# Chemical Tagging N-rich Field Stars with High-resolution Spectroscopy

Jincheng Yu School of Physics and Astronomy, Sun Yat-sen University, Zhuhai
519082, China  Baitian Tang School of Physics and Astronomy, Sun Yat-sen
University, Zhuhai 519082, China  J. G. Fernandez-Trincado Instituto de
Astronomía y Ciencias Planetarias, Universidad de Atacama, Copayapu 485,
Copiapó, Chile  Centro de Investigación en Astronomía, Universidad Bernardo O
Higgins, Avenida Viel 1497, Santiago, Chile  Douglas Geisler Departamento de
Astronomía, Casilla 160-C, Universidad de Concepción, Concepción, Chile
Instituto de Investigación Multidisciplinario en Ciencia y Tecnología,
Universidad de La Serena. Avenida Raúl Bitrán S/N, La Serena, Chile
Departamento de Astronomía, Facultad de Ciencias, Universidad de La Serena.
Av.  Hongliang Yan  Key Lab of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100101,
China  M. Soto  Instituto de Astronomía y Ciencias Planetarias, Universidad de
Atacama, Copayapu 485, Copiapó, Chile

###### Abstract

We measure chemical abundances for over 20 elements of 15 N-rich field stars
with high resolution ($R\sim 30000$) optical spectra. We find that Na, Mg, Al,
Si, and Ca abundances of our N-rich field stars are mostly consistent with
those of stars from globular clusters (GCs). Seven stars are estimated to have
[Al/Fe$]>0.5$, which is not found in most GC “first generation” stars. On the
other hand, $\alpha$ element abundances (especially Ti) could show
distinguishable differences between in situ stars and accreted stars. We
discover that one interesting star, with consistently low [Mg/Fe], [Si/Fe],
[Ca/Fe], [Ti/Fe], [Sc/Fe], [V/Fe], and [Co/Fe], show similar kinematic and
[Ba/Eu] as other stars from the dissolved dwarf galaxy “$Gaia$-Sausage-
Enceladus”. The $\alpha$-element abundances and the iron-peak element
abundances of the N-rich field stars with metallicities
$-1.25\leq{\rm[Fe/H]}\leq-0.95$ show consistent values with Milky Way field
stars 111We refer Milky Way field stars as Milky Way halo field stars unless
specified in this paper. rather than stars from dwarf galaxies, indicating
that they were formed in situ. In addition, the neutron capture elements of
N-rich field stars show that most of them could be enriched by asymptotic
giant branch (AGB) stars with masses around $3-5\,M_{\odot}$.

stars: chemically peculiar – stars: abundances – stars: evolution

##  1 Introduction

Globular clusters (GCs) were traditionally considered as simple stellar
population systems, i.e., all member stars originate from the same molecular
cloud, share the same age and abundances. However, an increasing number of
studies (see e.g. Gratton et al., 2012; Piotto et al., 2015; Milone et al.,
2017; Tang et al., 2017, 2018, and references therein) show that almost all
GCs host two or more groups of stars with different chemical abundances, which
is so-called multiple populations (MPs). Stars with enhanced N, Na, (sometimes
He, Al, and Si), but depleted C, O, (sometimes Mg), are traditionally called
“second generation” (SG) stars (or “second population” stars), distinct from
the primordial “first generation” (FG) stars. SG stars and FG stars possibly
differ in age by a few hundred Myr (e.g., Conroy, 2012; Bekki et al., 2017)
but have the same [Fe/H] abundance (e.g., Carretta et al., 2010b).
Furthermore, SG stars have different radial distribution as FG stars in most
GCs, generally being more centrally concentrated (e.g Lardo et al., 2011;
Simioni et al., 2016), although a few exceptions has also been reported: some
GCs show that FG stars concentrated more in the inner region (e.g. Larsen et
al., 2015; Vanderbeke et al., 2015; Lim et al., 2016), while some GCs show
similar spatial distribution between these two populations (e.g. Dalessandro
et al., 2014; Miholics et al., 2015).

Most scenarios trying to explain the MP phenomenon assume that chemically
enriched SG stars were formed in light-element-polluted environment formed
partially from the ejecta of FG stars. These stars are generally enriched in
light elements, such as C, N, Na, etc, which are also known as enriched
populations. The proposed astrophysical sites include very massive stars
(VMSs), fast rotating massive stars (FRMSs), asymptotic giant branch (AGB)
stars, etc, although none is currently able to explain all observational
constraints (e.g., Bastian & Lardo, 2018).

On one hand, stellar evolution of massive stars leads to the mass loss of GCs,
especially for FG stars(e.g., D’Ercole et al., 2008; Schaerer & Charbonnel,
2011). On the other hand, dynamical evolution, including internal two-body
interaction and external tidal stripping, also contribute to the stellar loss
of GCs (Lamers et al., 2010). FG stars are chemically similar to normal field
stars, while SG stars show peculiar abundances in several light elements.
Thus, identifying field stars with similar chemical pattern as GC SG stars
could potentially add important observational constraints to help motivate
theoretical scenarios.

With the help of large spectroscopic surveys, the search for these chemically
peculiar stars is becoming more efficient. N is usually enhanced in GC SG
stars, and relatively easier to measure in low-resolution spectroscopic
survey, therefore, N-rich field stars found in early surveys are usually
linked to stars escaped from GCs (e.g., Martell & Grebel, 2010; Martell et
al., 2011; Koch et al., 2019). Recently, high resolution spectroscopic
surveys, like the Apache Point Observatory Galactic Evolution Experiment
(APOGEE; Majewski et al., 2017) have led to the discovery of a large group of
N-rich field stars (Fernández-Trincado et al., 2016, 2017, 2019; Martell et
al., 2016; Schiavon et al., 2017). In this respect, the unprecedented large
number (more than $10^{6}$) of low-resolution stellar spectra available in the
Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) survey
(Zhao et al., 2012) greatly increase the efficiency and accuracy of such a
search. Tang et al. (2019) (hereafter Paper I, ) and Tang et al. (2020)
(hereafter Paper II, ) have identified $\sim 100$ N-rich stars from a sample
of metal-poor, red giant branch (RGB) field stars based on CN3839, CN4142, and
CH4300 spectral indices using LAMOST DR3 (data release) and DR5. These N-rich
field stars show similar kinematics as Galactic GCs, which are a mixture of
GCs formed in situ and ex situ (e.g., Massari et al., 2019). To further
explore the connection between N-rich field stars and GCs using the idea of
“chemical tagging”, we investigate in optical high resolution spectra to
carefully examine more than 20 elements in this work. Several elements that we
shown in this work (e.g., Na, Ba, Eu) are important for constraining the
nature of MPs, but rarely available in near-infrared studies.

The paper is organized as follows. In Section 2, we describe our data and
reduction processes. We carefully compare our derived abundances with
literature values of Milky Way (MW) GCs, MW (disc and halo stars) and dwarf
galaxies to investigate the origin of N-rich field stars in Section 3 and 4.
Our final conclusion is given in Section 5.

##  2 The Data

###  2.1 Follow-up observations and data reduction

We started the selection of observable targets from our LAMOST N-rich field
star sample (Paper I and Paper II), which are mostly located in the northern
sky. Fifteen stars with DEC $<10^{\circ}$ and a reasonable airmass were
selected. Our follow-up observations were taken using the MIKE (Magellan
Inamori Kyocera Echelle, Bernstein et al., 2003) spectrograph on the Magellan
Clay telescope (Shectman & Johns, 2003) at Las Campanas Observatory, Chile.
The data were taken in two runs, 2019/07 and 2020/02. The blue and red
detectors were used simultaneously to cover $3200<\lambda<5000$ Å (blue side)
and $4900<\lambda<10000$ Å (red side). We used the 0.7” slit and 2x2 spatial
on-chip binning to achieve nominal spectral resolution of 35k/28k on the blue
side and red side, respectively. Depending on the observation schedule and the
brightness of the star, we took several exposures to ensure SNR (signal-to-
noise ratio) $>50$ over most wavelength of the red detector (Table 1).

We reduced the observational data using the _CarPy_ package (Kelson et al.,
2000; Kelson, 2003). The observed spectral images were bias-subtracted, flat-
field corrected, wavelength calibrated, scatter-light and sky subtracted. The
multi-order reduced spectra were then merged into a single spectrum.

###  2.2 Stellar parameters and chemical abundances

Stellar parameters and chemical abundances were analyzed using Brussels
Automatic Stellar Parameter (BACCHUS) code (Masseron et al., 2016). We briefly
describe the basic procedures here, readers are referred to Masseron et al.
(2016) for detailed descriptions.

The equivalent widths (EW) of Fe I and Fe II absorption lines are first
measured. To derive stellar parameters, several stellar parameters are
attempted before they converge: $T_{\rm eff}$ is determined by obtaining null
trend for the abundance of iron lines against excitation potential (see Figure
11 in Appendix); $\log g$ is obtained by ionisation equilibrium between Fe I
and Fe II lines; metallicity is obtained by the mean of individual line
abundances relative to the solar value; microturbulence velocity is determined
by obtaining null trend for the abundance of iron lines against equivalent
widths. For initial guess, we used photometric $T_{\rm eff}$ calculated from
2MASS photometry (González Hernández & Bonifacio, 2009) and $\log g$, [Fe/H]
from LAMOST suggested values. Our derived stellar parameters using BACCHUS are
shown in Table 1. Note that most of our stars show $\log g$ greater than 2.
Because the red clump feature is located at $\log g\sim 2$ (Paper II), our
sample stars are mostly RGB stars.

Table 1: Stellar parameters

Id | RA | DEC | $T_{\rm eff}$ | $\log g$ | [Fe/H] | $G_{\rm mag}$ | microturbulence | $RV$ | airmass | exposure time
---|---|---|---|---|---|---|---|---|---|---
(K) | (dex) | (dex) | (mag) | ${\rm\,(km\,s^{-1})}$ | ${\rm\,(km\,s^{-1})}$ | (s)
9 (II) | 58.221019 | 7.203154 | 4312 | 0.65 | -1.34 | 13.63 | 1.78 | 69.0 | 1.42 | 4800
38 (II) | 258.215699 | 8.130489 | 4604 | 0.73 | -1.62 | 12.57 | 2.04 | -146.0 | 1.90 | 3600
47 (II) | 214.054033 | -2.145359 | 4953 | 2.14 | -1.38 | 12.06 | 1.95 | 66.5 | 1.80 | 3600
58 (II) | 197.656039 | -6.979531 | 5213 | 2.84 | -1.45 | 14.45 | 1.78 | -27.0 | 1.10 | 5400
62 (II) | 122.120480 | 1.946907 | 5020 | 2.36 | -1.42 | 12.87 | 1.56 | 147.0 | 1.17 | 5400
65 (II) | 233.551930 | 0.326274 | 4806 | 1.69 | -1.63 | 12.76 | 1.45 | -72.5 | 1.57 | 3600
67 (I) | 317.852325 | -2.385546 | 5265 | 2.12 | -0.97 | 13.06 | 2.46 | -5.0 | 1.14 | 3600
69 (I) | 150.981537 | 1.298311 | 5129 | 3.02 | -1.07 | 12.22 | 1.26 | 50.0 | 1.12 | 3600
80 (I) | 134.988876 | 1.055260 | 4800 | 1.37 | -1.88 | 13.79 | 1.35 | 179.0 | 1.00 | 5400
82 (I) | 195.094070 | -7.636771 | 5017 | 1.90 | -1.06 | 12.59 | 1.60 | -6.0 | 1.50 | 5400
86 (I) | 239.791901 | 10.020472 | 5149 | 2.91 | -0.98 | 14.10 | 1.37 | -141.0 | 1.47 | 3600
88 (I) | 244.844238 | 9.523706 | 4879 | 2.31 | -1.02 | 13.16 | 1.51 | -127.0 | 1.88 | 3600
94 (I) | 166.054535 | 8.444023 | 5161 | 2.88 | -1.18 | 14.44 | 1.33 | 51.0 | 2.00 | 3600
97 (I) | 222.293640 | 10.045577 | 4762 | 2.20 | -1.25 | 13.12 | 1.34 | -31.0 | 1.49 | 3600
98 (I) | 199.145416 | 10.957638 | 4529 | 0.77 | -1.07 | 10.59 | 2.03 | 84.0 | 1.30 | 3600

The Id is the corresponding id in Paper II (table 3), with (I) and (II)
indicate that they are from Paper I and Paper II, respectively.

To derive chemical abundance for a given absorption line, a sigma-clipping is
first applied on the selected continuum points around the targeted line, then
a linear fit is used for the remaining points as the continuum. Therefore, the
code can detect significantly bad fits, like a sudden drop in the spectrum due
to bad pixels in the detectors. Observed spectra and model spectra are
compared in four different methods to determine abundances: $\chi^{2}$
minimization, line intensity, equivalent width, and spectral synthesis (see
Figure 12 in Appendix). Furthermore, the code gives each of them a flag to
indicate the estimation quality. When multiple transition lines are detected
for a given element, the lines that satisfy the followings are chosen: 1)
$\lambda>5000\,\text{\AA}$ (for better SNR, see Figure 10 in Appendix); 2) all
four determination methods are flagged as good ($=1$); 3) SNR$>40$. Then, the
mean value of the chemical abundances of different absorption lines from
$\chi^{2}$ minimization is given as the estimated abundance for the given
element. If no line meets the above criteria, the abundance is not estimated.
The derived chemical abundances of our targets are given in Table 2.

Table 2: Chemical abundances Element | 9 (II) | 38 (II) | 47 (II) | 58 (II) | 62 (II) | 65 (II) | 67 (I) | 69 (I)
---|---|---|---|---|---|---|---|---
${\rm[O/Fe]}$ | $<0.14$ | $<-0.02$ | 0.57 | $<0.41$ | $<-0.18$ | $<0.16$ | 0.49 | $<0.37$
${\rm[Na/Fe]}$ | 0.14 | 0.38 | 0.33 | 0.52 | 0.01 |  $\cdots$ | 0.16 | 0.16
${\rm[Mg/Fe]}$ | 0.19 | 0.31 | 0.45 | 0.31 | -0.38 | 0.31 | 0.41 | 0.41
${\rm[Al/Fe]}$ | 0.21 | 1.29 | 0.28 | 0.39 | 0.68 | 0.83 | 0.26 |  $\cdots$
${\rm[Si/Fe]}$ | 0.11 | 0.51 | 0.14 | 0.30 | 0.45 | 0.45 | 0.41 | 0.35
${\rm[S/Fe]}$ |  $\cdots$ | 0.55 |  $\cdots$ | 0.59 |  $\cdots$ |  $\cdots$ | 0.20 | 0.52
${\rm[Ca/Fe]}$ | 0.11 | 0.44 | 0.27 | 0.33 | 0.38 | 0.39 | 0.10 | 0.41
${\rm[Sc/Fe]}$ | -0.23 | 0.09 | 0.19 | 0.15 | 0.34 | -0.04 | -0.17 | 0.41
${\rm[Ti/Fe]}$ | -0.02 | 0.21 | 0.41 | 0.31 | 0.17 | 0.35 | 0.22 | 0.34
${\rm[V/Fe]}$ | -0.22 | 0.05 | 0.22 | 0.23 | -0.09 | 0.10 | 0.10 | 0.03
${\rm[Cr/Fe]}$ | -0.12 | -0.15 | -0.06 | -0.14 | -0.24 |  $\cdots$ | -0.28 | -0.03
${\rm[Mn/Fe]}$ | -0.60 | -0.36 | -0.37 | -0.39 | -0.60 | -0.63 | -0.36 | -0.50
${\rm[Co/Fe]}$ | -0.26 | -0.02 | 0.08 | 0.10 | -0.01 |  $\cdots$ | 0.13 | -0.05
${\rm[Ni/Fe]}$ | -0.29 | -0.02 | 0.02 | 0.04 | -0.14 | -0.07 | 0.13 | -0.01
${\rm[Y/Fe]}$ | -0.08 | -0.09 | -0.05 | -0.04 | -0.01 | 0.04 | -0.10 | 0.22
${\rm[Zr/Fe]}$ | -0.00 | 0.22 | 0.42 |  $\cdots$ | 0.35 |  $\cdots$ | 0.31 |  $\cdots$
${\rm[Ba/Fe]}$ | -0.17 | -0.20 | -0.02 | 0.22 | 0.23 | 0.89 | 0.28 | 0.38
${\rm[La/Fe]}$ | 0.07 | 0.08 | 0.30 | 0.33 | 0.31 | 0.71 | 0.26 | 0.38
${\rm[Ce/Fe]}$ | -0.09 | -0.05 | 0.15 | 0.35 | 0.24 |  $\cdots$ | 0.12 | 0.20
${\rm[Nd/Fe]}$ | 0.18 | 0.18 | 0.39 | 0.76 | 0.37 |  $\cdots$ | 0.28 |  $\cdots$
${\rm[Eu/Fe]}$ | 0.41 | 0.29 | 0.65 | 0.55 | 0.53 |  $\cdots$ | 0.28 | 0.27
Element | 80 (I) | 82 (I) | 86 (I) | 88 (I) | 94 (I) | 97 (I) | 98 (I) |
${\rm[O/Fe]}$ | 0.26 | 0.46 | 0.95 | 0.56 | $<0.39$ | $<0.63$ | 0.61 |
${\rm[Na/Fe]}$ | 0.96 | 0.34 | 0.22 | 0.23 | 0.22 | 0.30 | 0.21 |
${\rm[Mg/Fe]}$ | 0.72 | 0.51 | 0.19 | 0.27 | 0.54 | 0.24 | 0.48 |
${\rm[Al/Fe]}$ | 0.74 | 0.20 | 0.21 | 0.14 | 0.74 | 1.01 |  $\cdots$ |
${\rm[Si/Fe]}$ | 0.48 | 0.55 | 0.16 | 0.26 | 0.42 | 0.54 | 0.43 |
${\rm[S/Fe]}$ | 0.82 | 0.27 |  $\cdots$ |  $\cdots$ | 0.59 |  $\cdots$ |  $\cdots$ |
${\rm[Ca/Fe]}$ | 0.63 | 0.41 | 0.32 | 0.23 | 0.54 | 0.60 | 0.16 |
${\rm[Sc/Fe]}$ | 0.10 | 0.08 | 0.30 | 0.20 |  $\cdots$ | 0.14 | -0.39 |
${\rm[Ti/Fe]}$ | 0.36 | 0.25 | 0.36 | 0.22 | 0.43 | 0.50 | 0.04 |
${\rm[V/Fe]}$ | 0.16 | 0.07 | 0.11 | -0.05 | 0.20 | 0.19 | -0.22 |
${\rm[Cr/Fe]}$ | -0.02 | -0.03 | 0.08 | -0.16 | 0.10 | 0.18 | -0.15 |
${\rm[Mn/Fe]}$ | -0.39 | -0.37 | -0.38 | -0.51 | -0.32 | -0.32 | -0.49 |
${\rm[Co/Fe]}$ | 0.04 | 0.01 | 0.10 | -0.17 | 0.05 | 0.06 | -0.01 |
${\rm[Ni/Fe]}$ | 0.15 | 0.03 | 0.04 | -0.09 | 0.08 | 0.21 | -0.01 |
${\rm[Y/Fe]}$ | -0.06 | 0.16 | 0.16 | 0.08 | 0.17 | 0.33 |  $\cdots$ |
${\rm[Zr/Fe]}$ | 0.34 | 0.43 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.67 |  $\cdots$ |
${\rm[Ba/Fe]}$ | 0.31 |  $\cdots$ | 0.44 | 0.46 | 0.40 | 0.76 |  $\cdots$ |
${\rm[La/Fe]}$ | 0.21 | 0.45 | 0.74 | 0.63 | 0.42 | 0.46 | -0.12 |
${\rm[Ce/Fe]}$ | 0.16 | 0.33 | 0.29 | 0.32 | 0.22 |  $\cdots$ | -0.14 |
${\rm[Nd/Fe]}$ | 0.29 | 0.35 | 0.82 | 0.49 | 0.51 | 0.48 | 0.02 |
${\rm[Eu/Fe]}$ |  $\cdots$ | 0.50 | 0.90 |  $\cdots$ | 0.38 |  $\cdots$ |  $\cdots$ |

In this work, we analyzed the internal errors by propagating the typical
errors in $T_{\rm eff}$, $\log g$, ${\rm[Fe/H]}$. The typical errors are set
as $\Delta T_{\rm eff}=50\,{\rm K}$, $\Delta\log g=0.1\,{\rm dex}$,
$\Delta{\rm[Fe/H]}=0.05\,{\rm dex}$. The total estimated error is thus
calculated as: $\sigma_{\rm tot}=\left((\sigma_{\Delta T_{\rm
eff}})^{2}+(\sigma_{\Delta\log
g})^{2}+(\sigma_{\Delta{\rm[Fe/H]}})^{2}\right)^{1/2}$. The abundance errors
of a typical star (star #88 from Paper II) are shown in Table 3 as an example.

Table 3: Errors of chemical abundances propagated from atmospheric parameters for star #88. Element | abundance | $\Delta T_{\rm eff}=50{\rm\,(K)}$ |  $\Delta\log(g)=0.1$ (dex) |  $\Delta{\rm[Fe/H]}=0.05$ (dex) | $\sigma{\rm tot}$
---|---|---|---|---|---
$\Delta({\rm[O/Fe]})$ | 0.56 | 0.00 | 0.03 | 0.03 | 0.04
$\Delta({\rm[Na/Fe]})$ | 0.23 | 0.03 | 0.00 | 0.08 | 0.09
$\Delta({\rm[Mg/Fe]})$ | 0.27 | 0.03 | 0.00 | 0.00 | 0.03
$\Delta({\rm[Al/Fe]})$ | 0.14 | 0.02 | 0.00 | 0.02 | 0.03
$\Delta({\rm[Si/Fe]})$ | 0.26 | 0.01 | 0.01 | 0.02 | 0.03
$\Delta({\rm[S/Fe]})$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
$\Delta({\rm[Ca/Fe]})$ | 0.23 | 0.04 | 0.00 | 0.00 | 0.04
$\Delta({\rm[Sc/Fe]})$ | 0.20 | 0.00 | 0.03 | 0.02 | 0.04
$\Delta({\rm[Ti/Fe]})$ | 0.22 | 0.04 | 0.01 | 0.01 | 0.04
$\Delta({\rm[V/Fe]})$ | -0.05 | 0.07 | 0.00 | 0.00 | 0.07
$\Delta({\rm[Cr/Fe]})$ | -0.16 | 0.03 | 0.02 | 0.04 | 0.06
$\Delta({\rm[Mn/Fe]})$ | -0.51 | 0.08 | 0.01 | 0.01 | 0.08
$\Delta({\rm[Co/Fe]})$ | -0.17 | 0.06 | 0.01 | 0.01 | 0.07
$\Delta({\rm[Ni/Fe]})$ | -0.09 | 0.03 | 0.01 | 0.03 | 0.04
$\Delta({\rm[Y/Fe]})$ | 0.08 | 0.04 | 0.04 | 0.05 | 0.07
$\Delta({\rm[Zr/Fe]})$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
$\Delta({\rm[Ba/Fe]})$ | 0.46 | 0.01 | 0.05 | 0.06 | 0.08
$\Delta({\rm[La/Fe]})$ | 0.63 | 0.01 | 0.04 | 0.01 | 0.05
$\Delta({\rm[Ce/Fe]})$ | 0.32 | 0.01 | 0.02 | 0.03 | 0.04
$\Delta({\rm[Nd/Fe]})$ | 0.49 | 0.03 | 0.04 | 0.07 | 0.09
$\Delta({\rm[Eu/Fe]})$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$

##  3 Results

###  3.1 Na-O relation

The anti-correlation between Na and O is observed in almost all GCs (Carretta
et al., 2009b, 2010b; Gratton et al., 2012), which is arguably the most
characteristic MP property of GC member stars. The Na-O anti-correlation is
suggested to originate from the CNO cycle and the NeNa cycle activated during
lower temperature H-burning. If our N-rich field stars show compatible Na and
O abundances as the GC enriched populations, it would be a supportive evidence
for their GC origin scenario. For the reference of chemical abundances of GC
stars, we used the observed data of 19 GCs from Carretta et al. (2009b). Based
on the Na and O abundances, they separated members of each GC into three
components, P (primordial) component, I (intermediate) component, and E
(extreme) component. FG stars (or P component) are assumed to have similar O
and Na abundances as field stars of the same metallicity, while I and E
components are all considered to be SG stars.

Following the convention to measure O abundances (e.g. Carretta et al.,
2009b), we only used [O I] $\lambda\lambda$6300, 6363$\text{\AA}$ forbidden
lines to minimize the effect of non-local thermodynamic equilibrium (NLTE)
correction. If no measurable line is found, the mean value from $\chi^{2}$
minimization is given as the upper limit. The Na and O abundances are shown in
Figure 1, where GC stars from Carretta et al. (2009b) are shown as background
dots and contours. In order to match the metallicities between our N-rich
field stars and GCs, only GCs with $-1.8\leq{\rm[Fe/H]}\leq-1.0$ are used.
Though a clear anti-correlation between Na and O with large spread is found
for GC stars, the separation between FG and SG stars is not clear-cut. Seven
stars have measurable O abundances. Most N-rich field stars basically show
consistent Na and O abundances with those in GCs, except for two stars: one
with [O/Fe$]\sim 1$ and the other with [Na/Fe$]\sim 1$. Based on their
measurable Na abundances, most N-rich field stars are located in the
transition region between FG and SG stars, thus it is difficult to verify
their SG origin.

Figure 1: [Na/Fe]-[O/Fe] relation. Na and O abundances of N-rich field stars
are shown as red symbols. Upper limits in O abundances are shown as triangles,
while detections are shown as stars with error bars. The Na and O abundances
of the individual stars in the GCs from Carretta et al. (2009b) are over
plotted for comparison. Green symbols indicate the GC E component. Blue
symbols indicate the GC I component, while orange symbols indicate the GC P
component. Upper limits in O abundances are shown as triangles, while
detections are shown as dots. The solid and dashed contours indicate the
number densities of I component and P component, respectively.

###  3.2 Mg-Al relation

Mg-Al anti-correlation is another common feature found in low metallicity GCs
(${\rm[Fe/H]}<-1.0$, Pancino et al. 2017), which is suggested to be the
consequence of Mg-Al cycle activated at higher core temperature during
H-burning (Arnould et al., 1999). Our N-rich field stars are shown as red
symbols in Figure 2. We note that the red pentagons are chemical abundances
derived in Paper II using APOGEE spectra. In the background, GC stars from
Carretta et al. (2009a) are separated in to FG (orange) and SG (blue) stars as
previously mentioned.

Though GC FG and SG stars are wide-spread and even overlap in [Al/Fe]
distribution, FG stars seldomly exceed ${\rm[Al/Fe]}\sim 0.5$ (excluding upper
limits). On the other hand, GC SG stars may show low [Al/Fe] down to 0, and
thus a lower [Al/Fe] cannot exclude the possibility that the star belongs to
SG. In this sense, almost half of our N-rich field star sample show
${\rm[Al/Fe]}>0.5$, which are probable SG stars. Those stars with
${\rm[Al/Fe]}>0.5$ also have compatible [Mg/Fe] to ensure that they are
covered by the Mg-Al anti-correlation of GC Al-enhanced stars. There are two
extremely Mg depleted N-rich field star (${\rm[Mg/Fe]}<-0.4$), but one of them
has been reported in Paper II and Fernández-Trincado et al. (2019). The Mg
depleted N-rich field stars were also discussed in Fernández-Trincado et al.
(2016, 2017). The Mg depletion in N-rich field stars is rare, it would be
interesting to know if these Mg-depleted N-rich field stars have gone through
additional nucleosynthesis. Meanwhile, the other half of N-rich field stars
(${\rm[Al/Fe]}<0.5$) show Mg and Al abundances consistent with GC FG/SG stars.
Though chemically being less distinguishable from FG stars based on their Mg
and Al abundances, we cannot rule out the possibilities that lower [Al/Fe]
N-rich field stars are SG stars, as more metal-rich GCs tend to show similar
Al abundances for FG and SG stars (Pancino et al., 2017). Furthermore, using
APOGEE GC stars (Mészáros et al., 2020), we found that lower Al abundances
(${\rm[Al/Fe]}<0.5$) N-rich field stars tend to be more metal-rich in the
sample (Figure 3), tentatively agree with the aforementioned GC behavior.

Figure 2: [Mg/Fe]-[Al/Fe] relation. Mg and Al abundances of N-rich field stars
are shown as red symbols: 15 N-rich stars observed with MIKE spectra are
marked as red stars with error bars, while seven N-rich stars with APOGEE
spectra (Paper II) are marked as red pentagons. Orange and blue symbols
correspond to GC first generation (P component) and second generation (I and E
component) stars, respectively (Carretta et al., 2009a). Upper limits in Al
abundances of GC stars are shown as triangles, while detections are shown as
small dots. Figure 3: [Al/Fe]-[Fe/H] relation. Al abundances of N-rich sample
with MIKE spectra are shown as red symbols (same meanings as mentioned in
Figure 2). Magenta squares correspond to MW stars from halo (92 out of 168
stars from Fulbright 2000, 35 stars from Cayrel et al. 2004, 234 out of 253
stars from Barklem et al. 2005, 131 out of 199 stars from Yong et al. 2013,
287 out of 313 stars from Roederer et al. 2014), while black dots for MW stars
from disc (174 out of 181 stars from Reddy et al. 2003, 153 out of 176 stars
from Reddy et al. 2006, 679 out of 714 stars from Bensby et al. 2014). Orange
circles correspond to Sagittarius (Hasselquist et al., 2017; Hayes et al.,
2020). Violin shaped symbols indicate GC stars from APOGEE (Mészáros et al.,
2020). In situ GCs are colored with yellow green, while accreted GCs are
colored with sky blue. Star #9 with high extragalactic GC origin possibility
are labeled with large red circle.

###  3.3 $\alpha$-elements

The $\alpha$-elements are mainly generated in massive stars through type II
supernovae (SNe II), consequently their recycle timescale in the interstellar
medium is much smaller than that of iron, which is mainly produced in type Ia
supernovae (SNe Ia). As a result, the $[\alpha/{\rm Fe}]$ abundance is
initially enhanced, and starts to decline with [Fe/H] after SN Ia explosion
rate reaches a maximum (Matteucci & Greggio, 1986). The decline point, or the
knee in $[\alpha/{\rm Fe}]$ vs. ${\rm[Fe/H]}$ trend, is related to star
formation rate and thus galaxy mass: the less massive the galaxy is, the more
metal-poor is the $[\alpha/{\rm Fe}]$ turnover. O and Mg are commonly
classified as hydrostatic $\alpha$-elements, while Si, Ca, and Ti are
classified as explosive $\alpha$-elements (Woosley & Weaver, 1995), implying
different nucleosynthetic processes.

The [Mg/Fe]-[Fe/H] relation is shown in the upper left panel of Figure 4. In
the background, we also include MW halo and disc stars, dwarf galaxy (Sculptor
and Sagittarius) stars, and APOGEE GC stars (violin shaped symbols).222There
are 31 GCs compiled by Mészáros et al. (2020) in total. In our Figure 3 and 4,
we do not show Omega cen due to its large metallicity spread. Four GCs, which
have metallicities smaller than -2.0, are not shown, neither. Among the rest
26 GCs, there are eight in situ GCs colored with yellow green and 18 accreted
GCs colored with sky blue. Each violin shaped symbol corresponds to one GC.
The center value and min-max values are determined by its GC members selected
from the APOGEE survey (e.g. Mészáros et al., 2020; Fernández-Trincado et al.,
2020). [Mg/Fe] of our N-rich field stars are inside the min-max range of
APOGEE GCs of similar metallicity, which does not contradict with the
hypothesis that N-rich field stars come from GCs. However, in situ GCs (GCs
that identified as main disc or main bulge) and accreted GCs are largely
overlapped especially in our chosen metallicity range
($-1.8<{\rm[Fe/H]}<-1.0$), which prevents us distinguishing their in situ or
accreted GC origin. Meanwhile, our N-rich field stars are mostly covered by
both data points of MW stars and dwarf galaxies, indicating a mixture of in
situ stars and extragalactic stars. This is consistent with their kinematics
in Paper II.

Figure 4: [Mg/Fe]-[Fe/H] (upper left), [Si/Fe]-[Fe/H] (bottom left),
[Ca/Fe]-[Fe/H] (upper right), and [Ti/Fe]-[Fe/H] (bottom right) relation.
Symbols are same as Figure 3. Blue circles represent stars from Sculptor (Hill
et al., 2019).

As the atomic number increases, Si, Ca, and Ti become more stable against
H-burning. Si abundance is suggested to vary by the “Si-leakage” during Mg-Al
cycle when the core temperature reaches as high as $T\sim 65\times
10^{6}\,{\rm K}$ (Carretta et al., 2009a). Most GCs do not show significant
variation in Ca (e.g., Carretta et al., 2010a), except for several massive GCs
with possible iron spread (Carretta & Bragaglia, 2020). No significant Ti
variation among GC members is reported to the best of our knowledge.

Interestingly, Horta et al. (2020) reported that in situ GCs may have higher
[Si/Fe] compared to accreted GCs for ${\rm[Fe/H]}>-1.5$. The [Si/Fe]
difference between two groups of GCs become less distinguishable as
metallicity decreases. The background MW stars, dwarf galaxy stars (Sculptor
and Sagittarius) of Figure 4 (bottom left panel) support similar statement of
Horta et al. (2020) for field stars, though the threshold metallicity can be
only vaguely located at $-1.5<{\rm[Fe/H]}<-1.0$. Upper right panel and bottom
right panel of Figure 4 shows the Ca and Ti abundances distributions of our
N-rich field stars, along with background MW stars, dwarf galaxy stars and GC
star distribution (violin shaped symbols), respectively. Jönsson et al. (2020)
warned against using APOGEE-derived Ti abundances, since the Ti I and Ti II
abundances from APOGEE pipeline may have unknown defects or large scatters, so
[Ti/Fe] for APOGEE GCs are not shown in this work.

Looking at Mg, Si, Ca, Ti abundances versus [Fe/H] simultaneously, interesting
results emerge: (1) The $\alpha$-element difference between in situ (MW) stars
and accreted (dwarf galaxy) stars is most significant for Ti, compared to Mg,
Si and Ca. It is promising to use [Ti/Fe] for future works to distinguish
accreted stars. (2) The [$\alpha$/Fe] from APOGEE survey (violin shaped
symbols) cover most of the N-rich field stars, again supporting their GC
origin. (3) Seven metal-rich ($-1.25\leq{\rm[Fe/H]}\leq-0.95$) N-rich field
stars show abundances more consistent with MW stars, indicating a higher
possibility of in situ origin. (4) At the lower end of the [$\alpha$/Fe]
distribution, star #9 (labeled with large red circle in Figure 4) shows
consistently low Mg, Si, Ca, and Ti abundances similar to stars in dwarf
galaxies, although it shows similar [Al/Fe] as MW disc and halo stars (Figure
3). However, Al can be enhanced in GCs without altering much of its
$\alpha$-abundances. Is it possible that this star has an extragalactic GC
origin? We will further discuss this with more chemical information below.

###  3.4 Iron-peak elements

Though Type Ia SNe, runaway deflagration obliterations of white dwarfs, have a
signature more tilted towards the iron-peak group (Nomoto et al., 1997), the
solar composition of the iron-peak elements are in fact a heterogeneous
combination of both Type Ia SNe and core collapse Type II SNe (Woosley &
Weaver, 1995). As dwarf galaxies and MW have different star formation
timescale, this discrepancy may manifest itself in iron-peak elements. Figure
5 show that MW stars (black dots and magenta squares) and dwarf galaxy stars
(blue circles) have appreciable different distributions in [Sc/Fe], [V/Fe],
and [Co/Fe] vs. [Fe/H] graphs over the given metallicity range, especially in
metal-rich part (${\rm[Fe/H]}\geq-1.25$). For these three elements, more
metal-rich N-rich field stars ($-1.25\leq{\rm[Fe/H]}\leq-0.95$) show
distributions more similar to MW stars, confirming a higher possibility of in
situ origin (Section 3.3). For [Cr/Fe], [Mn/Fe] and [Ni/Fe], MW stars and
dwarf galaxy stars show less distinguishable difference. The discrepancies in
iron-peak element of MW stars and dwarf galaxies are not evident in metal-poor
part (especially for $-1.25<{\rm[Fe/H]}<-1.5$) due to lack of data. However,
if a linear relation of iron-peak abundances and metallicity is assumed for
both MW (halo and disc) and dwarf galaxies, the dwarf galaxies would have
lower iron-peak abundances. Therefore, star #9 with low $\alpha$-element
abundances also show lower abundances in [Sc/Fe], [V/Fe], [Co/Fe], agree with
dwarf galaxies, which supports its possible extragalactic GC origin.

Figure 5: Abundances of iron peak elements vs. [Fe/H]. Abundances of N-rich
stars are shown as red stars, while magenta squares and black dots correspond
to MW stars from halo and disc, respectively (same sources as in Figure 3).
Blue circles correspond to stars from dwarf galaxies (Sagittarius, Sculptor,
Fornax, Carina, Leo I, Hasselquist et al., 2017; Hill et al., 2019; Shetrone
et al., 2003).

###  3.5 Neutron capture elements

Elements heavier than iron are generated via neutron-capture processes.
Depending on the relative speed of neutron capture compared to $\beta$-decay,
the neutron-capture processes are divided into rapid (r-) and slow (s-) ones.
The main s-process elements are synthesized by AGB stars during thermal
pulsations (Busso et al., 2001; Karakas & Lattanzio, 2014). The astrophysical
sites of producing r-process elements have been debated over the past sixty
years (e.g., Thielemann et al., 2011; Kajino et al., 2019). The more popular
models includes core collapse SNe (e.g., Woosley et al., 1994) and neutron
star mergers (e.g., Côté et al., 2018; Watson et al., 2019).

Y, Zr, Ba, La, Ce, Nd and Eu are neutron-capture elements detectable in most
of our sample stars across the observed wavelength. According to the recent
work of Kobayashi et al. (2020), Y, Zr, Ba, La, Ce and Nd are mostly produced
through s-process in the current Universe, while Eu is produced through
r-process. The neutron capture element abundances of our N-rich field stars
are compared with MW stars in Figure 6. We see that the scatter of abundances
in MW stars increases as metallicity decreases, given that feature lines are
weaker in more metal-poor stars. Generally, we see consistency between our
N-rich field stars and MW stars, but several stars with enhanced [Ba/Fe],
[La/Fe] and [Eu/Fe] are also noticed.

Figure 6: Abundances of neutron capture elements vs. [Fe/H]. Symbols are same
as Figure 5.

Two nuclear reactions are the major neutron excess sources in AGB stars:
$\rm{{}^{13}C(\alpha,n)^{16}O}$ and $\rm{{}^{22}Ne(\alpha,n)^{25}Mg}$. The
first reaction dominates the low mass AGB stars, while the latter one is
mainly found in massive AGB stars (Cristallo et al., 2015). As the number of
free neutrons per iron seed increases, the s-process flow first seeds the
light s-process peak (Sr$-$Y$-$Zr), extending to $\rm{}^{136}Ba$, and then
reaches the heavy s-process peak (Ba$-$La$-$Ce$-$Pr$-$Nd), extending to
$\rm{}^{204}Pb-^{207}Pb$ (Bisterzo et al., 2014). Therefore, the heavy
s-process element to light s-process element ratio ([hs/ls]) is closely
related to metallicity and initial stellar mass. In this work, [hs/ls] is
defined as ${\rm[hs/Fe]-[ls/Fe]}$, where
${\rm[hs/Fe]=([Ba/Fe]+[La/Fe]+[Nd/Fe])/3}$, and
${\rm[ls/Fe]=([Y/Fe]+[Zr/Fe])/2}$. In our sample, there are six stars with all
five s-process elements (Figure 7). Five stars show consistent [hs/ls] values
as other MW stars in the same metallicity range. The grid lines of Cristallo
et al. (2015) indicate these stars are enriched by AGB stars of masses around
$3-5\,M_{\odot}$.

Figure 7: [hs/ls] vs. [Fe/H]. Abundances of N-rich stars are shown as red
stars, while black dots correspond to Milky Way stars from halo and disc (same
sources as black dots in Figure 2). The grid lines with different colors and
line styles indicate stars enriched by AGB stars of different masses
(Cristallo et al., 2015).

The ratio between s-process element and r-process element is an indicator of
the neutron-capture speed compared to $\beta$-decay, which could be related to
the contribution of AGB stars (thus SNe Ia) over SNe II and neutron star
mergers. Ba is commonly used to represent the heavy s-process elements, while
Y for the light s-process elements. A gradual rise in [Y/Eu] and [Ba/Eu] with
increasing metallicity is seen in Figure 8, which was addressed in e.g.,
McWilliam (1998). The upper panel of Figure 8 shows that MW field stars, MW GC
stars and dwarf galaxy stars overlap in the [Ba/Eu]-[Fe/H] space; In the lower
panel, [Y/Eu] of dwarf galaxies from Shetrone et al. (2003) and Venn et al.
(2012) are slightly smaller compared to MW field stars and MW GC stars at
similar metallicity range, but with substantial overlap (also see Figure 19 of
Venn et al. 2012). Though most N-rich field stars show consistent abundances
with other studies, we are not able to distinguish their in situ or
extragalactic origin here. Bisterzo et al. (2014) suggested that
${\rm[Ba/Eu]}\sim-0.7$ is the typical value for a pure r-process. Most N-rich
field stars show [Ba/Eu] abundances higher than the pure r-process value
(green solid line in Figure 8), indicating a heterogeneous mixture of r- and
s- processes. Star #9 and star #47 show [Ba/Eu] abundances close to the pure
r-process value, indicating a strong r-process contribution.

Figure 8: [Ba/Eu] vs. [Fe/H] and [Y/Eu] vs. [Fe/H]. Abundances of N-rich stars
are shown as red stars. Magenta squares correspond to MW halo stars, while
black dots correspond to MW disc stars. The green solid line indicates that a
typical value of a pure r-process by (Bisterzo et al., 2014). Green diamonds
correspond to MW GCs (McWilliam et al., 1992; James et al., 2004; Muñoz et
al., 2013; Johnson et al., 2017; Massari et al., 2017). Blue squares
correspond to dwarf galaxies from Sagittarius (Hasselquist et al., 2017),
Sculptor (Hill et al., 2019) (for [Ba/Eu]), Carina (Venn et al., 2012), and 19
red giants from Sagittarius, Fornax, Carina, Leo I (Shetrone et al., 2003).
Star #9 with high extragalactic GC origin possibility are labeled with large
red circle.

##  4 Discussion

###  4.1 Chemical similarity between N-rich field stars and GC stars

Though our sample stars have been confirmed to be mostly N-rich using CN-CH
features around 4000 Å (Paper I), further chemical tagging with more elements
requires high-resolution spectra. Using the MIKE and APOGEE spectra, we find
that most N-rich field stars show consistent Na, O, Mg, Al, Si, and Ca with GC
stars of similar metallicities. This chemical similarity supports the GC
origin of these N-rich field stars. Two stars with strong Mg-depletion are
also found. Strong Mg-depletion is usually found in very metal-poor
([Fe/H$]<-2.0$) GCs (e.g., M15, M92). Two N-rich field stars show strong Mg-
depletion at [Fe/H$]\sim-1.3$ is somewhat puzzling. One possible scenario is:
accreted materials from companion stars may change the Mg abundances of these
two N-rich field stars dramatically (Fernández-Trincado et al., 2017).

Based on the Na abundances, we find that most N-rich field stars are located
in the transition region between FG and SG stars, thus it is difficult to
verify their SG origin (see Figure 1). On the other hand, the Mg-Al figure
(Figure 2) shows that half of our sample have [Al/Fe$]>0.5$, which is largely
different than GC FG stars. Interestingly, Fernández-Trincado et al. (2020)
identified 29 mildly metal-poor ([Fe/H$]<-0.7$) field stars with
[Al/Fe$]>0.5$. They suggested that these stars were ejected into the bulge and
inner halo from GCs formed in situ and/or GCs formed in dissolved dwarf galaxy
progenitors.

###  4.2 In situ or ex situ?

As our understanding of the MW formation rapidly improves in the _Gaia_ era,
GCs are now believed to formed in both MW and dwarf satellite galaxies, and
later merged together as the current Galactic GCs (e.g., Massari et al., 2019;
Myeong et al., 2019). GCs formed in situ and ex situ tend to show different
chemical and dynamical signatures. If the N-rich field stars were escaped from
GCs, they should carry similar chemo-dynamical information as the host GCs.
Paper II showed that our parent N-rich field sample ($\sim 100$) have both
stars formed in situ and stars formed ex situ using kinematics. In this work,
one interesting star (star #9, large circles in Figures 4 and 5) show
consistently low [Mg/Fe], [Si/Fe], [Ca/Fe], [Ti/Fe], [Sc/Fe], [V/Fe], and
[Co/Fe] compared to MW stars at similar metallicities, which is a strong
evidence that this star was formed in dissolved dwarf galaxies. The relatively
higher [Al/Fe] of star #9 compared to dwarf galaxy field stars indicates that
it is possibly enriched in GC environment, as Al-rich stars can be found in
accreted GCs. On the other hand, stars with $-1.25\leq{\rm[Fe/H]}\leq-0.95$
show [$\alpha$/Fe] and iron-peak abundances consistent with other MW field
stars, indicating that they were formed in situ. Moreover, we checked their
kinematic information in Paper II, and found that star #9 shows $<E>$ and
$<Lz>$ consistent with $Gaia$-Sausage-Enceladus (GSE) stars (Figure 9), while
stars with $-1.25\leq{\rm[Fe/H]}\leq-0.95$ show in situ disk or halo
kinematics (e.g. Naidu et al., 2020). Interestingly, star #9 shows [Ba/Eu]
close to the r-process limit, which is consistent with other GSE stars
observed in Aguado et al. (2020). This is a straightforward example to
demonstrate that chemical tagging is possible to help decipher the origin of
field stars. We also acknowledge the difficulties in judging the origin of
field stars based on several chemical and dynamical observation evidences,
which are not always consistent with each other. A more sophisticated
probability estimation of their origin is left for further studies.

Figure 9: Mean $z$-direction angular momentum $<L_{\rm z}>$ vs. mean orbital
energy $<E>$. Star #9 is labelled as red dots with error bars, while stars
with $-1.25\leq{\rm[Fe/H]}\leq-0.95$ are labelled as blue dots with error
bars. The dotted rectangle shows the GSE regiona given by Massari et al.
(2019).

  * •

a Positive $<L_{\rm z}>$ indicate retrograde and negative $<L_{\rm z}>$
indicate prograde, which is the opposite in Massari et al. (2019). This is
related to the Galactocentric coordinate system that we used (Paper II).

##  5 Conclusions

We used MIKE spectra of 15 stars, which are selected from the LAMOST N-rich
field star sample, to analyse their chemical abundances for more than 20
elements. We find that the Na, O, Mg, Al, Si, and Ca abundances of the N-rich
field stars are consistent with GC stars, which supports their GC origin.
Given that GC FG and SG stars have large overlap in Na-O and Mg-Al parameter
space, it is difficult to confirm the SG origin of N-rich field stars. But we
do find seven stars with ${\rm[Al/Fe]}>0.5$, which is typical for GC SG stars.

We notice that one star (#9) with consistently low [Mg/Fe], [Si/Fe], [Ca/Fe],
[Ti/Fe], [Sc/Fe], [V/Fe], and [Co/Fe] show similar kinematic and [Ba/Eu] as
other stars from GSE. On the other hand, more metal-rich stars
($-1.25\leq{\rm[Fe/H]}\leq-0.95$) show $\alpha$-elements abundances and iron-
peak abundances more consistent with MW field stars rather than dwarf
galaxies, indicating likely in situ origin.

The ratio between heavy s-process elements and light s-process elements
reveals that most N-rich field stars could be enriched by AGB stars with
masses around $3-5\,M_{\odot}$. A detailed comparison between different
chemical models and the obtained chemical patterns of our N-rich field stars
would be fruitful for future discussion (e.g., Masseron et al., 2020).

## Acknowledgments

We thank Ian Thompson, Yang Huang for helpful discussions. We thank the
anonymous referee for insightful comments. J.Y. and B.T. gratefully
acknowledges support from the National Natural Science Foundation of China
under grant No. U1931102. D.G. gratefully acknowledges support from the
Chilean Centro de Excelencia en Astrofísica y Tecnologías Afines (CATA) BASAL
grant AFB-170002. D.G. also acknowledges financial support from the Dirección
de Investigación y Desarrollo de la Universidad de La Serena through the
Programa de Incentivo a la Investigación de Académicos (PIA-DIDULS).
Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic
Telescope LAMOST) is a National Major Scientific Project built by the Chinese
Academy of Sciences. Funding for the project has been provided by the National
Development and Reform Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences.

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Spectra of star #88 of both blue side ($3950-4050$ Å) and red side
($5950-6050$ Å) are shown in Figure 10. We adopt red side spectra
($\lambda>5000\,\text{\AA}$) in abundances determination when considering
their better SNR. $T_{\rm eff}$ is determined by obtaining null trend for the
abundance of iron lines against excitation potential is shown in Figure 11. An
example of abundances determination with four different methods is shown in
Figure 12 as mentioned in Section 2.

Figure 10: Spectra of star #88 of both blue side and red side. Figure 11:
Illustration of $T_{\rm eff}$ determination with iron abundances and
excitation potential. Figure 12: Comparison of four different methods
($\chi^{2}$ minimization, line intensity, equivalent width, and spectral
synthesis) in determining Na abundances for star #88. The black solid line
shows the observed normalized spectrum, while colored lines show synthesis
obtained by different methods in both upper and bottom panel. The inserted
panel in the bottom right corner of the upper panel shows the diagram for the
four methods for abundances determination. Each of the method is represented
by a different color ($\chi^{2}$ minimization: magenta, line intensity: blue,
equivalent width: black, spectral synthesis: red). (See more detailed
descriptions in (Masseron et al., 2016).)


1112020 _Mathematics Subject Classification_ : Primary 30H20

# Contraction property of Fock type space of log-subharmonic functions in
$\mathbb{R}^{m}$

David Kalaj  University of Montenegro, Faculty of Natural Sciences and
Mathematics, Cetinjski put b.b. 81000 Podgorica, Montenegro<EMAIL_ADDRESS>
###### Abstract.

We prove a contraction property of Fock type spaces $\mathcal{L}_{\alpha}^{p}$
of log-subharmonic functions in $\mathbb{R}^{n}$. To prove the result, we
demonstrate a certain monotonic property of measures of the superlevel set of
the function $u(x)=|f(x)|^{p}e^{-\frac{\alpha}{2}p|x|^{2}}$, provided that $f$
is a certain log-subharmonic function in $\mathbb{R}^{m}$. The result recover
a contraction property of holomorphic functions in the Fock space
$\mathcal{F}_{\alpha}^{p}$ proved by Carlen in [4].

###### Key words and phrases:

Holomorphic functions, isoperimetric inequality, Fock space, contraction

###### Contents

  1. 1 Introduction
  2. 2 Motivation and the Main Results
  3. 3 Proof of Theorem 2.2
  4. 4 Proof of Theorem 2.3
  5. 5 Some additional properties of Fock space and proof of Theorem 2.7

##  1\. Introduction

Let $m\geqslant 1$ and let $\mathbb{R}^{m}$ be the Euclidean space endowed
with the Euclidean norm: $|x|=\sqrt{\left<x,x\right>}$, where
$\left<x,y\right>=\sum_{i=1}^{m}x_{i}y_{i}$, and
$x=(x_{1},\dots,x_{m}),y=(y_{1},\dots,y_{n})\in\mathbb{R}^{m}$. If $\alpha>0$
and $p>0$ and $m=2n$ is an even integer, we define the Fock space or Segal-
Bargmann space $\mathcal{F}_{\alpha}^{p}$ (cf. [1, 2, 20]) of entire
holomorphic functions $f$ in $\mathbb{C}^{n}=\mathbb{R}^{2n}$ so that:

$\|f\|^{p}_{p,\alpha}:=c_{p,\alpha}\int_{\mathbb{R}^{m}}|f(x)|^{p}e^{-\frac{\alpha}{2}p|x|^{2}}dA(x)<\infty,$

where

(1.1) $c_{p,\alpha}=\left(\frac{\alpha p}{2\pi}\right)^{\frac{m}{2}},$

and $dA(x)$ is Lebesgue measure on $\mathbb{R}^{m}$. Note that
$c_{p,\alpha}e^{-\frac{\alpha}{2}p|x|^{2}}dA(x)$ is the Gaussian probabily
measure in $\mathbb{R}^{m}$.

Assume now that $m\in\mathbb{N}$ is an arbitrary integer. We say that a real
twice differentiable function $f$ defined in a domain
$\Omega\subset\mathbb{R}^{m}$ is subharmonic if $\Delta f(x)\geqslant 0$ for
$x\in\Omega$. Here $\Delta$ is the Laplacian. This definition can also be
extended to not necessary double differentiable functions, by using the sub-
mean value property ([7]). We say that a mapping $f$ is log-subharmonic, if
$\log|f(x)|$ is subharmonic in $\Omega\setminus f^{-1}(0)$. We denote by
$\mathcal{L}^{p}_{\alpha}$ the space of complex-valued, real-analytic
functions whose absolute value is a log-subharmonic function, defined in
$\mathbb{R}^{m}$, with a finite $\|f\|_{p,\alpha}$ norm as defined in (1).
Here $m$ is an arbitrary positive integer. Observe that for $m=2n$ we have
$\mathcal{F}_{\alpha}^{p}\subset\mathcal{L}_{\alpha}^{p}$: If $f$ is
holomorphic in $\Omega$, then $|f(z)|$ is log-subharmonic. Indeed

$\Delta\log|f(z)|=\sum_{k=1}^{n}\Delta_{z_{k}}\log|f(z)|=0,$

where $z=(z_{1},\dots,z_{n})$, and

$\Delta_{z_{k}}=\frac{\partial^{2}}{(\partial_{x_{k}})^{2}}+\frac{\partial^{2}}{(\partial_{y_{k}})^{2}},$

$z_{k}=x_{k}+iy_{k}$ for $k=1,\dots,n$ and $z\in\Omega\setminus f^{-1}(0).$

##  2\. Motivation and the Main Results

Carlen, in his paper [4] proved the following result:

######  Theorem 2.1.

If $0<p<q<\infty$, then
$\mathcal{F}^{p}_{\alpha}(\mathbb{C}^{n})\subset\mathcal{F}^{q}_{\alpha}(\mathbb{C}^{n})$
and the inclusion is proper and continuous. Moreover

$\|f\|_{q,\alpha}\leq\|f\|_{p,\alpha}.$

Theorem 2.1 is applied in [4] to the coherent state transform in a new proof
of Wehrl’s entropy conjecture [12]. In this paper, among other results, we
recover Theorem 2.1 and provide a proof that works for a more general class of
mappings, namely real analytic complex mappings whose absolute value is a log-
subharmonic function in $\mathbb{R}^{m}$ and belongs to the Fock-type space
$\mathcal{L}^{p}_{\alpha}$.

Let $f$ be a real analytic complex-valued function defined in the Euclidean
space $\mathbb{R}^{m}$, such that $v=|f|$ is a log-subharmonic function in
$\mathbb{R}^{m}$ and such that $u(x)=v(x)^{p}e^{-\alpha p/2|x|^{2}}$ is
bounded and goes to $0$ uniformly as $|x|\to\infty$. Then the superlevel sets
$A_{t}=\\{x:u(x)>t\\}$ for $t>0$ are compactly embedded in $\mathbb{R}^{m}$
and thus have finite Lebesgue measure $\mu(t)=|A_{t}|$.

Those are the main results

######  Theorem 2.2.

Let $\alpha>0$ and $p>0$ and assume that $f$ is a real analytic complex valued
function such that $v=|f|:\mathbb{R}^{m}\to[0,+\infty)$ is a log-subharmonic
function. Assume further that the function $u(x)=|f(x)|^{p}e^{-\frac{\alpha
p}{2}|x|^{2}}$ is bounded and $u(x)$ tends to $0$ uniformly as $|x|\to\infty$.
Then the function

$g(t)=t\exp\left[\frac{\alpha p(\Gamma(m/2))^{2/m}}{2\pi}\mu^{2/m}(t)\right],$

is decreasing on the interval $(0,t_{\circ})$, where
$t_{\circ}=\max_{x\in\mathbb{R}^{m}}u(x)$.

If $f(x)\equiv 1$, the function $g$ turns out to be constant and this is an
important property of $g$.

The proof of this theorem is mostly based on the methods developed by Nicola
and Tilli in [16] (see also the subsequent papers where similar methods are
used: [10], [9], [19], [11], and [5]).

By using Theorem 2.2 we will prove the following theorem:

######  Theorem 2.3.

Let $p>0$ and $\alpha>0$. Let $G:[0,\infty)\to\mathbb{R}$ be a convex
function. Then the maximum value of

(2.1) $\int_{\mathbb{R}^{m}}G(|f(x)|^{p}e^{-\frac{\alpha}{2}p|x|^{2}})dA(x)$

is attained for

$f_{a}(x)=e^{\alpha\left<a,x\right>-\frac{\alpha}{2}|a|^{2}},$

where $a\in\mathbb{C}^{n}$ is arbitrary, subject to the condition that
$f\in\mathcal{L}^{p}_{\alpha}$ and $\|f\|_{p,\alpha}=1$.

Applying Theorem 2.3 to the convex and increasing function $G(t)=t^{q/p}$, we
get the extension of theorem [4, Theorem 2] by proving:

######  Theorem 2.4.

For all $0<p<q<\infty$ and $0<\alpha$ and for
$f\in\mathcal{L}^{p}_{\alpha}(\mathbb{R}^{m})$, we have
$f\in\mathcal{L}^{q}_{\alpha}(\mathbb{R}^{m})$ and

$\|f\|_{q,\alpha}\leq\|f\|_{p,\alpha}$

with equality for
$f_{a}(x)=e^{\alpha\left<a,x\right>-\frac{\alpha}{2}|a|^{2}}$, where
$a\in\mathbb{R}^{m}$ is arbitrary.

###### Proof of Theorem 2.4.

For $\|f\|_{p,\alpha}=N$, $\|f/N\|_{p,\alpha}=1$ and from Theorem 2.3 we have

$\int_{\mathbb{R}^{m}}|f(x)/N|^{q}e^{-\frac{\alpha}{2}q|x|^{2}}dA(x)\leq\int_{\mathbb{R}^{m}}e^{-\frac{\alpha}{2}q|x|^{2}}dA(x)=1/c_{q,\alpha}.$

Thus

$c_{q,\alpha}\int_{\mathbb{R}^{m}}|f(x)|^{q}e^{-\frac{\alpha}{2}q|x|^{2}}dA(x)\leq
N^{q},$

or what is the same

$\|f\|_{q,\alpha}\leq\|f\|_{p,\alpha}.$

The equality statement follows from the equality statement of Theorem 2.4, but
can be proved by using the same approach as in monograph of Zhu [20, Lemma
2.33]. ∎

######  Remark 2.5.

The last theorem is an extension of Theorem 2.1. Moreover, its proof is
different from the proof in [4] and seems to be simpler. We refer to the paper
[6] for some related inequalities for log-subharmonic functions in
$\mathbb{R}^{n}$.

Theorem 2.4 is a counterpart of a similar contraction property of Bergman
spaces $\mathbf{B}^{p}_{\alpha}$ ([8, p. 2]), proved by Kulikov in [10] for
holomorphic functions in the unit disk and for $\mathcal{M}-$log-subharmonic
functions in the unit ball in $\mathbb{R}^{n}$ by the author in [9]. It is
known that

$\mathbf{B}^{p}_{\alpha}\subset\mathbf{B}^{q}_{\beta},\quad\frac{p}{\alpha}=\frac{q}{\beta}=r,\quad
p<q.$

For $n=2$, it was asked whether these embeddings are contractions, that is
whether the norm $\|f\|_{\mathbf{B}^{r\alpha}_{\alpha}}$ is decreasing in
$\alpha$. In the case of Bergman spaces, this question was asked by Lieb and
Solovej [13]. They proved that such contractivity implies their Wehrl-type
entropy conjecture for the $SU(1,1)$ group. In the case of contractions from
the Hardy spaces to the Bergman spaces, it was asked by Pavlović in [18] and
by Brevig, Ortega-Cerdà, Seip, and Zhao [3] concerning the estimates for
analytic functions. The mentioned contraction property proved by Kulikov
confirmes these conjectures. An interesting application of Kulikov result has
been given by Melentijević in [14].

We end this paper with the construction of a new normed Fock type space:

######  Definition 2.6 (Fock limit space).

Let $f$ be a holomorphic function in $\mathbb{C}^{n}$. Then for $\alpha>0$ we
say $f\in\mathcal{F}_{\alpha}$ if $\displaystyle
f\in\bigcap_{p>0}\mathcal{F}_{\alpha}^{p}$. Then we define

$\|f\|_{\alpha}:=\inf_{p>0}\|f\|_{p,\alpha}.$

For $\alpha>0$ define as in [20, eq. 2.2] the following Banach norm

$\|f\|_{\infty,\alpha}:=\mathrm{esssup}\\{|f(z)|e^{-\frac{\alpha}{2}|z|^{2}},z\in\mathbb{C}^{n}\\}.$

Then we prove

######  Theorem 2.7.

For every $\alpha>0$ we have

$\|f\|_{\alpha}=\|f\|_{\infty,\alpha}.$

In particular $(\mathcal{F}_{\alpha},\|\cdot\|_{\alpha})$ is a normed subspace
of Banach space $\mathcal{F}_{\alpha}^{\infty}.$

##  3\. Proof of Theorem 2.2

###### Proof of Theorem 2.2.

We start with the formula

$\mu(t)=|A_{t}|=\int_{A_{t}}dx=\int_{t}^{\max
u}\int_{|u(x)|=\kappa}d\mathcal{H}^{m-1}(x)d\kappa.$

Then we get

(3.1) $-\mu^{\prime}(t)=\int_{u=t}|\nabla u|^{-1}d\mathcal{H}^{m-1}(x)$

along with the claim that $\\{x:u(x)=t\\}=\partial A_{t}$ and that this set is
a smooth hypersurface for almost all $t\in(0,t_{\circ})$. Here
$dS=d\mathcal{H}^{m-1}$ is $m-1$ dimensional Hausdorff measure. These
assertions follow the proof of [16, Lemma 3.2]. We point out that, since $u$
is real analytic, then it is a well-known fact from measure theory that the
level set $\\{x:u(x)=t\\}$ has a zero measure ([15]), and this is equivalent
to the fact that the $\mu$ is continuous.

Following the approach from [16], our next step is to apply the Cauchy–Schwarz
inequality to the $m-1$ dimensional measure of $\partial A_{t}$:

(3.2) $|\partial A_{t}|^{2}=\left(\int_{\partial
A_{t}}dS\right)^{2}\leq\int_{\partial A_{t}}|\nabla u|^{-1}dS\int_{\partial
A_{t}}|\nabla u|dS.$

Let $\nu=\nu(x)$ be the outward unit normal to $\partial A_{t}$ at a point
$x$. Note that, $\nabla u$ is parallel to $\nu$, but directed in the opposite
direction. Thus we have $|\nabla u|=-\left<\nabla u,\nu\right>$. Also, we note
that since for $x\in\partial A_{t}$ we have $u(x)=t$, we obtain for
$x\in\partial A_{t}$ that

$\frac{|\nabla u(x)|}{t}=\frac{|\nabla u(x)|}{u}=\left<\nabla\log
u(x),\nu\right>.$

Now the second integral on the right-hand side of (3.2) can be evaluated by
Gauss’s divergence theorem:

$\begin{split}\int_{\partial A_{t}}{|\nabla
u||dS|}&=-t\int_{A_{t}}\mathrm{div}\left({\nabla\log u(x)}\right)dA(x)\\\
&=-t\int_{A_{t}}\Delta{\log u(x)}dA(x).\end{split}$

Now we plug $u=|f(x)|^{p}e^{-\frac{\alpha}{2}p|x|^{2}}$, and calculate

$-t\Delta\log(|f(x)|^{p}e^{-\frac{\alpha}{2}p|x|^{2}})=-(pt\Delta\log
v-t\frac{\alpha}{2}p\Delta|x|^{2})\leq 0+mt\alpha p.$

By using (3.1) and (3.2) we obtain

$\begin{split}|\partial A_{t}|^{2}&\leq(-\mu^{\prime}(t))\int_{\partial
A_{t}}{|\nabla u|dS}.\\\ &\leq-mt\alpha p\mu^{\prime}(t)\mu(t).\end{split}$

Now we use the isoperimetric inequality for the space:

$|\partial A_{t}|^{2}\geqslant\pi
m^{2}|A_{t}|^{\frac{2(m-1)}{m}}\left({\Gamma(m/2)}\right)^{-\frac{2}{m}},$

which implies that

(3.3) $mt\alpha
p\mu^{\prime}(t)\mu(t)+m^{2}\pi\mu(t)^{\frac{2(m-1)}{m}}\left({\Gamma(m/2)}\right)^{-\frac{2}{m}}\leq
0$

with equality in (3.3) if and only if
$v(x)=e^{\alpha\left<x,a\right>-\frac{\alpha}{2}|a|^{2}}$ because in that case
$A_{t}$ is a ball centered at $a$. So

(3.4) $M(t):=\alpha
p\mu^{\prime}(t)\mu(t)^{\frac{2-m}{m}}+\frac{m\pi\left({\Gamma(m/2)}\right)^{-\frac{2}{m}}}{t}\leq
0.$

Since $\mu(t^{\circ})=0$, we obtain that

$G(t)=\int_{t_{\circ}}^{t}M(t)dt=m\pi(\Gamma(m/2))^{-2/m}\log\frac{t}{t_{\circ}}+\frac{m}{2}\alpha
p\mu^{\frac{2}{m}}(t)$

is a non-increasing function for $0\leq t<t_{\circ}$.

In the case $v(x)\equiv e^{\alpha\left<a,x\right>-\frac{\alpha}{2}|a|^{2}}$,
$t_{\circ}=1$ and $\mu(t_{\circ})=0$. Moreover

$g(t):=\exp(G(t))=t\exp\left[\frac{\alpha
p(\Gamma(m/2))^{2/m}}{2\pi}\mu^{2/m}(t)\right]$

is non-increasing for $0\leq t<t_{\circ}$.

∎

######  Remark 3.1.

Note that for the function $f(x)\equiv 1$ or

$f(x)=e^{-\frac{\alpha}{2}|a|^{2}}e^{\alpha\left<a,x\right>},$

for a fixed $a$, everywhere in the proof above we have equalities for all
values of $p$ and $\alpha$. Moreover in this case the maximum of $u(x)$ is
equal to $1$ and achieved for $x=a$.

##  4\. Proof of Theorem 2.3

We need the following lemma

######  Lemma 4.1.

[9] Assume that $\Phi,\Psi$ are positive increasing functions and $g$ positive
non-increasing such that

$\int_{0}^{t_{\circ}}\Phi\left({g(t)}/{t}\right)dt=\int_{0}^{t_{\circ}}\Phi\left({1}/{t}\right)dt=c.$

Then

$\int_{0}^{t_{\circ}}\Phi\left({g(t)}/{t}\right)\Psi(t)dt\leq\int_{0}^{t_{\circ}}\Phi\left({1}/{t}\right)\Psi(t)dt.$

As in [10, 9], where is treated Bergman version of this theorem, we restrict
ourselves to the only nontrivial case $\lim_{t\to 0^{+}}G(t)=0$. Let
$\mu(t)=\mu(\\{x:u(x)>t\\})$ be the Lebesgue measure in $\mathbb{R}^{m}$,
where $u(x)=|f(x)|^{p}e^{-\frac{\alpha p}{2}|x|^{2}}$. Applying Theorem 2.2 to
$f$, we get that the function

$g(t)=t\exp\left[\frac{\alpha(\Gamma(m/2))^{2/m}}{2\pi}\mu^{2/m}(t)\right],$

is decreasing on $(0,t_{\circ})$ with
$t_{\circ}=\max_{x\in\mathbb{R}^{m}}u(x)$. Proposition 5.1 below ensures the
existence of $t_{\circ}$.

For $f\equiv 1$, $g$ is a constant function equal to $1$.

Then

$\mu(t)=\left(\frac{2\pi}{\alpha(\Gamma(m/2))^{2/m}}\log\frac{g(t)}{t}\right)^{\frac{m}{2}}.$

We assume that $\|f\|_{p,\alpha}=1$, that is

$I_{1}=c_{p,\alpha}\int_{0}^{t_{\circ}}\mu(t)dt=c_{p,\alpha}\int_{0}^{t_{\circ}}\left(\frac{2\pi}{\alpha(\Gamma(m/2))^{2/m}}\log\frac{g(t)}{t}\right)^{m/2}dt=1.$

Now the integral in (2.1) can be rewritten as

$I_{2}=c_{p,\alpha}\int_{0}^{t_{\circ}}\left(\frac{2\pi}{\alpha(\Gamma(m/2))^{2/m}}\log\frac{g(t)}{t}\right)^{m/2}G^{\prime}(t)dt.$

Then by Lemma 4.1, by taking
$\Phi(s)=c_{p,\alpha}\left(\frac{2\pi}{\alpha(\Gamma(m/2))^{2/m}}\log
s\right)^{\frac{m}{2}}$ and $\Psi(t)=G^{\prime}(t)$, the maximum of $I_{2}$
under $I_{1}=1$ is attained for $g\equiv 1$.

##  5\. Some additional properties of Fock space and proof of Theorem 2.7

Now we prove the following proposition used in the proof of our main result.

######  Proposition 5.1.

Assume that $f$ is a real-analytic log-subharmonic function in
$\mathbb{R}^{m}$ belonging to the Fock type space. Then for every $x$,

(5.1) $|f(x)|^{p}e^{-\frac{\alpha p}{2}|x|^{2}}\leq
c_{p,\alpha}\int_{\mathbb{R}^{m}}|f(y)|^{p}e^{-\frac{\alpha
p}{2}|y|^{2}}dA(y).$

Moreover

(5.2) $\lim_{|x|\to\infty}|f(x)|e^{-\frac{\alpha}{2}|x|^{2}}=0.$

Notice that (5.1) extends [20, Theorem 2.7] and the relation (5.2) extends
corresponding relation in [20, p. 38].

###### Proof.

Let $g(y)=|f(x+y)|^{p}e^{-\alpha p\left<(y+x),x\right>}.$ Now use the mean
value property to the log-subharmonic function $g$ (it is also subharmonic).

$|g(0)|\leq c_{p,\alpha}\int_{\mathbb{R}^{m}}|g(y)|e^{-\frac{\alpha
p}{2}|y|^{2}}dA(y).$

Then we have

$g(0)=|f(x)|^{p}e^{-\alpha p|x|^{2}}\leq
c_{p,\alpha}\int_{\mathbb{R}^{m}}f^{p}(y+x)e^{-\frac{\alpha}{2}p\left<(x+y),x\right>}e^{-\frac{\alpha
p}{2}|y|^{2}}dA(y).$

Therefore

$|f(x)|^{p}e^{-{\alpha}p|x|^{2}}\leq
c_{p,\alpha}\int_{\mathbb{R}^{m}}f^{p}(y)e^{-\alpha
p\left<y,x\right>}e^{-\frac{\alpha p}{2}|y-x|^{2}}dA(y).$

So

$|f(x)|^{p}e^{-\frac{\alpha p}{2}|x|^{2}}\leq
c_{p,\alpha}\int_{\mathbb{R}^{m}}f^{p}(y)e^{-\frac{\alpha p}{2}|y|^{2}}dA(y).$

Now, to prove (5.2), we use the following inequality, which is also a
consequence of the sub-mean value property of subharmonic functions. Let
$B_{1}(x)=\\{y\in\mathbb{R}^{m}:|y-x|<1\\}$. Then for every subharmonic
function $g$ we have

$|g(0)|\leq\frac{n}{\omega_{n}}\int_{B_{1}(0)}|g(y)|dA(y).$

Thus

(5.3) $|g(0)|e^{-\frac{\alpha
p}{2}}\leq\frac{n}{\omega_{n}}\int_{B_{1}(0)}|g(y)|e^{-\frac{\alpha
p}{2}|y|^{2}}dA(y).$

By applying the previous inequality for $g(y)=|f(x+y)|^{p}e^{-\alpha
p\left<(y+x),x\right>},$ we obtain from (5.3) that

$\begin{split}|f(x)|^{p}e^{-\alpha p|x|^{2}}e^{-\frac{\alpha
p}{2}}&\leq{\frac{n}{\omega_{n}}}\int_{B_{1}(0)}|f(x+y)|^{p}e^{-\alpha
p\left<(y+x),x\right>}e^{-\frac{\alpha p}{2}|y|^{2}}dA(y)\\\
&=\frac{n}{\omega_{n}}\int_{B_{1}(x)}|f(y)|^{p}e^{-\alpha
p\left<y,x\right>}e^{-\frac{\alpha p}{2}|y-x|^{2}}dA(y)\\\
&=\frac{n}{\omega_{n}}e^{-\frac{\alpha
p}{2}|x|^{2}}\int_{B_{1}(x)}|f(y)|^{p}e^{-\frac{\alpha
p}{2}|y|^{2}}dA(y).\end{split}$

Thus

$|f(x)|^{p}e^{-\frac{\alpha p}{2}|x|^{2}}e^{-\frac{\alpha
p}{2}}\leq\frac{n}{\omega_{n}}\int_{B_{1}(x)}|f(y)|^{p}e^{-\frac{\alpha
p}{2}|y|^{2}}dA(y).$

Since $f\in\mathcal{L}_{\alpha}^{p}$, it follows that

$\lim_{|x|\to\infty}\frac{n}{\omega_{n}}\int_{B_{1}(x)}|f(y)|^{p}e^{-\frac{\alpha
p}{2}|y|^{2}}dA(y)=0.$

This implies (5.2). ∎

It follows from the following lemma that $\|f\|_{\alpha}$ is a norm on
$\mathcal{F}_{\alpha}$. Theorem 2.7 is a direct application of the following
lemma

######  Lemma 5.2.

a) If $f,g\in\mathcal{F}_{\alpha}$, then
$\|f+g\|_{\alpha}\leq\|f\|_{\alpha}+\|g\|_{\alpha}$.

b) For every $\alpha>0$ and $f\in\mathcal{F}_{\alpha}$ and
$x\in\mathbb{C}^{m}$ we have
$|f(x)|e^{-\frac{\alpha}{2}|x|^{2}}\leq\|f\|_{\alpha}$.

c) For every $\alpha>0$ and $f\in\mathcal{F}_{\alpha}$,
$\|f\|_{\alpha}=\sup_{x\in\mathbb{C}^{n}}\left(|f(x)|e^{-\frac{\alpha}{2}|x|^{2}}\right).$

###### Proof.

Let us restrict ourselves to the case $n=1$. The general case is a trivial
modification of this case.

a) Let $f,g\in\mathcal{F}_{\alpha}$. Then for every $\alpha>0$,
$f,g\in\mathcal{F}^{p}_{\alpha}$ and by the triangle inequality for the norm
in $\mathcal{F}^{p}_{\alpha}$ we obtain

$\begin{split}\|f+g\|_{\alpha}&=\lim_{p\to\infty}\|f+g\|_{p,\alpha}\\\
&\leq\lim_{p\to\infty}\|f\|_{p,\alpha}+\lim_{p\to\infty}\|g\|_{p,\alpha}\\\
&=\|f\|_{\alpha}+\|g\|_{\alpha}.\end{split}$

b) This follows from Proposition 5.1.

c) It follows from (5.1) that

$|f(x)|e^{-\frac{\alpha}{2}|x|^{2}}\leq\|f\|_{p,\alpha}.$

By letting $p\to\infty$ we obtain

$|f(x)|e^{-\frac{\alpha}{2}|x|^{2}}\leq\|f\|_{\alpha}.$

Thus

$\mathrm{ess}\,\sup|f(x)|e^{-\frac{\alpha}{2}|x|^{2}}\leq\|f\|_{\alpha}.$

To prove the converse, fix an $R>0$ and assume first that $f=P$ is a
polynomial. Then

$\|P\|^{p}_{p,\alpha}=\int_{|x|\leq
R}|P(x)|^{p}e^{-\frac{\alpha}{2}p|x|^{2}}dx+\int_{|x|>R}|P(x)|^{p}e^{-\frac{\alpha}{2}p|x|^{2}}dx.$

Moreover for sufficiently large $R$

$I(R):=\int_{|x|>R}|P(x)|^{p}e^{-\frac{\alpha}{2}p|x|^{2}}dx\leq
c_{P}\int_{|x|>R}|z|^{n_{P}p}e^{-\frac{\alpha}{2}p|x|^{2}}dx$

and the last expression is smaller than $\|F\|^{p}_{\infty,\alpha}$. In fact
the last expression tends to zero as $R\to\infty$. Therefore

$\|P\|_{p,\alpha}\leq(\|P\|^{p}_{\infty,\alpha}R^{n}\omega_{n}+\|P\|^{p}_{\infty,\alpha})^{1/p},$

where $\omega_{n}$ is the meausre of the unit sphere. Thus

$\|P\|_{\alpha}=\lim_{p\to\infty}\|P\|_{p,\alpha}\leq\|P\|_{\infty,\alpha}.$

Thus if $f$ is a polynomial, then

(5.4) $\|f\|_{\alpha}=\|f\|_{\infty,\alpha}.$

Further, if $f$ is not a polynomial and $\epsilon>0$ is arbitrary, then for
$p=2$, there exists a polynomial $P$ so that $\|P-f\|_{p,\alpha}<\epsilon$.
Moreover,

$\|f\|_{\alpha}\leq\|P\|_{\alpha}+\|f-P\|_{\alpha}=\|P\|_{\infty,\alpha}+\|f-P\|_{\alpha}\leq\|P\|_{\alpha}+\epsilon.$

Since $\epsilon$ is arbitrary, we conclude that (5.4) hold for every function
$f\in\mathcal{F}_{\alpha}$.

∎

######  Remark 5.3.

One can ask, given a holomorphic function $f$, when this

$\lim_{p\to 0}\|f\|_{\alpha,p}$

exists. The answer is that limit is infinity except for the case when $f\equiv
const$, so we cannot produce a Hardy type space for holomorphic mappings in
$\mathbb{C}^{n}$.

### Acknowledgments

After I proved my main results, Kehe Zhu directed me to the reference by
Carlen [4], for which I am thankful. I am also grateful to Djordjije
Vujadinovic for his valuable insights and discussions on the topic.

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