|z_{p}|^{2}|z_{q}^{\prime}|^{2}-|z_{1}\cdots
z_{m}|^{2})|z_{1}\cdots z_{m}|^{2k}du(z).$

Passing through polar coordinates we get, for $p\neq q,$ that

$\displaystyle\int_{{\mathbb{B}_{n}}}|z_{p}|^{2}|z_{q}^{\prime}|^{2}$
$\displaystyle|z_{1}\cdots z_{m}|^{2k}du(z)$ $\displaystyle\asymp
2n(n-1)!\frac{[(k+1)!]^{m-1}k!}{(mk+n+m-1)!}\frac{k+2}{2km+2n+2m},$

and

$\displaystyle\int_{{\mathbb{B}_{n}}}|z_{1}\cdots z_{m}|^{2(k+1)}$
$\displaystyle du(z)$
$\displaystyle=2n(n-1)!\frac{[(k+1)!]^{m-1}k!}{(mk+n+m-1)!}\frac{k+1}{2km+2n+2m}.$

Hence we obtain

$\displaystyle\int_{{\mathbb{B}_{n}}}(|z_{p}|^{2}|z_{q}^{\prime}|^{2}-|z_{1}\cdots
z_{m}|^{2})$ $\displaystyle|z_{1}\cdots z_{m}|^{2k}du(z)$
$\displaystyle\asymp\frac{[(k+1)!]^{m-1}k!}{(mk+n+m-2)!(k+1)^{2}}.$

Again, applying the Stirling formula to the one above estimates we obtain

$I_{p,q}\asymp(1-r^{m})^{2}\sum_{k=0}^{\infty}(k+1)r^{2mk}.$

This proves the assertion made about $I_{p,q}.$ ∎

##  5\. Sufficient conditions for non-cyclicity via Cauchy transforms and
$\alpha$-capacities

We consider the Cauchy transform of a complex Borel measure $\mu$ on the unit
sphere by

$C_{[\mu]}(z)=\int_{\mathbb{S}_{n}}\frac{1}{(1-\langle
z,\bar{\zeta}\rangle)^{n}}d\mu(\zeta),\quad z\in{\mathbb{B}_{n}}.$

Note that this definition differs from the classical one.

Let $f\in D_{\alpha}({\mathbb{B}_{n}})$ and put a measure $\mu$ on
$\mathcal{Z}(f^{*})$: the zero set in the sphere of the radial limits of $f$.
The results in [21] about Cauchy transforms and non-cyclicity are valid in our
settings. We deduce that $[f]\neq D_{\alpha}({\mathbb{B}_{n}})$, and hence
non-cyclicity, whenever $C_{[\mu]}\in D_{-\alpha}({\mathbb{B}_{n}}).$ Thus, it
is important to compute the Dirichlet-type norm of the Cauchy transform.

Let $\mu$ be a Borel measure on $\mathbb{S}_{n}$ and set
$\mu^{*}(j)=\int_{\mathbb{S}_{n}}\zeta^{j}d\mu(\zeta),$
$\bar{\mu}^{*}(j)=\int_{\mathbb{S}_{n}}\bar{\zeta}^{j}d\mu(\zeta).$ We have
the following.

######  Lemma 13.

Let $\mu$ be a Borel measure on $\mathbb{S}_{n}.$ Then

$||C_{[\mu]}||_{-\alpha}^{2}\asymp\sum_{k=0}^{\infty}\sum_{|j|=k}\frac{(k+1)^{n-1-\alpha}k!}{j!}|\bar{\mu}^{*}(j)|^{2}.$

###### Proof.

Our Cauchy integral of $\mu$ on ${\mathbb{B}_{n}}$ has the following expansion

$C_{[\mu]}(z)=\sum_{k=0}^{\infty}\sum_{|j|=k}\frac{\Gamma(k+n)}{\Gamma(n)j!}\bar{\mu}^{*}(j)z^{j}.$

Therefore, one can compute the norm of $C_{[\mu]}$ in the space
$D_{-\alpha}({\mathbb{B}_{n}}).$ The assertion follows. ∎

The following lemma is crucial in the sequel. It is probably known, but we
were not able to locate it in the literature, and hence we include its proof.

######  Lemma 14.

Let $j_{1},...,j_{n},k$ be non-negative integers satisfying
$j_{1}+...+j_{n}=nk.$ Then

$j_{1}!\cdots j_{n}!\geq(k!)^{n}.$

###### Proof.

The $\Gamma$-function is logarithmically convex, and hence, we may apply the
Jensen inequality to it:

$\log\Gamma\Big{(}\frac{x_{1}}{n}+...+\frac{x_{n}}{n}\Big{)}\leq\frac{\log\Gamma(x_{1})}{n}+...+\frac{\log\Gamma(x_{n})}{n}.$

Set $x_{i}:=j_{i}+1,$ $i=1,...,n.$ Since $j_{1}+...+j_{n}=nk,$ the assertion
follows. ∎

We may identify non-cyclicity for model polynomials via Cauchy transforms.

######  Lemma 15.

Consider the polynomial $p(z)=1-m^{m/2}z_{1}\cdots z_{m},$ where $1\leq m\leq
n.$ Then $p$ is not cyclic in $D_{\alpha}({\mathbb{B}_{n}})$ whenever
$\alpha>\frac{2n+1-m}{2}.$

###### Proof.

Recall that the model polynomials vanish in the closed unit ball along
analytic sets of the form:

$\mathcal{Z}(p)\cap\mathbb{S}_{n}=\\{1/\sqrt{m}(e^{i\theta_{1}},..,e^{i\theta_{m-1}},e^{-i(\theta_{1}+...+\theta_{m-1})},0,..,0):\theta_{i}\in\mathbb{R}\\}.$

It is easy to see that for a proper measure $\mu$, $\mu^{*}(j)$ is non-zero
when $mj_{m}=k$ and $\mu^{*}(j)\asymp m^{-k/2}.$ By Stirling’s formula and
Lemma 14 we get that

$\displaystyle||C_{[\mu]}||_{-\alpha}^{2}$ $\displaystyle\leq
C\sum_{k=0}^{\infty}\frac{(mk+1)^{n-1-\alpha}(mk)!}{(k!)^{m}m^{mk}}\asymp\sum_{k=0}^{\infty}(k+1)^{1/2(2n-m-1)-\alpha}.$

Thus, $p$ is not cyclic in $D_{\alpha}({\mathbb{B}_{n}})$ for
$\alpha>\frac{2n+1-m}{2}.$ ∎

We consider Riesz $\alpha$-capacity for a fixed parameter $\alpha\in(0,n)$
with respect to the _anisotropic distance_ in $\mathbb{S}_{n}$ given by

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

and the non-negative _kernel_ $K_{\alpha}:(0,\infty)\rightarrow[0,\infty)$
given by

$K_{\alpha}(t)=\begin{dcases}t^{\alpha-n},&\alpha\in(0,n)\\\
\log(e/t),&\alpha=n\end{dcases}.$

Note that we may extend the definition of $K$ to $0$ by defining
$K(0):=\lim_{t\rightarrow 0^{+}}K(t).$

Let $\mu$ be any Borel probability measure supported on some Borel set
$E\subset\mathbb{S}_{n}.$ Then the Riesz $\alpha$-energy of $\mu$ is given by

$I_{\alpha}[\mu]=\iint_{\mathbb{S}_{n}}K_{\alpha}(|1-\langle\zeta,\eta\rangle|)d\mu(\zeta)d\mu(\eta)$

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

$\mathrm{cap}_{\alpha}(E)=\inf\\{I_{\alpha}[\mu]:\mu\in\mathcal{P}(E)\\}^{-1},$

where $\mathcal{P}(E)$ is the set of all Borel probability measures supported
on $E.$ Note that if $\text{cap}_{\alpha}(E)>0,$ then there exist at least one
probability measure supported on $E$ having finite Riesz $\alpha$-energy.
Moreover, any $f\in D_{\alpha}({\mathbb{B}_{n}})$ has finite radial limits
$f^{*}$ on $\mathbb{S}_{n},$ except possibly, on a set $E$ having
$\text{cap}_{\alpha}(E)=0.$ Theory regarding to the above standard
construction in potential theory can be found in [1], [9], [12], [17].

The relation between non-cyclicity of a function and the Riesz
$\alpha$-capacity of the zeros of its radial limits follows.

######  Theorem 16.

Fix $\alpha\in(0,n]$ and let $f\in D_{\alpha}({\mathbb{B}_{n}}).$ If
$\mathrm{cap}_{\alpha}(\mathcal{Z}(f^{*}))>0,$ then $f$ is not cyclic in
$D_{\alpha}({\mathbb{B}_{n}}).$

###### Proof.

Let $\mu$ be a probability measure supported in $\mathcal{Z}(f^{*}),$ with
finite Riesz $n$-energy. If $r\in(0,1),$ then

$\displaystyle\log\frac{e}{|1-r\langle\zeta,\eta\rangle|}$
$\displaystyle=1+\textrm{Re}\Big{(}\log\frac{1}{1-r\langle\zeta,\eta\rangle}\Big{)}$
$\displaystyle=1+\textrm{Re}\sum_{k=1}^{\infty}\sum_{|j|=k}\frac{r^{k}k!}{kj!}\zeta^{j}\overline{\eta}^{j}.$

Note that $\mu$ is a probability measure and hence

$\displaystyle\iint_{\mathbb{S}_{n}}\log\frac{e}{|1-r\langle\zeta,\eta\rangle|}d\mu(\zeta)d\mu(\eta)=1+\sum_{k=1}^{\infty}\sum_{|j|=k}\frac{r^{k}k!}{kj!}|\mu^{*}(j)|^{2}.$

Since $|1-w|/|1-rw|\leq 2$ for $r\in(0,1)$ and $w\in\overline{{\mathbb{D}}},$
the dominated convergence theorem and Lemma 13 give

$||C_{[\mu]}||^{2}_{-n}\asymp\sum_{k=1}^{\infty}\sum_{|j|=k}\frac{k!}{kj!}|\mu^{*}(j)|^{2}<\infty.$

The assertion follows.

We continue setting a probability measure $\mu,$ supported in
$\mathcal{Z}(f^{*}),$ with finite Riesz $\alpha$-energy, where
$\alpha\in(0,n).$ If $r\in(0,1),$ then

$\displaystyle\frac{1}{(1-r\langle\zeta,\eta\rangle)^{n-\alpha}}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{|j|=k}\frac{\Gamma(k+n-\alpha)k!r^{k}}{k!\Gamma(n-\alpha)j!}\zeta^{j}\overline{\eta}^{j}.$

Similar arguments to the one above show that

$\displaystyle I_{\alpha}[\mu]$
$\displaystyle\geq\Big{|}\iint_{\mathbb{S}_{n}}\textrm{Re}\Big{(}\frac{1}{(1-r\langle\zeta,\eta\rangle)^{n-\alpha}}\Big{)}d\mu(\zeta)d\mu(\eta)\Big{|}$
$\displaystyle=\Big{|}\sum_{k=0}^{\infty}\sum_{|j|=k}\frac{\Gamma(k+n-\alpha)k!r^{k}}{k!\Gamma(n-\alpha)j!}\iint_{\mathbb{S}_{n}}\zeta^{j}\overline{\eta}^{j}d\mu(\zeta)d\mu(\eta)\Big{|}$
$\displaystyle\asymp\sum_{k=0}^{\infty}\sum_{|j|=k}\frac{(k+1)^{n-1-\alpha}k!}{j!}r^{k}|\mu^{*}(j)|^{2}.$

Again, letting $r\rightarrow 1^{-}$ by Lemma 13 we obtain that $C_{[\mu]}\in
D_{-\alpha}({\mathbb{B}_{n}}).$ The assertion follows. ∎

######  Remark 17.

According to [14] one can expect that the cyclicity problem of polynomials in
the unit ball of ${\mathbb{C}}^{n}$ depends on the real dimension of their
zero set restricted on the unit sphere:
$\text{dim}_{{\mathbb{R}}}(\mathcal{Z}(p)\cap\mathbb{S}_{n}).$

Let us point out the nature of the boundary zeros of a polynomial non-
vanishing in the ball. See [14] for the two dimensional case where had been
used the Curve Selection Lemma of [10].

Let $p\in{\mathbb{C}}[z_{1},...,z_{n}]$ be a polynomial non-vanishing in the
ball. Looking at $\mathcal{Z}(p)\cap\mathbb{S}_{n}$ as at a semi-algebraic
set, we conclude that it is the disjoint union of a finite number of Nash
manifolds $M_{i},$ each Nash diffeomorphic to an open hypercube
$(0,1)^{\textrm{dim}(M_{i})}.$ Note that the Nash diffeomorphisms over the
closed field of the real numbers satisfy some additional properties (see [7],
Proposition 2.9.10).

One can expect then that the characterization of cyclicity and the nature of
the boundary zeros of the model polynomials, as well as, the unitary
invariance of the Dirichlet norm and the sufficient capacity condition, will
be crucial in the characterization of cyclic polynomials in arbitrary
dimension.

Acknowledgments. I would like to thank Ł. Kosiński for the helpful
conversations during the preparation of the present work. I would like to
thank also the anonymous referee for numerous remarks that substantially
improved the shape of the paper.

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