1Department of Physics and Astronomy, National Institute of Technology,
Rourkela, Odisha - 769 008, India.
2Indian Institute of Astrophysics, Koramangala II-Block, Bangalore - 560 034,
India.
3 Tata Institute of Fundamental Research (TIFR), Homi Bhabha Road, Mumbai -
400 005, India.
4INAF - Astronomical Observatory of Abruzzo, Via M. Maggini, sn. 64100 Teramo,
Italy.
5INFN - Sezione di Pisa, Università di Pisa, Largo Pontecorvo 3, 56127 Pisa,
Italy.

# UVIT study of UV bright stars in the globular cluster NGC 4147

Ranjan Kumar1* Ananta C. Pradhan1* Mudumba Parthasarathy2 Devendra K. Ojha3
Abhisek Mohapatra1 Jayant Murthy2 and Santi Cassisi4,5

###### Abstract

We present far ultraviolet (FUV) observations of globular cluster NGC 4147
using three FUV filters, BaF2 (F154W), Sapphire (F169M), and Silica (F172M) of
Ultra-Violet Imaging Telescope (UVIT) on-board the AstroSat satellite. We
confirmed the cluster membership of the UVIT observed sources using proper
motions from Gaia data release 2 (GAIA DR2). We identified 37 blue horizontal
branch stars (BHBs), one blue straggler star (BSS) and 15 variable stars using
UV-optical color magnitude diagrams (CMDs). We find that all the FUV bright
BHBs are second generation population stars. Using UV-optical CMDs, we
identify two sub-populations, BHB1 and BHB2, among the UV-bright BHBs in the
cluster with stars count ratio of 24:13 for BHB1 and BHB2. The effective
temperatures (Teff) of BHB1 and BHB2 were derived using color-temperature
relation of BaSTI-IAC zero-age horizontal branch (ZAHB). We found that BHB1
stars are more centrally concentrated than BHB2 stars. We also derive physical
parameters of the detected FUV bright BSS by fitting younger age BaSTI-IAC
isochrones on optical and UV-optical CMDs.

###### keywords:

(Galaxy:) globular clusters: individual: NGC 4147 - stars: horizontal-branch,
(stars:) blue stragglers \- (stars:) Hertzsprung-Russell and colour-magnitude
diagrams.

<EMAIL_ADDRESS><EMAIL_ADDRESS>
07 Nov 202013 Dec 2020

12.3456/s78910-011-012-3 #### 000 0000 1– 1

##  1 Introduction

The glimpse of ultraviolet (UV) light in the old stellar population of the
Galactic globular clusters (GGCs) is dominated by hot luminous UV-bright stars
which are mostly the stars of blue horizontal branch (BHB), blue-stragglers
(BS), post-asymptotic giant branch (pAGB), and extreme horizontal branch (EHB)
phases having temperature more than 7,000 K. Various physical properties of
these UV-bright populations have been explored using large sample of GGCs
observed by Galaxy Evolution Explorer (GALEX, Schiavon $et~{}al$., 2012) and
Hubble space telescope (HST, Nardiello $et~{}al$., 2018). Ultraviolet Imaging
Telescope (UVIT) on-board the AstroSat satellite (Kumar $et~{}al$., 2012) has
also performed the imaging observations of several GGCs in UV with a better
spatial resolution than GALEX enabling in resolving the core of the clusters
and also in distinguishing UV bright stars of different evolutionary stages to
study their physical parameters in a great detail (Subramaniam $et~{}al$.,
2017; Sahu $et~{}al$., 2019a; Jain $et~{}al$., 2019; Kumar $et~{}al$., 2020a,
b; Rani $et~{}al$., 2020; Singh $et~{}al$., 2020). Using the spectroscopic
data of a large number of stars in many GGCs along with the UV photometric
observations with HST, the evidence for the presence of multiple stellar
populations in GGCs is now well established (see reviews Bastian & Lardo,
2018; Gratton $et~{}al$., 2019; Cassisi & Salaris, 2020, and references
therein). The different chemical evolution within the GGCs seems to be the
origin of two distinct sub-population of stars (the first generation (1G) and
second generation (2G)) characterized by significant abundance variations in
light elements. The stellar populations enriched in He, N, and depleted in O
and C are 2G stellar population while stellar populations enriched in C and O
and depleted in N and Na with a primordial He-abundance are termed as normal
or 1G stellar population (see Milone $et~{}al$., 2017; Marino $et~{}al$.,
2019, and references therein).

We present here the UV study of an old age, metal poor, low density GC NGC
4147, located at a distance of 21 kpc from the Galactic center and 19 kpc from
the Sun ($l=252.84^{\circ}$, $b=+77.18^{\circ}$). We have observed this
cluster using three far-UV (FUV) filters of UVIT. Prior to this, the UV
imaging and color-magnitude diagrams (CMDs) of the cluster were presented by
Schiavon $et~{}al$. (2012) using GALEX observations. Their study was
restricted to the sources in the outer region of the cluster due to the lower
resolution ($\sim 5^{\prime\prime}$) of GALEX. However, the cluster is well
studied both photometrically and spectroscopically in the optical bands of the
electromagnetic spectrum. Several of the photometric observations in the
optical bands have explored about specific sources of the cluster such as
BHBs, variable stars, red giant branch stars (RGBs), etc., using their optical
CMDs (Auriere & Lauzeral, 1991; Arellano Ferro $et~{}al$., 2004; Stetson
$et~{}al$., 2005; Arellano Ferro $et~{}al$., 2018; Lata $et~{}al$., 2019).
Similarly, many low resolution spectroscopic observations of the cluster have
been performed to find its metallicity ($[Fe/H]=-1.85$ dex), and chemical
abundances of various alpha-elements (Suntzeff $et~{}al$., 1988; Martell
$et~{}al$., 2008; Ivans, 2009). The latest high-resolution spectroscopy of 18
RGB stars by Villanova $et~{}al$. (2016) has revealed that most of the RGB
stars of the cluster are of 2G population type with 2G to 1G RGB stars’ ratio
of 85:15. They found that the red-HB stars (RHBs) are progeny of the 1G
population and the BHBs are progeny of the 2G population. They also confirmed
an alpha-enhancement of 0.38 dex and a mono-metallicity of $[Fe/H]=-1.84$ dex
among 1G and 2G stars in the cluster.

In section 2, we present the observation details, data reduction process and
photometry of the UVIT observation of the cluster. In section 3, we show the
CMDs of all the detected sources. In section 4, we discuss the sub-population
among the UVIT detected 2G BHBs. In section 5, we estimate various physical
properties of the UV bright blue straggler star (BSS). Finally, we summarize
our results in section 6.

##  2 Data Reduction

We obtained the UVIT observation of the cluster NGC 4147 in three FUV filters,
BaF2, Sapphire and Silica, having wide, medium and narrow bandwidths,
respectively. The UVIT data of the cluster was obtained from AstroSat archival
web-page111https://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp in
level1 format. The instrument and calibration details of UVIT can be found in
Tandon $et~{}al$. (2017, 2020). We reduced the level1 data into science image
using a customized software package CCDLAB (Postma & Leahy, 2017), specially
designed for the UVIT data reduction. The UVIT observations have been
performed in several orbits and we get science image for each orbit. In order
to get a good signal to noise ratio (SNR), we combined all the observed orbits
into a single image and then performed the photometry. The UVIT observation
details are given in Table 1.

Table 1: Observational details of NGC 4147.

Date of observation | 2017, April 17
---|---
Telescope pointing | RA $=182.5263^{\circ}$, DEC $=18.5426^{\circ}$
Channel | Filters | mean $\lambda$ | $\Delta\lambda$ | Exp. Time | No. of Orbits
|  | (Å) | (Å) | (sec.) |
FUV | BaF2 | 1541 | 380 | 1536 | 5
Sapphire | 1608 | 290 | 1648 | 3
Silica | 1717 | 125 | 1209 | 2

In Figure 1, we provide a color image of the cluster using multi-wavelength
observations from infrared (IR) to UV. The sources in blue color are observed
in UV using the UVIT BaF2 filter, the green color sources are from the
archival catalog of CFHT-4m $V$ band observation (Stetson $et~{}al$., 2019),
and the red color sources are obtained from 2MASS $J$ band
observation222https://irsa.ipac.caltech.edu/applications/2MASS of the cluster.
We can see that the hot sources, observed in UV (blue sources) are bright
enough and easily distinguishable in the outer region of the cluster. The
central region of the cluster also contains many hot sources as visible in
cyan color (mixture of blue and green) due to crowding effect at the center.

Figure 1: Color image of NGC 4147; UV: UVIT BaF2 (blue), Visible: CFHT-4m V
band (green), Infrared: 2MASS J band (red).

We applied DAOPHOT point spread function (PSF) photometry (Stetson, 1987) on
the science images to obtain the source positions and their respective
apparent magnitudes. The aperture and saturation corrections were applied on
the detected sources in all the three FUV filters following the suggestion by
Tandon $et~{}al$. (2017). A typical PSF of 1.5′′ was obtained in all the
science images of the FUV filters. We have excluded the sources within 10′′
from the cluster center to avoid contamination. We detected 114, 92, and 65
sources in the BaF2, Sapphire, and Silica filters, respectively. The AB-
magnitude limits of the detected sources in BaF2, Sapphire and Silica filters
are 23.0, 22.5, and 21.0, respectively. The magnitudes were corrected for
extinction value using $E(B-V)=0.0221$ mag (Schlafly & Finkbeiner, 2011) along
the cluster direction and extinction law of Cardelli $et~{}al$. (1989) in the
UV filters.

##  3 Color-Magnitude Diagrams

Figure 2: Optical and UV-optical CMDs of the UVIT observed sources of NGC
4147. Panel (a): The B$-$V vs V CMD where ground-based detections from Stetson
$et~{}al$. (2019) are shown in gray solids, the UVIT observed BHBs are shown
in blue and green asterisks. The BSS is shown in the brown solid. BaSTI-IAC
isochrones and ZAHB with cluster parameter, $[Fe/H]=-1.896$,
$[\alpha/Fe]=0.40$, Age = 13.0 Gyr, and distance-modulus $(m-M)=16.40$, are
over-plotted on the observed sources.Panels (b), (c), and (d): BaF2$-$V vs
BaF2, Sapphire$-$V vs Sapphire, and Silica$-$V vs Silica CMDs, respectively,
are shown for the UVIT observed sources with the same colors as mentioned in
panel (a), and BaSTI-IAC ZAHB with the same cluster parameters as mentioned in
panel (a) are over-plotted on all CMDs.

To study UV and UV-optical CMDs of the cluster members, we cross-matched the
UVIT observed sources with GAIA data release 2 catalog (Gaia DR2, Gaia
Collaboration $et~{}al$., 2018) with a matching radius of 1.5′′. The confirmed
cluster members were separated out from the field stars and the background
sources (Galaxies, quasars, etc.) using Gaia DR2 proper motions. Once the
sources were confirmed as cluster members, we cross-matched them to deeper
photometric observations, the latest released archival catalog of ground-based
observations in UBVRI filters (Stetson $et~{}al$., 2019). An optical CMD,
B$-$V vs V, is shown in panel (a) of Figure 2 for all the ground-based
observed sources within the UVIT field of view (FoV $\sim 30^{\prime}$) in
gray solids. The UVIT detections are over-plotted in the CMD and are shown in
blue and green asterisks and in brown solid circle.

In order to obtain the cluster properties, we used the Bag of Stellar Tracks
and Isochrones models (BaSTI-IAC333http://basti-iac.oa-
abruzzo.inaf.it/index.html, Hidalgo $et~{}al$., 2018) and generated isochrones
and zero-age horizontal branch (ZAHB) with cluster parameters,
$[Fe/H]=-1.896$, age = 13.0 Gyr (Harris, 2010) and $[\alpha/Fe]=0.40$
(Villanova $et~{}al$., 2016). Since the alpha-enhanced set of the BaSTI-IAC
library has not been still published, we relied on the extension of the BaSTI-
IAC library to alpha-enhanced mixture (Pietrinferni $et~{}al$., 2020) for the
present analysis. In panel (a) of Figure 2, BaSTI-IAC isochrones are over-
plotted on the B$-$V vs V CMD in red line using a distance-modulus of 16.40,
which are fitting very well with the ground-based observations. In Panels (b),
(c), and (d) of Figure 2, we have shown the UV-optical CMDs of the cluster
using BaF2, Sapphire, and Silica filters, respectively, in combination with V
magnitude from the optical bands. The ZAHB from BaSTI-IAC library is over
plotted on all the UV-optical CMDs. Although all the UV-optical CMDs look
similar we have shown them to see the morphological distribution of UV bright
stars of the cluster in all the FUV filters. We can see in Figure 2 that BHBs
are major contributors to the UV bright sources of the cluster.

If we look at the optical CMD in panel (a) of Figure 2, we notice that the
UVIT observed BHBs have spread along the horizontal line of the HB phase. This
spread is better visible in UV-optical CMDs along the diagonal HB line than
the optical CMD. In fact, the spread among BHBs in BaF2$-$V vs BaF2 CMD (panel
(b)) becomes more clear with a gap of 0.8 magnitude in the BaF2$-$V color
within BHBs. Based upon the gap observed in BaF2$-$V vs BaF2 CMD, we divide
the UVIT observed BHBs in two groups: BHBs with BaF2$-$V color brighter than
2.8 magnitude as BHB1 (blue asterisks) and BHBs with BaF2$-$V color fainter
than 3.6 magnitude as BHB2 (green asterisks). These BHBs are lying in left and
right portions of the BHB region in B$-$V vs V CMD (panel (a)) and they are
clearly identified in Sapphire$-$V vs Sapphire and Silica-V vs Silica CMDs in
panels (c) and (d), respectively. There are 24 sources in BHB1 and 13 sources
in BHB2 groups.

Villanova $et~{}al$. (2016) have made a chemical abundance analysis of 18 RGB
stars of the cluster NGC 4147 and found that the cluster is composed of 1G
stars with $[Na/Fe]\sim+0.0$ and $[O/Fe]\sim+0.3$ and 2G stars with
$[Na/Fe]\sim+0.5$ and $[O/Fe]\sim-0.2$. In Figure 1 of their paper they have
shown that the 2G HB stars are located at $B-V<0.3$. If we compare the UVIT
observations with optical data, it appears that all the observed BHBs seem to
belong to the 2G population (panel (a) of Figure 2). In the same figure, one
can notice that the BHBs are separated in two sub-groups, which could point to
the presence of two sub-populations in the observed 2G BHBs. We discuss this
possibility in the following sections.

Apart from the BHBs, we have also detected one UV bright BSS in the cluster.
The BSS was observed in BaF2 and Sapphire filters but not in the Silica
filter. The brightness of the BBS may be fainter than the detection limit of
Silica filter (21.0 AB-magnitude). The BSS is shown in Figure 2 with brown
solid. We also found 15 variable stars by cross-matching the UVIT observed
sources with the updated variable stars catalog of GCs (Clement, 2017).
However, we have not shown these variable stars on the CMDs due to the severe
crowding conditions affecting these stars, and the fact that we do not have a
complete sampling of their light curves.

##  4 Sub-population in BHBs

We derive the effective temperature (Teff) of the observed BHBs using color-
temperature relation of the BaSTI-IAC ZAHB in Figure 3. The best fitted ZAHB
from Figure 2 is taken to derive the color-temperature relation of the UVIT
observed BHBs. As shown in Figure 3, the Teff of BHB1 is ranging between
10,000 K and 11,300 K and Teff of BHB2 ranges between 8,000 K and 9,500 K. The
comparison between observations and models suggests the presence of a gap
between BHB1 and BHB2 stars with a temperature difference of about 500 K.
However, we caution that this width of the observed temperature gap and its
actual existence may be affected by the limited number of sample stars (number
of BHBs detected) in this cluster. Although a detailed statistical analysis of
the reality of such a gap will be carried out in a future work, we note that a
puzzling gap or discontinuity in the distribution of BHBs is seen in the
color-magnitude diagram (CMD) of many GGCs (Catelan $et~{}al$., 1998; Piotto
$et~{}al$., 1999; Behr $et~{}al$., 2000; Brown $et~{}al$., 2016). It was
initially believed to be due to statistically significant under-population of
stars but its appearance at similar location for different clusters has
substantiated the claim that it is indeed a real feature and the satisfactory
explanation for its origin is not yet known. However, speculation is that the
gaps may be demarcating the boundaries between separate, discrete populations
of HB stars, which differ in their origin or evolution.

Figure 3: BaF2$-$V vs BaF2 CMD of the observed BHBs. We use BaSTI-IAC ZAHB
color-temperature relation to derive the range of Teff of BHB1 and BHB2. The
Teff derived from the ZAHB is given in the upper x-axis of the plot.

By relying on the BaF2$-$V vs BaF2 CMDs obtained in the present work, we
collect some evidence suggesting the possible presence of a bi-modality among
the BHB population. In the following, we provide some suggestions about the
possible origin for the observed discontinuity in the color distribution of
BHB stars in NGC 4147. The main empirical findings are that: i) the BHB1 and
BHB2 are clustered in the optical plane in two quite clear separate regions -
although in that plane these two regions appear quite adjacent (panel (a) of
Figure 2); ii) in the UV-optical plane there exists an evident gap of about
500 K whose width has been evaluated by over-imposing the models. These
empirical evidence suggests that BHB1 stars could be slightly He enhanced with
respect to BHB2 ones and in such a case they would represent two distinct sub-
populations in the 2G population; or alternatively they could share with BHB1
more or less the same He abundance, and have experienced different mass loss
efficiency during the previous RGB stage (see, e.g., Tailo $et~{}al$., 2020).

In Figure 4, we have shown the radial distribution of BHB1 and BHB2 stars. We
see that the BHB1 stars are spread within 15 pc (with an exception of one BHB1
star at a distance of 43 pc) from the cluster center. However, BHB2 stars show
their radial distribution up to 25 pc. We find that the BHB1 stars are having
almost double counts than the BHB2 stars up to 10 pc. This indicates that the
hotter sources are relatively concentrated at the center than the lower Teff
BHBs.

Figure 4: The radial distribution of UVIT detected BHBs. The histogram of BHB1
stars is shown in blue line and histogram of BHB2 stars is shown in green
line. The lower and upper x-axes is the distance from the cluster center in
parsec and arcminutes, respectively. The y-axes repesents the number of stars
present in each distance bin of 5 parsec from the cluster center. The vertical
dashed line is the tidal radius of the cluster.

##  5 FUV bright Blue Straggler Star

The blue-stragglers (BS) are generally formed in the old age clusters through
two formation channels: mass-transfer in binary system in low density
environment (Knigge $et~{}al$., 2009; Leigh $et~{}al$., 2013) and merger due
to collision between stars in dense environment (Chatterjee $et~{}al$., 2013).
They are relatively more massive hydrogen burning MS stars than the normal MS
population of the cluster owning to the higher mass and consequently shifted
in the upper MS branch in the H-R diagram. The UV photometry is a great tool
to identify the BS population in a cluster. The data from UV telescopes (HST,
GALEX and UVIT) have been excellent tool to explore BS population and their
formation scenario (Ferraro $et~{}al$., 1997, 2003; Dieball $et~{}al$., 2010;
Schiavon $et~{}al$., 2012; Gosnell $et~{}al$., 2015; Subramaniam $et~{}al$.,
2017; Sahu $et~{}al$., 2019a; Kumar $et~{}al$., 2020a, b; Rani $et~{}al$.,
2020). In particular, UVIT observations of star clusters are able to identify
the hot binary companions (WD, HB, EHB, etc.) of BS stars (Subramaniam
$et~{}al$., 2016; Sindhu $et~{}al$., 2019; Sahu $et~{}al$., 2019b; Jadhav
$et~{}al$., 2019; Singh $et~{}al$., 2020).

Figure 5: Optical and UV-optical CMDs are shown in left and right panels,
respectively. The FUV bright BSS of NGC 4147 is shown in the black solid in
both CMDs. The BaSTI-IAC isochrones with the same parameters as mentioned in
Figure 2 but with younger ages, i.e 2 Gyr (dashed line), 3 Gyr (solid line), 4
Gyr (dash-dotted line), and 5 Gyr (dotted line), are over-plotted in both
CMDs.

Present observational data-set allows us to identify one FUV bright BSS in the
outer region of the cluster, at a distance of 2.15′ from the center. The
respective colors and magnitudes of the BSS in UV-optical and optical CMDs are
shown in Figure 2. The BS are generally more massive than the stars currently
evolving at the MS turn-off and are located along the brighter and hotter
extension of the MS locus, mimicking the location of intermediate-age stars.
Therefore, we generated relatively younger ages BaSTI-IAC isochrones by
adopting the same assumptions concerning metallicity and alpha-element
enhancement as the old stellar component. The comparison between model
predictions and the empirical data in the B$-$V vs V and BaF2$-$V vs BaF2 CMDs
is shown in Figure 5. We see that the isochrone with age 3 Gyr is lying closer
to the observed BSS. Based on the observed location of the BSS in optical and
UV-optical CMDs, we extracted the mass, luminosity, effective temperature of
the BSS from the 3 Gyr BaSTI-IAC isochrone which are $M=1.14\pm
0.04M_{\odot}$, $\log L/L_{\odot}=0.9\pm 0.1$, and $\log
T_{\mathrm{eff}}/K=3.93\pm 0.01$, respectively.

Table 2: List of the telescopes and their filters used in the SED fit.

Telescope Filters Wavelength range in Å Reference UVIT/AstroSat BaF2, Sapphire
1350 - 1,800 This paper GALEX FUV, NUV 1350 - 3000 Schiavon $et~{}al$. (2012)
CFHT-3.6m U, B, V, R, I 3000 - 11800 Stetson $et~{}al$. (2019) SDSS u, g, r,
i, z 3000 - 10800 DR9, Ahn $et~{}al$. (2012) GAIA G, BP 3300 - 10600 Gaia
Collaboration $et~{}al$. (2018) PAN-STARRS g, r, i, z, y 3900 - 10800 Chambers
$et~{}al$. (2016) CTIO/DECam g, r 3925 - 7233 DECam Legacy Survey DR3 a

  * a

https://www.legacysurvey.org/dr3/description/

We obtained optical photometry of the FUV bright BSS from various archival
catalogs and performed spectral energy distribution (SED) fitting on the
observed fluxes to the model generated fluxes for different filters from UV to
near-IR bandwidth. We used an online SED fitting tool, VO SED analyszer (VOSA,
Bayo $et~{}al$., 2008), to perform the SED fit. It uses the chi-square
minimisation technique to fit the observed fluxes on the model fluxes
generated from the theoretical spectra incorporating various stellar
atmosphere models. We used the Kurucz stellar atmosphere model, ATLAS9 (Kurucz
model, Castelli & Kurucz, 2003) grids (spectra) with model parameters in the
following range, Teff: 3,500 K to 50,000 K, $[Fe/H]:-2.0$ and $-1.5$ (nearest
to the cluster metallicity), and log (g): 0.0 to 5.0, respectively. The
observed fluxes of different filters used in the SED fit is given in Table 2.
We have used observed fluxes from 23 filters in the wavelength range 1,800 Å
to 11,800 Å to perform the SED fit.

Figure 6: The SED fit of BSS using the Kurucz model is shown in the upper
panel. The bottom panel shows the residue (deviation) of the observed fluxes
from the Kurucz model fluxes. Table 3: Various physical parameters of the BSS
derived from the SED fit and BaSTI-IAC isochrones are listed in the table.

ID RAJ2000 DEJ2000 fit Teff Luminosity Radius log(g) Mass degree degree K L⊙
R⊙ dex M⊙ BSS01 182.5633 18.5480 Isochrone 8511 $\pm$ 196 7.94 $\pm$ 1.83 1.30
$\pm$ 0.07 - 1.14$\pm$0.04 SED 8500 $\pm$ 250 10.74 $\pm$ 0.35 1.51 $\pm$ 0.05
4.0$\pm$0.5 1.23$\pm$0.02

In Figure 6, we have shown the best fitted Kurucz model spectra of 8,500 K on
the observed fluxes in gray line. The model fluxes calculated for various
filters using the Kurucz model spectra are shown in solid black circles and
the observed fluxes from the UVIT, GALEX, CFHT-3.6m, SDSS, GAIA, PAN-STARRS,
and CTIO/DECam telescopes are shown in green, red, blue, brown, violet,
magenta, and yellow diamonds, respectively. In the lower panel of Figure 6, we
have shown the fractional deviation of the observed flux from the model flux
and indicated as residual [(=
$F_{\mathrm{obs.}}-F_{\mathrm{mod}})/F_{\mathrm{mod}})$]. The best fitted
Kurucz model spectra with physical parameters Teff = 8,500 K, $[Fe/H]=-2.0$
and log (g) = 4.0, was obtained with a reduced chi-square value of 9.3. The
deviation of the observed fluxes from the theoretical fluxes for most of the
filters is less than 15%. But for a few filters (e.g., CTIO/DECam g and Gaia G
filters) the deviation is more than or around 15% which might be the reason
for the high value of $\chi^{2}$. However, we can see that the overall
photometric magnitudes are well within the maximum photometric errors of 20%
(0.2 mag). We extracted bolometric luminosity and radius of the FUV bright BSS
from the slope of the best fitted spectrum using scaling relation $(R/d)^{2}$
used in the SED fit, where R is the radius of the source and $d$ is the
distance. The distance of the cluster = 19.0 kpc and extinction value
A${}_{\mathrm{v}}=0.068$ magnitude were used in the SED fit. We have also
estimated the mass of the BSS using the Teff, luminosity and radius derived
from the SED analysis. The derived physical parameters from the SED fit and
the BaSTI-IAC isochrone fitting are listed in Table 3. We find that the Teff,
mass and luminosity of the BSS, derived from the SED fit and BaSTI-IAC
isochrone fitting on CMDs are matching well within the error range.

##  6 Summary and Conclusion

UV analysis of the GGC NGC 4147 is presented using observations in three FUV
filters of UVIT. A catalog of the UVIT detected BHBs, and BSS along with their
FUV magnitudes in three UVIT filters is provided in Table 4. It is found that
all the FUV bright BHBs belong to the 2G population as defined by (Villanova
$et~{}al$., 2016) and they lie in two sub-populations on the UV-optical CMDs
with a count ratio of 24:13 (BHB1:BHB2). The derived ranges of Teff of BHB1
and BHB2 are 10,000 - 11,300 K and 8,000 - 9,500 K, respectively with a gap of
500 K between both the sub-populations. Spectroscopic analysis of the sources
is required to investigate further on the distinction between the two sub-
population of BHBs. The physical parameters of the FUV bright BSS were derived
by fitting younger BaSTI-IAC isochrones in optical and UV-optical CMDs, and
SED fitting of the observed fluxes from UV to near-IR wavelengths with the
Kurucz model. The derived physical parameters of the FUV bright BSS are listed
in Table 3.

## Acknowledgements

We would like to thank the referee for her/his valuable suggestions and
comments. RK would like to acknowledge CSIR Research Fellowship (JRF) Grant
No.09/983(0034)/2019-EMR-1 for the financial support. ACP would like to
acknowledge the support by Indian Space Research Organization, Department of
Space, Government of India (ISRO RESPOND project No.ISRO/RES/2/409/17-18). ACP
thanks Inter University centre for Astronomy and Astrophysics (IUCAA), Pune,
India for providing facilities to carry out his work. DKO acknowledges the
support of the Department of Atomic Energy, Government of India, under Project
Identification No. RTI 4002. AM thanks DST-INSPIRE (IF150845) for the funding.
This publication uses the data from the AstroSat mission of the Indian Space
Research Organisation (ISRO), archived at the Indian Space Science Data Center
(ISSDC). The UVIT data used here was processed by the Payload Operations
Centre at IIA. The UVIT is built in collaboration between IIA, IUCAA, TIFR,
ISRO and CSA.

## References

  * Ahn $et~{}al$. (2012) Ahn, C. P., Alexandroff, R., Allende Prieto, C., $et~{}al$. 2012, ApJS , 203, 21
  * Arellano Ferro $et~{}al$. (2004) Arellano Ferro, A., Arévalo, M. J., Lázaro, C., $et~{}al$. 2004, Revista Mexicana de Astronomia y Astrofisica , 40, 209
  * Arellano Ferro $et~{}al$. (2018) Arellano Ferro, A., Rojas Galindo, F. C., Muneer, S., & Giridhar, S. 2018, Revista Mexicana de Astronomia y Astrofisica , 54, 325
  * Auriere & Lauzeral (1991) Auriere, M., & Lauzeral, C. 1991, A&A , 244, 303
  * Bastian & Lardo (2018) Bastian, N., & Lardo, C. 2018, ARA&A , 56, 83
  * Bayo $et~{}al$. (2008) Bayo, A., Rodrigo, C., Barrado Y Navascués, D., $et~{}al$. 2008, A&A , 492, 277
  * Behr $et~{}al$. (2000) Behr, B. B., Djorgovski, S. G., Cohen, J. G., $et~{}al$. 2000, ApJ , 528, 849
  * Brown $et~{}al$. (2016) Brown, T. M., Cassisi, S., D’Antona, F., $et~{}al$. 2016, ApJ , 822, 44
  * Cardelli $et~{}al$. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ , 345, 245
  * Cassisi & Salaris (2020) Cassisi, S., & Salaris, M. 2020, A&AR , 28, 5
  * Castelli & Kurucz (2003) Castelli, F., & Kurucz, R. L. 2003, in Modelling of Stellar Atmospheres, ed. N. Piskunov, W. W. Weiss, & D. F. Gray, Vol. 210, A20
  * Catelan $et~{}al$. (1998) Catelan, M., Borissova, J., Sweigart, A. V., & Spassova, N. 1998, ApJ , 494, 265
  * Chambers $et~{}al$. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., $et~{}al$. 2016, arXiv e-prints, arXiv:1612.05560
  * Chatterjee $et~{}al$. (2013) Chatterjee, S., Rasio, F. A., Sills, A., & Glebbeek, E. 2013, ApJ , 777, 106
  * Clement (2017) Clement, C. 2017, in European Physical Journal Web of Conferences, Vol. 152, European Physical Journal Web of Conferences, 01021
  * Dieball $et~{}al$. (2010) Dieball, A., Long, K. S., Knigge, C., Thomson, G. S., & Zurek, D. R. 2010, ApJ , 710, 332
  * Ferraro $et~{}al$. (2003) Ferraro, F. R., Sills, A., Rood, R. T., Paltrinieri, B., & Buonanno, R. 2003, ApJ , 588, 464
  * Ferraro $et~{}al$. (1997) Ferraro, F. R., Paltrinieri, B., Fusi Pecci, F., $et~{}al$. 1997, A&A , 324, 915
  * Gaia Collaboration $et~{}al$. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., $et~{}al$. 2018, A&A , 616, A1
  * Gosnell $et~{}al$. (2015) Gosnell, N. M., Mathieu, R. D., Geller, A. M., $et~{}al$. 2015, ApJ , 814, 163
  * Gratton $et~{}al$. (2019) Gratton, R., Bragaglia, A., Carretta, E., $et~{}al$. 2019, A&AR , 27, 8
  * Harris (2010) Harris, W. E. 2010, arXiv e-prints, arXiv:1012.3224
  * Hidalgo $et~{}al$. (2018) Hidalgo, S. L., Pietrinferni, A., Cassisi, S., $et~{}al$. 2018, ApJ , 856, 125
  * Ivans (2009) Ivans, I. I. 2009, in American Astronomical Society Meeting Abstracts, Vol. 213, American Astronomical Society Meeting Abstracts #213, 341.06
  * Jadhav $et~{}al$. (2019) Jadhav, V. V., Sindhu, N., & Subramaniam, A. 2019, ApJ , 886, 13
  * Jain $et~{}al$. (2019) Jain, R., Vig, S., & Ghosh, S. K. 2019, MNRAS , 485, 2877
  * Knigge $et~{}al$. (2009) Knigge, C., Leigh, N., & Sills, A. 2009, Nature , 457, 288
  * Kumar $et~{}al$. (2012) Kumar, A., Ghosh, S. K., & et al. 2012, in Proc. SPIE , Vol. 8443, Space Telescopes and Instrumentation 2012: Ultraviolet to Gamma Ray, 84431N
  * Kumar $et~{}al$. (2020a) Kumar, R., Pradhan, A. C., Mohapatra, A., $et~{}al$. 2020a, MNRAS , revision submitted
  * Kumar $et~{}al$. (2020b) Kumar, R., Pradhan, A. C., Parthasarathy, M., $et~{}al$. 2020b, in Star Clusters: From the Milky Way to the Early Universe, ed. A. Bragaglia, M. Davies, A. Sills, & E. Vesperini, Vol. 351, 464–467
  * Lata $et~{}al$. (2019) Lata, S., Pandey, A. K., Pandey, J. C., $et~{}al$. 2019, AJ , 158, 51
  * Leigh $et~{}al$. (2013) Leigh, N., Knigge, C., Sills, A., $et~{}al$. 2013, MNRAS , 428, 897
  * Marino $et~{}al$. (2019) Marino, A. F., Milone, A. P., Renzini, A., $et~{}al$. 2019, MNRAS , 487, 3815
  * Martell $et~{}al$. (2008) Martell, S. L., Smith, G. H., & Briley, M. M. 2008, AJ , 136, 2522
  * Milone $et~{}al$. (2017) Milone, A. P., Piotto, G., Renzini, A., $et~{}al$. 2017, MNRAS , 464, 3636
  * Nardiello $et~{}al$. (2018) Nardiello, D., Libralato, M., Piotto, G., $et~{}al$. 2018, MNRAS , 481, 3382
  * Pietrinferni $et~{}al$. (2020) Pietrinferni, A., Hidalgo, S., Cassisi, S., $et~{}al$. 2020, ApJ , submitted
  * Piotto $et~{}al$. (1999) Piotto, G., Zoccali, M., King, I. R., $et~{}al$. 1999, AJ , 118, 1727
  * Postma & Leahy (2017) Postma, J. E., & Leahy, D. 2017, PASP , 129, 115002
  * Rani $et~{}al$. (2020) Rani, S., Pandey, G., Subramaniam, A., Sahu, S., & Kameswara Rao, N. 2020, arXiv e-prints, arXiv:2010.00569
  * Sahu $et~{}al$. (2019a) Sahu, S., Subramaniam, A., Côté, P., Rao, N. K., & Stetson, P. B. 2019a, MNRAS , 482, 1080
  * Sahu $et~{}al$. (2019b) Sahu, S., Subramaniam, A., Simunovic, M., $et~{}al$. 2019b, ApJ , 876, 34
  * Schiavon $et~{}al$. (2012) Schiavon, R. P., Dalessandro, E., Sohn, S. T., $et~{}al$. 2012, AJ , 143, 121
  * Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ , 737, 103
  * Sindhu $et~{}al$. (2019) Sindhu, N., Subramaniam, A., Jadhav, V. V., $et~{}al$. 2019, ApJ , 882, 43
  * Singh $et~{}al$. (2020) Singh, G., Sahu, S., Subramaniam, A., & Yadav, R. K. S. 2020, arXiv e-prints, arXiv:2010.06979
  * Stetson (1987) Stetson, P. B. 1987, PASP , 99, 191
  * Stetson $et~{}al$. (2005) Stetson, P. B., Catelan, M., & Smith, H. A. 2005, PASP , 117, 1325
  * Stetson $et~{}al$. (2019) Stetson, P. B., Pancino, E., Zocchi, A., Sanna, N., & Monelli, M. 2019, MNRAS , 485, 3042
  * Subramaniam $et~{}al$. (2016) Subramaniam, A., Sindhu, N., Tandon, S. N., $et~{}al$. 2016, ApJL , 833, L27
  * Subramaniam $et~{}al$. (2017) Subramaniam, A., Sahu, S., Postma, J. E., $et~{}al$. 2017, AJ , 154, 233
  * Suntzeff $et~{}al$. (1988) Suntzeff, N. B., Kraft, R. P., & Kinman, T. D. 1988, AJ , 95, 91
  * Tailo $et~{}al$. (2020) Tailo, M., Milone, A. P., Lagioia, E. P., $et~{}al$. 2020, MNRAS , 498, 5745
  * Tandon $et~{}al$. (2017) Tandon, S. N., Subramaniam, A., Girish, V., & et. al. 2017, AJ , 154, 128
  * Tandon $et~{}al$. (2020) Tandon, S. N., Postma, J., Joseph, P., $et~{}al$. 2020, AJ , 159, 158
  * Villanova $et~{}al$. (2016) Villanova, S., Monaco, L., Moni Bidin, C., & Assmann, P. 2016, MNRAS , 460, 2351

Table 4: UVIT photometry table of the BHB1, BHB2 and FUV bright BSS containing
their respective positions, the extinction-corrected magnitudes and the
magnitude errors. IDs HB01 to HB24 belong to the BHB1 group and IDs HB25 to
HB37 belong to the BHB2 group. The BSS is listed at the end of the table with
ID BSS01.

ID | RAJ2000 | DEJ2000 | BaF2 | eBaF2 | Sapphire | eSapphire | Silica | eSilica
---|---|---|---|---|---|---|---|---
HB01 | 182.5358 | 18.5215 | 19.57 | 0.01 | 19.37 | 0.02 | 19.35 | 0.02
HB02 | 182.5117 | 18.5330 | 19.79 | 0.02 | 19.62 | 0.02 | 19.44 | 0.02
HB03 | 182.5076 | 18.5353 | 19.83 | 0.02 | 19.55 | 0.02 | 19.41 | 0.02
HB04 | 182.5239 | 18.5356 | 19.43 | 0.01 | 19.23 | 0.02 | 19.55 | 0.03
HB05 | 182.5231 | 18.5393 | 19.46 | 0.01 | 19.13 | 0.01 | 19.31 | 0.03
HB06 | 182.5243 | 18.5396 | 19.62 | 0.01 | 19.48 | 0.02 | 19.29 | 0.03
HB07 | 182.5302 | 18.5413 | 19.45 | 0.01 | 19.27 | 0.01 | 19.45 | 0.02
HB08 | 182.5229 | 18.5422 | 19.53 | 0.01 | 19.22 | 0.01 | 19.23 | 0.03
HB09 | 182.5309 | 18.5458 | 20.05 | 0.01 | 19.84 | 0.02 | 20.08 | 0.03
HB10 | 182.5159 | 18.5455 | 19.41 | 0.02 | 19.13 | 0.02 | 19.15 | 0.02
HB11 | 182.5288 | 18.5465 | 19.70 | 0.01 | 19.47 | 0.02 | 19.31 | 0.03
HB12 | 182.5297 | 18.5473 | 19.97 | 0.01 | 19.59 | 0.02 | 19.70 | 0.03
HB13 | 182.5111 | 18.5564 | 19.95 | 0.01 | 20.07 | 0.02 | 19.73 | 0.02
HB14 | 182.5240 | 18.5567 | 19.85 | 0.02 | 19.40 | 0.01 | 19.84 | 0.03
HB15 | 182.5394 | 18.5611 | 19.74 | 0.01 | 19.65 | 0.01 | 19.41 | 0.03
HB16 | 182.5644 | 18.5639 | 19.49 | 0.01 | 19.42 | 0.02 | 19.44 | 0.02
HB17 | 182.5427 | 18.6696 | 19.52 | 0.01 | 19.23 | 0.02 | 19.16 | 0.02
HB18 | 182.5103 | 18.5344 | 19.54 | 0.02 | 19.26 | 0.01 | 19.33 | 0.03
HB19 | 182.5291 | 18.5376 | 19.22 | 0.03 | 19.09 | 0.03 | 18.98 | 0.03
HB20 | 182.5228 | 18.5379 | 19.51 | 0.01 | 19.22 | 0.01 | 19.09 | 0.03
HB21 | 182.5315 | 18.5503 | 19.49 | 0.01 | 19.43 | 0.02 | 19.36 | 0.02
HB22 | 182.5528 | 18.5609 | 19.36 | 0.01 | 19.11 | 0.01 | 19.33 | 0.02
HB23 | 182.5381 | 18.5407 | 19.28 | 0.01 | 19.14 | 0.01 | 19.21 | 0.03
HB24 | 182.5279 | 18.5506 | 19.44 | 0.02 | 19.23 | 0.04 | 19.38 | 0.02
HB25 | 182.5417 | 18.5302 | 20.63 | 0.03 | 20.54 | 0.03 | 20.05 | 0.03
HB26 | 182.5150 | 18.5352 | 21.78 | 0.04 | 21.13 | 0.03 | 20.53 | 0.03
HB27 | 182.5268 | 18.5388 | 20.77 | 0.02 | 20.60 | 0.03 | 20.26 | 0.04
HB28 | 182.5312 | 18.5420 | 21.26 | 0.03 | 20.84 | 0.03 | 20.82 | 0.06
HB29 | 182.5171 | 18.5441 | 21.84 | 0.04 | 21.21 | 0.04 | 20.26 | 0.04
HB30 | 182.5097 | 18.5476 | 20.59 | 0.02 | 20.15 | 0.02 | 19.89 | 0.05
HB31 | 182.5226 | 18.5489 | 20.75 | 0.03 | 20.13 | 0.02 | 19.91 | 0.02
HB32 | 182.4897 | 18.5769 | 21.73 | 0.04 | 21.17 | 0.04 | 20.27 | 0.04
HB33 | 182.5668 | 18.5771 | 21.60 | 0.03 | 21.29 | 0.03 | 20.59 | 0.04
HB34 | 182.5318 | 18.5426 | 20.99 | 0.03 | 21.05 | 0.04 | 20.07 | 0.03
HB35 | 182.5307 | 18.5430 | 21.96 | 0.04 | 21.55 | 0.05 | - | -
HB36 | 182.5292 | 18.5481 | 21.69 | 0.04 | 21.45 | 0.06 | - | -
HB37 | 182.5123 | 18.6111 | 21.48 | 0.05 | 21.68 | 0.05 | - | -
BSS01 | 182.5633 | 18.5480 | 21.80 | 0.05 | 21.81 | 0.05 | - | -


# Wavelet approach to operator-valued Hardy spaces

Guixiang Hong  Laboratoire de Mathématiques, Université de Franche-Comté,
25030 Besançon Cedex, France
_E-mail address<EMAIL_ADDRESS>and  Zhi Yin  Laboratoire de
Mathématiques, Université de Franche-Comté,
25030 Besançon Cedex, France
_E-mail address<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the study of operator-valued Hardy spaces via wavelet
method. This approach is parallel to that in noncommutative martingale case.
We show that our Hardy spaces defined by wavelet coincide with those
introduced by Tao Mei via the usual Lusin and Littlewood-Paley square
functions. As a consequence, we give an explicit complete unconditional basis
of the Hardy space $H_{1}(\mathbb{R})$ when $H_{1}(\mathbb{R})$ is equipped
with an appropriate operator space structure.

##  1\. Introduction

In this paper, we exploit Meyer’s wavelet methods to the study of the
operator-valued Hardy spaces. We are motivated by two rapidly developed
fields. The firs one is the theory of noncommutative martingales inequalities.
This theory had been already initiated in the 1970’s. Its modern period of
development has begun with Pisier and Xu’s seminal paper [20] in which the
authors established the noncommutative Burkholder-Gundy inequalities and
Fefferman duality theorem between $H_{1}$ and $BMO$. Since then many classical
results have been successfully transferred to the noncommutative world (see
[11], [14], [15], [1]). In particular, motivated by [9], Mei [15] developed
the theory of Hardy spaces on $\mathbb{R}^{n}$ for operator-valued functions.

Our second motivation is the theory of wavelets founded by Meyer. It is
nowadays well known that this theory is important for many domains, in
particular in harmonic analysis. For instance, it provides powerful tools to
the theory of Calderón-Zygmund singular integral operators. More recently,
Meyer’s wavelet methods were extended to study more sophistical subjects in
harmonic analysis. For example, the authors of [5] exploited the properties of
Meyer’s wavelets to give a characterization of product $BMO$ by commutators;
[17] deals with the estimates of bi-parameter paraproducts.

It is in this spirit that we wish to understand how useful wavelet methods are
for noncommutative analysis. The most natural and possible way would be first
to do this in the semi-commutative case. This is exactly the purpose of the
present paper which could be viewed as the first attempt towards the
development of wavelet techniques for noncommutative analysis.

A wavelet basis of $L_{2}({\mathbb{R}})$ is a complete orthonormal system
$(w_{I})_{I\in\mathcal{D}}$, where $\mathcal{D}$ denotes the collection of all
dyadic intervals in $\mathbb{R}$, $w$ is a Schwartz function satisfying the
properties needed for Meryer’s construction in [16], and

$w_{I}(x)\doteq\frac{1}{|I|^{\frac{1}{2}}}w\big{(}\frac{x-c_{I}}{|I|}\big{)},$

where $c_{I}$ is the center of $I$. The central facts that we will need about
the wavelet basis are the orthogonality between different $w_{I}$’s,
$\|w\|_{L_{2}({\mathbb{R}})}=1$ and the regularity of $w$,

$\max(|w(x)|,|w^{\prime}(x)|)\precsim(1+|x|)^{-m},\quad\forall m\geq 2.$

The analogy between wavelets and dyadic martingales is well known. The key
observation is the following parallelism:

$\sum_{|I|=2^{-n+1}}\langle f,w_{I}\rangle w_{I}\sim df_{n},$

where $df_{n}$ denotes $n$-th dyadic martingale difference of $f$. As dyadic
martingales are much easier to handle, this parallelism explains why wavelet
approach to many problems in harmonic analysis is usually simple and
efficient. On the other hand, it also indicates that martingale methods may be
used to deal with wavelets. With this in mind, we develop the operator-valued
Hardy spaces based on the wavelet methods in the way which is well known in
the noncommutative martingales case. Then we show that our Hardy and BMO
spaces coincide with Mei’s. In other words, we provide another approach, which
is much simpler than Mei’s original one, to recover all the results of [15].

This paper is organized as follows. In section 1, we will give some
preliminaries on noncommutative analysis, the definition of
$\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})$ with $1\leq p<\infty$ and
$L_{q}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ with $2<q\leq\infty$ in our
setting. In section 2, we are concerned with three duality results. The most
important one is the noncommutative analogue of the famous Fefferman duality
theorem between $\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$ and
$\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})$. The second one is the duality
between $\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ and
$L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ with $1<p<2$,
where we need the noncommutative Doob’s inequality, this is why we consider
the case $1<p<2$ independently. The last one is the duality between
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ and
$\mathcal{H}^{c}_{p^{\prime}}({\mathbb{R}},{\mathcal{M}})$ with $1<p<\infty$.
As a corollary of the last two results, we identify
$\mathcal{H}^{c}_{q}({\mathbb{R}},{\mathcal{M}})$ and
$L^{c}_{q}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ with $2<q<\infty$.
Section 3 deals with the interpolation of our Hardy spaces. In the last
section, we show that our Hardy spaces coincide with those of [15]. So, we can
give an explicit completely unconditional basis for the space
$H_{1}({\mathbb{R}})$, when $H_{1}({\mathbb{R}})$ is equipped with an
appropriate operator space structure.

We end this introduction by the convention that throughout the paper the
letter $c$ will denote an absolute positive constant, which may vary from
lines to lines, and $c_{p}$ a positive constant depending only on $p$.

##  2\. Preliminaries

###  2.1. Operator-valued noncommutative $L_{p}$-spaces

Let $\mathcal{M}$ be a von Neumann algebra equipped with a normal semifinite
faithful trace $\tau$ and $S^{+}_{\mathcal{M}}$ be the set of all positive
element $x$ in $\mathcal{M}$ with $\tau(s(x))<\infty$, where $s(x)$ is the
smallest projection $e$ such that $exe=x$. Let $S_{\mathcal{M}}$ be the linear
span of $S^{+}_{\mathcal{M}}$. Then any $x\in S_{\mathcal{M}}$ has finite
trace, and $S_{\mathcal{M}}$ is a $w^{*}$-dense $*$-subalgebra of
$\mathcal{M}$.

Let $1\leq p<\infty$. For any $x\in S_{\mathcal{M}}$, the operator $|x|^{p}$
belongs to $S^{+}_{\mathcal{M}}$ ($|x|=(x^{*}x)^{\frac{1}{2}}$). We define

$\|x\|_{p}=\big{(}\tau(|x|^{p})\big{)}^{\frac{1}{p}},\qquad\forall x\in
S_{\mathcal{M}}.$

One can check that $\|\cdot\|_{p}$ is well defined and is a norm on
$S_{\mathcal{M}}$. The completion of $(S_{\mathcal{M}},\|\cdot\|_{p})$ is
denoted by $L_{p}({\mathcal{M}})$ which is the usual noncommutative $L_{p}$\-
space associated with $({\mathcal{M}},\tau)$. For convenience, we usually set
$L_{\infty}({\mathcal{M}})={\mathcal{M}}$ equipped with the operator norm
$\|\cdot\|_{{\mathcal{M}}}$. The elements of $L_{p}({\mathcal{M}},\tau)$ can
be described as closed densely defined operators on $H$ ($H$ being the Hilbert
space on which ${\mathcal{M}}$ acts). We refer the reader to [21] for more
information on noncommutative $L_{p}$-spaces.

In this paper, we are concerned with three operator-valued noncommutative
$L_{p}$-spaces. The first one is the Hilbert-valued noncommutative space
$L_{p}(\mathcal{M};H^{c})$ (resp. $L_{p}(\mathcal{M};H^{r})$), which is
studied at length in [9]. For this space, we need the following properties. In
the sequel, $p^{\prime}$ will always denote the conjugate index of $p$.

######  Lemma 2.1.

Let $1\leq p<\infty$. Then

(2.1) $(L_{p}(\mathcal{M};H^{c}))^{*}=L_{p^{\prime}}(\mathcal{M};H^{c}).$

Thus, for $f\in L_{p}(\mathcal{M};H^{c})$ and $g\in
L_{p^{\prime}}(\mathcal{M};H^{c})$, we have

$|\tau(\langle
f,g\rangle)|\leq\|f\|_{L_{p}(\mathcal{M};H^{c})}\|g\|_{L_{p^{\prime}}(\mathcal{M};H^{c})},$

where $\langle,\rangle$ denotes the inner product of $H$.

######  Lemma 2.2.

Let $1\leq p_{0}<p<p_{1}\leq\infty$, $0<\theta<1$,
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$. Then

(2.2)
$[L_{p_{0}}(\mathcal{M};H^{c}),L_{p_{1}}(\mathcal{M};H^{c})]_{\theta}=L_{p}(\mathcal{M};H^{c}).$

A same equality holds for row spaces.

The second one is the $\ell_{\infty}$-valued noncommutative space
$L_{p}(\mathcal{M};\ell_{\infty})$, which is studied by Pisier [19] for an
injective ${\mathcal{M}}$ and Junge [8] for a general ${\mathcal{M}}$ (see
also [11] and [13] for more properties). About this one, we need the following
property:

######  Lemma 2.3.

Let $1\leq p<\infty$. Then

$(L_{p}(\mathcal{M};\ell_{1}))^{*}=L_{p^{\prime}}(\mathcal{M};\ell_{\infty}).$

Thus, for $x=(x_{n})_{n}\in L_{p}(\mathcal{M};\ell_{1})$ and $y=(y_{n})_{n}\in
L_{p^{\prime}}(\mathcal{M};\ell_{\infty})$, we have

(2.3) $\big{|}\sum_{n\geq
1}\tau(x_{n}y_{n})\big{|}\leq\|x\|_{L_{p}(\mathcal{M};\ell_{1})}\|y\|_{L_{p^{\prime}}(\mathcal{M};\ell_{\infty})}.$

The third one is $L_{p}(\mathcal{M};\ell^{c}_{\infty})$ for $2\leq
p\leq\infty$, which was introduced in [4] and is related with the second one
by

$\|(x_{n})_{n}\|_{L_{p}(\mathcal{M};\ell^{c}_{\infty})}=\|(|x_{n}|^{2})_{n}\|_{L_{\frac{p}{2}}({\mathcal{M}};\ell_{\infty})}.$

And these are normed spaces by the following characterization

$\|(x_{n})_{n}\|_{L_{p}(\mathcal{M};\ell^{c}_{\infty})}=\inf_{x_{n}=y_{n}a}\|(y_{n})\|_{\ell_{\infty}(L_{\infty}({\mathcal{M}}))}\|a\|_{L_{p}({\mathcal{M}})}.$

We need the interpolation results about these spaces (see [18]):

######  Lemma 2.4.

Let $2\leq p_{0}<p<p_{1}\leq\infty$, $0<\theta<1$,
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$. Then

(2.4)
$[L_{p_{0}}(\mathcal{M};\ell^{c}_{\infty}),L_{p_{1}}(\mathcal{M};\ell^{c}_{\infty})]_{\theta}=L_{p}(\mathcal{M};\ell^{c}_{\infty}).$

###  2.2. Operator-valued Hardy spaces

In this paper, for simplicity, we denote
$L_{\infty}({\mathbb{R}})\bar{\otimes}{\mathcal{M}}$ by $\mathcal{N}$. As
indicated in the introduction, one can observe that we have the following
operator-valued Calderón identity

(2.5) $f(x)=\sum_{I\in\mathcal{D}}\langle f,w_{I}\rangle w_{I}(x),$

which holds when $f\in L_{2}({\mathcal{N}})$. As in the classical case, for
$f\in S_{{\mathcal{N}}}$, we define the two Littlewood-Paley square functions
as

(2.6) $S_{c}(f)(x)=\Big{(}\sum_{I\in\mathcal{D}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}.$ (2.7)
$S_{r}(f)(x)=\Big{(}\sum_{I\in\mathcal{D}}\frac{|\langle
f^{*},w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}.$

For $1\leq p<\infty$, define

$\|f\|_{\mathcal{H}^{c}_{p}}=\|S_{c}(f)\|_{L_{p}({\mathcal{N}})},$
$\|f\|_{\mathcal{H}^{r}_{p}}=\|S_{r}(f)\|_{L_{p}({\mathcal{N}})}.$

These are norms, which can be seen easily from the space
$L_{p}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))$. So we define the spaces
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ (resp.
$\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})$) as the completion of
$(S_{\mathcal{N}},\|\cdot\|_{\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})})$
(resp.
$(S_{\mathcal{N}},\|\cdot\|_{\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})})$.
Now, we define the operator-valued Hardy spaces as follows: for $1\leq p<2,$

(2.8)
$\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})+\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})$

with the norm

$\|f\|_{\mathcal{H}_{p}}=\inf\\{\|g\|_{\mathcal{H}^{c}_{p}}+\|h\|_{\mathcal{H}^{r}_{p}}:f=g+h,g\in\mathcal{H}^{c}_{p},h\in\mathcal{H}^{r}_{p}\\}$

and for $2\leq p<\infty$,

(2.9)
$\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})\cap\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})$

with the norm defined as

$\|f\|_{\mathcal{H}_{p}}=\max\\{\|f\|_{\mathcal{H}^{c}_{p}},\|f\|_{\mathcal{H}^{r}_{p}}\\}.$

We can identify $\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ as a
subspace of $L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$, which is related
with the two maps below.

######  Lemma 2.5.

$\rm(i)$ The embedding map $\Phi$ is defined from
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ to
$L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ by

(2.10) $\Phi(f)=\Big{(}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\Big{)}_{I\in\mathcal{D}}.$

$\rm(ii)$ The projection map $\Psi$ is defined from
$L_{2}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ to
$\mathcal{H}^{c}_{2}({\mathbb{R}},{\mathcal{M}})$ by

(2.11)
$\Psi((g_{I}))=\sum_{I\in\mathcal{D}}\int\frac{g_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}dy\cdot
w_{I}.$

###  2.3. Operator-valued $\mathcal{BMO}$ spaces

For $\varphi\in
L_{\infty}({\mathcal{M}};L^{c}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}}))$, set

(2.12)
$\|\varphi\|_{{\mathcal{BMO}}^{c}}=\sup_{J\in\mathcal{D}}\Big{\|}\big{(}\frac{1}{|J|}\sum_{I\subset
J}|\langle\varphi,w_{I}\rangle|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{{\mathcal{M}}}$

and

$\|\varphi\|_{\mathcal{BMO}^{r}}=\|\varphi^{*}\|_{\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})}.$

These are again norms modulo constant functions. Define

${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})=\\{\varphi\in
L_{\infty}({\mathcal{M}};L^{c}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}})):\|\varphi\|_{{\mathcal{BMO}}^{c}}<\infty\\}$

and

${\mathcal{BMO}}^{r}({\mathbb{R}},{\mathcal{M}})=\\{\varphi\in
L_{\infty}({\mathcal{M}};L^{r}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}})):\|\varphi\|_{{\mathcal{BMO}}^{r}}<\infty\\}$

Now we define

$\mathcal{BMO}({\mathbb{R}},{\mathcal{M}})=\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})\cap\mathcal{BMO}^{r}({\mathbb{R}},{\mathcal{M}}).$

As in the martingale case [11], we can also define
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ for all $2<p\leq\infty$.
For $\varphi\in
L_{p}({\mathcal{M}};L^{c}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}}))$, set

(2.13)
$\|\varphi\|_{L^{c}_{p}{\mathcal{MO}}}=\Big{\|}(\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2})_{k}\Big{\|}_{L_{\frac{p}{2}}({\mathcal{N}};\ell_{\infty})}$

and

$\|\varphi\|_{L^{r}_{p}{\mathcal{MO}}}=\|\varphi^{*}\|_{L^{c}_{p}{\mathcal{MO}}},$

where $I^{x}_{k}$ denotes the unique dyadic interval with length $2^{-k+1}$
that containing $x$. We will use the convention adopted in [13] for the norm
in $L_{\frac{p}{2}}({\mathcal{N}};\ell_{\infty}).$ Thus

$\Big{\|}(\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2})_{k}\Big{\|}_{L_{\frac{p}{2}}({\mathcal{N}};\ell_{\infty})}=\Big{\|}{\sup_{k}}^{+}\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2}\Big{\|}_{L_{\frac{p}{2}({\mathcal{N}})}}$

These are norms, which can be seen from the Banach spaces
$L_{p}(\mathcal{N}\bar{\otimes}B(\ell_{2}(\mathcal{D}));\ell^{c}_{\infty})$.
Again, we can define

$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})=\\{\varphi\in
L_{p}({\mathcal{M}};L^{c}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}})):\|\varphi\|_{L^{c}_{p}{\mathcal{MO}}}<\infty\\}$

and

$L^{r}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})=\\{\varphi\in
L_{p}({\mathcal{M}};L^{r}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}})):\|\varphi\|_{L^{c}_{r}{\mathcal{MO}}}<\infty\\}$

Define

$L_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})=L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})\cap
L^{r}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}}).$

Note that
$L^{c}_{\infty}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})={\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})$.
It is easy to check all the spaces we defined here respect to the relevant
norms are Banach spaces.

##  3\. Duality

To prove the first two duality results in this section, we need the following
noncommutative Doob’s inequality from [8].

Let $(\mathcal{E}_{n})_{n}$ be the conditional expectation with respect to a
filtration $({{\mathcal{N}}}_{n})_{n}$ of ${\mathcal{N}}$.

######  Lemma 3.1.

Let $1<p\leq\infty$ and $f\in L_{p}({\mathcal{N}})$. Then

(3.1) $\|{\sup_{n}}^{+}\mathcal{E}_{n}(f)\|_{L_{p}({\mathcal{N}})}\leq
c_{p}\|f\|_{L_{p}({\mathcal{N}})}.$

######  Theorem 3.1.

We have

(3.2)
$(\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}}))^{*}={\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})$

with equivalent norms. That is, every
$\varphi\in\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})$ induces a continuous
linear functional $l_{\varphi}$ on
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$ by

(3.3) $l_{\varphi}(f)=\tau\int\varphi^{*}f,\quad\forall f\in S_{\mathcal{N}}.$

Conversely, for every
$l\in(\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}}))^{*}$, there exits a
$\varphi\in\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})$ such that
$l=l_{\varphi}$. Moreover,

$c^{-1}\|\varphi\|_{\mathcal{BMO}^{c}}\leq\|l_{\varphi}\|_{(\mathcal{H}^{c}_{1})^{*}}\leq
c\|\varphi\|_{\mathcal{BMO}^{c}}$

where $c>0$ is a universal constant.

Similarly, the duality holds between $\mathcal{H}^{r}_{1}$ and
$\mathcal{BMO}^{r}$, between $\mathcal{H}_{1}$ and $\mathcal{BMO}$ with
equivalent norms.

In order to adapt the arguments in the martingale case, we need to define the
truncated square functions for $n\in\mathbb{Z}$,

$S_{c,n}(f)(x)=\Big{(}\sum^{n}_{k=-\infty}\sum_{|I|=2^{-k+1}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}.$

###### Proof.

Since $S_{\mathcal{N}}$ is dense in
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$, by an approximation
argument, we only need to prove the inequality

$|l_{\varphi}(f)|\leq
c\|\varphi\|_{\mathcal{BMO}^{c}}\|f\|_{\mathcal{H}^{c}_{1}}$

for $f\in S_{\mathcal{N}}$. By approximation we may assume that
$S_{c,n}(f)(x)$ is invertible in ${\mathcal{M}}$ for all $x\in{\mathbb{R}}$
and $n\in\mathbb{Z}$. Then we have

$\displaystyle|l_{\varphi}(f)|$ $\displaystyle=|\tau\int\varphi^{*}fdx|$
$\displaystyle=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\langle\varphi,w_{I}\rangle^{*}w_{I}\sum_{|I^{\prime}|=2^{-n+1}}\langle
f,w_{I^{\prime}}\rangle w_{I^{\prime}}dx\Big{|}$
$\displaystyle=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\sum_{|I^{\prime}|=2^{-n+1}}\frac{\langle
f,w_{I^{\prime}}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I^{\prime}}dx\Big{|}$
$\displaystyle\leq\sum_{n}\Big{(}\tau\int\big{|}\sum_{|I|=2^{-n+1}}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{|}^{2}S^{-1}_{c,n}(f)\Big{)}^{\frac{1}{2}}$
$\displaystyle\qquad\cdot\Big{(}\tau\int\big{|}\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{|}^{2}{S_{c,n}(f)}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\Big{(}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}S^{-1}_{c,n}(f)\Big{)}^{\frac{1}{2}}$
$\displaystyle\qquad\cdot\Big{(}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}{S_{c,n}(f)}\Big{)}^{\frac{1}{2}}$
$\displaystyle=A\cdot B.$

In the above estimates, the first equality has used the orthogonality of the
$w_{I}$’s on different levels, the second one the orthogonality of the
$w_{I}$’s on the same level and the disjoint of different dyadic $I$’s on the
same level; the first inequality has used the Hölder inequality in Lemma 2.1,
and the second one the Cauchy-Schwarz inequality and the disjointness of
different $I$’s on the same level.

Now, let us estimate $A$:

$\displaystyle\begin{split}A^{2}&=\sum_{n}\tau\int(S^{2}_{c,n}(f)-S^{2}_{c,n-1}(f))S^{-1}_{c,n}(f)\\\
&=\sum_{n}\tau\int(S_{c,n}(f)-S_{c,n-1}(f))(1+S_{c,n-1}(f)S^{-1}_{c,n}(f))\\\
&\leq\sum_{n}\tau\int(S_{c,n}(f)-S_{c,n-1}(f))\|1+S_{c,n-1}(f)S^{-1}_{c,n}(f)\|_{\infty}\\\
&\leq 2\sum_{n}\tau\int(S_{c,n}(f)-S_{c,n-1}(f))\\\
&=2\|f\|_{\mathcal{H}^{c}_{1}}.\end{split}$

For the first inequality, we have used the Hölder inequality and the
positivity of $S_{c,n}(f)-S_{c,n-1}(f)$.

The second term is estimated as follows:

$\displaystyle\begin{split}B^{2}&=\sum_{k}\tau\int(S_{c,k}(f)-S_{c,k-1}(f))\sum_{n\geq
k}\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\\\
&=\sum_{k}\tau\sum_{j}(S_{c,k}(f)-S_{c,k-1}(f))\int_{I^{j}_{k}}\sum_{n\geq
k}\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\\\
&=\sum_{k}\tau\sum_{j}\int_{I^{j}_{k}}(S_{c,k}(f)-S_{c,k-1}(f))\frac{1}{|I^{j}_{k}|}\sum_{I\subset
I^{j}_{k}}|\langle\varphi,w_{I}\rangle|^{2}\\\
&\leq\sum_{k}\sum_{j}\tau\int_{I^{j}_{k}}(S_{c,k}(f)-S_{c,k-1}(f))\Big{\|}\frac{1}{|I^{j}_{k}|}\sum_{I\subset
I^{j}_{k}}|\langle\varphi,w_{I}\rangle|^{2}\Big{\|}_{\infty}\\\
&\leq\|\varphi\|^{2}_{{\mathcal{BMO}}^{c}}\sum_{k}\sum_{j}\tau\int_{I^{j}_{k}}(S_{c,k}(f)-S_{c,k-1}(f))\\\
&=\|\varphi\|^{2}_{{\mathcal{BMO}}^{c}}\|f\|_{\mathcal{H}^{c}_{1}}\\\
\end{split}$

The fist equality has used the Fubini theorem, the second one the fact that
$S_{c,k-1}(f)$ and $S_{c,k}(f)$ are constant on the dyadic interval
$I^{j}_{k}=[j2^{-k+1},(j+1)2^{-k+1})$; the first inequality has used the
Hölder inequality and the positivity of $S_{c,n}(f)-S_{c,n-1}(f)$.

Now, let us begin to deal with another direction, i.e. suppose that $l$ is a
bounded linear functional on
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$, we want to find an
operator-valued function $\varphi$ in
${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})$, such that $l=l_{\varphi}$
and $l_{\varphi}(f)=\tau\int\varphi^{*}f$ for $f\in
S_{\mathcal{{\mathcal{N}}}}$. By the embedding operator $\Phi$ in (2.10) and
by the Banach-Hahn theorem, $l$ extends to a bounded continuous functional on
$L_{1}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ of the same norm. Then by the
results in Lemma 2.1 there exists $g=(g_{I})_{I\in\mathcal{D}}$ such that
$\|g\|_{L_{\infty}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))}=\|l\|$, and

$l(f)=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I},\quad\forall f\in
S_{\mathcal{N}}.$

Now let $\varphi=\Psi(g)$, where $\Psi$ is defined as (2.11). The
orthogonality of the $w_{I}$’s yields

$\displaystyle\begin{split}\big{\|}\sum_{I\subset
J}|\langle\varphi,w_{I}\rangle|^{2}\big{\|}_{{\mathcal{M}}}&=\big{\|}\sum_{I\subset
J}|\int\frac{g_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|^{2}\big{\|}_{{\mathcal{M}}}\leq\big{\|}\sum_{I\subset
J}\int_{J}|g_{I}|^{2}\big{\|}_{{\mathcal{M}}}\\\
&\leq|J|\big{\|}\sum_{I\subset
J}|g_{I}|^{2}\big{\|}_{L_{\infty}({\mathcal{N}})}\leq|J|\big{\|}(g_{I})_{I}\big{\|}_{L_{\infty}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))},\\\
\end{split}$

where the first inequality has used the Kadison-Schwartz inequality. Also
thanks to the orthogonality of the $w_{I}$’s, we get

$\displaystyle\begin{split}l(f)&=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}=\tau\int\varphi^{*}f\\\
\end{split}$

for all $f\in S_{\mathcal{N}}$. Therefore, we complete the proof about
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$ and
$\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})$. Passing to adjoint, we have
the conclusion concerning $\mathcal{H}^{r}_{1}$ and $\mathcal{BMO}^{r}$.
Finally, by the classical fact that the dual of a sum space is the
intersection space, we obtain the duality between $\mathcal{H}_{1}$ and
$\mathcal{BMO}$. ∎

######  Theorem 3.2.

Let $1<p<2$. We have

(3.4)
$(\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}))^{*}=L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$

with equivalent norms. That is, every $\varphi\in
L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ induces a
continuous linear functional $l_{\varphi}$ on
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ by

(3.5) $l_{\varphi}(f)=\tau\int\varphi^{*}f,\quad\forall f\in S_{\mathcal{N}}.$

Conversely, for every
$l\in(\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}))^{*}$, there exists an
operator-valued function $\varphi\in
L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ such that
$l=l_{\varphi}$ and

$c_{p}^{-1}\|\varphi\|_{L^{c}_{p^{\prime}}{\mathcal{MO}}}\leq\|l_{\varphi}\|_{(\mathcal{H}^{c}_{p})^{*}}\leq\sqrt{2}\|\varphi\|_{L^{c}_{p^{\prime}}{\mathcal{MO}}}$

Similarly, the duality holds between $\mathcal{H}^{r}_{p}$ and
$L^{r}_{p^{\prime}}$, between $\mathcal{H}_{p}$ and
$L_{p^{\prime}}{\mathcal{MO}}$ with equivalent norms.

We need the following lemma of [11]. We write it down for the reader’s
convenience but without proof.

######  Lemma 3.2.

Let $s,t$ be two real numbers such that $s<t$ and $0\leq s\leq 1\leq t\leq 2$.
Let $x,y$ be two positive operators such that $x\leq y$ and
$x^{t-s},y^{t-s}\in L_{1}({\mathcal{N}})$. Then

$\tau\int y^{-s/2}(y^{t}-x^{t})y^{-s/2}\leq 2\tau\int
y^{-(s+1-t)/2}(y-x)y^{-(s+1-t)/2}.$

###### Proof.

We need only to prove the first assertion on $\mathcal{H}^{c}_{p}$. Since
$S_{\mathcal{N}}$ is dense in
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$, by an approximation
argument, we only need to prove the inequality

$|l_{\varphi}(f)|\leq
c\|\varphi\|_{L^{c}_{p^{\prime}}{\mathcal{MO}}}\|f\|_{\mathcal{H}^{c}_{p}}$

for $f\in S_{\mathcal{N}}$. By approximation we may assume that
$S_{c,n}(f)(x)$ is invertible in ${\mathcal{M}}$ for all $x\in{\mathbb{R}}$
and $n\in\mathbb{Z}$. By the similar principle in the noncommutative
martingale case as in [11], we have

$\displaystyle\begin{split}|l_{\varphi}(f)|&=|\tau\int\varphi^{*}fdx|\\\
&=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\langle\varphi,w_{I}\rangle^{*}w_{I}\sum_{|I^{\prime}|=2^{-n+1}}\langle
f,w_{I^{\prime}}\rangle w_{I^{\prime}}dx\Big{|}\\\
&=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\sum_{|I^{\prime}|=2^{-n+1}}\frac{\langle
f,w_{I^{\prime}}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I^{\prime}}dx\Big{|}\\\
&\leq\sum_{n}\Big{(}\tau\int\big{|}\sum_{|I|=2^{-n+1}}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{|}^{2}S^{p-2}_{c,n}(f)\Big{)}^{\frac{1}{2}}\\\
&\qquad\cdot\Big{(}\tau\int\big{|}\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{|}^{2}{S^{2-p}_{c,n}(f)}\Big{)}^{\frac{1}{2}}\\\
&\leq\Big{(}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}S^{p-2}_{c,n}(f)\Big{)}^{\frac{1}{2}}\\\
&\qquad\cdot\Big{(}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}{S^{2-p}_{c,n}(f)}\Big{)}^{\frac{1}{2}}\\\
&=A\cdot B.\\\ \end{split}$

Now we need the above lemma to estimate the first term. Take $s=2-p$ and
$t=2$, the lemma yields

$\displaystyle A^{2}$
$\displaystyle=\sum_{n}\tau\int(S^{2}_{c,n}(f)-S^{2}_{c,n-1}(f))S^{p-2}_{c,n}(f)$
$\displaystyle=\sum_{n}\tau\int
S^{-(2-p)/2}_{c,n}(f)(S^{2}_{c,n}(f)-S^{2}_{c,n-1}(f))S^{-(2-p)/2}_{c,n}(f)$
$\displaystyle\leq 2\sum_{n}\tau\int
S^{-(1-p)/2}_{c,n}(f)(S_{c,n}(f)-S_{c,n-1}(f))S^{-(1-p)/2}_{c,n}(f)$
$\displaystyle=2\sum_{n}\tau\int S_{c,n}(f)-S_{c,n-1}(f)S^{p-1}_{c,n}(f)$
$\displaystyle\leq 2\sum_{n}\tau\int S^{p}_{c,n}(f)-S^{p}_{c,n-1}(f)$
$\displaystyle=2\|f\|^{p}_{\mathcal{H}^{c}_{p}}.$

The last inequality has used two elementary inequalities: $0\leq
S_{c,n-1}(f)\leq S_{c,n}(f)$ implies $S^{p-1}_{c,n-1}(f)\leq S^{p-1}_{c,n}(f)$
for $0<p-1<1$; and
$\tau(S^{p}_{c,n-1}(f))\leq\tau(S^{\frac{1}{2}}_{c,n-1}(f)S^{p-1}_{c,n}(f)S^{\frac{1}{2}}_{c,n-1}(f)).$

The second term can be deduced from the nontrivial duality results in Lemma
2.3 for $1<p<\infty$ as follows.

$\displaystyle\begin{split}B^{2}&=\sum_{k}\tau\int
S^{2-p}_{c,k}(f)-S^{2-p}_{c,k-1}(f)\sum_{n\geq
k}\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\\\
&=\sum_{k}\tau\sum_{j}S^{2-p}_{c,k}(f)-S^{2-p}_{c,k-1}(f)\int_{I^{j}_{k}}\sum_{n\geq
k}\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\\\
&=\sum_{k}\tau\sum_{j}\int\mathds{1}_{I^{j}_{k}}(x)S^{2-p}_{c,k}(f)(x)-S^{2-p}_{c,k-1}(f)(x)\frac{1}{|I^{j}_{k}|}\sum_{I\subset
I^{j}_{k}}|\langle\varphi,w_{I}\rangle|^{2}dx\\\ &=\sum_{k}\tau\int
S^{2-p}_{c,k}(f)(x)-S^{2-p}_{c,k-1}(f)(x)\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2}dx\\\
&\leq\|\sum_{k}S^{2-p}_{c,k}(f)-S^{2-p}_{c,k-1}(f)\|_{L_{({p^{\prime}}/2)^{\prime}}}\Big{\|}\sup_{k}\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2}\Big{\|}_{L_{{p^{\prime}}/2}}\\\
&=\|\varphi\|^{2}_{L^{c}_{p^{\prime}}{\mathcal{MO}}}\|f\|^{2-p}_{\mathcal{H}^{c}_{p}}\end{split}$

The fist equality has used the Fubini theorem, the second one the fact that
$S_{c,k-1}(f)$ and $S_{c,k}(f)$ are constant on the dyadic intervals with
length $2^{-k+1}$.

For another direction, we can carry out the proof as that in the case $p=1$.
Suppose that $l$ is a bounded linear functional on
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$. By the embedding operator
$\Phi$ and by Hahn-Banach theorem, and the results in Lemma 2.1, we can find
$g=(g_{I})_{I\in\mathcal{D}}$ such that
$\|g\|_{L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))}=\|l\|$ and

$l(f)=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I},\forall f\in
S_{\mathcal{N}}.$

Now let $\varphi=\Psi(g)$ defined in (2.11), the orthogonality of the
$w_{I}$’s yields

$\displaystyle\begin{split}\big{\|}{\sup_{n}}^{+}&\frac{1}{|I^{x}_{n}|}\sum_{I\subset
I^{x}_{n}}|\langle\varphi,w_{I}\rangle|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&=\big{\|}{\sup_{n}}^{+}\frac{1}{|I^{x}_{n}|}\sum_{I\subset
I^{x}_{n}}|\int\frac{g_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&\leq\big{\|}{\sup_{n}}^{+}\frac{1}{|I^{x}_{n}|}\sum_{I\subset
I^{x}_{n}}\int_{I^{x}_{n}}|g_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&\leq\big{\|}{\sup_{n}}^{+}\frac{1}{|I^{x}_{n}|}\int_{I^{x}_{n}}\sum_{I\subset
I^{x}_{n}}|g_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&\leq\big{\|}{\sup_{n}}^{+}\frac{1}{|I^{x}_{n}|}\int_{I^{x}_{n}}\sum_{I\in\mathcal{D}}|g_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&\leq
c\big{\|}\sum_{I\in\mathcal{D}}|g_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&=c\big{\|}(g_{I})_{I}\big{\|}_{L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))},\\\
\end{split}$

where for the first inequality we have used the Kadison-Schwartz inequality,
and the last inequality is (3.1). Also due to the orthogonality of the
$w_{I}$’s, we get

$\displaystyle\begin{split}l(f)&=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}=\tau\int\varphi^{*}f,\\\
\end{split}$

for all $f\in S_{\mathcal{N}}$. Therefore, we complete the proof about
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ and
$L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$. ∎

Instead of using the noncommutative Doob’s inequality, we will use the
following noncommutative Stein inequality from [20] to prove the duality
between the spaces $\mathcal{H}^{c}_{p}$, $1<p<\infty$.

Let $(\mathcal{E}_{n})_{n}$ be the conditional expectation with respect to a
filtration $({{\mathcal{N}}}_{n})_{n}$ of ${\mathcal{N}}$.

######  Lemma 3.3.

Let $1<p<\infty$ and $a=(a_{n})_{n}\in L_{p}({\mathcal{N}};\ell^{c}_{2})$.
Then there exists a constant depending only on $p$ such that

(3.6)
$\Big{\|}\big{(}\sum_{n}|\mathcal{E}_{n}a_{n}|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}({\mathcal{N}})}\leq
c_{p}\Big{\|}\big{(}\sum_{n}|a_{n}|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}({\mathcal{N}})}.$

######  Theorem 3.3.

For any $1<p<\infty$, we have

(3.7)
$(\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}))^{*}=\mathcal{H}^{c}_{p^{\prime}}({\mathbb{R}},{\mathcal{M}}),$

###### Proof.

By a similar reason as in the corresponding part of the proof of Theorem 3.1,
we can carry out the following calculation,

$\displaystyle\begin{split}|l_{\varphi}(f)|&=|\tau\int\varphi^{*}fdx|\\\
&=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\langle\varphi,w_{I}\rangle^{*}w_{I}\sum_{|I^{\prime}|=2^{-n+1}}\langle
f,w_{I^{\prime}}\rangle w_{I^{\prime}}dx\Big{|}\\\
&=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}dx\Big{|}\\\
&\leq\big{\|}\big{(}\sum_{I\in\mathcal{D}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\big{)}^{\frac{1}{2}}\big{\|}_{L_{p}({\mathbb{R}},{\mathcal{M}})}\cdot\big{\|}\big{(}\sum_{I\in\mathcal{D}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\big{)}^{\frac{1}{2}}\big{\|}_{L_{p^{\prime}}({\mathbb{R}},{\mathcal{M}})}.\\\
\end{split}$

Now, we turn to the proof of the inverse direction. Take a bounded linear
functional $l\in(\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}))^{*}$, by the
embedding operator $\Phi$ and the Hahn-Banach extension theorem, $l$ extends
to a bounded linear functional on $L_{p}({\mathcal{N}};\ell^{c}_{2})$ with the
same norm. Thus by (2.1), there exists a sequence $g=(g_{I})_{I}$ such that

$\|g\|_{L_{q}({\mathcal{N}};l^{c}_{2}(\mathcal{D}))}=\|l\|$

and

$l(f)=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{p}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I},\forall f\in
S_{\mathcal{N}}.$

Now let $\varphi=\Psi(g)$ where $\Psi$ is defined in (2.11), then applying the
Stein inequality (3.3) to the conditional expectation

$\mathcal{E}_{I}(h)=\sum_{J}\frac{1}{|J|}\int_{J}h(y)dy\cdot\mathds{1}_{J},$

where $J$ is dyadic interval with the same length as $I$, we get

$\displaystyle\begin{split}\|\varphi\|_{\mathcal{H}^{c}_{p^{\prime}}({\mathbb{R}},{\mathcal{M}})}&=\|\big{(}\sum_{I\in\mathcal{D}}|\frac{1}{|I|}\int_{I}g_{I}dy\cdot\mathds{1}_{I}|^{2}\big{)}^{\frac{1}{2}}\|_{L_{p^{\prime}}({\mathcal{N}})}\\\
&\leq\|\big{(}\sum_{I\in\mathcal{D}}|\mathcal{E}_{I}(g_{I})|^{2}\big{)}^{\frac{1}{2}}\|_{L_{p^{\prime}}({\mathcal{N}})}\\\
&\leq
c_{p^{\prime}}\|\big{(}\sum_{I\in\mathcal{D}}|g_{I}|^{2}\big{)}^{\frac{1}{2}}\|_{L_{p^{\prime}}({\mathcal{N}})}.\\\
\end{split}$

By the orthogonality of the $w_{I}$’s, we have

$\displaystyle\begin{split}l(f)=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}=\tau\int\varphi^{*}f,\\\
\end{split}$

for all $f\in S_{\mathcal{N}}$. ∎

From the proof of the second part of Theorem 3.1, Theorem 3.2 and Theorem 3.3,
we state the boundedness of $\Psi$ as a corollary.

######  Corollary 3.1.

$\rm(i)$ Let $1<p<\infty$, $\Psi$ is a projection map from
$L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ onto
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ if we identify the latter as
a subspace of the former.

$\rm(ii)$ Let $2<p\leq\infty$, $\Psi$ is also a bounded map from
$L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ to
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$.

Theorem 3.2 and Theorem 3.3 immediately imply the following corollary:

######  Corollary 3.2.

Let $2<p<\infty$. Then

$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})=L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}}),\quad\forall
2<p<\infty$

with equivalent norms.

However, for the part
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})\subset\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$,
we can give another proof. The idea is essentially similar to that in [15],
the good news is that in our case, the argument seems very elegant. Now we
give the detailed proof.

###### Proof.

Our tent space is defined as

$\displaystyle T^{c}_{p}=\Big{\\{}f=\\{f_{I}\\}_{I}\in
L_{p}({\mathcal{M}};\ell^{c}_{2}(\mathcal{D})):\quad\tau\int\big{(}\sum_{I\in\mathcal{D}}\frac{f^{2}_{I}}{|I|}\mathds{1}_{I}\big{)}^{\frac{p}{2}}<\infty\Big{\\}}$

We claim that every $\varphi\in
L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ induces a bounded linear
functional on $T^{c}_{p^{\prime}}$,

$\displaystyle
l_{\varphi}(f)=\tau\int\sum_{I\in\mathcal{D}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\frac{f_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}dx$

and
$\|l_{\varphi}\|\leq\|\varphi\|_{L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})}$.
The proof is just the copy of the proof of the first part in the last theorem.
Now $T^{c}_{p^{\prime}}$ is naturally embedded into
$L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ by
$(f_{I})_{I}\rightarrow(\frac{f_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I})_{I}$.
So by the Hahn-Banach extension theorem, $l_{\varphi}$ extends to an bounded
linear functional on $L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$
with the same norm. Then by the duality between

$(L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D})))^{*}=L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D})).$

there exists a unique $h=(h_{I})_{I}$ such that
$\|h\|_{L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))}\leq\|l_{\varphi}\|$
and for $f=(f_{I})_{I}\in T^{c}_{p^{\prime}}$,

$\displaystyle
l_{\varphi}(f)=\tau\int\sum_{I\in\mathcal{D}}h^{*}_{I}\frac{f_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}dx$

So we get

$\frac{\langle\varphi,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}=h_{I},$

thus,

$\displaystyle\begin{split}\|\varphi\|_{\mathcal{H}^{c}_{p}}&=\Big{\|}\big{(}\sum_{I\in\mathcal{D}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}({\mathcal{N}})}\\\
&=\|h_{I}\|_{L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))}\leq\|l_{\varphi}\|\end{split}$

∎

##  4\. Interpolation

This section is devoted to the interpolation of our wavelet Hardy spaces. The
interpolation results below will be needed in the next section to compare our
Hardy spaces with those of Mei.

######  Lemma 4.1.

Let $1<p_{0}<p<p_{1}<\infty$, we have

(4.1)
$[\mathcal{H}^{c}_{p_{0}}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{p_{1}}({\mathbb{R}},{\mathcal{M}})]_{\theta}=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$

with equivalent norms, where $\theta$ satisfies
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$.

###### Proof.

The embedding map $\Phi$ yields

$[\mathcal{H}^{c}_{p_{0}},\mathcal{H}^{c}_{p_{1}}]_{\theta}\subset\mathcal{H}^{c}_{p}.$

On the other hand, it is the boundedness of the projection map $\Psi$ from
$L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ to
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ stated in Corollary 3.1 that
yields the inverse direction. ∎

######  Theorem 4.1.

Let $1\leq q<p<\infty$, we have

(4.2)
$[\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{q}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$

with equivalent norms.

###### Proof.

We will prove the theorem by a general strategy as appeared in [18].

Step 1: We prove the conclusion for $2<q<p<\infty$:

(4.3)
$[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{q}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}).$

The identity can be seen easily from the following two inclusions. On one
hand, the operator $\Phi$ which in (2.10), together with (2.2) yields

$[\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{q^{\prime}}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}\subset\mathcal{H}^{c}_{p^{\prime}}({\mathbb{R}},{\mathcal{M}}).$

Then by duality and Corollary 3.2, we have

(4.4)
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})\subset[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),L^{c}_{q}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}.$

On the other hand, the operator $\mathcal{T}$ identifying
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ as a subspace of
$L_{p}(L_{\infty}(\mathcal{N}\bar{\otimes}B(\ell_{2}(\mathcal{D}));\ell^{c}_{\infty})$
defined by

(4.5) $\mathcal{T}(\varphi)={\langle
f,w_{I}\rangle}{|I^{t}_{k}|^{-\frac{1}{2}}}\mathds{1}_{I\subset
I^{t}_{k}}(I)\otimes e_{I,1},$

together with Lemma 2.4 yields

(4.6)
$[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),L^{c}_{q}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}\subset
L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}}).$

Step 2: we prove the conclusion for $1<q<p<\infty$. This step can be divided
into two substeps.

Substep 21: $p>2$. Let $p<s<\infty$. By Step 1, we have

$[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})]_{\frac{p}{s}}=\mathcal{H}^{c}_{s}({\mathbb{R}},{\mathcal{M}}).$

On the other hand, by Theorem 4.1, we have

$[\mathcal{H}^{c}_{q},\mathcal{H}^{c}_{s}]_{\theta}=\mathcal{H}^{c}_{p},$

where(and in the rest of the paper) $\theta$ denote the interpolation
parameter. Then Wolff’s interpolation theorem yields the result.

Substep 22: $p\leq 2$. Let $s>2$, then by Substep 21, we have

$[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})]_{\frac{p}{s}}=\mathcal{H}^{c}_{s}({\mathbb{R}},{\mathcal{M}}).$

Then together with Lemma 4.1, Wolff’s interpolation theorem yields the result.

Step 3: we prove the conclusion for $1=q<p<\infty$. Take $s>\max(p,2)$. By
Step 2 and duality [2, Theorem 4.3.1], we get

$[\mathcal{H}^{c}_{1},\mathcal{H}^{c}_{s}]_{\theta}=\mathcal{H}^{c}_{p}.$

Then together with Step 2, Wolff’s interpolation yields the conclusion. ∎

######  Remark 4.1.

If one can directly prove Lemma 4.1 for $p_{0}=1$, we can prove the above
theorem without the help of
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ for $2<p<\infty$ as
carried out in [1], where one needs an auxiliary space.

######  Theorem 4.2.

For $1<p<\infty$, we have

$\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})=L_{p}({\mathcal{N}})$

with equivalent norms.

###### Proof.

There are several ways to prove this result. One can prove it by the strategy
in [20] together with Stein’s inequality (3.3). Here, we just use the fact
that $L_{p}({\mathcal{M}})$ with $1<p<\infty$ is a UMD space and our
$(w_{I})_{I}$ is an complete orthonormal basis. So by Theorem 3.8 in [7], we
have

$\|f\|_{L_{p}({\mathcal{N}})}\simeq\Big{(}\mathbb{E}\Big{\|}\sum_{I\in\mathcal{D}}\varepsilon_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\Big{\|}^{p}_{L_{p}({\mathcal{N}})}\Big{)}^{\frac{1}{p}}.$

Then we complete the proof for $2\leq p<\infty$ by Khintchine’s inequalities.
Now, let us prove the case $1<p<2$. Let
$f\in\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})$, then for any $\epsilon>0$,
by the definition of $\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})$, there
exists a decomposition $f=f_{c}+f_{r}$ such that

$\|f_{c}\|_{\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})}+\|f_{r}\|_{\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})}\leq\|f\|_{\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})}+\epsilon.$

Take any $g\in L_{p^{\prime}}(\mathcal{N})$, by the results for
$p^{\prime}>2$, the operator-valued Calderón identity (2.5) yields

$\displaystyle|\tau\int gf^{*}|$
$\displaystyle=|\sum_{I\in\mathcal{D}}\tau\int\frac{\langle
g,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\cdot\frac{\langle
f^{*},w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|$
$\displaystyle\leq|\sum_{I\in\mathcal{D}}\tau\int\frac{\langle
g,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\cdot\frac{\langle
f_{c}^{*},w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|$
$\displaystyle\qquad+|\sum_{I\in\mathcal{D}}\tau\int\frac{\langle
g,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\cdot\frac{\langle
f_{r}^{*},w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|$
$\displaystyle\leq\|S_{c}(g)\|_{L_{p^{\prime}}(\mathcal{N})}\|S_{c}(f_{c})\|_{L_{p}(\mathcal{N})}+|S_{r}(g)\|_{L_{p^{\prime}}(\mathcal{N})}\|S_{r}(f_{r})\|_{L_{p}(\mathcal{N})}$
$\displaystyle\leq
c_{p^{\prime}}\|g\|_{L_{p^{\prime}}}(\|f\|_{\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})}+\epsilon).$

Taking $\sup$ and let $\epsilon\rightarrow 0$, we get the required result.

Finally, we prove the inverse inequality. Let $f\in L_{p}(\mathcal{N})$, by
duality, we can find two sequences of functions $(F_{c,I})_{I}\in
L_{p}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))$ and $(F_{r,I})_{I}\in
L_{p}(\mathcal{N};\ell^{r}_{2}(\mathcal{D}))$ such that
$F_{c,I}+F_{r,I}=\langle f,w_{I}\rangle|I|^{-\frac{1}{2}}\mathds{1}_{I}$ and

$\|(F_{c,I})_{I}\|_{L_{p}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))}+\|(F_{r,I})_{I}\|_{L_{p}(\mathcal{N};\ell^{r}_{2}(\mathcal{D}))}\leq\|f\|_{L_{p}(\mathcal{N})}.$

Let $f_{c}=\Psi(({F_{c,I}})_{I})$ and $f_{r}=\Psi(({F_{r,I}})_{I})$, by
identity (2.5), we have $f=f_{c}+f_{r}$. On the other hand, by the Stein
inequality (3.3), we have
$\|f_{c}\|_{\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})}\leq\|(F_{c,I})_{I}\|_{L_{p}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))}$
and
$\|f_{r}\|_{\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})}\leq\|(F_{r,I})_{I}\|_{L_{p}(\mathcal{N};\ell^{r}_{2}(\mathcal{D}))}$.
So we have found the desired decomposition of $f$. ∎

######  Theorem 4.3.

The following results hold with equivalent norms:

$\rm(i)$ Let $1\leq q<p<\infty$, we have

(4.7)
$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{q}({\mathcal{N}})]_{\frac{q}{p}}=L_{p}({\mathcal{N}}).$

$\rm(ii)$ Let $1<q<p\leq\infty$, we have

(4.8)
$[\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}}),L_{p}({\mathcal{N}})]_{\frac{p^{\prime}}{q^{\prime}}}=L_{q}({\mathcal{N}}).$

$\rm(iii)$ Let $1<p<\infty$, we have

(4.9)
$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})]_{\frac{1}{p}}=L_{p}({\mathcal{N}}).$

In order to prove this theorem, we need the following result from the theory
of interpolation. We formulate it here without proof.

######  Lemma 4.2.

Let $A_{0},B_{0},A_{1},B_{1}$ be four Banach spaces satisfying the property
needed for interpolation. Then

$[A_{0}+B_{0},A_{1}+B_{1}]_{\theta}\supset[A_{0},A_{1}]_{\theta}+[B_{0},B_{1}]_{\theta}$

and

$[A_{0}\cap B_{0},A_{1}\cap
B_{1}]_{\theta}\subset[A_{0},A_{1}]_{\theta}\cap[B_{0},B_{1}]_{\theta}.$

###### Proof.

$\rm(i)$ We also exploit the similar but different strategy with that in the
proof of Theorem 4.1.

Step 1: we prove the results for $2\leq q<p<\infty$. By Theorem 4.2, Theorem
4.1 and the lemma, we have

$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{q}({\mathcal{N}})]_{\frac{q}{p}}\subset
L_{p}({\mathcal{N}}).$

The inverse direction follows from
$L_{\infty}({\mathcal{N}})\subset\mathcal{BMO}({\mathbb{R}},{\mathcal{M}})$,

$\displaystyle\begin{split}L_{p}({\mathcal{N}})&=[L_{\infty}({\mathcal{N}}),L_{q}({\mathcal{N}})]_{\frac{q}{p}}\\\
&\subset[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{q}({\mathcal{N}})]_{\frac{q}{p}}\\\
\end{split}$

Step 2: we prove the results for $1\leq q<2\leq p<\infty$. By Step 1, we have

$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{2}({\mathcal{N}})]_{\frac{2}{p}}=L_{p}({\mathcal{N}}).$

Together with

$L_{2}({\mathcal{N}})=[L_{p}({\mathcal{N}}),L_{q}({\mathcal{N}})]_{\theta},$

Wolff’s interpolation yields the conclusion.

Step 3: we prove the results for $1\leq q<p<2$. By Step 2, we have

$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{p}({\mathcal{N}})]_{\frac{p}{2}}=L_{2}({\mathcal{N}}).$

Together with

$L_{p}({\mathcal{N}})=[L_{2}({\mathcal{N}}),L_{q}({\mathcal{N}})]_{\theta},$

Wolff’s interpolation yields the conclusion.

$\rm(ii)$ The results for $1<q<p<\infty$ can be immediately proved by duality
and the partial results in $(i)$. For $p=\infty$, take $q<s<\infty$, then by
Wolff’s argument, we get the conclusion.

$\rm(iii)$ First, we prove conclusion for $p<2$. Then by $\rm(i)$ and
$\rm(ii)$, we have

$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{p}({\mathcal{N}})]_{\frac{p}{p^{\prime}}}=L_{p^{\prime}}({\mathcal{N}})$

and

$[\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}}),L_{p^{\prime}}({\mathcal{N}})]_{\frac{p}{p^{\prime}}}=L_{p}({\mathcal{N}}).$

Therefore, we end with Wolff’s argument. Second, the proof for $p>2$ is the
same. At last, when $p=2$, we can take $s>2$, by the results for $p\neq 2$ and
reiteration theorem in [2, Theorem 4.6.1], we get

$\displaystyle\begin{split}L_{2}&=[L_{s},L_{s^{\prime}}]_{\theta}=[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})]_{\frac{1}{s}},\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})]_{\frac{1}{s^{\prime}}}]_{\theta}\\\
&=[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})]_{\theta}.\\\
\end{split}$

∎

##  5\. Comparison with Mei’s results

We denote the column Hardy space by $H^{c}_{p}({\mathbb{R}},{\mathcal{M}})$
and the bounded mean oscillation space by
$BMO^{c}({\mathbb{R}},{\mathcal{M}})$ in [15]. We have the following result.

######  Theorem 5.1.

We have

${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})=BMO^{c}({\mathbb{R}},{\mathcal{M}})$

with equivalent norms. Similar results holds for the row spaces. Consequently,
${\mathcal{BMO}}({\mathbb{R}},{\mathcal{M}})={\mathcal{BMO}}({\mathbb{R}},{\mathcal{M}})$
with equivalent norms.

The theorem can be easily seen from the corresponding
$BMO({\mathbb{R}},H)$-spaces. However, we can exploit the idea of [7] to prove
our ${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})$ also coincide with that
defined by the mean oscillation $BMO({\mathbb{R}},H)$.

###### Proof.

${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})\subset
BMO^{c}({\mathbb{R}},{\mathcal{M}}).$ Let
$\varphi\in\mathcal{BMO}_{c}(\mathbb{R},\mathcal{M})$. As in [7], fix a finite
interval $I\subset\mathbb{R}$, and consider the collections of dyadic
intervals

  1. (1)

$\mathcal{D}_{1}:=\\{J\in\mathcal{D};2|J|>|I|\\}$’

  2. (2)

$\mathcal{D}_{2}:=\\{J\in\mathcal{D};2|J|\leq|I|,2J\cap 2I=\emptyset\\}$,

  3. (3)

$\mathcal{D}_{3}:=\\{J\in\mathcal{D};2|J|\leq|I|,2J\cap 2I\neq\emptyset\\}$.

Let $a_{J}=\langle\varphi,\omega_{J}\rangle$, then we have a priori formal
series

$\varphi_{1}(x)=\sum_{J\in\mathcal{D}_{1}}a_{J}[\omega_{J}(x)-\omega_{J}(c_{I})],\varphi_{i}(x)=\sum_{J\in\mathcal{D}_{i}}a_{J}\omega_{J}(x),i=2,3,$

where $c_{I}$ is the center of the interval $I$. Denote
$\varphi_{I}=\varphi_{1}+\varphi_{2}+\varphi_{3}$, by a similar discussion in
[7], we only need to prove:

$\|\frac{1}{|I|}\int_{I}|\varphi_{I}(x)|^{2}dx\|_{\mathcal{M}}<\infty.$

By scaling we can assume:

$\sup_{I}\frac{1}{|I|}\|\sum_{J\subset I}|a_{J}|^{2}\|=1.$

Then we have the obvious bound for individual terms
$\|a_{J}\|\leq|J|^{\frac{1}{2}}$.

Estimates for $\varphi_{1}$:

$\begin{split}\|\frac{1}{|I|}\int_{I}|\varphi_{1}(x)|^{2}dx\|&\leq\frac{1}{|I|}(\sum_{J\in\mathcal{D}_{1}}\|a_{J}\||\omega_{J}(x)-\omega_{J}(c_{I})|)^{2}dx\\\
&\leq
c\frac{1}{|I|}\int_{I}[\sum_{J\in\mathcal{D}_{1}}|J|^{\frac{1}{2}}|I||J|^{-\frac{3}{2}}(1+\frac{dist(I,J)}{|J|})^{-2}]^{2}dx\\\
&=c[\sum_{j=0}^{\infty}\sum_{|J|\in(2^{j-1},2^{j}]|I|}|I||J|^{-1}(1+\frac{dist(I,J)}{|J|})^{-2}]^{2}<\infty.\end{split}$

Estimates for $\varphi_{2}$:

$\begin{split}\|\frac{1}{|I|}\int_{I}|\varphi_{2}(x)|^{2}dx\|&\leq\frac{1}{|I|}\int_{I}\|\sum_{\mathcal{D}_{2}}a_{J}\omega_{J}(x)\|^{2}dx\\\
&\leq\frac{1}{|I|}\int_{I}(\sum_{\mathcal{D}_{2}}\|a_{J}\||\omega_{J}(x)|)^{2}dx\\\
&\leq
c\frac{1}{|I|}\int_{I}[\sum_{\mathcal{D}_{2}}|J|^{\frac{1}{2}}|J|^{-\frac{1}{2}}(\frac{dist(I,J)}{|J|})^{-2}]^{2}dx\\\
&=c[\sum_{j=1}^{\infty}\sum_{|J|\in(2^{-j-1},2^{-j})|I|,dist(I,J)>2^{-1}|I|}(\frac{dist(I,J)}{|J|})^{-2}]^{2}<\infty.\end{split}$

Estimates for $\varphi_{3}$:

$\begin{split}\|\frac{1}{|I|}\int_{I}|\varphi_{3}(x)|^{2}dx\|&\leq\frac{1}{|I|}\|\sum_{J\in\mathcal{D}_{3}}|a_{J}|^{2}\|\leq\frac{1}{|I|}\|\sum_{J\subset
4I}|a_{J}|^{2}\|<\infty\end{split}$

Hence we deduce that:

$\|\int_{I}|\varphi_{I}(x)|^{2}dx\|_{\mathcal{M}}\leq
c\sum_{i=1}^{3}\|\int_{I}|\varphi_{i}(x)|^{2}dx\|_{\mathcal{M}}\leq c|I|$

Now we turn to the proof of inverse direction
$BMO^{c}({\mathbb{R}},{\mathcal{M}})\subset{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}).$
Let $\varphi\in BMO^{c}(\mathbb{R},\mathcal{M})$. The proof is very similar to
that in Mei’s work [15]. For any dyadic interval $I\subset\mathbb{R}$, write
$\varphi=\varphi_{1}+\varphi_{2}+\varphi_{3}$, where
$\varphi_{1}=(\varphi-\varphi_{2I})\chi_{2I},\varphi_{2}=(\varphi-\varphi_{2I})\chi_{2I^{c}},\varphi_{3}=\varphi_{2I}$.

Thus

$\begin{split}\sum_{J\subset I}|\langle\varphi,\omega_{J}\rangle|^{2}\leq
2(\sum_{J\subset I}|\langle\varphi_{1},\omega_{J}\rangle|^{2}+\sum_{J\subset
I}|\langle\varphi_{2},\omega_{J}\rangle|^{2})\end{split}$

Estimates for $\varphi_{1}$:

$\begin{split}\|\sum_{J\subset
I}|\langle\varphi_{1},\omega_{J}\rangle|^{2}\|\leq\|\int|\varphi_{1}(x)|^{2}dx\|\leq
c\|\int_{2I}|\varphi-\varphi_{2I}|^{2}\|\leq c|I|\end{split}$

Estimates for $\varphi_{2}$:

$\begin{split}\|\sum_{J\subset
I}|\langle\varphi_{2},\omega_{J}\rangle|^{2}\|&=\|\sum_{J\subset
I}|\sum_{k=1}^{\infty}\int_{2^{k+1}I/2^{k}I}\varphi_{2}\omega_{J}dx|^{2}\|\\\
&\leq\|\sum_{J\subset
I}(\sum_{k=1}^{\infty}\frac{1}{2^{2k}}\int_{2^{k+1}I/2^{k}I}|\varphi_{2}|^{2})(\sum_{k=1}^{\infty}2^{2k}\int_{2^{k+1}I/2^{k}I}|\omega_{J}|^{2})\|\\\
&\leq
c(\sum_{k=1}^{\infty}\frac{1}{2^{2k}}\|\int_{2^{k+1}I}|\varphi-\varphi_{2I}|^{2}\|)\\\
&\qquad\qquad(\sum_{J\subset
I}\sum_{k=1}^{\infty}2^{2k}\int_{2^{k+1}I/2^{k}I}|\omega_{J}|^{2})\\\ &\leq
c|I|\|\varphi\|^{2}_{{\mathcal{BMO}}_{c}}\sum_{j=0}^{\infty}2^{j}\sum_{k=1}^{\infty}\int_{2^{k+1}I/2^{k}I}2^{2k}\frac{|2^{-j}I|^{3}}{|2^{k}I|^{4}}\\\
&\leq c|I|\end{split}$

Therefore $\|\sum_{J\subset I}|\langle\varphi,\omega_{J}\rangle|^{2}\|\leq
c|I|$, which completes our proof. ∎

Combined with Theorem 3.2 and Theorem 4.1, we have the following corollary

######  Corollary 5.1.

For $1\leq p<\infty$, we have

$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})=H^{c}_{p}({\mathbb{R}},{\mathcal{M}}).$

Similar results hold for $\mathcal{H}^{r}_{p}$ and ${H}^{r}_{p}$, and
$\mathcal{H}_{p}$ and ${H}_{p}$.

If ${\mathcal{M}}=\mathbb{C},$ $\mathcal{H}_{1}({\mathbb{R}},\mathbb{C})$ is
just the usual Hardy space $H_{1}({\mathbb{R}})$ on $\mathbb{R}.$
$H_{1}({\mathbb{R}})$ also has the following characterization:

$H_{1}({\mathbb{R}})=\\{f\in L_{1}({\mathbb{R}}):H(f)\in
L_{1}({\mathbb{R}})\\},$

where $H$ is the Hilbert transform. For any $f\in H_{1}({\mathbb{R}})$,

$\|f\|_{H_{1}({\mathbb{R}})}\approx\|f\|_{L_{1}({\mathbb{R}})}+\|H(f)\|_{L_{1}({\mathbb{R}})}.$

Thus $H_{1}({\mathbb{R}})$ can be viewed as a subspace of
$L_{1}({\mathbb{R}})\oplus_{1}L_{1}({\mathbb{R}})$. The latter direct sum has
its natural operator structure as an $L_{1}$ space. This induces an operator
space structure on $H_{1}({\mathbb{R}}).$ Although $(w_{I})_{I\in\mathcal{D}}$
is a unconditional basis of $H_{1}({\mathbb{R}})$, Ricard [22] (see also [23])
proved that $H_{1}({\mathbb{R}})$ does not have complete unconditional basis.
However, in noncommutative analysis, one can introduce another natural
operator space structure on $H_{1}({\mathbb{R}})$ as follows:
$S_{1}(H_{1}({\mathbb{R}}))=\mathcal{H}_{1}({\mathbb{R}},B(\ell_{2})),$ where
$S_{1}$ is the trace class on $\ell_{2}.$ Then we have the following result.
Note that Ricard [23] obtained a similar result using Hilbert space
techniques.

######  Corollary 5.2.

The complete orthogonal systems $(w_{I})_{I\in\mathcal{D}}$ of
$L_{2}({\mathbb{R}})$ is a completely unconditional basis for
$H_{1}({\mathbb{R}})$ if we define the operator space structure imposed on
$H_{1}({\mathbb{R}})$ by
$\mathcal{S}_{1}(H_{1}({\mathbb{R}}))=\mathcal{H}_{1}({\mathbb{R}},B(\ell_{2}))$.

###### Proof.

Fix a finite subset $\mathcal{I}\subset\mathcal{D}$. Let
$T_{\varepsilon}f\doteq\sum_{I\in\mathcal{I}}\varepsilon_{I}\langle
f,w_{I}\rangle w_{I}$, where $\varepsilon_{I}=\pm 1$. By the definition of
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$, the orthogonality of
$(w_{I})_{I\in\mathcal{D}}$ yields immediately that

$\displaystyle\begin{split}\|T_{\varepsilon}f\|_{\mathcal{H}^{c}_{1}}&=\Big{\|}\Big{(}\sum_{I\in\mathcal{I}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}\Big{\|}_{{L_{1}({\mathcal{N}})}}\\\
&\leq\Big{\|}\Big{(}\sum_{I\in\mathcal{D}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}\Big{\|}_{{L_{1}({\mathcal{N}})}}=\|f\|_{\mathcal{H}^{c}_{1}}\\\
\end{split}$

Similarly, the above inequality holds for
$\mathcal{H}^{r}_{1}({\mathbb{R}},{\mathcal{M}})$. Now, let
$f\in\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})$, then for any $\epsilon>0$,
there exists a decomposition $f=g+h$ such that

$\|g\|_{\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})}+\|h\|_{\mathcal{H}^{r}_{1}({\mathbb{R}},{\mathcal{M}})}\leq\|f\|_{\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})}+\epsilon.$

Therefore

$\displaystyle\begin{split}\|T_{\varepsilon}f\|_{\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})}&\leq\|T_{\varepsilon}g\|_{\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})}+\|T_{\varepsilon}h\|_{\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})}\\\
&\leq\|g\|_{\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})}+\|h\|_{\mathcal{H}^{r}_{1}({\mathbb{R}},{\mathcal{M}})}\leq\|f\|_{\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})}+\epsilon.\\\
\end{split}$

Let $\epsilon\rightarrow 0$, we get the result. ∎

## References

  * [1] T. Bekjan, Z. Chen, M. Perrin, Z. Yin, Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales. J. Funct. Analysis., 258(7):2483-2505, 2010.
  * [2] J. Bergh, J. Löfström, Interpolation Spaces. An Introduction, Springer, New York, 1976.
  * [3] I. Cuculescu, Martingales on von Neumann algebras. J. Multiv. Anal., 1:17-27, 1971.
  * [4] A. Defant, M. Junge, Maximal theorems of Menchoff-Rademacher type in non-commutative $L_{q}$-spaces. J. Funct. Analysis., 206:322-355, 2004.
  * [5] S.H. Ferguson and M.T. Lacey, A characterization of product BMO by commutators, Acta Math. 189(2):143-160, 2002.
  * [6] C. Fefferman, E.M. Stein, Hp spaces of several variables, Acta Math., 129:137-193, 1972.
  * [7] T. Hytönen, O. Salinas, and B. Viviani, Wavelet expansions for weighted vector-valued BMO functions, arXiv:0901.1577v1 [math.FA] 12 Jan 2009.
  * [8] M. Junge, Doob’s Inequality for Non-commutative Martingales, J. Reine Angew. Math. 549:149-190, 2002.
  * [9] M. Junge, C. Le Merdy and Q. Xu, $H^{\infty}$ functional calculus and square functions on noncommutative Lp-spaces. Asterisque 305, 2006\.
  * [10] M. Junge, M. Musat, Non-commutative John-Nirenberg theorem, Trans. Amer. Math. Soc. 359(1):115-142, 2007.
  * [11] M. Junge, Q. Xu, Non-commutative Burkholder/Rosenthal Inequalities, Ann. Prob. 31(2):948-995, 2003.
  * [12] M. Junge and Q. Xu, On the best constants in some non-commutative martingale inequalities, Bull. London Math. Soc. 37:243 C253, 2005.
  * [13] M. Junge and Q. Xu, Noncommutative maximal ergodic theorems, J.Amer.Math.Soc. 20:385-439, 2006.
  * [14] M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities II: applications, Israel J. Math. 167:227-282, 2008.
  * [15] T. Mei, Operator valued Hardy spaces, Mem. Amer. Math. Soc. 188, 2007.
  * [16] Y. Meyer, Wavelets and Operators. Translated from the 1990 French original by D. H. Salinger. Cambridge Stud. Adv. Math., 37. Cambridge Univ. Press, Cambridge, 1992.
  * [17] C. Mucalu, J. Pipher, T. Tao, and C. Thiele, Bi-parameter paraproducts, Acta Math. 193:269-296, 2004.
  * [18] M. Musat, Interpolation Between Non-commutative BMO and Non-commutative $Lp$-spaces, J. Funct. Analysis., 202(1):195-225, 2003\.
  * [19] G. Pisier, Non-commutative Vector Valued Lp-Spaces and Completely p-Summing Maps, Soc. Math. France. Astérisque 237, 1998\.
  * [20] G. Pisier, Q. Xu, Non-commutative Martingale Inequalities, Comm. Math. Phys. 189:667-698, 1997.
  * [21] G. Pisier, and Q. Xu, _Non-commutative $L_{p}$-spaces,_ pp. 1459-1517 in ”Handbook of the Geometry of Banach Spaces”, Vol. II, edited by W.B. Johnson and J. Lindenstrauss, Elsevier, 2003.
  * [22] E. Ricard, L’espace $H^{1}$ n’a pas de base complètement inconditionnelle, C. R. Acad. Sci. Paris 331 (2000), 625-628.
  * [23] E. Ricard, Décomposition de $H^{1}$, Multiplicateurs de Schur et Espaces d’Operateurs, Thèse de Doctorat de l’Université Paris VI. 2001.
  * [24] T. Wolff, A note on Interpolation Spaces, Harmonic Analysis, Lec. Notes in Math., 1568, Spring-Verlag, Berlin-Heidelberg-New York, 1994.