# An Investigation of New Brown Dwarf Spectral Binary Candidates From the
Backyard Worlds: Planet 9 Citizen Science Initiative

Alexia Bravo United States Naval Observatory, Flagstaff Station, 10391 West
Naval Observatory Rd., Flagstaff, AZ 86005, USA  Adam C. Schneider United
States Naval Observatory, Flagstaff Station, 10391 West Naval Observatory Rd.,
Flagstaff, AZ 86005, USA  Daniella Bardalez Gagliuffi Department of Physics &
Astronomy, Amherst College, 25 East Drive, Amherst, MA 01003, USA  Adam J.
Burgasser Department of Astronomy & Astrophysics, University of California San
Diego, La Jolla, CA 92093, USA  Aaron M. Meisner NSF’s National Optical-
Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719,
USA  J. Davy Kirkpatrick IPAC, Mail Code 100-22, Caltech, 1200 E. California
Blvd., Pasadena, CA 91125, USA  Jacqueline K. Faherty Department of
Astrophysics, American Museum of Natural History, Central Park West at 79th
St., New York, NY 10024, USA  Marc J. Kuchner Exoplanets and Stellar
Astrophysics Laboratory, NASA Goddard Space Flight Center, 8800 Greenbelt
Road, Greenbelt, MD 20771, USA  Dan Caselden Department of Astrophysics,
American Museum of Natural History, Central Park West at 79th St., New York,
NY 10024, USA  Arttu Sainio Backyard Worlds: Planet 9, USA  Les Hamlet
Backyard Worlds: Planet 9, USA  The Backyard Worlds: Planet 9 Collaboration

###### Abstract

We present three new brown dwarf spectral binary candidates: CWISE
J072708.09$-$360729.2, CWISE J103604.84$-$514424.4, and CWISE
J134446.62$-$732053.9, discovered by citizen scientists through the Backyard
Worlds: Planet 9 project. Follow-up near-infrared spectroscopy shows that each
of these objects is poorly fit by a single near-infrared standard. We
constructed binary templates and found significantly better fits, with
component types of L7+T4 for CWISE J072708.09$-$360729.2, L7+T4 for CWISE
J103604.84$-$514424.4, and L7+T7 for CWISE J134446.62$-$732053.9. However,
further investigation of available spectroscopic indices for evidence of
binarity and large amplitude variability suggests that CWISE
J072708.09$-$360729.2 may instead be a strong variability candidate. Our
analysis offers tentative evidence and characterization of these peculiar
brown dwarf sources, emphasizing their value as promising targets for future
high-resolution imaging or photometric variability studies.

††software: SPLAT (Burgasser & Splat Development Team, 2017), Spextool (Vacca
et al., 2003; Cushing et al., 2004)††facilities: SOAR, WISE

##  1 Introduction

Brown dwarfs occupy a unique space between stars and planets, possessing
masses below the threshold required for sustained hydrogen fusion in their
cores (Kumar, 1962, 1963; Hayashi & Nakano, 1963). Their cool temperatures and
intrinsic faintness make them difficult to detect, and nearby brown dwarfs are
continuing to be discovered (e.g., Marocco et al. 2019; Best et al. 2020;
Meisner et al. 2020a, b; Bardalez Gagliuffi et al. 2020; Kirkpatrick et al.
2021a, b; Schneider et al. 2021; Kota et al. 2022; Lodieu et al. 2022;
Schapera et al. 2022). The Backyard Worlds: Planet 9 project (Kuchner et al.,
2017) leverages the collaboration between citizen and professional scientists
to locate nearby substellar objects by identifying and analyzing moving
objects in images from the Wide-field Infrared Survey Explorer (Wright et al.,
2010; Mainzer et al., 2014). Multi-epoch WISE images and motion measurements
from WISE data (CatWISE 2020; Marocco et al. 2021) have allowed for numerous
discoveries missed by previous surveys, many of which were based solely on
infrared colors or limited to nearby objects with very large proper motions.
In addition to finding new nearby brown dwarfs, the Backyard Worlds project
has been adept at discovering unusual substellar objects, such as old, low-
metallicity subdwarfs (Schneider et al., 2020; Meisner et al., 2021; Brooks et
al., 2022) and co-moving companions (Faherty et al., 2020; Rothermich et al.,
2021; Jalowiczor et al., 2021; Faherty et al., 2021; Kiwy et al., 2021;
Softich et al., 2022; Kiwy et al., 2022; Gramaize et al., 2022). In this
study, we examine three spectrally peculiar brown dwarfs found in the Backyard
Worlds program: CWISE J072708.09$-$360729.2 (W0727$-$3607), CWISE
J103604.84$-$514424.4 (W1036$-$5144), and CWISE J134446.62$-$732053.9
(W1344$-$7320). These objects were discovered by citizen scientists Dan
Caselden, Arttu Sainio, and Les Hamlet. Each were flagged as high-priority for
follow-up observations because their photometry and estimated spectral types
indicated that they may be nearby ($d<$ 25 pc). Properties of these three
sources are provided in Table 1.

Table 1: Candidate Spectral Binary PropertiesaaAll near-infrared photometry comes from the Vista Hemisphere Survey (VHS; McMahon et al. 2013), with the exception of the $J$-band magnitude of CWISE J134446.62$-$732053.9, which comes from 2MASS (Skrutskie et al., 2006). All proper motions listed are taken from the CatWISE 2020 catalog (Marocco et al., 2021). CatWISE Name | $J$ | $K_{\rm S}$ | $\mu_{\alpha}$ | $\mu_{\delta}$ | SpT | SpT
---|---|---|---|---|---|---
| (mag) | (mag) | (mas yr-1) | (mas yr-1) | (single) | (binary)
CWISE J072708.09$-$360729.2 | 16.521$\pm$0.017 | 15.344$\pm$0.026 | -54.0$\pm$8.3 | 233.6$\pm$9.3 | T4 | L7+T4
CWISE J103604.84$-$514424.4 | 15.977$\pm$0.012 | 14.776$\pm$0.015 | -108.1$\pm$6.8 | -62.7$\pm$8.0 | T2 | L7+T4
CWISE J134446.62$-$732053.9 | 15.890$\pm$0.087 | 14.185$\pm$0.010 | -34.3$\pm$5.4 | 143.3$\pm$6.7 | T4 | L7+T7

##  2 Observations

W0727$-$3607, W1036$-$5144, and W1344$-$7320 were observed with the
TripleSpec4.1 near-infrared spectrograph (Schlawin et al., 2014; Herter et
al., 2020) located at the Southern Astrophysical Research (SOAR) telescope.
TripleSpec4.1 simultaneously covers 0.8$-$2.4 $\mu$m using six cross-dispersed
orders with a resolving power of $\sim$3500\. The observations were conducted
in AEON queue mode and took place on 16 April 2023 (UT), 10 February 2023
(UT), and 14 March 2023 (UT) for W0727$-$3607, W1036$-$5144, and W1344$-$7320,
respectively. The spectra were obtained in an ABBA pattern with 120 second
exposures. The total integration times for W0727$-$3607, W1036$-$5144, and
W1344$-$7320 were 2880, 2400, and 1920 seconds, respectively. A0 stars were
observed immediately after each science target for telluric correction and
flux calibration. Data reduction and telluric correction was performed using a
modified version of the Spextool data reduction package (Vacca et al., 2003;
Cushing et al., 2004). The final reduced spectra are shown in Figure 1.

Table 2: Properties of Objects Used to Create Binary Templates

SpT | Disc. | Disc. | Spec. | $J$ | Ref | $\varpi$ | Ref
---|---|---|---|---|---|---|---
| Name | Ref | Ref | (mag) |  | (mas) |
L0 | 2MASP J0345432$+$254023 | 1 | 15 | 13.924$\pm$0.003 | 23 | 37.89$\pm$0.26 | 27
L1 | 2MASS J21304464$-$0845205 | 2 | 16 | 14.059$\pm$0.002 | 24 | 37.47$\pm$0.31 | 27
L2 | 2MASS J04082905$-$1450334aaThis object replaces the L2 near-infrared standard (Kelu-1), which is a resolved binary (Liu & Leggett, 2005). | 3 | 16 | 14.128$\pm$0.002 | 24 | 45.57$\pm$0.27 | 27
L3 | 2MASSW J1506544$+$132106 | 4 | 17 | 13.211$\pm$0.002 | 25 | 85.43$\pm$0.19 | 27
L4 | 2MASS J21580457$-$1550098 | 2 | 16 | 14.794$\pm$0.004 | 24 | 43.11$\pm$0.91 | 27
L5 | 2MASS J21373742$+$0808463bbThis object replaces the L5 near-infrared standard (2MASS J08350622$+$1953050), which has no measured parallax. | 5 | 18 | 14.644$\pm$0.005 | 25 | 66.37$\pm$0.66 | 27
L6 | 2MASSI J1010148$-$040649 | 6 | 19 | 15.372$\pm$0.005 | 24 | 57.7$\pm$3.6 | 28
L7 | 2MASSI J0825196$+$211552 | 7 | 18 | 15.014$\pm$0.005 | 25 | 93.19$\pm$0.59 | 29
L8 | 2MASSW J1632291$+$190441 | 8 | 17 | 15.823$\pm$0.010 | 25 | 66.29$\pm$1.61 | 29
L9 | DENIS J025503.3$-$470049 | 9 | 20 | 13.122$\pm$0.001 | 24 | 205.43$\pm$0.19 | 27
T0 | WISEPA J015010.86$+$382724.3ccThis object replaces the T0 near-infrared standard (2MASS J12074717$+$0244249), which is a suspected spectral binary (Burgasser et al., 2010). | 10 | 10 | 15.901$\pm$0.010 | 25 | 44.6$\pm$3.2 | 28
T1 | SDSS J083717.21$-$000018.0 | 11 | 20 | 16.929$\pm$0.007 | 26 | 29.8$\pm$2.7 | 30
T2 | SDSS J125453.90$-$012247.5 | 11 | 21 | 14.694$\pm$0.002 | 26 | 78.34$\pm$1.07 | 29
T3 | WISEPC J223937.55$+$161716.2ddThis object replaces the T3 near-infrared standard (2MASS J12095613$-$1004008), which is a resolved binary (Liu et al., 2010) | 10 | 10 | 15.995$\pm$0.010 | 25 | 42.9$\pm$3.0 | 30
T4 | 2MASSI J2254188$+$312349 | 12 | 21 | 15.000$\pm$0.005 | 25 | 72.0$\pm$3.0 | 31
T5 | 2MASS J15031961$+$2525196 | 13 | 21 | 13.621$\pm$0.003 | 25 | 155.78$\pm$0.76 | 27
T6 | SDSSp J162414.37$+$002915.6 | 14 | 22 | 15.187$\pm$0.006 | 25 | 90.9$\pm$1.2 | 32
T7 | 2MASSI J0727182$+$171001 | 12 | 22 | 15.210$\pm$0.006 | 25 | 112.5$\pm$0.9 | 33
T8 | 2MASSI J0415195$-$093506 | 12 | 21 | 15.327$\pm$0.004 | 24 | 175.2$\pm$1.7 | 33

References. — (1) Kirkpatrick et al. (1997); (2) Kirkpatrick et al. (2008);
(3) Wilson et al. (2003); (4) Gizis et al. (2000); (5) Reid et al. (2008); (6)
Cruz et al. (2003); (7)Kirkpatrick et al. (2000); (8) Kirkpatrick et al.
(1999); (9) Martín et al. (1999); (10) Kirkpatrick et al. (2011); (11) Leggett
et al. (2000); (12) Burgasser et al. (2002); (13) Burgasser et al. (2003);
(14) Strauss et al. (1999); (15) Burgasser & McElwain (2006); (16) Bardalez
Gagliuffi et al. (2014); (17) Burgasser (2007); (18) Burgasser et al. (2010);
(19) Reid et al. (2006); (20) Burgasser et al. (2006b); (21) Burgasser et al.
(2004); (22) Burgasser et al. (2006a); (23) Lawrence et al. (2007); (24)
McMahon et al. (2013); (25) Dye et al. (2018); (26) Edge et al. (2013); (27)
Gaia Collaboration et al. (2023); (28) Kirkpatrick et al. (2021a); (29) Dahn
et al. (2017); (30) Best et al. (2020); (31) Manjavacas et al. (2013); (32)
Tinney et al. (2003); (33) Dupuy & Liu (2012)

##  3 Analysis

To determine preliminary spectral types, we compared to the near-infrared L
and T dwarf standards from Kirkpatrick et al. (2010) and Burgasser et al.
(2006b), with the exception of the L7 standard, where we use 2MASSI
J0825196$+$211552 as recommended in Cruz et al. (2018), as the original L7
near-infrared standard appears to have an unusually low surface gravity and
age. The best fitting standards for each object are shown in the left column
of Figure 1, and match reasonably well in the $J$-band region, but show
significant discrepancies beyond $\sim$1.5 $\mu$m. To ensure that differences
between standards and the observed spectra were not due to uncertainties
incurred when stitching orders or some other observational systematic, we
calculated synthetic photometric $J-K_{S}$ colors using the VISTA filter
response curves. We find synthetic colors consistent with the photometric
colors in Table 1 to within 1$\sigma$. For this reason, we constructed binary
templates in an attempt to better fit the observed spectra.

We utilized the same spectral standards as above, with a few exceptions to
account for known distance and multiplicity. The L2 near-infrared standard is
a resolved binary (Liu & Leggett, 2005), so we instead use the L2 object 2MASS
J04082905$-$1450334 (Wilson et al., 2003; Bardalez Gagliuffi et al., 2014).
The L5 near-infrared standard 2MASS J08350622$+$1953050 has no measured
parallax, so we instead use the L5 2MASS J21373742$+$0808463 (Reid et al.,
2008; Burgasser et al., 2010). The T0 near-infrared standard is suspected to
be a spectral binary (Burgasser et al., 2010; Ashraf et al., 2022), thus we
employed the T0 WISEPA J015010.86$+$382724.3 (Kirkpatrick et al., 2011).
Lastly, the T3 standard 2MASS J12095613$-$1004008 is a resolved binary (Liu et
al., 2010), and we therefore use the T3 WISEPC J223937.55$+$161716.2
(Kirkpatrick et al., 2011). The objects used to create spectral binary
templates are summarized in Table 2.

To generate spectral binary templates, we acquired near-infrared spectra from
the the SpeX Prism Library Analysis Toolkit (SPLAT; Burgasser & Splat
Development Team 2017). We absolute flux calibrated each spectrum using UKIDSS
Hemisphere Survey (UHS; Dye et al. 2018) or Vista Hemisphere Survey (VHS;
McMahon et al. 2013) $J$-band photometry, and measured parallaxes (Tinney et
al., 2003; Dupuy & Liu, 2012; Manjavacas et al., 2013; Dahn et al., 2017; Best
et al., 2020; Kirkpatrick et al., 2021a; Gaia Collaboration et al., 2023).
Finally, we added the spectra together and normalized the result to the
$J$-band peak between 1.27–1.29 $\mu$m. We compared the resulting templates to
each observed spectrum and found the best fits by calculating $\chi^{2}_{\nu}$
values following Burgasser et al. (2010).

###  3.1 Spectral Index Calculations

Spectral indices are also used to identify potential objects of interest
(e.g., young brown dwarfs, spectral binaries, photometric variables). We
follow the methods and calculate the indices defined in Burgasser et al.
(2002, 2006b, 2010) and Bardalez Gagliuffi et al. (2014) for our observed
objects (Table 3.1) and a large comparison sample from the SPLAT library.
Uncertainties were determined in a Monte Carlo fashion. We use binary index
regions from Burgasser et al. (2010) ’and variability regions from Ashraf et
al. (2022) in the following sections.

Table 3: Spectral Index Values

Spectral | Object | Ref. aaGiven reference corresponds to the work where the index was originally defined.
---|---|---
Index | W0727$-$3607 | W1036$-$5144 | W1344$-$7320 |
H2O$-J$ | 0.392$\pm$0.009 | 0.546$\pm$0.009 | 0.467$\pm$0.017 | 1
CH4$-J$ | 0.542$\pm$0.006 | 0.586$\pm$0.004 | 0.579$\pm$0.010 | 1
$J$-Curve | 4.121$\pm$0.084 | 2.970$\pm$0.042 | 3.286$\pm$0.104 | 2
CH4$-H$ | 0.999$\pm$0.011 | 0.822$\pm$0.005 | 0.858$\pm$0.011 | 1
$H$-Bump | 0.852$\pm$0.012 | 1.120$\pm$0.008 | 1.056$\pm$0.016 | 2
$H$-Dip | 0.518$\pm$0.006 | 0.459$\pm$0.003 | 0.451$\pm$0.006 | 3
CH4$-K$ | 0.607$\pm$0.010 | 0.801$\pm$0.009 | 0.909$\pm$0.013 | 1
$K$-Slope | 0.960$\pm$0.010 | 0.975$\pm$0.008 | 0.973$\pm$0.012 | 4
$K/J$ | 0.453$\pm$0.005 | 0.433$\pm$0.003 | 0.559$\pm$0.007 | 1
H2O$-H$ | 0.414$\pm$0.010 | 0.572$\pm$0.006 | 0.622$\pm$0.013 | 1
BinaritybbBased on the index criteria defined in Burgasser et al. (2010). | weak | strong | strong |
VariabilityccBased on the index criteria defined in Ashraf et al. (2022). | strong | … | … |

References. — (1) Burgasser et al. (2006b); (2) Bardalez Gagliuffi et al.
(2014); (3) Burgasser et al. (2010) Burgasser et al. (2002)

##  4 Results and Discussion

The observed discrepancies between observed spectra and existing spectral
standards can be attributed to various factors, such as binarity, variability,
youth, and metallicity, or a combination of these influences, as discussed
below.

###  4.1 Binarity

####  4.1.1 Analysis of Binary Templates

The right column of Figure 1 compares the best-fitting binary templates to our
sources. The discrepancies observed with the single fits beyond approximately
1.5 $\mu$m are greatly reduced with the binary templates, as verified by lower
$\chi^{2}_{\nu}$ values. A discussion of the results for each individual
object follows.

Figure 1:  Observed SOAR/TripleSpec4.1 spectra for W0727$-$3607 (top),
W1036$-$5144 (middle), and W1344$-$7320 (bottom), with gray lines showing the
full resolution and black lines showing the spectra smoothed to the resolution
of the standards and binary templates (R$\approx$150) according to their
parallax-based absolute fluxes. The left-hand column depicts the best-fitting
single standards (orange line) for the spectra of the three candidates while
the right-hand column shows both the binary template (orange line) and the
standard spectra that compose it (purple line for primary, blue line for
secondary). Reduced $\chi^{2}_{\nu}$ values are reported for each of the fits.

CWISE J072708.09$-$360729.2: The best-fitting T4 single standard exhibits a
significantly higher $\chi^{2}_{\nu}$ = 503 compared to the L7+T4 binary
template, which has $\chi^{2}_{\nu}$ = 39.7. Following Burgasser et al.
(2010), we evaluate our fits with a one-sided $F$-test, finding $\eta_{\rm
SB}$ = 12.7, well above the $\eta_{\rm SB}$ $>$1.34 spectral binary confidence
threshold given in that work. Although the binary template provides a superior
fit overall, discrepancies persist, particularly at the $H$-band peak.We
retain W0727$-$3607 as a spectral binary candidate, but discuss below that it
is also a potential variable source (see Section 5).

CWISE J103604.84$-$514424.4: W1036$-$5144 shows a lower-than-expected peak in
the $H$-band and more flux across the $K$-band compared to the T2 standard.
These discrepancies are largely resolved in the L7+T4 binary fit, with only
minor deviations in the $J$-band and in the longest wavelengths of the
$K$-band. The binary fit yields a significantly lower $\chi^{2}_{\nu}$ = 16.8,
compared to the $\chi^{2}_{\nu}$ = 107.4 for the single fit, and yields
$\eta_{\rm SB}$ = 6.4, consistent with the spectral binary criteria specified
in Burgasser et al. (2010).

CWISE J134446.62$-$732053.9: The single standard fit to W1344$-$7320 has $H$\-
and $K$-bands that are underluminous, leading to a relatively poor
$\chi^{2}_{\nu}$ = 610. The L7+T7 binary template yields a significantly
improved fit with $\chi^{2}_{\nu}$ = 13.7. Although there is a slight
discrepancy in $Y$-band, where the template is too bright compared to the
observed spectrum, the remainder of the $J$-, $H$-, and $K$-band regions
exhibit a well-matched morphology. Our $\chi^{2}_{\nu}$ values correspond to
$\eta_{\rm SB}$ = 44.5, which again satisfies the spectral binary criteria
specified in Burgasser et al. (2010).

####  4.1.2 Analysis of Binary Indices

The spectral indices shown to be indicative of binarity in Burgasser et al.
(2010) are shown in Figure 2. That study designated sources that satisfied at
least three index criteria as “strong” candidates, and two index criteria as
“weak” candidates.

CWISE J072708.09$-$360729.2: W0727$-$3607 falls within two of five regions
indicative of binarity, and are very close to the edges of these regions. This
classifies the source as a weak candidate. Note that we exclude for
consideration the H2O$-J$/H2O$-H$ versus spectral type comparison as it is not
applicable for spectral types later than T3.5. Along with the mediocre fit of
the L7+T4 binary template, this result suggests a factor other than binarity
is responsible for this object’s unusual spectrum.

CWISE J103604.84$-$514424.4: W1036$-$5144 satisfies five of six regions,
making it a strong binary candidate. Considering the well-fit binary template,
binarity is a strong possibility for this source.

CWISE J134446.62$-$732053.9: W1344$-$7320 satisfies five of five binary
regions (excluding H2O$-J$/H2O$-H$ versus spectral type), making it a strong
binary candidate. Again, the well-fit binary template of L7+T7 makes this
object a likely spectral binary.

Figure 2: Index-index plots highlighting regions defined in Burgasser et al.
(2010); Bardalez Gagliuffi et al. (2015) as indicative of spectral binarity
(red shaded regions). Background grey points are SPLAT objects with spectral
types between L3 and T8. The three targets of this study are shown in the
legend.

####  4.1.3 Distance Estimation

If these objects are indeed spectral binaries, then updated distance estimates
can be obtained by scaling the best-fitting absolutely flux calibrated binary
templates to each object’s observed photometric magnitudes. Using $J$-band
photometry from VHS for W0727$-$3607 and W1036$-$5144 and 2MASS (Skrutskie et
al., 2006) for W1344$-$7320, we find estimated distances of 34$\pm$7 pc,
27$\pm$5 pc, and 20$\pm$4 pc, respectively.

###  4.2 Variability

There is evidence that some spectral binary candidates are instead single
stars with inhomogeneous atmospheres as indicated via their photometric or
spectroscopic variability (e.g., 2MASS J21392676$+$0220226; Radigan et al.
2012; Khandrika et al. 2013). We applied the spectral index criteria
introduced in Ashraf et al. (2022) to analyze the spectra of W0727$-$3607,
W1036$-$5144, and W1344$-$7320 and evaluate their potential as strong variable
sources. This method uses single low-resolution spectra to empirically
identify spectral indices that may be indicative of variability based on known
variable sources. The idea is that cloudy or patchy layers at different
temperatures in a brown dwarf atmosphere that lead to photometric variability
may have measurable effects in the emergent spectra of these objects. Note
that we do not examine any index-index criteria that use the H2O$-$K index, as
the region of the numerator range (1.975$-$1.995 $\mu$m) is especially noisy
in our TripleSpec4.1 spectra. In Figure 3, there are nine total index-index
correlation plots, as discussed below. Ashraf et al. (2022) only considered
objects that satisfied all of their variability index criteria as “strong”
candidates, and those that satisfied all but one of their outlined criteria as
“weak” candidates.

Figure 3: Index-index plots highlighting the regions indicative of photometric
variability defined in Ashraf et al. (2022) (green shaded regions). Background
grey points are SPLAT objects with spectral types between L7 and T3, with our
three observed targets indicated shown in the legend.

####  4.2.1 Analysis of Variable Indices

CWISE J072708.09$-$360729.2: W0727$-$3607 falls within all of the designated
regions indicating potential variability. Coupled with the weak spectral
binary designation and less accurate match to binary templates, variability
appear to be at least partly responsible for the unusual shape of this
object’s spectrum. We investigated potential variability using WISE single-
exposure photometry following the methods in Schneider et al. (2023). We found
no clear signs of variability in the single-exposure WISE data, with W1 and W2
photometric standard deviation values within 1$\sigma$ of the median value for
objects with similar magnitudes. However, we note that with magnitudes of W1 =
14.639$\pm$0.018 mag and W2 = 13.862$\pm$0.15 mag, W0727$-$3607 may be too
faint to detect significant variability in WISE data, as no variables were
identified in Schneider et al. (2023) with magnitudes as faint as these.

CWISE J103604.84$-$514424.4: W1036$-$5144 consistently lies outside the range
indicative of potential variability, supporting the spectral binary
hypothesis.

CWISE J134446.62$-$732053.9: W1344$-$7320 satisfies four of the nine
variability regions, below the weak candidate threshold of Ashraf et al.
(2022) Again, the spectral binary hypothesis is favored for this object.

##  5 Conclusions

In this study, we presented three new brown dwarf spectral binary candidates:
CWISE J072708.09$-$360729.2, CWISE J103604.84$-$514424.4, and CWISE
J134446.62$-$732053.9. By constructing binary templates and comparing them to
the observed spectra, we found significantly better fits, revealing component
types of T0+T7 for CWISE J072708.09$-$360729.2, L7+T4 for CWISE
J103604.84$-$514424.4, and L7+T7 for CWISE J134446.62$-$732053.9. However, our
investigation of variability indices suggests that CWISE J072708.09$-$360729.2
is a strong variability candidate.

The rarity of brown dwarf-brown dwarf binary systems, as consistently
highlighted by statistical studies (Burgasser, 2007; Radigan et al., 2013;
Aberasturi et al., 2014; Opitz et al., 2016; Fontanive et al., 2018),
underscores the importance of pinpointing more such systems. Moreover, the
ability to identify spectral binaries offers the advantage that their orbital
period are sufficiently short to be amenable to dynamical mass follow-up
(Burgasser et al., 2012; Bardalez Gagliuffi et al., 2015; Burgasser et al.,
2016; Sahlmann et al., 2020). Similarly, variable brown dwarfs provide
opportunities to study their rotational and atmospheric dynamics, particularly
through multi-wavelength studies (Buenzli et al., 2012; Apai et al., 2013;
Radigan et al., 2013). Future work includes confirming the binary and/or
variability nature of these objects through high resolution imaging and
photometry, astrometric, and radial velocity monitoring. Increasing the sample
of close binary and variable brown dwarfs has the potential to advance our
understanding of substellar formation, evolution, and dynamics.

The Backyard Worlds: Planet 9 team would like to thank the many Zooniverse
volunteers who have participated in this project. We would also like to thank
the Zooniverse web development team for their work creating and maintaining
the Zooniverse platform and the Project Builder tools. This work is based on
observations obtained at the Southern Astrophysical Research (SOAR) telescope,
which is a joint project of the Ministério da Ciência, Tecnologia e Inovações
(MCTI/LNA) do Brasil, the US National Science Foundation’s NOIRLab, the
University of North Carolina at Chapel Hill (UNC), and Michigan State
University (MSU). This material is based upon work supported by the National
Science Foundation under Grant No. 2007068, 2009136, and 2009177. This
publication makes use of data products from the Wide-field Infrared Survey
Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration. This publication
also makes use of data products from NEOWISE, which is a project of the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
Planetary Science Division of the National Aeronautics and Space
Administration.

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# A fractal uncertainty principle for Bergman spaces and analytic wavelets

Luis Daniel Abreu<EMAIL_ADDRESS>Faculty of Mathematics,
University of Vienna, Oskar-Morgenstern-Platz 1, A-1090, Vienna, Austria  ,
Zouhair Mouayn<EMAIL_ADDRESS>Department of Mathematics, Faculty of
Sciences and Technics (M’Ghila), Sultan Moulay Slimane University, P.O. Box.
523, Béni Mellal, Morocco
Acoustics Research Institute, Wohllebengasse 12-14, A-1040, Vienna, Austria
and  Felix Voigtlaender<EMAIL_ADDRESS>Faculty of Mathematics,
University of Vienna, Oskar-Morgenstern-Platz 1, A-1090, Vienna, Austria

###### Abstract.

Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor
sets and of Knutsen, for joint time-frequency representations (STFT with
Gaussian, equivalent to Fock spaces), we suggest a general setting relating
localization and uncertainty and prove, within this context, an uncertainty
principle for Cantor sets in Bergman spaces of the unit disc, defined as
unions of annuli equidistributed in the hyperbolic measure. The result can be
written in terms of analytic Cauchy wavelets. As in the case of the STFT
considered by Knutsen, our result consists of a double side bound for the norm
of a localization operator involving the fractal dimension log2/log3 in the
exponent. As in the STFT case and in Dyatlov´s fractal uncertainty principle,
the norm of the dilated iterates of the Cantor set in the disc tends to
infinite, while the corresponding norm of the localization operator tends to
zero.

The authors were supported by the Austrian Science Fund (FWF) via the project
(P31225-N32).

##  1\. Introduction

###  1.1. Fractal uncertainty principles for the Fourier transform

The uncertainty principle is a collection of statements in harmonic analysis,
each of them quantifying in some form the fundamental duality between a
function $f$ and its Fourier transform $\mathcal{F}f$, which prevents both
representations from being ‘simultaneously concentrated in small sets’ [25].
Let us consider the Fourier transform
$\mathcal{F}:L^{2}\left(\mathbb{R}\right)\rightarrow
L^{2}\left(\mathbb{R}\right)$

$\mathcal{F}f(\xi)=(2\pi)^{-1/2}\int_{\mathbb{R}}e^{-ix\xi}f(x)dx$

and the $h$-dilated Fourier transform
$\mathcal{F}_{h}:L^{2}\left(\mathbb{R}\right)\rightarrow
L^{2}\left(\mathbb{R}\right)$

$\mathcal{F}_{h}f(\xi)=h^{-1/2}\mathcal{F}f(\xi h^{-1})\text{.}$

Several mathematical impersonations of the uncertainty principle consist of
bounds on the norm of the operator which concentrates the energy of $f$ in a
set $X$ and the energy of $\mathcal{F}f$ in a set $Y$. Following Dyatlov’s
definition in [12], one can resort to the $h$-dilated Fourier transform, and
declare a pair of real $h$-dependent sets $X$ and $Y$ to satisfy an
uncertainty principle with exponent $\beta>0$ if, as $h\rightarrow 0$,

(1.1) $\left\|1_{X}\mathcal{F}_{h}1_{Y}\right\|_{op}=O(h^{\beta})\rightarrow
0\text{.}$

For instance, if $X,Y=[0,h]$ then Hölder inequality gives the uncertainty
principle

(1.2)
$\left\|1_{X}\mathcal{F}_{h}1_{Y}\right\|_{op}=O(h^{\frac{1}{2}})\rightarrow
0\text{.}$

Given $R>0$, set $h=R^{-2}$. Then

$\left\|1_{X}\mathcal{F}_{h}1_{Y}\right\|_{op}=\left\|1_{X/\sqrt{h}}\mathcal{F}1_{Y/\sqrt{h}}\right\|_{op}=\left\|1_{RX}\mathcal{F}1_{RY}\right\|_{op}\text{.}$

Now, (1.2) becomes an uncertainty principle for the Fourier transform and the
dilated sets $RX$ and $RY$: as $R\rightarrow\infty$,

$\left\|1_{RX}\mathcal{F}1_{RY}\right\|_{op}=O(R^{-1})\rightarrow 0\text{.}$

Moreover, as $R\rightarrow\infty$, (1.1) becomes

$\left\|1_{X}\mathcal{F}_{h}1_{Y}\right\|_{op}=\left\|1_{X/\sqrt{h}}\mathcal{F}1_{Y/\sqrt{h}}\right\|_{op}=\left\|1_{RX}\mathcal{F}1_{RY}\right\|_{op}=O(R^{-2\beta})\rightarrow
0\text{.}$

We want to emphasize the following aspect of _the_ _fractal uncertainty
principle_[12, 13, 14]: it covers situations where $RX$ and $RY$ are close to
have a fractal structure (they depend on $R$ and approach fractals when
$R\rightarrow\infty$), where their volume approaches$\ \infty$ as
$R\rightarrow\infty$, but nevertheless $RX$ and $RY$ satisfy an uncertainty
principle (their operator norm decays like $O(R^{-\beta})$ for some
$\beta>0$). This can roughly be described by saying that _no function can be
localized close to a fractal set in both time and frequency_.

For illustration and motivation, consider the dilated real Cantor set
$C(R;\mathbb{R})\subset[0,R]$ defined as

$C(R;\mathbb{R})=\bigcap_{n=1}^{\infty}C_{n}(R;\mathbb{R})=\bigcap_{n=1}^{\infty}RC_{n}\text{,}$

where $C_{n}$ is the $n$-iterated set obtained from $[0,1]$ by removing, in
each iteration step, the middle thirds of the intervals. The set
$C_{n}(R;\mathbb{R})$ consists of $2^{n}$ disjoint intervals
$I_{n,j}\subset[0,R]$, each with measure$\ vol(I_{n,j})=3^{-n}R$. Taking $n$
such that $vol(I_{n,j})\asymp 1/R$, then $3^{n/2}\asymp R$ and

$vol(C_{n}(R;\mathbb{R}))=2^{n}vol(I_{n,j})\asymp R^{-1+2\frac{\ln 2}{\ln
3}}\rightarrow\infty\text{,}$

as $R\rightarrow\infty$. Using this volume bound and Hölder inequality as in
(1.2), leads to

$\left\|1_{C(R;\mathbb{R})}\mathcal{F}1_{C(R;\mathbb{R})}\right\|_{op}\asymp
R^{-1/2+\frac{\ln 2}{\ln 3}}\rightarrow\infty\text{,}$

therefore not enough to assure that $X=Y=C(R;\mathbb{R})$ satisfy an
uncertainty principle according to Dyatlov’s definition. However, from Theorem
2.12 and 2.13 in [12] it follows that there exists an exponent $\beta>0$ such
that, as $R\rightarrow\infty$, (1.1) holds for $RX,RY=C_{n}(R;\mathbb{R})$.
Since $R\asymp 3^{n/2}$, as $n\rightarrow\infty$, also $R\rightarrow\infty$
and

(1.3)
$\left\|1_{C_{n}(R;\mathbb{R})}\mathcal{F}1_{C_{n}(R;\mathbb{R})}\right\|_{op}=O(R^{-2\beta})\rightarrow
0\text{.}$

###  1.2. Uncertainty and localization

The time-frequency localization operator of the previous paragraph suggests a
construction in a general Lebesgue space $L^{2}\left(\Lambda\right)$ (with
$\Lambda$ metric and topological). If $L^{2}\left(\Lambda\right)$ has a
reproducing kernel $K(z,w)$, one can define a localization/Toeplitz operator
$P_{\Omega}$ mapping the function $f\in L^{2}\left(\Lambda\right)$ to a smooth
function $P_{\Omega}f\in L^{2}\left(\Lambda\right)$ essentially concentrated
in a bounded region $\Omega\subset\Lambda$. The operator $P_{\Omega}$ is
explicitly defined as

$(P_{\Omega}f)(w)=\int_{\Omega}f(z)\overline{K(z,w)}d\mu(z)\text{.}$

In this context, we say that $\Omega(R)$ ($R>0$) satisfies an uncertainty
principle if, as $R\rightarrow\infty$,

$\left\|P_{\Omega}\right\|_{op}=O(R^{-\beta})\text{.}$

As in the previous paragraph, one can look for bounds of
$\left\|P_{\Omega}\right\|_{op}$ when $\Omega$ is a fractal set. In [4], the
case of the Fock space setting has been considered, in the equivalent
formulation provided by the short-time-Fourier transform [15, Chapter 3]. For
the analogy with our results, it will be convenient to rephrase the results in
the Fock space. This corresponds to the choice of measure
$d\mu(z)=e^{-\pi\left|z\right|^{2}}dz$, where $dz$ is Lebesgue measure, of the
kernel

$K(z,w)=K_{Fock}(z,w)=e^{\pi z\overline{w}}$

and of $\Omega\subset\mathbb{C}$ as the $nth$ iteration of the _planar Cantor
set_

$C_{n}(R;\mathbb{C}):=\\{z\in\mathbb{C}\colon|z|^{2}\in
C_{n}(R^{2};\mathbb{R})\\}.$

The planar Cantor set is a disjoint union of $2^{n}$ annuli
$I_{n,j}^{\mathbb{C}}$, each of measure
$vol(I_{n,j}^{\mathbb{C}}(R^{2}))=\frac{\pi\,R^{2}}{3^{n}}$, so that for each
annulus we consider a $1/3^{n}$ part of the initial disc with area
$\pi\,R^{2}$. For this measure to be well distributed among $\left[0,\pi
R^{2}\right]$, one takes $vol(I_{n,j}^{\mathbb{C}}(R^{2}))=1/(\pi\,R^{2})$,
yielding $(\pi\,R^{2})^{2}\asymp 3^{n}$ and

$vol(C_{n}(R;\mathbb{C}))\asymp R^{-2+4\frac{\ln 2}{\ln
3}}\rightarrow\infty\text{,}$

as $R\rightarrow\infty$. Moreover, as $R\rightarrow\infty$, it is proven in
[4, Corollary 4.1] that

(1.4) $\left\|P_{C_{n}(R;\mathbb{C})}\right\|_{op}\asymp R^{-2+2\frac{\ln
2}{\ln 3}}\rightarrow 0\text{.}$

Thus, in the Fock case, the measure of the dilates of the iterated Cantor set
tends to $\infty$, while the norm of the operator tends to zero.

Small operator norms facilitate recovery in signal analysis problems [2]. The
operator norm of $P_{\Omega}$ is maximized when $\Omega$ is a disc [24], while
there exist sets $\Omega$ with infinite Lebesgue measure such that the
operator norm of $P_{\Omega}$ is arbitrarily small [16].

We will show in this paper that, in the case of the disc, we have a similar
situation: one can define _a Cantor set in the disc, whose hyperbolic measure
tends to infinity,_ and an associated Toeplitz operator _, whose operator norm
tends to zero._

Our contributions are organized as follows. A disc version of the Cantor set
is considered in the next section. The main result on fractal uncertainty
principle on Bergman spaces is stated and translated to the language of
analytic wavelets. More details and proofs are given in the remaining two
sections, with the most technical estimations in the last section of the
paper.

##  2\. Fractal uncertainty principles for the Bergman space

In the present paper we consider the measure $dA(z)=\frac{dz}{\pi}$, where
$dz$ is planar Lebesgue measure, and, given $\alpha\in(0,+\infty)$,
$dA_{\alpha}(z)=2\alpha(1-\left|z\right|^{2})^{2\alpha-1}dA(z)$ and the
reproducing kernel of the _weighted analytic Bergman space,_ provided
explicitly by [18]:

(2.1)
$\mathcal{K}_{\mathbb{D}}^{\alpha}(z,w)=\frac{1}{(1-z\overline{w})^{2\alpha+1}}=\sum_{n=0}^{\infty}e_{n}^{\alpha}(z)\overline{e_{n}^{\alpha}(w)}$

where

$e_{n}^{\alpha}(z)=\sqrt{\gamma_{n}}z^{n},\text{ \ \ \
}\gamma_{n}=\gamma_{n}^{\alpha}:=\frac{\Gamma(n+1+2\alpha)}{\Gamma(n+1)\Gamma(1+2\alpha)}=\frac{1}{2\alpha}[B(k+1,2\alpha)]^{-1}\text{.}$

Consider the localization operator

$(P_{C_{n}(R;\mathbb{D})}^{(\alpha)}f)(w)=\int_{C_{n}(R;\mathbb{D})}f(z)\overline{\mathcal{K}_{\mathbb{D}}^{\alpha}(z,w)}dA_{\alpha}(z)\text{,}$

where the fractal localization region is now the following disc version of the
iterates of the Cantor set

$C_{n}(R;\mathbb{D}):=\left\\{z\in\mathbb{D}:\frac{|z|^{2}}{1-|z|^{2}}\in
C_{n}(R;\mathbb{R})\right\\}\text{.}$

We will show that $C_{n}(R;\mathbb{D})$ is a disjoint union of $2^{n}$ annuli
$D_{l}^{(n)}(0,R)$, each of hyperbolic measure
$\mu_{\mathbb{D}}\left(D_{l}^{(n)}(0,R)\right)=\frac{\,R}{3^{n}}$. Thus, as in
the previous examples, for this measure to be well distributed among
$\left[0,R\right]$ we take $n$ such that
$\mu_{\mathbb{D}}\left(D_{l}^{(n)}(0,R)\right)\sim 1/R$, leading to
$3^{n}\asymp R^{2}$ and to $R^{2\frac{\ln 2}{\ln 3}}\asymp 2^{n}$, so that

$\mu_{\mathbb{D}}(C_{n}(R;\mathbb{D}))=R\left(\frac{2}{3}\right)^{n}\asymp
R^{2\frac{\ln 2}{\ln 3}-1}\rightarrow\infty\text{,}$

as $R,n\rightarrow\infty$. So far everything is perfectly tuned with our model
Fourier and time-frequency/Fock cases. However, the analogue of the conditions
(1.3) only holds in the asymptotic case. The bounds on the non-asymptotic case
depend on the size of $R$.

######  Theorem 1.

Given $\alpha\in(0,\infty)$, there are constants $0<C_{1}\leq C_{2}<\infty$
(which only depend on $\alpha$) such that the operator norm of the time-scale
localization operator $P_{C_{n}(R;\mathbb{D})}^{(\alpha)}$ satisfies,
$\forall\,n\in\mathbb{N}$ and $R\in(0,\infty)$,

$C_{1}\begin{cases}\left(\frac{2}{3}\right)^{n}R,&\text{if }0<R\leq 1,\\\
\left(\frac{2}{3}\right)^{n}R^{1-\frac{\ln 2}{\ln 3}},&\text{if }1\leq R\leq
3^{n},\\\ 1,&\text{if }R\geq
3^{n}.\end{cases}\leq\left\|P_{C_{n}(R;\mathbb{D})}^{(\alpha)}\right\|_{op}\leq
C_{2}\begin{cases}\left(\frac{2}{3}\right)^{n}R,&\text{if }0<R\leq 1,\\\
\left(\frac{2}{3}\right)^{n}R^{1-\frac{\ln 2}{\ln 3}},&\text{if }1\leq R\leq
3^{n},\\\ 1,&\text{if }R\geq 3^{n}.\end{cases}$

Furthermore, if $R$ is chosen so that $R^{2}\sim 3^{n}$, then

$\left\|P_{C_{n}(R;\mathbb{D})}^{(\alpha)}\right\|_{op}\asymp\left(\frac{2}{3}\right)^{\frac{n}{2}}\asymp
R^{\frac{\ln 2}{\ln 3}-1}\rightarrow 0\text{,}$

as $R\rightarrow\infty$.

We note that the term

$\delta=\delta_{C(R;\mathbb{R})}=\frac{\ln 2}{\ln 3}$

appearing in the exponent of $R^{-1+\frac{\ln 2}{\ln 3}}$ is the Hausdorff
dimension of the Cantor set. The uncertainty principles in [12] consider more
general fractal sets and the results are obtained in terms of their Hausdorff
dimensions. See [5] for new developments in this direction in the planar case.
As in the joint time-frequency case, following the suggestion in the comments
after [4, Corollary 4.1], where the bound $R^{-2+2\delta}$ is obtained for the
planar Cantor set, this opens interesting problems, if one looks at more
general fractal sets and looks for bounds of the associated Toeplitz operator
in terms of the Hausdorff dimension of the sets.

###  2.1. Fractal uncertainty principle for analytic wavelets

In this section we outline how our result can be written in terms of analytic
wavelets. We will use the basic notation for
$\mathcal{H}^{2}(\mathbb{C}^{+})$, the Hardy space in the upper half plane, as
the space of analytic functions in $\mathbb{C}^{+}$, $f$ , such that

$\text{
}\sup_{0<s<\infty}\int_{-\infty}^{\infty}\left|f(x+is)\right|^{2}dx<\infty\text{.}$

To simplify the computations it is often convenient to use the equivalent
definition (since the Paley-Wiener theorem [11] gives
$\mathcal{F}(\mathcal{H}^{2}(\mathbb{C}^{+}))=L^{2}(0,\infty)$)

$\mathcal{H}^{2}(\mathbb{C}^{+})=\left\\{f\in
L^{2}(\mathbb{R}):(\mathcal{F}f)(\xi)=0\text{ for }\xi<0\right\\}\text{.}$

The _wavelet transform_ of a function $f$ on $\mathcal{H}^{2}(\mathbb{C}^{+})$
with _mother wavelet_ $\psi\in\mathcal{H}^{2}(\mathbb{C}^{+})$, such that its
_admissible constant_
$C_{\psi}=\left\|\mathcal{F}\psi\right\|_{L^{2}(\mathbb{R}^{+},t^{-1}dt)}^{2}$
is finite, is defined as

$W_{\psi}f(z):=\int_{\mathbb{R}}f(t)\overline{s^{-\frac{1}{2}}\psi(s^{-1}(t-x))}dt=\sqrt{s}\int_{0}^{\infty}\widehat{f}(\xi)\overline{\widehat{\psi}\left(s\xi\right)}e^{ix\xi}{d\xi}\text{,
}z=x+is$

The _analytic wavelets_ are the functions $\psi_{0}^{\alpha}$ defined via
their Fourier transforms by

(2.2)
$(\mathcal{F}\psi_{0}^{\alpha})(\xi)=\xi^{\frac{\alpha}{2}}e^{-\xi},\quad\xi>0\text{.}$

As proven recently in [19], $W_{\psi}f(z)$ only leads to analytic (Bergman)
phase spaces for this special choice of $\psi$ (up to a phase factor).
Nevertheless, it became usual to name $W_{\psi}f(z)$ as the analytic wavelet
transform, due to the discard of negative frequencies. We will write

(2.3) $d\mu^{+}(z)=\left(\mathop{\mathrm{I}m}z\right)^{-2}dz\text{,}$

where $dz$ is the Lebesgue measure on $\mathbb{C}^{+}$. The orthogonality
relations for the wavelet transform

(2.4)
$\int_{\mathbb{C}^{+}}W_{\psi_{1}}f_{1}(x,s)\overline{W_{\psi_{2}}f_{2}(x,s)}d\mu^{+}(z)=\left\langle\mathcal{F}\psi_{1},\mathcal{F}\psi_{2}\right\rangle_{L^{2}(\mathbb{R}^{+},t^{-1}dt)}\left\langle
f_{1},f_{2}\right\rangle_{\mathcal{H}^{2}\left(\mathbb{C}^{+}\right)}\text{,}$

are valid for all $f_{1},f_{2}\in\mathcal{H}^{2}\left(\mathbb{C}^{+}\right)$
and $\psi_{1},\psi_{2}\in\mathcal{H}^{2}\left(\mathbb{C}^{+}\right)$
admissible. Then, setting $\psi_{1}=\psi_{2}=\psi$ and $f_{1}=f_{2}$ in (2.4),
gives

$\int_{\mathbb{C}^{+}}\left|W_{\psi}f(x,s)\right|^{2}d\mu^{+}(z)=C_{\psi}\left\|f\right\|_{\mathcal{H}^{2}\left(\mathbb{C}^{+}\right)}^{2}$

and the continuous wavelet transform provides an isometric inclusion
$W_{\psi}:\mathcal{H}^{2}\left(\mathbb{C}^{+}\right)\rightarrow
L^{2}(\mathbb{C}^{+}\mathbf{,}d\mu^{+}(z))$. Setting $\psi_{1}=\psi_{2}=\psi$
and $f_{2}=\pi(z)\psi$, where
$\pi(z)\psi(t):=s^{-\frac{1}{2}}\psi(s^{-1}(t-x))$, in (2.4), then for every
$f\in\mathcal{H}^{2}(\mathbb{C}^{+})$ one has

(2.5)
$W_{\psi}f(z)=\frac{1}{C_{\psi}}\int_{\mathbb{C}^{+}}W_{\psi}f(w)\langle\pi(w)\psi,\pi(z)\psi\rangle
d\mu^{+}(z),\quad z\in\mathbb{C}^{+}\text{.}$

Thus, the range of the wavelet transform

$\mathcal{W}_{\psi}:=\\{F\in L^{2}(\mathbb{C}^{+},d\mu^{+}):\ F=W_{\psi}f,\
f\in\mathcal{H}^{2}(\mathbb{C}^{+})\\}$

is a subspace of $L^{2}(\mathbb{C}^{+},\mu)$ with reproducing kernel

(2.6)
$k_{\psi}(z,w)=\frac{1}{C_{\psi}}\langle\pi(w)\psi,\pi(z)\psi\rangle_{\mathcal{H}^{2}\left(\mathbb{C}^{+}\right)}=\frac{1}{C_{\psi}}W_{\psi}\psi({w}^{-1}z),\quad\text{and}\quad
k_{\psi}(z,z)=\frac{\|\psi\|_{2}^{2}}{C_{\psi}}\text{.}$

For $z=x+is,w=y+iv\in{\mathbb{C}}^{+}$, the kernel
$k_{\psi_{0}^{2\alpha}}(z,w)$ is

(2.7)
$k_{\psi_{0}^{2\alpha}}(z,w)=\frac{1}{C_{\psi_{0}^{2\alpha}}}W_{\psi_{0}^{2\alpha}}\psi_{0}^{2\alpha}({w}^{-1}\cdot
z)=2^{2\alpha+1}\alpha\left(\frac{\sqrt{\mathrm{Im}\,z\,\mathrm{Im}\,w}}{-i(z-\overline{w})}\right)^{2\alpha+1}\text{.}$

This is a multiple of the reproducing kernel of the Bergman space __

$A^{2\alpha-1}({\mathbb{C}}^{+}):=\left\\{f:{\mathbb{C}}^{+}\rightarrow{\mathbb{C}}\text{
holomorphic}:\int_{{\mathbb{C}}^{+}}|f(z)|^{2}\text{Im}(z)^{2\alpha-1}dz<\infty\right\\}\text{,}$

Therefore,
$(\mathop{\mathrm{I}m}.)^{-\alpha-1/2}\mathcal{W}_{\psi_{0}^{2\alpha}}:\mathcal{H}^{2}(\mathbb{C}^{+})\rightarrow
A^{2\alpha-1}({\mathbb{C}}^{+})$ is an isometric inclusion. Moreover,
$A^{2\alpha-1}({\mathbb{C}}^{+})$ is conformal equivalent to
$A^{2\alpha-1}({\mathbb{D}})$ under the transformation
$z\in{\mathbb{C}}^{+}\rightarrow$ $\xi(z)=\frac{z-i}{z+i}\in\mathbb{D}$ [11].
It follows that $P_{C_{n}(R;\mathbb{D})}^{(\alpha)}$ can be unitarily mapped
to a constant multiple of the operator

(2.8)
$(P_{\xi^{-1}(C_{n}(R;\mathbb{D}))}f)(w)=\int_{\xi^{-1}(C_{n}(R;\mathbb{D}))}f(z)\overline{k_{\psi_{0}^{2\alpha}}(z,w)}dz\text{.}$

Thus, Theorem 1 is equivalent to an uncertainty principle for wavelet
representations on the Cantor set of ${\mathbb{C}}^{+}$ defined by
$\xi^{-1}(C_{n}(R;\mathbb{D}))$. The analyzing wavelet $\psi_{0}^{2\alpha}$ is
the only leading to an analytic structure [19]. Considering more general class
of analyzing wavelets, as those leading to a polyanalytic decomposition in
[27, 1], or the slightly different ones connected to Maass forms and
hyperbolic Landau levels [23, 22], may be a natural extension of the problem
we have considered here. The Toeplitz operator (2.8) has been first considered
by Daubechies and Paul [9] for $n=0$. The approach based on double
orthogonality is due to Seip [26] and provides some insight on why the
approach using circular symmetric sets in the plane and in the disc,
considerably simplifies the problem. With square time-frequency regions, as
required by the Fourier transform approach of Dyatlov, one has no access to
explicit eigenvalue formulas. The general eigenvalues problem of Gabor and
wavelet localization operators has been considered in [10].

##  3\. The Cantor set

###  3.1. The Cantor set in the line

Given $R>0$, consider the real Cantor set $C(R;\mathbb{R})\subset[0,R]$
defined as

$C(R;\mathbb{R})=\bigcap_{n=1}^{\infty}C_{n}(R;\mathbb{R})\text{,}$

where

(3.1) $C_{n}(R;\mathbb{R}):=R\cdot C_{n}\quad\text{with}\quad
C_{n}:=\biguplus_{a\in\\{0,2\\}^{n}}\left[\,\sum_{j=1}^{n}\frac{a_{j}}{3^{j}}\,,\quad
3^{-n}+\sum_{j=1}^{n}\frac{a_{j}}{3^{j}}\,\right]\text{.}$

In the iterative construction of $C_{n}$, the middle thirds of the interval of
$C_{n-1}$ are removed. For instance, $C_{0}=[0,1]$,
$C_{1}=[0,\tfrac{1}{3}]\cup[\tfrac{2}{3},1]$ and
$C_{2}=[0,\tfrac{1}{9}]\cup[\tfrac{2}{9},\tfrac{1}{3}]\cup[\tfrac{2}{3},\tfrac{7}{9}]\cup[\tfrac{8}{9},1]$.
Let $\Omega^{(n)}:=\\{0,2\\}^{n}$, $a=(a_{1},\dots,a_{n})\in\Omega^{(n)}$ and

$b_{a}=b_{a}^{(n)}:=\sum_{j=1}^{n}\frac{a_{j}}{3^{j}},$

so that

$C_{n}:=\biguplus_{a\in\\{0,2\\}^{n}}\left[\,b_{a}^{(n)}\,,3^{-n}+b_{a}^{(n)}\right]\text{.}$

For analyzing the size of the endpoints $b_{a}$, it is convenient to introduce
for $a\in\Omega^{(n)}\setminus\\{0\\}$ the _order_ of its index as

$\omega_{a}:=\min\\{\ell\in\\{1,\dots,n\\}\colon a_{\ell}\neq 0\\}\text{,}$

which gives rise to the restricted index sets

$\Omega_{m}^{(n)}:=\left\\{a\in\Omega^{(n)}\setminus\\{0\\}\quad\colon\quad\omega_{a}=m\right\\}\quad\text{for}\quad
m\in\\{1,\dots,n\\}\text{.}$

It is straightforward to verify that
$\Omega_{m}^{(n)}=\\{0\\}^{m-1}\times\\{2\\}\times\\{0,2\\}^{n-m}$, and hence

(3.2) $\\#\Omega_{m}^{(n)}=2^{n-m}\text{.}$

We will frequently use the estimate

(3.3) $\frac{2}{3^{m}}\leq
b_{a}=\sum_{j=m}^{n}\frac{a_{j}}{3^{j}}\leq\frac{2}{3^{m}}\sum_{\ell=0}^{\infty}3^{-\ell}=\frac{3}{3^{m}}\qquad\forall\,a\in\Omega_{m}^{(n)}.$

###  3.2. The Cantor set in the plane

The key properties of the Cantor-type set defined in the complex plane in [4]
will be preserved in the construction of the next section (modulo the required
adaptations to the disc). First, since $C_{n}(R^{2};\mathbb{R})$ is a disjoint
union of $2^{n}$ intervals, $C_{n}(R;\mathbb{C})$ is a disjoint union of
$2^{n}$ annuli. Furthermore, the condition $|z|^{2}\in
C_{n}(R^{2};\mathbb{R})$ is chosen since it ensures that all of the $2^{n}$
annuli have the same Lebesgue measure.

###  3.3. The Cantor set in the disc

The Cantor-type set in the disc is an union of $2^{n}$ disjoint annuli with
the same hyperbolic measure, which is defined, for a disc
${D(0,r)=\\{w\in\mathbb{C}}:{|w|<r\\}}$ with $r\in[0,1)$ as

$\mu_{\mathbb{D}}\left({D(0,r)}\right)=\int_{{D(0,r)}}(1-\left|z\right|^{2})^{-2}dA(z)=\frac{r^{2}}{1-r^{2}}\text{.}$

This suggests defining the $n$-th Cantor set as the set of points
$w\in\mathbb{D}$ such that $\varphi\left(|w|\right)\in C_{n}(R;\mathbb{R})$,
where

$\varphi:[0,1)\rightarrow[0,\infty),\text{ \ \
}r\mapsto\varphi(r)=\frac{r^{2}}{1-r^{2}}\text{.}$

More explicitly,

$C_{n}(R;\mathbb{D}):=\left\\{w\in\mathbb{D}\colon\frac{|w|^{2}}{1-|w|^{2}}\in
C_{n}(R;\mathbb{R})\right\\}\text{.}$

Our next result shows that this construction indeed yields a set
$C_{n}(R;\mathbb{D})$ that behaves similarly to the set $C_{n}(R;\mathbb{C})$
from [4] discussed in Remark 3.2 and Section 1.2. Let

$D_{l}^{(n)}(0,R)=\left\\{w\in\mathbb{D}:\varphi(|w|)\in{I_{\ell}^{(n)}=}\left[\,\sum_{j=1}^{n}\frac{a_{j}}{3^{j}}\,,\quad
3^{-n}+\sum_{j=1}^{n}\frac{a_{j}}{3^{j}}\,\right]\right\\}\text{.}$

######  Proposition 1.

The $n$-th disc Cantor set $C_{n}(R;\mathbb{D})$ is a disjoint union of$\
2^{n}$annuli $D_{l}^{(n)}(0,R)=\\{w\in\mathbb{D}\colon\varphi(|w|)\in
I_{\ell}^{(n)}\\}$:

$C_{n}(R;\mathbb{D})=\biguplus_{\ell=1}^{2^{n}}D_{l}^{(n)}(0,R)\text{. }$

Each annuli has hyperbolic measure
$\mu_{\mathbb{D}}\left(D_{l}^{(n)}(0,R)\right)=\frac{\,R}{3^{n}}$. Therefore,

$\mu_{\mathbb{D}}\left(C_{n}(R;\mathbb{D})\right)=\left(\frac{2}{3}\right)^{n}R$

###### Proof.

Given an arbitrary measurable function $F:[0,\infty)\rightarrow[0,\infty]$,
write the measure $dA(z)$ in polar coordinates, set $s=r^{2}$ and then
$t=\frac{s}{1-s}$ to yield

$\displaystyle\int_{\mathbb{D}}F\\!\left(\frac{|z|^{2}}{1-|z|^{2}}\right)dA(z)$
$\displaystyle=2\int_{0}^{1}\\!F\left(\frac{r^{2}}{1-r^{2}}\right)\frac{r}{(1-r^{2})^{2}}\,dr$
$\displaystyle=\int_{0}^{1}\\!F\left(\frac{s}{1-s}\right)(1-s)^{-2}\,ds=\int_{0}^{\infty}F(t)\,dt\text{.}$

Since $C_{n}(R;\mathbb{R})=\biguplus_{\ell=1}^{2^{n}}I_{\ell}^{(n)}$, where
the intervals
${I_{\ell}^{(n)}=[a_{\ell}^{(n)},b_{\ell}^{(n)}]\subset[0,\infty)}$ have
length $\left|I_{\ell}^{(n)}\right|\,=3^{-n}\,R$, then

$C_{n}(R;\mathbb{D})=\biguplus_{\ell=1}^{2^{n}}D_{l}^{(n)}(0,R)=\biguplus_{\ell=1}^{2^{n}}\\{w\in\mathbb{D}:\varphi^{-1}(a_{\ell}^{(n)})\leq\left|w\right|\leq\varphi^{-1}(b_{\ell}^{(n)})\\}$

is a disjoint union of $2^{n}$ annuli, each of which has
$\mu_{\mathbb{D}}$-measure

$\displaystyle\mu_{\mathbb{D}}\left(D_{l}^{(n)}(0,R)\right)$
$\displaystyle=\int_{\mathbb{D}}1_{I_{\ell}^{(n)}(R)}\left(|z|^{2}/(1-|z|^{2})\right)\,dA(z)$
$\displaystyle=\int_{0}^{\infty}1_{I_{\ell}^{(n)}(R)}(t)\,dt=\frac{\,R}{3^{n}}.$

##  4\. Localization to the Cantor set in the disc

In this section, we study the spectral properties of the time-scale
localization operators

(4.1)
$P_{C_{n}(R;\mathbb{D})}^{(\alpha)}:=P_{1_{C_{n}(R;\mathbb{D})}}^{(\alpha)}\quad\text{for}\quad
R\in(0,\infty),\,\,n\in\mathbb{N},\text{ and }\alpha\in(0,\infty).$

###  4.1. Eigenvalues of the Cantor-type localization operator

######  Proposition 2.

Let $\alpha,R\in(0,\infty)$ and $n\in\mathbb{N}$. Then the eigenvalues of the
operator $P_{C_{n}(R;\mathbb{D})}^{(\alpha)}$ introduced in Equation (4.1) are
given by

$\lambda_{k}=\lambda_{k}^{(\alpha)}\left(C_{n}(R;\mathbb{D})\right)=\,\int_{0}^{\infty}g_{k}(t;\alpha)\,1_{C_{n}(R;\mathbb{R})}(t)dt\text{,}$

where $g_{k}(\cdot\,;\alpha):(0,\infty)\rightarrow(0,\infty)$ is given by

$g_{k}(t;\alpha)=B(k+1,2\alpha)^{-1}\cdot\left(\frac{t}{1+t}\right)^{k}(1+t)^{-(1+2\alpha)}.$

###### Proof.

Observe that the orthogonality of the basis functions
$e_{n}^{\alpha}(z)=\sqrt{\gamma_{n}}z^{n}$ in the reproducing kernel expansion
(2.1) holds for any radial measure $\mu(\left|z\right|)dA(z)$, with
$\mu:[0,\infty)\rightarrow[0,\infty)$ satisfying
$\int_{0}^{1}\mu(r)dr<\infty$. This follows from the calculation, for $n\neq
m$:

$\displaystyle\int_{\mathbb{D}}z^{n}\overline{z^{m}}\mu(\left|z\right|)dA(z)$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{1}\int_{0}^{2\pi}(re^{i\theta})^{n}(re^{-i\theta})^{m}\mu(\left|re^{i\theta}\right|)rd\theta
dr$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{1}r^{n+m+1}\mu(r)\int_{0}^{2\pi}e^{i\theta(n-m)}d\theta
dr=0\text{.}$

Thus, we can set

$\mu(\left|z\right|)=2\alpha(1-\left|z\right|^{2})^{2\alpha-1}1_{C_{n}(R;\mathbb{D})}(z)=2\alpha(1-\left|z\right|^{2})^{2\alpha-1}1_{C_{n}(R;\mathbb{R})}\left(\frac{|z|^{2}}{1-|z|^{2}}\right)$

and combine (2.1) with the orthogonality of $e_{n}^{\alpha}(z)$ with respect
to $\mu(\left|z\right|)dA(z)$ to obtain

$P_{C_{n}(R;\mathbb{D})}^{(\alpha)}(e_{k}^{\alpha})(w)=\int_{\mathbb{D}}e_{k}^{\alpha}(z)\overline{\mathcal{K}_{\mathbb{D}}^{\alpha}(z,w)}\,\mu(\left|z\right|)dA(z)=\left[\int_{\mathbb{D}}\left|e_{k}^{\alpha}(z)\right|^{2}\,\mu(\left|z\right|)dz\right]e_{k}^{\alpha}(w)\text{.}$

Thus, $e_{n}^{\alpha}$ is an eigenfunction of
$P_{C_{n}(R;\mathbb{D})}^{(\alpha)}$ with eigenvalue

$\displaystyle\lambda_{k}^{(\alpha)}(C_{n}(R;\mathbb{D}))$ $\displaystyle=$
$\displaystyle\int_{\mathbb{D}}\left|e_{k}^{\alpha}(z)\right|^{2}\,\mu(\left|z\right|)dA(z)$
$\displaystyle=$ $\displaystyle
2\alpha\gamma_{k}\int_{\mathbb{D}}\left|z\right|^{2k}(1-\left|z\right|^{2})^{2\alpha-1}\,\,1_{C_{n}(R;\mathbb{D})}(z)dA(z)$
$\displaystyle=$ $\displaystyle
2\alpha\gamma_{k}\int_{\mathbb{D}}\left|z\right|^{2k}(1-\left|z\right|^{2})^{2\alpha-1}1_{C_{n}(R;\mathbb{R})}\left(\frac{|z|^{2}}{1-|z|^{2}}\right)dA(z)$
$\displaystyle=$ $\displaystyle
4\alpha\gamma_{k}\int_{0}^{1}1_{C_{n}(R;\mathbb{R})}\left(\frac{r^{2}}{1-r^{2}}\right)r^{2k+1}(1-r^{2})^{2\alpha-1}\,dr\text{.}$

Using the substitutions $s=r^{2}$ and $t=\frac{s}{1-s}=\frac{1}{1-s}-1$,

$\displaystyle\lambda_{k}^{(\alpha)}(C_{n}(R;\mathbb{D}))$
$\displaystyle=2\alpha\gamma_{k}\int_{0}^{1}1_{C_{n}(R;\mathbb{R})}\left(\frac{s}{1-s}\right)s^{k}(1-s)^{2\alpha-1}\,ds$
$\displaystyle=2\alpha\gamma_{k}\int_{0}^{\infty}1_{C_{n}(R;\mathbb{R})}(t)\left(\frac{t}{1+t}\right)^{k}(1+t)^{-(2\alpha+1)}\,dt$
$\displaystyle=\int_{0}^{\infty}1_{C_{n}(R;\mathbb{R})}(t)g_{k}(t;\alpha)\,dt\text{.}$

######  Remark 1.

We observe that the function $g_{k}(\cdot\,;\alpha)$ is the density function
of the _Beta prime distribution_ $\beta^{\prime}(k+1,2\alpha)$ with form
parameters $k+1$ and $2\alpha$; see [21, Equation (25.79)]. In particular,
this implies

(4.2) $\int_{0}^{\infty}g_{k}(t;\alpha)\,dt=1$

and the following probabilistic interpretation of the eigenvalues

$\lambda_{k}^{(\alpha)}(C_{n}(R;\mathbb{D}))=\mathbb{E}\left(1_{C_{n}(R;\mathbb{R})}(X)\right)\quad\text{if
}X\sim\beta^{\prime}(k+1,2\alpha)\text{.}$

Recalling the definition of $g_{k}(\cdot\,;\alpha)$ and applying [17, Section
3.197.8] (with $\mu=1$, $\alpha=1$, $\lambda=-(k+1+2\alpha)$ and $u=y$) shows
that the cumulative distribution function of $\beta^{\prime}(k+1,2\alpha)$ is
given by

$F_{k,\alpha}(y)=\int_{0}^{y}g_{k}(t;\alpha)\,dt=\frac{y^{1+k}}{(1+k)\cdot
B(1+k,2\alpha)}\cdot_{2}F_{1}(k+1+2\alpha,k+1;k+2;-y)\text{,}$

in terms of the ordinary hypergeometric function
${}_{2}F_{1}\left[a,b,c;z\right]=\sum_{j=0}^{\infty}\frac{(a)_{j}(b)_{j}}{j!(c)_{j}}z^{j}$.
When $n=0$, $1_{C_{0}(R;\mathbb{R})}=1_{[0,R]}$ and

$\lambda_{k}^{(\alpha)}(C_{0}(R;\mathbb{D}))=\frac{R^{1+k}}{(1+k)\cdot
B(1+k,2\alpha)}\cdot_{2}F_{1}(k+1+2\alpha,1+k;k+2;-R)\text{.}$

###  4.2. Upper bounding the eigenvalues

The upper bounds for the eigenvalues depend on the following pointwise
estimate for the density functions $g_{k}(\cdot\,;\alpha)$. The details of the
proof are in Appendix 5.

######  Lemma 1.

For each $\alpha\in(0,\infty)$, there is a constant $C=C(\alpha)>0$ satisfying

$g_{k}(x;\alpha)\leq C\cdot(1+x)^{-1}\qquad\forall\,x\in(0,\infty)\text{ and
}k\in\mathbb{N}_{0}.$

In this section, we prove the following upper bound for the eigenvalues
$\lambda_{k}^{(\alpha)}(C_{n}(R;\mathbb{D}))$:

######  Proposition 3.

For each $\alpha\in(0,\infty)$, there is a constant $C=C(\alpha)>0$ satisfying

$\lambda_{k}^{(\alpha)}(C_{n}(R;\mathbb{D}))\leq
C\begin{cases}(2/3)^{n}R,&\text{if }0<R\leq 1,\\\ (2/3)^{n}R^{1-\frac{\ln
2}{\ln 3}},&\text{if }1\leq R\leq 3^{n},\\\ 1,&\text{if }R\geq
3^{n}\end{cases}$

for all $k\in\mathbb{N}_{0}$, $n\in\mathbb{N}$, and $R\in(0,\infty)$.

###### Proof.

First of all, recall from Equation (4.2) that each $g_{k}(\cdot\,;\alpha)$ is
a probability density function on $(0,\infty)$, so that

$\lambda_{k}^{(\alpha)}(C_{n}(R;\mathbb{D}))=\int_{C_{n}(R;\mathbb{R})}g_{k}(y;\alpha)\,dy\leq\int_{0}^{\infty}g_{k}(y;\alpha)\,dy=1.$

This establishes the desired estimate in case of $R\geq 3^{n}$. Next, recall
that the Lebesgue measure $\lambda(C_{n}(R;\mathbb{R}))$ of the Cantor set
$C_{n}(R;\mathbb{R})$ is $\lambda(C_{n}(R;\mathbb{R}))=(2/3)^{n}R$.
Furthermore, Lemma 1 yields a constant $C_{1}=C_{1}(\alpha)>0$ satisfying
$g_{k}(x;\alpha)\leq C\cdot(1+x)^{-1}\leq C$ for all $x\in(0,\infty)$ and
$k\in\mathbb{N}_{0}$. Therefore,

$\lambda_{k}^{(\alpha)}(C_{n}(R;\mathbb{D}))=\int_{C_{n}(R;\mathbb{R})}g_{k}(y;\alpha)\,dy\leq
C\cdot\lambda(C_{n}(R;\mathbb{R}))=C\cdot(2/3)^{n}R.$

This proves the desired estimate for the case that $0<R\leq 1$. Finally, let
us consider the case $1\leq R\leq 3^{n}$. Due to this assumption on $R$, there
is $t\in\\{0,\dots,n-1\\}$ such that

$3^{t}\leq R\leq 3^{t+1}\quad\text{and hence}\quad\frac{\ln R}{\ln 3}-1\leq
t\leq\frac{\ln R}{\ln 3}.$

Using the representation ((3.1)) of the Cantor set $C_{n}(R;\mathbb{R})$ and
$\Omega^{(n)}\setminus\\{0\\}=\biguplus_{m=1}^{n}\Omega_{m}^{(n)}$, we see

$\lambda_{k}^{(\alpha)}(C_{n}(R;\mathbb{D}))=\int_{0}^{R/3^{n}}g_{k}(y;\alpha)\,dy+\sum_{m=1}^{n}\,\sum_{a\in\Omega_{m}^{(n)}}\,\int_{R\cdot
b_{a}}^{R\cdot(b_{a}+3^{-n})}g_{k}(y;\alpha)\,dy.$

If $a\in\Omega_{m}^{(n)}$, then ((3.3)) implies $b_{a}\geq 2/3^{m}\geq 3^{-m}$
and $R\cdot b_{a}\geq 3^{t-m}$. Moreover, $g_{k}(x;\alpha)\leq
C\cdot(1+x)^{-1}\leq C$ gives $\int_{0}^{R/3^{n}}g_{k}(y;\alpha)\,dy\leq
C\frac{R}{3^{n}}$. Thus, for $y\in[R\cdot b_{a},R\cdot(b_{a}+3^{-n})]$ that

$(1+y)^{-1}\leq(1+3^{t-m})^{-1}\leq 3^{-(t-m)_{+}}\quad\text{where}\quad
x_{+}:=\max\\{0,x\\},$

and hence

$\int_{R\cdot b_{a}}^{R\cdot(b_{a}+3^{-n})}g_{k}(y;\alpha)\,dy\leq
C\int_{R\cdot b_{a}}^{R\cdot(b_{a}+3^{-n})}(1+y)^{-1}\,dy\leq
C\cdot\frac{R}{3^{n}}\cdot 3^{-(t-m)_{+}}.$

Recalling, from ((3.2)) that $|\Omega_{m}^{(n)}|=2^{n-m}$, we conclude

$\sum_{m=1}^{n}\sum_{a\in\Omega_{m}^{(n)}}\int_{R\cdot
b_{a}}^{R\cdot(b_{a}+3^{-n})}g_{k}(y;\alpha)\,dy\leq
C\,(2/3)^{n}\,R\,2^{-t}\sum_{m=1}^{n}2^{t-m}3^{-(t-m)_{+}}$

Introducing the new summation index $\ell=t-m$, we see that

$\sum_{m=1}^{n}2^{t-m}3^{-(t-m)_{+}}\leq\sum_{\ell\in\mathbb{Z}}2^{\ell}3^{-\ell_{+}}=\sum_{\ell=-\infty}^{-1}2^{\ell}+\sum_{\ell=0}^{\infty}(2/3)^{\ell}=1+\frac{1}{1-\frac{2}{3}}=4.$

Furthermore, since $t\geq\frac{\ln R}{\ln 3}-1$, we have $2^{-t}\leq 2\cdot
2^{-\ln R/\ln 3}=2\cdot R^{-\ln 2/\ln 3}$. By combining the estimates that we
collected, we finally conclude that

$4\,\lambda_{k}^{(\alpha)}\left(C_{n}(R;\mathbb{D})\right)\leq
C\frac{R}{3^{n}}+8C\,(2/3)^{n}R^{1-\frac{\ln 2}{\ln 3}}.$

It remains to observe that $R/3^{n}\leq(2/3)^{n}R^{1-\frac{\ln 2}{\ln 3}}$,
which easily follows from the condition $R\leq 3^{n}$.

###  4.3. Lower bounding the eigenvalue

In this subsection we prove that the eigenvalue
$\lambda_{0}^{(\alpha)}(C_{n}(R;\mathbb{D}))$ of
$P_{C_{n}(R;\mathbb{D})}^{(\alpha)}$ fulfills a lower bound which matches the
upper bound from Proposition 3.

######  Proposition 4.

For each $\alpha\in(0,\infty)$, there is a constant $C=C(\alpha)>0$ satisfying

$\lambda_{0}^{(\alpha)}(C_{n}(R;\mathbb{D}))\geq
C\begin{cases}(2/3)^{n}R,&\text{if }0<R\leq 1,\\\ (2/3)^{n}R^{1-\frac{\ln
2}{\ln 3}},&\text{if }1\leq R\leq 3^{n},\\\ 1,&\text{if }R\geq
3^{n}\end{cases}$

for all $n\in\mathbb{N}$ and $R\in(0,\infty)$.

###### Proof.

In case of $R\geq 3^{n}$, we have
$C_{n}(R;\mathbb{R})\supset[0,R/3^{n}]\supset[0,1]$, and hence

$\lambda_{0}^{(\alpha)}(C_{n}(R;\mathbb{D}))\geq\int_{0}^{1}g_{0}(y;\alpha)\,dy=:C_{1}(\alpha)>0,$

since $g_{0}(\cdot\,;\alpha)$ is a positive continuous function. Likewise, if
$0<R\leq 1$, then ${C_{n}(R;\mathbb{R})\subset[0,1]}$. Since the continuous,
positive function $g_{0}(\cdot\,;\alpha)$ is lower bounded on the compact set
$[0,1]$ (say, $g_{0}(x;\alpha)\geq 4C_{2}$ for $x\in[0,1]$ with
$C_{2}=C_{2}(\alpha)>0$), we thus see

$\lambda_{0}^{(\alpha)}(C_{n}(R;\mathbb{D}))=\int_{C_{n}(R;\mathbb{R})}g_{0}(y;\alpha)\,dy\geq
C_{2}\int_{C_{n}(R;\mathbb{R})}1\,dy=C_{2}\,(2/3)^{n}\,R,$

proving the desired bound for the case $0<R\leq 1$. Finally, consider the case
$1\leq R\leq 3^{n}$. For brevity, define $C_{3}:=[4\,B(1,2\alpha)]^{-1}$.
Choose $t\in\\{0,\dots,n-1\\}$ such that $3^{t}\leq R\leq 3^{t+1}$ and
$\frac{\ln R}{\ln 3}-1\leq t\leq\frac{\ln R}{\ln 3}$. Using the representation
(3.1) of the Cantor set, and observing that
$\Omega_{t+1}^{(n)}\subset\Omega^{(n)}$ since $t+1\in\\{1,\dots,n\\}$,

$\lambda_{0}^{(\alpha)}(C_{n}(R;\mathbb{D}))\geq
C_{3}\sum_{a\in\Omega_{t+1}^{(n)}}\int_{R\cdot
b_{a}}^{R\cdot(b_{a}+3^{-n})}(1+y)^{-(1+2\alpha)}\,dy.$

Now, note that if $a\in\Omega_{t+1}^{(n)}$ and $y\leq R\cdot(b_{a}+3^{-n})$,
then Equation (3.3) shows that

$y\leq R\cdot(b_{a}+3^{-n})\leq R\cdot(3\cdot 3^{-(t+1)}+3^{-n})\leq 4R\cdot
3^{-(t+1)}\leq 4,$

and hence $(1+y)^{-(1+2\alpha)}\geq 4^{-(1+2\alpha)}=:C_{4}$. Next, recall
from Equation (3.2) that ${|\Omega_{t+1}^{(n)}|=2^{n-t-1}}$. Overall, we thus
see as desired that

$\displaystyle\lambda_{0}^{(\alpha)}(C_{n}(R;\mathbb{D}))$ $\displaystyle\geq
C_{3}C_{4}\cdot
2^{n-t-1}\frac{R}{3^{n}}=\frac{C_{3}C_{4}}{2}\cdot(2/3)^{n}\cdot R\cdot
2^{-t}$ $\displaystyle\geq\frac{C_{3}C_{4}}{2}\cdot(2/3)^{n}\cdot
2^{-\frac{\ln R}{\ln 3}}\cdot R=\frac{C_{3}C_{4}}{2}\cdot(2/3)^{n}\cdot
R^{1-\frac{\ln 2}{\ln 3}}\text{.}$

###  4.4. Proof of Theorem 1

Combining Propositions 2, 3 and 4, leads to

(4.3) $C_{1}\begin{cases}(2/3)^{n}R,&\text{if }0<R\leq 1,\\\
(2/3)^{n}R^{1-\frac{\ln 2}{\ln 3}},&\text{if }1\leq R\leq 3^{n},\\\
1,&\text{if }R\geq
3^{n}.\end{cases}\leq\left\|P_{C_{n}(R;\mathbb{D})}^{(\alpha)}\right\|_{op}\leq
C_{2}\begin{cases}(2/3)^{n}R,&\text{if }0<R\leq 1,\\\ (2/3)^{n}R^{1-\frac{\ln
2}{\ln 3}},&\text{if }1\leq R\leq 3^{n},\\\ 1,&\text{if }R\geq
3^{n}.\end{cases}\text{.}$

The condition $3^{n}\asymp R^{2}$ implies for large $R$ that $1\leq R\leq
3^{n}$ $3^{n/2}\asymp R$, $R^{\frac{\ln 2}{\ln 3}}\asymp 2^{n/2}$ and
$R^{2\frac{\ln 2}{\ln 3}}\sim 2^{n}$, leading to, as $R\rightarrow\infty$,

$(2/3)^{n}R^{1-\frac{\ln 2}{\ln 3}}\asymp(2/3)^{n/2}\asymp R^{-1+\frac{\ln
2}{\ln 3}}$

and the result follows from (4.3).

##  5\. Proof of Lemma 1

The proof will be given in three steps.

Step 1 _(Estimating the Beta function):_ By definition, the Beta function can
be written in terms of the Gamma function as
${B(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}}$ for $x,y>0$. Furthermore,
a precise form or Stirling’s formula (see [20]) shows that the Gamma function
satisfies

$\Gamma(x)=\sqrt{2\pi/x}\cdot(x/e)^{x}\cdot e^{\mu(x)}\quad\text{for}\quad
x>0\quad\text{where}\quad 0\leq\mu(x)\leq(12\,x)^{-1}.$

Note that $0\leq\mu(k+1+2\alpha)\leq\frac{1}{12(k+1+2\alpha)}\leq\frac{1}{12}$
and $0\leq\mu(k+1)\leq\frac{1}{12(k+1)}\leq\frac{1}{12}$, since
$k+1+2\alpha\geq k+1\geq 1$. Therefore, we see that

$\displaystyle B(k+1,2\alpha)^{-1}$
$\displaystyle=\frac{\Gamma(k+1+2\alpha)}{\Gamma(2\alpha)\,\Gamma(1+k)}$
$\displaystyle=\frac{\sqrt{\frac{2\pi}{k+1+2\alpha}}\left(\frac{k+1+2\alpha}{e}\right)^{k+1+2\alpha}\,e^{\mu(k+1+2\alpha)}}{\Gamma(2\alpha)\,\sqrt{\frac{2\pi}{k+1}}\,\left(\frac{k+1}{e}\right)^{k+1}\,e^{\mu(k+1)}}$
$\displaystyle\leq
C_{1}\sqrt{\frac{k+1}{k+1+2\alpha}}\left(\frac{k+1+2\alpha}{e}\right)^{2\alpha}\left(1+\frac{2\alpha}{k+1}\right)^{k+1},$

for a suitable constant $C_{1}=C_{1}(\alpha)$. Next, choose
$\beta\in\mathbb{N}$ such that $2\alpha\leq\beta$. It is well-known (see for
instance [3, Chapter II, Section 4, Exercise 3]) that
$(1+\frac{\beta}{n})^{n}\rightarrow e^{\beta}$ as $n\rightarrow\infty$. Since
convergent sequences are bounded, this implies that there is a constant
$C_{2}=C_{2}(\alpha)>0$ satisfying
$\left({1+\frac{2\alpha}{k+1}}\right){{}^{k+1}\leq(1+\frac{\beta}{k+1})^{k+1}\leq
C_{2}}$ for all $k\in\mathbb{N}$. Moreover, since $\alpha>0$, we have
$\sqrt{(k+1)/(k+1+2\alpha)}\leq 1$. Finally, we can clearly choose
$C_{3}=C_{3}(\alpha)>0$ such that $\left((k+1+2\alpha)/e\right)^{2\alpha}\leq
C_{3}\,k^{2\alpha}$ for all $k\in\mathbb{N}$ (but not for $k=0$). Gathering it
all,

$B(k+1,2\alpha)^{-1}\leq C_{1}C_{2}C_{3}\cdot
k^{2\alpha}\qquad\forall\,k\in\mathbb{N}.$

Step 2 _(Estimating $k^{2\alpha}\cdot y^{k}$):_ Let $y\in(0,1)$ be fixed, and
define

$f:[0,\infty)\rightarrow[0,\infty),\text{ \ \ }t\mapsto t^{2\alpha}\cdot
y^{t}\text{.}$

Then $f$ is differentiable on $(0,\infty)$ with derivative

$f^{\prime}(t)=2\alpha\,t^{2\alpha-1}y^{t}+t^{2\alpha}\,\ln(y)\,y^{t}=t^{2\alpha-1}\,y^{t}(2\alpha+t\ln(y))\text{.}$

From this, it follows that if we define $t_{0}:=-\frac{2\alpha}{\ln
y}=\frac{2\alpha}{\ln(1/y)}\in(0,\infty)$, then $f^{\prime}(t)>0$ for
$t\in(0,t_{0})$ and $f^{\prime}(t)<0$ for $t\in(t_{0},\infty)$. Hence, $f$ has
a global maximum in $t=t_{0}$, showing that

$t^{2\alpha}y^{t}=f(t)\leq
f(t_{0})=\left(\frac{2\alpha}{\ln(1/y)}\right)^{2\alpha}y^{-\frac{2\alpha}{\ln(y)}}=\left(\frac{2\alpha}{\ln(1/y)}\right)^{2\alpha}e^{-2\alpha}$

for all $t\in[0,\infty)$.

Step 3 _(Completing the proof):_ If we apply the estimate from the preceding
step for $t=k$ and $y=\frac{x}{1+x}\in(0,1)$ (where $x\in(0,\infty)$), then we
see that there is a constant $C_{4}=C_{4}(\alpha)>0$ such that

$k^{2\alpha}\left(\frac{x}{1+x}\right)^{k}\leq
C_{4}\left[\ln\frac{1+x}{x}\right]^{-2\alpha}=C_{4}\ln(1+x^{-1})^{-2\alpha}\quad\forall\,x\in(0,\infty).$

Now, for $x\geq 1$ note that $1+x^{-1}\leq 2$, and hence

$\ln(1+x^{-1})=\int_{1}^{1+x^{-1}}t^{-1}\,dt\geq\frac{1}{2}\,x^{-1}$

from which we see—because of $\alpha>0$—that

$\ln(1+x^{-1})^{-2\alpha}\leq 2^{2\alpha}\cdot x^{2\alpha}\leq
2^{2\alpha}\cdot(1+x)^{2\alpha}.$

Likewise, if $0<x\leq 1$ then $\ln(1+x^{-1})\geq\ln(2)$ and hence

$\ln(1+x^{-1})^{-2\alpha}\leq[\ln(2)]^{-2\alpha}\leq[\ln(2)]^{-2\alpha}\cdot(1+x)^{2\alpha}.$

All in all, we have thus shown that there is a constant
$C_{5}=C_{5}(\alpha)>0$ satisfying

$k^{2\alpha}\left(\frac{x}{1+x}\right)^{k}\leq
C_{5}\cdot(1+x)^{2\alpha}\qquad\forall\,x\in[0,\infty)\text{ and
}k\in\mathbb{N}_{0}.$

Combining this with the estimate from Step 1, we see for $k\in\mathbb{N}$ that

$\displaystyle g_{k}(y;\alpha)$
$\displaystyle=B(k+1,2\alpha)^{-1}\left(\frac{x}{1+x}\right)^{k}(1+x)^{-(1+2\alpha)}$
$\displaystyle\leq C_{1}C_{2}C_{3}\cdot
k^{2\alpha}\left(\frac{x}{1+x}\right)^{k}\cdot(1+x)^{-2\alpha}\,(1+x)^{-1}\leq
C_{1}C_{2}C_{3}C_{5}\cdot(1+x)^{-1}.$

Finally, in case of $k=0$, we see directly from the definition of the Beta
function that

$B(k+1,2\alpha)=B(1,2\alpha)=\int_{0}^{1}(1-t)^{2\alpha-1}\,dt=\int_{0}^{1}s^{2\alpha-1}\,ds=\frac{1}{2\alpha},$

and hence

$g_{0}(y;\alpha)=2\alpha(1+x)^{-(1+2\alpha)}\leq 2\alpha\cdot(1+x)^{-1},$

since $1+x\geq 1$ and $1+2\alpha\geq 1$.

######  Acknowledgement 1.

We thank Helge Knutsen for several comments and corrections on an earlier
version of the manuscript.

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