# HUVECs-encapsulation via Millimeter-sized Alginate Droplets ††thanks:
HUVECs-encapsulation via Millimeter-sized Alginate Droplets: Tran K, et al.,
DOI:000000/11111.

Khanh Tran1; Brenda A.A.B. Ametepe1; Erika Gomez1;
Daniel Ramos1; Clare Kim1; Ga-Young Kelly Suh1;
Siavash Ahrar1,†; Perla Ayala 1,†

1 Department of Biomedical Engineering, CSU Long Beach, CA, USA
†Co-corresponding authors.

> Abstract
>
>
> Droplet microfluidics are a powerful approach for hydrogel cell
> encapsulations. Much of the field has focused on single-cell encapsulations
> with pico-nanoliter droplet volumes necessary for single-cell sequencing or
> high-throughput screening. These small volumes, however, limit the use of
> hydrogel droplets for tissue engineering or cell therapies. We describe
> simple droplet microfluidics to generate millimeter-sized alginate droplets
> and demonstrate their use for cell encapsulations. This effort builds on our
> recent efforts [1], specifically by replacing the glass slide forming the
> bottom layer of the chamber with a more hydrophobic acrylic (PMMA) layer to
> improve the alginate-in-oil droplet formation. Using glass layer and PMMA
> layer devices, we characterized the tunable production of water-in-oil
> droplets (average droplet lengths ranged from 0.8 to 3.7 mm). Next, PMMA
> layer devices were used to demonstrate the tunable generation of alginate-
> in-oil droplets (average droplet lengths ranged from 3-6 mm). Increasing the
> flow ratio (Q.ratio = Q.oil/Q.alginate) led to more uniform droplets as
> measured by the coefficient of variance (COV%), which was approximately 5%.
> Finally, a proof-of-use experiment used HUVEC-encapsulated alginate droplets
> as part of a scratch-healing assay. Specifically, HUVEC-encapsulated
> droplets (AH droplets) led to recovery of 3T3 fibroblast monolayers compared
> to no droplets or cell-free droplets (A droplets). Our results extended the
> use of simple microfluidics to generate and retrieve millimeter-sized
> alginate droplets for effective cell encapsulations.
> Key words: Millimeter-sized droplets, Hydrogel encapsulations, Alginate.

##  1 Introduction

Hydrogel encapsulation of cells is an attractive approach for the sustained
release of bioactive agents for tissue engineering in the laboratory and
clinic [2, 3]. Hydrogels provide 3D microenvironments supporting donor cells
and prevent potential immune rejection. Droplet microfluidics has emerged as a
valuable tool for hydrogel encapsulation of cells [4, 5]. Yet, driven by
single-cell sequencing and other high throughput screenings, most droplets
have focused on picoliter to nanoliter volumes [6, 7, 8, 9, 10]. These pL-nL
volumes are essential for accelerating reactions or enabling sorting
strategies. However, the limited reagent volumes (e.g., media for cells),
preclude long-term cell culture applications via droplets due to waste buildup
or depletion of reagents. To address this limitation, microfluidics or off-
the-shelf droplet generators have been demonstrated to produce organoid and
other 3D cultures [11, 12, 13]. However, most existing systems maintain the
generated droplets inside the tubing (in-situ) for long-term monitoring or
provide limited opportunities to modify device properties [13, 11]. Therefore,
developing alternative strategies to produce, modify device geometries, and
retrieve millimeter-sized droplets for embedding cells via hydrogels is of
great interest.

Various hydrogels (e.g., alginate [14, 15], Matrigel[13], and polyethylene
glycol diacrylate[4]) have been used for droplet cell encapsulations [16]. In
this effort, we used alginate, due to its robust and tunable mechanical
properties [17], common use in tissue engineering, and broad availability.
Previously Trivedi et al. [11], using off-the-shelf components, had
successfully demonstrated the generation of large cell-encapsulate alginate
capsules/droplets that were up to 1.5 mm long. However, alginate capsules were
permanently stored inside the tubing and were not retrieved. In another study,
Zhang et al. used simple microfluidics combined with tubing to generate
uniform Matrigel organoids up to 500 $\mu$m diameter)[13]. Here the organoids
were solidified inside the tubing (temperature gelatin at 37∘C) and then
collected for further investigations. Informed by these efforts, we sought to
extend the use of mm-sized droplets by generating and retrieving alginate
droplets. Additionally, we selected Human umbilical vein endothelial cells
(HUVECs) as our target cells for encapsulation. Since HUVECs have been
effectively use as a model for human endothelium and angiogenesis studies[18,
19].

We demonstrated millimeter-sized HUVECs-encapsulated alginate droplets (Figure
1 A). To this aim, first, our recent co-flow [1] microfluidics were updated by
replacing the glass slides forming their bottom layer with the more
hydrophobic acrylic (PMMA) slides. Using water-in-oil experiments, glass-slide
and PMMA-slide devices were characterized, and tunable droplet generations
were verified (average droplet lengths ranged from 0.8 to 3.7 mm). Next, the
PMMA-slide chips were used to generate tunable alginate-in-oil (average
droplet lengths ranged from 3-6 mm). Increasing the flow ratios led to more
uniform droplets as measured by the coefficient of variance (COV%), which was
approximately 5%. Finally, HUVEC-encapsulated alginate droplets were generated
and used as part of a scratch-healing assay. Specifically, HUVEC-encapsulated
droplets led to faster healing of 3T3 fibroblast monolayers compared to no
droplets or cell-free droplets. Collectively, these results extended the use
of co-flow microfluidics to generate and retrieve millimeter-sized alginate
droplets for effective cell encapsulations.

##  2 Materials and Methods

###  2.1 Microfluidics design and fabrication

Co-flow microfluidics were designed similarly to our recent effort [1]. Mold
for the channels was designed with 2D CAD software. Figure 1. C demonstrates
the design and critical dimensions of the channels. Briefly, the channels were
2 mm wide and 55 mm long. The diameter of the inlet was designed at 3.6 mm,
and the outlet was 5 mm (Figure 1 C). Rectangular PMMA slides (68mm x 25mm)
were also cut for experiments requiring a PMMA slide. These channels and the
slides were cut from PMMA sheets (1/16" height, McMaster-Carr part number
8560K171) via a 40 W desktop CO2 laser cutter (Glowforge Plus, Glowforge,
Seattle, WA) following the manufacturer’s recommendations for power and speed.

After laser cutting, PMMA parts were retrieved and glued to a glass microscope
slide (68 x 25 mm) using a strong adhesive (Super Glue, Gorilla) to create the
fluidic molds. Molds were rested for approximately 30 minutes to allow for
strong adhesion. 77 g of Polydimethylsiloxane (PDMS) (Dow Sylgard™ 184
Silicone 0.5 Kg Elastomer Kit) was prepared using the standard 10:1 w/w ratio
(i.e., 70 g base and 7 g curing agent). Molds were placed inside a petri dish,
and the PDMS mixture was added. Molds were placed inside a vacuum chamber (60
minutes or until clear of bubbles) to remove dissolved air bubbles. PDMS and
the molds were baked for approximately 10 hours at 90∘C. Once the PDMS was
fully cured, the microchannels were cut out with a border frame (15mm x 60mm)
and separated from the remaining PDMS and glass slide. Care was taken to
prevent detachment between the plastic mold and the glass slide. A first hole
was made in the center of the inlet using a 2.5 mm diameter biopsy punch. A
second 2.5 mm hole was cut inside the channel, 15 mm from the first hole. The
first inlet was used for the continuous phase (oil), while the second was for
the dispersed phase (water or alginate solution). The outlet was cut using a 5
mm biopsy punch. The larger outlet was helpful in maintaining droplet shape as
the droplets exited the microfluidic channel. PDMS parts were permanently
attached to glass slides via a standard plasma treatment (Harrick Plasma,
Basic Plasma Cleaner, maximum RF power 18W) for the glass-slide systems. For
the PMMA-slide devices, PDMS parts were attached using uncured PDMS. The
finalized chip was cured at 90 ∘ C for a minimum of 3 hours to ensure robust
attachment. Care was taken to prevent any PDMS from coating the channel
surfaces. Figure 1 D demonstrates the simplified workflow of building the
PMMA-slide chambers.

###  2.2 Cell culture and alginate-encapsulation

3T3 fibroblasts were obtained from ATCC (American Type Culture Collection,
USA) and cultured in DMEM supplemented with calf bovine serum. Human umbilical
vein endothelial cells (HUVECs) were obtained from Lonza and cultured in EBM-2
media, supplemented with 10% fetal bovine serum (Lonza) according to the
provider’s instructions. Cells were maintained at 37 ∘C, 5% CO2, and the
culture media was changed every 2-3 days. HUVECs from passages 5 to 12 were
used in this study. Before cell-encapsulation experiments, HUVECs were
collected by standard trypsin-EDTA treatment and suspended in the hydrogel
culture and media. A 2% (w/v) Sodium alginate solution was prepared by
dissolving Sodium alginate powder (alginic acid sodium salt, Sigma Aldrich) in
1X Dulbecco’s phosphate-buffered saline (1X PBS). HUVECs were seeded within
these 2% alginate solutions and cultured for 2 days at 10000 cells per mL
density. A homogeneous hydrogel precursor solution containing 2% (w/v)
alginate and HUVECs were used as the dispersed phase for cell encapsulation.
Food-grade olive oil containing 1% (v/v) surfactant (SPAN 80) was used for the
continuous phase. For water-in-oil characterization experiments, DI water with
trace food dye was used.

Cell and alginate (1 mL syringe) and oil (10 mL syringe) were each loaded
immediately before an experiment. Two independent syringe pumps (NE-300 Just
Infusion - New Era) were used to deliver and control each solution. Silicone
tubes (inner diameter of 1.5 mm) were used to connect each syringe (via a
dispensing needle) to the chip. In cell-encapsulation experiments, the flow
rate of the cell solution was kept constant at 0.5 mL/hr to avoid cell damage
(e.g., due to shearing force during the injection). The flow rate of the
continuous phase was set to 10 mL/hr. Outlet tubes were placed inside a
graduated cylinder containing a 10 wt% CaCl2 solution. After the microfluidic
production of the alginate droplets, droplets were delivered to the CaCl2
solution. Droplets were incubated in the solution for 20 minutes before their
collection for wound scratch assays. The system was housed inside a biosafety
cabinet to prevent contamination.

###  2.3 Imaging and data analysis

For characterization experiments, the operation of the devices was recorded
using a standard cellphone camera held via a tripod (UBeesize 10" Ring Light
with Tripod Stand). Droplet dimensions were measured using NIH ImageJ
software. In some experiments, since the droplets were elongated, the ratio of
length over width was considered instead of the diameter. An EVOS Cell-Imaging
system (bright-field images) was used for wound healing assays.

##  3 Results

###  3.1 Characterizing PMMA-slide channels

Droplet generation is informed by the channel’s surface proprieties (e.g.,
hydrophobicity) added to considerations such as flow rates and the shape of
the chamber. Most microfluidics use microscope glass slides as their base.
Glass has ideal imaging properties and provides sufficent mechanical
durability. However, for aqueous-in-oil droplets (particularly hydrogels) the
glass surfaces may need to be more hydrophobic. To this end, chemical
treatments such as silanization or rain repellent have been used. These
chemical treatments may need to be reapplied or could introduce other
complications during fabrication. Moreover, for hydrogels, other rheological
properties, for example, higher viscosity (e.g., alginate has a viscosity of
157 mPa.s compared to 1 mPa.s of water) could further complicate droplet
formations. Therefore, improvement to the system for increasing the channel
hydrophobicity are desirable. We considered replacing the standard glass layer
with a more hydrophobic surface. PMMA and PDMS (both more hydrophobic than
glass) were considered for this aim. While PDMS is more hydrophobic than PMMA,
it was determined that attaching the PDMS channels to PMMA slides were more
reliable. PMMA slides were also more mechanically durable as compared to PDMS.
To verify the hydrophobic properties of PMMA, contact angles for water and 1%
alginate solutions on both glass and PMMA were measured (supplemental
section). Experiments confirmed that both solutions had a higher contact angle
on PMMA as compared to glass. It was hypothesized that the PMMA chips, due to
their hydrophobicity, would produce more uniform droplets as compared to the
standard chips. To test this hypothesis first, water-in-oil experiments were
conducted.

In this report, PDMS chambers with the exact dimensions were made. The first
chamber was attached to a glass slide via plasma treatment. The second chamber
was attached to a PMMA slide. Both chambers were tested under the same
volumetric flow conditions to determine the role of surfaces in droplet
formation. Specifically, four flow ratios (Q.ratio = Q.oil/Q.water) were
tested for each chip. These were Q.ratio 2, 3, 4, and 5. For these
experiments, the volumetric flow rate of oil was fixed (Q.oil = 5 mL/hr), and
the Q.water was adjusted. These flow rates were selected based on their
success in our preliminary experiments. After the successful generation of
water-in-oil droplets, tunable generation of alginate droplets were
investigated. Alginate is a linear polyanionic block copolymer composed of two
units; its gelation is mediated through cations - typically calcium ions. The
glass layer devices resulted in threading or merging droplets. Therefore their
use were abandoned. Here the results from PMMA-slide devices are shared.
Figure 3 demonstrates the tunable production of droplets for four Q.ratios of
2, 3, 4, and 5. The volumetric flow rate of the oil was fixed (Q.oil = 5
mL/hr). In this experiment, each flow condition produced a statistically
different droplet size. The length/width ratio ranged from 2-4 for the
alginate droplets. Histogram of droplet size and the coefficient of variance
(COV%) for each condition is reported. After the successful generation of
water-in-oil droplets, tunable generation of alginate droplets were
investigated. Alginate is a linear polyanionic block copolymer composed of two
units; its gelation is mediated through cations - typically calcium ions. The
glass layer devices resulted in threading or merging droplets. Therefore their
use were abandoned. Here the results from PMMA-slide devices are shared.
Figure 3 demonstrates the tunable production of droplets for four Q.ratios of
2, 3, 4, and 5. The volumetric flow rate of the oil was fixed (Q.oil = 5
mL/hr). In this experiment, each flow condition produced a statistically
different droplet size. The length/width ratio ranged from 2-4 for the
alginate droplets. Histogram of droplet size and the coefficient of variance
(COV%) for each condition is reported. Droplet generation was recorded, and
ImageJ was used to measure the properties of all droplets. For each droplet,
specifically, length and width were measured. The length-to-width ratio was
considered as the critical metric to characterize each droplet. Figure 2
summarizes the results of these experiments. As expected, increasing the
Q.ratio led to smaller droplets. Channels with PMMA layers generated more
uniform droplets as measured by their standard deviation. For both systems the
length over width ratio ranged from 1 - 3. Histogram of droplet properties and
the coefficient of variance (COV%) for each condition is reported. The
frequency of droplet generation was similar for both devices.

###  3.2 Tunable generation of alignate droplets

After the successful generation of water-in-oil droplets, tunable generation
of alginate droplets were investigated. Alginate is a linear polyanionic block
copolymer composed of two units; its gelation is mediated through cations -
typically calcium ions. The glass layer devices resulted in threading or
merging droplets. Therefore their use were abandoned. Here the results from
PMMA-slide devices are shared. Figure 3 demonstrates the tunable production of
droplets for four Q.ratios of 2, 3, 4, and 5. The volumetric flow rate of the
oil was fixed (Q.oil = 5 mL/hr). In this experiment, each flow condition
produced a statistically different droplet size. The length/width ratio ranged
from 2-4 for the alginate droplets. Histogram of droplet size and the
coefficient of variance (COV%) for each condition is reported.

###  3.3 Wound-scratch Assay

Next, we encapsulated HUVECs via uniform droplets and demonstrated their use
as proof-of-concept cell therapeutics. HUVECs were cultured and pre-mixed with
alginate, as described in the methods section. The alginate-cell solution and
the oil were loaded syringes immediately before the experiments. Syringes were
attached to a chip via 1.5 mm silicone tubing. Care was taken to prevent
bubbles inside the tubes. The chips were primed with oil before the
experiments. The flow rate for the cell solution was set to 0.5 mL/hr, and the
flow rate of the oil was set to 10 mL/hr (leading to Q.ratio = 20). Alginate
droplets generated inside the chip were delivered to an 20 mL graduated
cylinder via tubing containing a 10 wt% CaCl2 solution for polymerization. The
droplets remained inside the solution for 20 minutes before their removal for
the wound-scratch assay.

For the wound scratch assay, monolayers of 3T3 fibroblast cells were prepared.
Cells were seeded on a 12 well-plate at a density of 6500 cells per cm2. A 10
$\mu$L pipette tip was used to generate the wound to remove the cells from the
monolayers. Cells were washed with PBS to remove any floating/non-adhering
cells. Using a marker, additionally, the positions of the wounds were marked
on the bottom of the 12 well-plate. The following conditions were considered
to determine the efficacy of the alginate-encapsulated HUVEC droplets (AH
droplets). The first control, only the monolayer (with no droplet) was used.
The second control used alginate droplets (A droplets) with no HUVECs.
Finally, for the experimental condition, AH droplets were used. The initial
gaps for the wounds were imaged and quantified via ImageJ. For AH and A
droplets, four droplets were placed at each corner of the wound on the
monolayer. The wound healing was assessed by measuring the gap after two days.
Moreover, to control for the effects of media, experiments were conducted for
both the 3T3 media and the HUVEC media. Figure 4 provides the results of the
experiments.

For both 3T3 and HUVEC media conditions, the initial gaps for No droplets
(first control) were reduced. 3T3 cells grew better in the 3T3 media and
reduced the gap more when compared to HUVEC media. For both media conditions,
the gap decreased in the presence of the AH droplets for both types of media.
The A beads (no HUVECs) with HUVEC media did not result in a statistically
significant change in the gap. The A beads with the 3T3 media increased the
gap. Collectively, these results provided a proof-of-concept application of
cell-encapsulated alginate beads in the context of a standard scratch assay
and wound healing. Experiments were again replicated, and gaps were measured
after 1 days. Similar results were observed.

##  4 Discussion

We investigated the use of co-flow microfluidics to generate millimeter-sized
alginate droplets. Applying the hydrophobic PMMA-layer was essential for this
goal, specifically compared to the standard glass-layer devices. We had
previously used co-flow microfluidics to generate millimeter-sized water-in-
oil and Matrigel-in-oil droplets [1]. We selected alginate and HUVECs
specifically for cell-therapeutics due to the well-established use of alginate
in wound healing, tissue engineering [20, 21], and versatile application of
HUVECs in tissue engineering [19]. At the start of this investigation, glass-
layered co-flow microfluidics was ineffective in the uniform production of
alginate-in-oil droplets. Various issues (threading, non-uniform productions,
and alginate droplets merging) were observed. To address these issues, we
sought to increase the hydrophobic properties of the channel. Therefore, we
replaced the glass layer with PMMA. The PMMA layer enabled the robust
production of large alginate droplets. Unlike other approaches that increase
the hydrophobic properties, for example, via a surface coating, there was no
need to reapply the treatment. The PMMA approach was selected due to its easy
fabrication and robust mechanical durability. However, if an increase in
hydrophobic properties were needed, other materials (including PDMS) can be
explored.

Previous studies have successfully used off-the-shelf components and capillary
glass tubes to establish droplet generators [22]. In one great example,
Trivedi et al. demonstrated a robust and practical approach for large droplets
(up to 1.5 mm) using off-the-shelf components such as tubing and connectors
[11]. Moreover, they were able to generate alginate droplets and in-situ
culture of cells (up to multiple days) inside the Teflon (PTFE) tubes. This
approach is practical and inexpensive for the production and storage of
droplets. However, as the paper describes, realizing complex geometries with
the system may be difficult (or often impossible). Additionally, the size of
the droplets depend on a tube’s dimensions (inner diameter) and their
availability. The use of additive manufacturing (3D printing) was also
considered. Indeed, 3D printing approaches are highly suitable for realizing
complex geometries. While significant progress has been made in addressing
potential challenges[23], some concerns remain regarding the biocompatibility
and potential toxicity of 3D printed parts (particularly the more desirable
STL systems) [24]. We sought to avoid these complications by using desktop
laser cutters. Similar to 3D printers, in a university setting, laser cutters
are broadly available in university (and even community) maker spaces.

##  5 Conclusion

The use of droplet microfluidics for single-cell assays, diagnostics, high-
throughput screening, and liquid handling systems has increased over the past
decade. Yet the typical small pL-nL volumes preclude the application of the
droplets for long-term cell cultures. Recent advances have been made to
generate mm-sized and in-situ cell cultures of organoids and capsules.
However, the ability to retrieve these cultures from the device/tube and
potential modifications to the channel geometries are highly desirable.
Specifically, the shape of the chambers and nozzles may be essential in
scaling and controlling the robust production of these larger droplets. The
approach and outcomes presented here provide a simple and clean-room free for
large droplet generation and seek to broaden access to these useful
technologies. The alginate-encapsulated HUVEC droplets system has obtained
good activity to promote wound healing by enhancing angiogenesis and cell
migration. This system could incorporate bioactive agents that facilitate
wound healing and promote angiogenesis (e.g., VEGF). Alginate droplets could
also be used for drug delivery vehicles and therapeutic development. This
research provides a novel and versatile strategy for fabricating cellularized
hydrogel constructs with clinically relevant dimensions.

## Acknowledgments

Authors express gratitude to members of Ahrar-Lab (specifically Brian Le).
This work was supported in part by CSULB startup funds and a CSUPERB new
investigator award to SA. Additionally, this work was supported by National
Science Foundation grant to PA.
Funding agencies are not responsible for the content of this manuscript.
KT was supported by the CSULB’s NIH BUILD Scholars program.

## Conflicts of Interest

The authors declare no conflict of interest.

## Additional Resources

## Corresponding Authors Addresses

Siavash Ahrar (Ph.D.)
Mail: Department of Biomedical Engineering, CSU Long Beach, 1250 Bellflower
Blvd.
VEC-404.A, Long Beach, CA 90840, E-mail<EMAIL_ADDRESS>

Perla Ayala (Ph.D.)
Mail: Department of Biomedical Engineering, CSU Long Beach, 1250 Bellflower
Blvd.
ET-108, Long Beach, CA 90840 E-mail<EMAIL_ADDRESS>

List of Figures:

  * •

Figure-1: Millimeter-sized Alginate Droplet generators.

  * •

Figure-2: Water-in-oil droplet production with PMMA-layer and Glass-layer
chambers.

  * •

Figure-3: Alginate-in-oil droplets with PMMA-layer devices.

  * •

Figure-4: The role of Alginate-encapsulated HUVECs droplets (AH Droplets) in
Wound healing (scratch assay).

List of Supplemental Figures:

  * •

S.Figure-1: Sample contact angles.

  * •

S.Figure-2: Alginate-in-oil droplets with glass-layer devices demonstrating
Alginate threading.

##  6 Supplemental Materials

###  6.1 Contact-angle characterization

Contact angle measurements were carried out with water and alginate to
determine the hydrophobicity and surface properties of untreated glass and
PMMA. Droplets were placed on the solid sample of Glass or PMMA. A high-
resolution camera of CellScale and software IC capture 2.5 were used to
capture the image of the droplets. ImageJ was used to calculate the angles.

###  6.2 Code and design availability

Analysis code developed for this manuscript is available upon request.

###  6.3 Statistical analysis

Statistical analysis and plots for the experiments were conducted via R. In
the analysis, the following p-value were considered:
ns: 5.00 $\times 10^{-2}$ < p <= 1.00
*: 1.00$\times 10^{-2}$< p <= 5.00$\times 10^{-2}$
**: 1.00$\times 10^{-3}$ < p <= 1.00$\times 10^{-2}$
***: 1.00$\times 10^{-4}$ < p <= 1.00$\times 10^{-3}$
****: p <= 1.00$\times 10^{-4}$

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Figure 1: Millimeter-sized Alginate Droplets. (A) Co-flow microfluidics for
the encapsulation of cells (HUVECs) inside millimeter-sized alginate droplets.
Cell were pre-mixed with alginate. Alginate-in-oil droplets are generated and
can be tuned. The droplets were polymerized inside the CaCl2 solutions. (B)
Image of the chip, production of water-in-oil droplets, and example of HUVECs
encapsulated droplets with cells stained. (C) Key dimensions and layout for
the droplet generator chip. (D) The workflow to manufacturing the mold and the
microfluidics device with a PMMA layer. Figure in part produced by BioRender.
Figure 2: Water-in-oil droplet production with Glass-layer and PMMA-layer
devices. (A) Tunable generation of water-in-oil droplets via glass-layer and
PMMA-layer devices. Increasing the flow ratio led to smaller droplets. Here
droplet size is characterized via the ratio of droplet length to width. (B)
The histogram of droplet length for Glass and PMMA layer devices. (C) Water-
in-oil droplets generated via glass-layer chambers. Water-in-oil droplets
generated via PMMA-layer chambers Examples of droplet generators. (Flow ratios
2,3,4, and 5). Figure 3: Alginate-in-oil droplet production with PMMA-layer
devices. (A) Tunable generation of alginate-in-oil droplets via PMMA-layer
devices. Increasing the flow ratio led to smaller droplets. The droplet size
is characterized via the ratio of droplet length to width (B) The histogram of
droplet length for PMMA layer devices. Increasing the Flow ratio leds to more
uniform droplets. Figure 4: Demonstration of the role of alginate-encapsulated
HUVECs droplets in wound healing via a scratch assay. (A) Characterization of
the wound gap initial day and day 2 with no droplets, alginate droplets with
no cell, and HUVECs encapsulated droplets when cells were cultured with 3T3
media. Droplets with cells and no droplet conditions led to wound healing. The
gap increased for alginate droplets without cells. Yellow lines indicate
distances that were used from sample images to quantify gaps. (B)
Characterization of the wound gap initial day and day 2 with no droplets,
alginate droplets with no cell, and HUVECs encapsulated droplets when cells
were cultured with HUVEC media. Similar to the previous experiments, droplets
with cells and no droplet conditions led to wound healing; gap increased for
alginate droplets without cells. Sample images for each condition
demonstrating measurments between boundaries/gaps at the start of the
experiment and day 2. Figure 5: Supplemental Materials Figure-1: Contact
Angles. (A) Table summary of contact angle measurements. PMMA is more
hydrophobic than glass. (B) Sample images of DI water and alginate solutions
on glass and PMAA.  Figure 6: Supplemental Materials Figure-2: Alginate-in-oil
droplet PMMA vs Glass. (A) Alginate-in-oil droplets on PMMA layer devices
(flow rations 2,3,4, and 5). Droplets form on PMMA surfaces without threading.
(B) Glass layer devices led to threading.


I(u_{j,k})_{k=1}^{M}\rVert^{2}_{H_{\vartheta^{0},\delta}}\leq\sum_{j^{\prime}=1}^{\infty}\big{(}4\delta^{-4}\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\lVert
I(u_{j^{\prime},k})_{k=1}^{M}\rVert^{2}_{H_{\vartheta^{0},\delta}}+2\lVert
I^{\prime}(u_{j^{\prime},k})_{k=1}^{M}\rVert^{2}_{H_{\vartheta^{0},\delta}}\big{)}.$

Applying again Theorem 4.2, we arrive at

$\displaystyle\sum_{j=1}^{\infty}\sigma_{j}^{-2}\lVert
I(u_{j,k})_{k=1}^{M}\rVert^{2}_{H_{\vartheta^{0},\delta}}$ $\displaystyle\leq
16\frac{\lVert\Delta
K\rVert^{4}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{8}}\lVert
I\rVert^{2}_{\operatorname{HS}(L^{2}([0,T])^{M})}$
$\displaystyle+16\frac{\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{4}}\lVert
I^{\prime}\rVert^{2}_{\operatorname{HS}(L^{2}([0,T])^{M})}+4\lVert
I^{\prime\prime}\rVert^{2}_{\operatorname{HS}(L^{2}([0,T])^{M})},$

Inserting

$\displaystyle\lVert
I^{(j)}\rVert^{2}_{\operatorname{HS}(L^{2}([0,T])^{M})}=\sum_{k,l=1}^{M}\lVert
I_{k,l}^{(j)}\rVert_{\operatorname{HS}(L^{2}([0,T]))}^{2},\qquad j=0,1,2,$

with

$\displaystyle\lVert
I_{k,k}^{(j)}\rVert_{\operatorname{HS}(L^{2}([0,T]))}^{2}=\int_{0}^{T}\int_{0}^{T}(\kappa_{k,k}^{(j)}(|t-t^{\prime}|))^{2}dtdt^{\prime}\leq
2T\lVert\kappa_{k,k}^{(j)}\rVert^{2}_{L^{2}([0,T])}$

for all $k=1,\dots,M$ and

$\displaystyle\lVert
I_{k,l}^{(j)}\rVert_{\operatorname{HS}(L^{2}([0,T]))}^{2}+\lVert
I_{l,k}^{(j)}\rVert_{\operatorname{HS}(L^{2}([0,T]))}^{2}$
$\displaystyle=\int_{0}^{T}\int_{0}^{t}(\kappa_{k,l}^{(j)}(t-t^{\prime}))^{2}\,dt^{\prime}dt+\int_{0}^{T}\int_{t}^{T}(\kappa_{l,k}^{(j)}(t^{\prime}-t))^{2}\,dt^{\prime}dt$
$\displaystyle+\int_{0}^{T}\int_{0}^{t}(\kappa_{l,k}^{(j)}(t-t^{\prime}))^{2}\,dt^{\prime}dt+\int_{0}^{T}\int_{t}^{T}(\kappa_{k,l}^{(j)}(t^{\prime}-t))^{2}\,dt^{\prime}dt$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}(\kappa_{k,l}^{(j)}(|t-t^{\prime}|))^{2}\,dt^{\prime}dt+\int_{0}^{T}\int_{0}^{T}(\kappa_{l,k}^{(j)}(|t^{\prime}-t|))^{2}\,dt^{\prime}dt$
$\displaystyle\leq
2T\lVert\kappa_{k,l}^{(j)}\rVert^{2}_{L^{2}([0,T])}+2T\lVert\kappa_{l,k}^{(j)}\rVert^{2}_{L^{2}([0,T])}$

for all $1\leq k\neq l\leq M$ (here, we used $I_{l,k}^{(0)}=I_{l,k}$,
$I_{l,k}^{(1)}=I_{l,k}^{\prime}$ and $I_{l,k}^{(2)}=I_{l,k}^{\prime\prime}$
and similar notation for the derivatives of $\kappa$), we arrive at

$\displaystyle\sum_{j=1}^{\infty}\frac{\lVert
Iu_{j}\rVert^{2}_{H_{\vartheta^{0},\delta}}}{\sigma_{j}^{2}}$
$\displaystyle\leq 32T\frac{\lVert\Delta
K\rVert^{4}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{8}}\sum_{k,l=1}^{M}\lVert\kappa_{k,l}\rVert^{2}_{L^{2}([0,T])}$
$\displaystyle+32T\frac{\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{4}}\sum_{k,l=1}^{M}\lVert\kappa_{k,l}^{\prime}\rVert^{2}_{L^{2}([0,T])}+8T\sum_{k,l=1}^{M}\lVert\kappa_{k,l}^{\prime\prime}\rVert^{2}_{L^{2}([0,T])}$

for all $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ and all $T\geq 1$. The claim
now follows as in the case $M=1$ by an interpolation argument. ∎

###### Proof of Lemma 6.9.

We apply Lemma A.2 to
$I=C_{\vartheta^{0},\delta}-C_{\vartheta^{1},\delta}:\mathcal{H}\rightarrow\mathcal{H}$.
This means that $I=(I_{k,l})_{k,l=1}^{M}$ with
$I_{k,l}f(t)=\int_{0}^{t}\kappa_{k,l}(t^{\prime}-t)f(t^{\prime})dt^{\prime}+\int_{t}^{T}\kappa_{l,k}(t-t^{\prime})f(t^{\prime})dt^{\prime}$
for the integral kernels
$\kappa_{k,l}(t)=c_{\vartheta^{0},\delta,k,l}(t)-c_{\vartheta^{1},\delta,k,l}(t)$,
$0\leq t\leq T$, with

$\displaystyle
c_{\vartheta,\delta,k,l}(t)=\operatorname{{\mathbb{E}}}_{\vartheta}[X_{\delta,k}(t)X_{\delta,l}(0)]=\int_{0}^{\infty}\langle
S_{\vartheta}^{*}(t+t^{\prime})K_{\delta,x_{k}},S_{\vartheta}^{*}(t^{\prime})K_{\delta,x_{l}}\rangle
dt^{\prime},$ (A.9)

where the last equality follows from Lemma 6.5(ii). From (A.9), we immediately
infer $\kappa_{k,l}(0)=\kappa_{l,k}(0)$. The first two derivatives of the
cross-covariance integral kernels for
$\vartheta\in\\{\vartheta^{0},\vartheta^{1}\\}$ are

$\displaystyle c^{\prime}_{\vartheta,\delta,k,l}(t)$
$\displaystyle=\int_{0}^{\infty}\langle
A^{*}_{\vartheta}S^{*}_{\vartheta}(t+t^{\prime})K_{\delta,x_{k}},S^{*}_{\vartheta}(t^{\prime})K_{\delta,x_{l}}\rangle
dt^{\prime},$ (A.10) $\displaystyle
c^{\prime\prime}_{\vartheta,\delta,k,l}(t)$
$\displaystyle=\int_{0}^{\infty}\langle(A^{*}_{\vartheta})^{2}S^{*}_{\vartheta}(t+t^{\prime})K_{\delta,x_{k}},S^{*}_{\vartheta}(t^{\prime})K_{\delta,x_{l}}\rangle
dt^{\prime}.$ (A.11)

We note that

$\displaystyle
c^{\prime}_{\vartheta,\delta,k,l}(0)+c^{\prime}_{\vartheta,\delta,l,k}(0)$
$\displaystyle\quad=\int_{0}^{\infty}(d/dt^{\prime})\langle
S^{*}_{\vartheta}(t^{\prime})K_{\delta,x_{k}},S^{*}_{\vartheta}(t^{\prime})K_{\delta,x_{l}}\rangle
dt^{\prime}=-\langle K_{\delta,x_{k}},K_{\delta,x_{l}}\rangle$

is independent of $\vartheta$, and hence,
$\kappa^{\prime}_{k,l}(0)+\kappa^{\prime}_{l,k}(0)=0$. ∎

###  A.5 Proof of Lemma 6.10

It is sufficient to upper bound the $L^{2}$-norms of the $\kappa_{k,l}$.
Indeed, the proof below for this remains valid if $K_{\delta,x_{k}}$ is
replaced by $\delta^{-4}(A_{\vartheta,\delta,x_{k}}^{2}K)_{\delta,x_{k}}$ and
so it yields also the wanted bound on the $L^{2}$-norm of
$\kappa_{k,l}^{\prime\prime}$, cf. (A.11).

Let us first define some notation. We write $\Delta$ and $e^{t\Delta}$ for the
Laplacian and its generated semigroup on $L^{2}(\Lambda)$, as well as
$\Delta_{\delta,x}$ and $e^{t\Delta_{\delta,x}}$ on
$L^{2}(\Lambda_{\delta,x})$ and $\Delta_{0}$ and $e^{t\Delta_{0}}$ on
$L^{2}(\operatorname{{\mathbb{R}}}^{d})$. From (6.1) we have

$S^{*}_{\vartheta^{0}}(t)=e^{t\Delta},\quad
S^{*}_{\vartheta^{1}}(t)=U_{\vartheta^{1}}^{-1}e^{t\vartheta_{1}\Delta}U_{\vartheta^{1}}e^{t\tilde{c}_{\vartheta^{1}}}$

with $U_{\vartheta^{1}}(x)=e^{-(2\vartheta_{1})^{-1}\vartheta_{2}b\cdot x}$
and
$\tilde{c}_{\vartheta^{1}}=\vartheta_{3}-(4\vartheta_{1})^{-1}\vartheta^{2}_{2}\leq
0$. From Lemma 6.1(iii), we also have

$S_{\vartheta^{0},\delta,x_{k}}(t)=e^{t\Delta_{\delta,x_{k}}},\quad
S^{*}_{\vartheta^{1},\delta,x_{k}}(t)=U_{\vartheta^{1},\delta,x_{k}}^{-1}e^{t\vartheta_{1}\Delta_{\delta,x_{k}}}U_{\vartheta^{1},\delta,x_{k}}e^{t\delta^{2}\tilde{c}_{\vartheta^{1}}}$

with $U_{\vartheta^{1},\delta,x_{k}}(x)=U_{\vartheta^{1}}(x_{k}+\delta x)$.
Note that

$e^{t\vartheta_{1}\Delta}=U_{\vartheta^{1}}(x_{k})^{-1}e^{t\vartheta_{1}\Delta}U_{\vartheta^{1}}(x_{k}).$

To get started, let $1\leq k,l\leq M$ and decompose
$\kappa_{k,l}=\sum_{j=1}^{6}\kappa_{k,l}^{(j)}$ with

$\displaystyle\kappa_{k,l}^{(1)}(t)$
$\displaystyle=\int_{0}^{\infty}\langle(e^{(t+t^{\prime})\Delta}-e^{(t+t^{\prime})\vartheta_{1}\Delta})K_{\delta,x_{k}},e^{t^{\prime}\Delta}K_{\delta,x_{l}}\rangle
dt^{\prime},$ $\displaystyle\kappa_{k,l}^{(2)}(t)$
$\displaystyle=\int_{0}^{\infty}\langle
e^{(t+t^{\prime})\vartheta_{1}\Delta}K_{\delta,x_{k}},(e^{t^{\prime}\Delta}-e^{t^{\prime}\vartheta_{1}\Delta})K_{\delta,x_{l}}\rangle
dt^{\prime},$ $\displaystyle\kappa_{k,l}^{(3)}(t)$
$\displaystyle=\int_{0}^{\infty}\langle
U_{\vartheta^{1}}(x_{k})^{-1}e^{(t+t^{\prime})\vartheta_{1}\Delta}(U_{\vartheta^{1}}(x_{k})-U_{\vartheta^{1}}e^{\tilde{c}_{\vartheta^{1}}(t+t^{\prime})})K_{\delta,x_{k}},e^{t^{\prime}\vartheta_{1}\Delta}K_{\delta,x_{l}}\rangle
dt^{\prime},$ $\displaystyle\kappa_{k,l}^{(4)}(t)$
$\displaystyle=\int_{0}^{\infty}\langle(U_{\vartheta^{1}}(x_{k})^{-1}-U_{\vartheta^{1}}^{-1})U_{\vartheta^{1}}S^{*}_{\vartheta^{1}}(t+t^{\prime})K_{\delta,x_{k}},e^{t^{\prime}\vartheta_{1}\Delta}K_{\delta,x_{l}}\rangle
dt^{\prime},$ $\displaystyle\kappa_{k,l}^{(5)}(t)$
$\displaystyle=\int_{0}^{\infty}\langle
S^{*}_{\vartheta^{1}}(t+t^{\prime})K_{\delta,x_{k}},U_{\vartheta^{1}}(x_{k})^{-1}e^{t^{\prime}\vartheta_{1}\Delta}(U_{\vartheta^{1}}(x_{k})-U_{\vartheta^{1}}e^{\tilde{c}_{\vartheta^{1}}t^{\prime}})K_{\delta,x_{l}}\rangle
dt^{\prime},$ $\displaystyle\kappa_{k,l}^{(6)}(t)$
$\displaystyle=\int_{0}^{\infty}\langle
S^{*}_{\vartheta^{1}}(t+t^{\prime})K_{\delta,x_{k}},(U_{\vartheta^{1}}(x_{k})^{-1}-U_{\vartheta^{1}}^{-1})e^{t^{\prime}\vartheta_{1}\Delta}U_{\vartheta^{1}}e^{\tilde{c}_{\vartheta^{1}}t^{\prime}}K_{\delta,x_{l}}\rangle
dt^{\prime}.$

We only show $\sum_{1\leq k,l\leq
M}\lVert\kappa_{k,l}^{(j)}\rVert^{2}_{L^{2}([0,T])}\leq
c\delta^{8}M(\delta^{-2}(1-\vartheta_{1})^{2}+\vartheta_{2}^{2}+\delta^{2}\vartheta_{3}^{2})$
for $j=1,3,4$. The proof that the same bound holds for $j=2,5,6$ follows from
similar arguments and is therefore skipped. Diagonal (i.e., $k=l$) and off-
diagonal (i.e., $k\neq l$) terms are treated separately. Set
$K_{k,l}=K(\cdot+\delta^{-1}(x_{k}-x_{l}))$.

##### Case $j=1$.

The scaling in Lemma 6.1 and changing variables yield

$\displaystyle\kappa_{k,l}^{(1)}(t\delta^{2})$
$\displaystyle=\delta^{2}\int_{0}^{\infty}\langle(e^{(t+t^{\prime})\Delta_{\delta,x_{k}}}-e^{(t+t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}})K,e^{t^{\prime}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}$
$\displaystyle=\delta^{2}\langle\int_{0}^{\infty}(e^{(t+2t^{\prime})\Delta_{\delta,x_{k}}}-e^{(t\vartheta_{1}+(t^{\prime}(1+\vartheta_{1}))\Delta_{\delta,x_{k}}})dt^{\prime}\,K,K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}$
$\displaystyle=\frac{\delta^{2}}{2}\langle(e^{t\Delta_{\delta,x_{k}}}-2(1+\vartheta_{1})^{-1}e^{t\vartheta_{1}\Delta_{\delta,x_{k}}})(-\Delta_{\delta,x_{k}})^{-1}K,K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}$
$\displaystyle=\frac{\delta^{2}}{2}\langle(e^{t\Delta_{\delta,x_{k}}}-e^{t\vartheta_{1}\Delta_{\delta,x_{k}}})(-\Delta_{\delta,x_{k}})^{-1}K,K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}$
$\displaystyle\quad-\frac{\delta^{2}(1-\vartheta_{1})}{2(1+\vartheta_{1})}\langle
e^{t\vartheta_{1}\Delta_{\delta,x_{k}}}(-\Delta_{\delta,x_{k}})^{-1}K,K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}.$

With

$e^{t\vartheta_{1}\Delta_{\delta,x_{k}}}-e^{t\Delta_{\delta,x_{k}}}=(1-\vartheta_{1})\int_{0}^{t}e^{s(\vartheta_{1}-1)\Delta_{\delta,x_{k}}}ds\,e^{t\Delta_{\delta,x_{k}}}(-\Delta_{\delta,x_{k}}),$

as follows from the variation of parameters formula, see [16, p. 162], the
identity
$e^{t\Delta_{\delta,x_{k}}}=e^{(t/2)\Delta_{\delta,x_{k}}}e^{(t/2)\Delta_{\delta,x_{k}}}$
and the Cauchy-Schwarz inequality, we get

$\displaystyle\left|\langle(e^{t\Delta_{\delta,x_{k}}}-e^{t\vartheta_{1}\Delta_{\delta,x_{k}}})(-\Delta_{\delta,x_{k}})^{-1}K,K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}\right|$
$\displaystyle=|1-\vartheta_{1}|\left|\int_{0}^{t}\langle
e^{s(\vartheta_{1}-1)\Delta_{\delta,x_{k}}}e^{(t/2)\Delta_{\delta,x_{k}}}K,e^{(t/2)\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}ds\right|$
(A.12) $\displaystyle\lesssim|1-\vartheta_{1}|t\lVert
e^{(t/2)\Delta_{\delta,x_{k}}}K\rVert_{L^{2}(\Lambda_{\delta,x_{k}})}\lVert
e^{(t/2)\Delta_{\delta,x_{k}}}K_{k,l}\rVert_{L^{2}(\Lambda_{\delta,x_{k}})}.$

In the same way, and using $K=\Delta^{2}\tilde{K}$,

$\displaystyle\left|\langle
e^{t\vartheta_{1}\Delta_{\delta,x_{k}}}(-\Delta_{\delta,x_{k}})^{-1}K,K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}\right|$
(A.13) $\displaystyle\quad\leq\lVert
e^{(t/2)\vartheta_{1}\Delta_{\delta,x_{k}}}\Delta\tilde{K}\rVert_{L^{2}(\Lambda_{\delta,x_{k}})}\lVert
e^{(t/2)\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rVert_{L^{2}(\Lambda_{\delta,x_{k}})}.$

Lemma 6.4 therefore gives
$\kappa_{k,l}^{(1)}(t\delta^{2})\lesssim\delta^{2}|1-\vartheta_{1}|(1\wedge
t^{-1-d/2+\varepsilon})$ for any $\varepsilon>0$. Changing variables one more
time and recalling that $\vartheta_{1}\geq 1$ already proves for the sum of
diagonal terms that $\sum_{1\leq k\leq
M}\lVert\kappa_{k,k}^{(1)}\rVert^{2}_{L^{2}([0,T])}\lesssim\delta^{6}(1-\vartheta_{1})^{2}M$,
and the implied constant depends only on $K$.

With respect to the off-diagonal terms, by exploring the different supports of
$K$ and $K(\cdot+\delta^{-1}(x_{k}-x_{l}))$, we can obtain a second bound for
$\kappa_{k,l}^{(1)}$. First, Lemma 6.3 gives

$\displaystyle\sup_{y\in\operatorname{supp}K}\left|(e^{t\Delta_{\delta,x_{k}}}\Delta^{2}\tilde{K}_{k,l})(y)\right|\leq\lVert
e^{t\Delta_{\delta,x_{k}}}\Delta^{2}\tilde{K}_{k,l}\rVert_{L^{\infty}(\Lambda_{\delta,x_{k}})}$
$\displaystyle\quad\lesssim\lVert\Delta^{2}\tilde{K}_{k,l}\rVert_{L^{\infty}(\operatorname{{\mathbb{R}}}^{d})}\wedge(t^{-2}\lVert\tilde{K}_{k,l}\rVert_{L^{\infty}(\operatorname{{\mathbb{R}}}^{d})})\lesssim
1\wedge t^{-2},$ (A.14)

while on the other hand Proposition 6.2(i) shows

$\displaystyle\sup_{y\in\operatorname{supp}K}|(e^{t\Delta_{\delta,x_{k}}}K_{k,l})(y)|\lesssim\sup_{y\in\operatorname{supp}K}|(e^{t\Delta_{0}}|K_{k,l}|)(y)|$
$\displaystyle=\sup_{y\in\operatorname{supp}K}\int_{\operatorname{{\mathbb{R}}}^{d}}(4\pi
t)^{-d/2}\exp(-|x-y|^{2}/(4t))|K_{k,l}(x)|\,dx$ $\displaystyle\leq(4\pi
t)^{-d/2}e^{-c^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}\|K\|_{L^{1}(\operatorname{{\mathbb{R}}}^{d})}\lesssim
t^{-d/2}e^{-c^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}},$ (A.15)

for some $c^{\prime}>0$. By applying the Hölder inequality and using the
results from the last two displays we thus obtain for (A.12) the upper bound

$\displaystyle(\vartheta_{1}-1)t\lVert
K\rVert_{L^{1}(\operatorname{{\mathbb{R}}})}\sup_{0\leq s\leq
t,y\in\operatorname{supp}K}|(e^{(t+s(\vartheta_{1}-1))\Delta_{\delta,x_{k}}}K_{k,l})(y)|^{1/2+1/2}$
$\displaystyle\lesssim(\vartheta_{1}-1)\sup_{y\in\operatorname{supp}K}|(e^{t\Delta_{0}}|K_{k,l}|)(y)|^{1/2}\lesssim(\vartheta_{1}-1)t^{-d/4}e^{-(c^{\prime}/2)\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}.$

The same upper bound (up to the factor $\vartheta_{1}-1$ and with $c^{\prime}$
instead of $c^{\prime}/2$) holds in (A.13). Hence, together with the bound
$\kappa_{k,l}^{(1)}(t\delta^{2})\lesssim\delta^{2}(\vartheta_{1}-1)t^{-(1+d/2-\varepsilon
d/4)(1-\varepsilon)^{-1}})$ from above (for sufficiently small $\varepsilon$)
we get

$\displaystyle|\kappa^{(1)}_{k,l}(t\delta^{2})|$
$\displaystyle\lesssim\delta^{2}(\vartheta_{1}-1)\min\Big{(}t^{-(1+d/2-\varepsilon
d/4)(1-\varepsilon)^{-1}},t^{-d/4}e^{-(c^{\prime}/2)\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}\Big{)}$
$\displaystyle\leq\delta^{2}(\vartheta_{1}-1)t^{-1-d/2}e^{-\varepsilon^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}$
(A.16)

for $\varepsilon^{\prime}=c^{\prime}\varepsilon/2$, where we have used the
inequality $\min(a,b)\leq a^{1-\varepsilon}b^{\varepsilon}$ valid for $a,b\geq
0$. Applying the bound

$\int_{0}^{\infty}t^{-p-1}e^{-a/t}dt=a^{-p}\int_{0}^{\infty}t^{-p-1}e^{-1/t}dt\lesssim
a^{-p}$

to $p=1+d>0$ and $a=2\varepsilon^{\prime}\delta^{-2}|x_{k}-x_{l}|^{2}$ this
means

$\displaystyle\int_{0}^{T}\kappa_{k,l}^{(1)}(t)^{2}dt$
$\displaystyle\lesssim\delta^{6}(1-\vartheta_{1})^{2}\int_{0}^{\infty}t^{-2-d}e^{-2\varepsilon^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}dt$
$\displaystyle\lesssim\delta^{6}(1-\vartheta_{1})^{2}\frac{\delta^{2+2d}}{|x_{k}-x_{l}|^{2+2d}}.$
(A.17)

Recalling that the $x_{k}$ are $\delta$-separated we obtain from Lemma A.3
below that

$\displaystyle\sum_{1\leq k\neq l\leq
M}\lVert\kappa_{k,l}^{(1)}\rVert^{2}_{L^{2}([0,T])}\lesssim\delta^{6}(1-\vartheta_{1})^{2}\sum_{k=1}^{M}\sum_{l=1,l\neq
k}^{M}\frac{\delta^{2+2d}}{|x_{k}-x_{l}|^{2+2d}}\lesssim\delta^{6}(1-\vartheta_{1})^{2}M.$

Together with the bounds for the diagonal terms this yields in all
$\sum_{1\leq k,l\leq M}\lVert\kappa_{k,l}^{(1)}\rVert^{2}_{L^{2}([0,T])}\leq
c\delta^{6}M(1-\vartheta_{1})^{2}$ for a constant $c$ depending only on $K$.

##### Case $j=3$.

We begin again with the scaling from Lemma 6.1 and changing variables such
that with the multiplication operators
$V_{t,t^{\prime},\delta}(x)=1-e^{\tilde{c}_{\vartheta^{1}}\delta^{2}(t+t^{\prime})-(2\vartheta_{1})^{-1}\vartheta_{2}\delta
b\cdot x}$

$\displaystyle\kappa_{k,l}^{(3)}(t\delta^{2})=\delta^{2}\int_{0}^{\infty}\langle
e^{(t+t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}V_{t,t^{\prime},\delta}K,e^{t^{\prime}\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}$
$\displaystyle\quad=\delta^{2}\int_{0}^{\infty}\langle
e^{(t/2+t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}V_{t,t^{\prime},\delta}K,e^{(t/2+t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}.$
(A.18)

Since $K$ is compactly supported and $\tilde{c}_{\vartheta^{1}}\leq 0$,
$V_{t,t^{\prime},\delta}$ can be extended to smooth multiplication operators
with operator norms bounded by
$v_{t,t^{\prime},\delta}=-\tilde{c}_{\vartheta^{1}}\delta^{2}(t+t^{\prime})+(2\vartheta_{1})^{-1}\delta\vartheta_{2}$.
Recalling $K=\Delta^{2}\tilde{K}$, Lemma 6.4 gives for any $\varepsilon>0$

$\displaystyle|\kappa_{k,l}^{(3)}(t\delta^{2})|\leq\delta^{2}\int_{0}^{\infty}\lVert
e^{(t/2+t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}V_{t,t^{\prime},\delta}K\rVert_{L^{2}(\Lambda_{\delta,x_{k}})}\lVert
e^{(t/2+t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rVert_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}$
$\displaystyle\quad\leq\delta^{2}\int_{0}^{\infty}v_{t,t^{\prime},\delta}(1\wedge(t+t^{\prime})^{-4-d/2+\varepsilon})dt^{\prime}\lesssim\delta^{3}(-\tilde{c}_{\vartheta^{1}}\delta+(2\vartheta_{1})^{-1}\vartheta_{2})(1\wedge
t^{-1-d/2})$
$\displaystyle\quad\leq\delta^{3}(\delta|\vartheta_{3}|+\vartheta_{2})(1\wedge
t^{-1-d/2}),$ (A.19)

recalling in the last line that $\vartheta_{1}\geq 1$ and $\vartheta_{2}\leq
1$. Changing variables therefore proves for the sum of diagonal terms
$\sum_{1\leq k\leq
M}\lVert\kappa_{k,k}^{(3)}\rVert^{2}_{L^{2}([0,T])}\lesssim\delta^{8}(\delta|\vartheta_{3}|+\vartheta_{2})^{2}M$.

With respect to the off-diagonal terms we have

$\displaystyle\kappa_{k,l}^{(3)}(t\delta^{2})=\delta^{2}\int_{0}^{\infty}\langle
V_{t,t^{\prime},\delta}K,e^{(t+2t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}.$

Write $K=\Delta\bar{K}$ for some compactly supported $\bar{K}$ and note that

$\displaystyle
V_{t,t^{\prime},\delta}K=V_{t,t^{\prime},\delta}\Delta\bar{K}=\Delta(V_{t,t^{\prime},\delta}\bar{K})-(\Delta
V_{t,t^{\prime},\delta})\bar{K}-2\nabla
V_{t,t^{\prime},\delta}\cdot\nabla\bar{K}.$

Similar to (A.14) we find from Lemma 6.3

$\displaystyle\sup_{y\in\operatorname{supp}\bar{K}}\left|(\Delta
e^{t\Delta_{\delta,x_{k}}}K_{k,l})(y)\right|\lesssim\lVert\Delta
K_{k,l}\rVert_{L^{\infty}(\operatorname{{\mathbb{R}}}^{d})}\wedge(t^{-3}\lVert\tilde{K}_{k,l}\rVert_{L^{\infty}(\operatorname{{\mathbb{R}}}^{d})})\lesssim
1\wedge t^{-3}.$

Together with the Hölder inequality and (A.15) this provides us for
sufficiently small $\varepsilon>0$ with

$\displaystyle\delta^{2}\left|\int_{0}^{\infty}\langle\Delta(V_{t,t^{\prime},\delta}\bar{K}),e^{(t+2t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}\right|$
$\displaystyle=\delta^{2}\left|\int_{0}^{\infty}\langle
V_{t,t^{\prime},\delta}\bar{K},\Delta
e^{(t+2t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}\right|$
$\displaystyle\lesssim\delta^{2}\int_{0}^{\infty}v_{t,t^{\prime},\delta}\lVert\bar{K}\rVert_{L^{1}(\operatorname{{\mathbb{R}}}^{d})}\sup_{y\in\operatorname{supp}K}\left|(\Delta
e^{(t+2t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l})(y)\right|dt^{\prime}$
$\displaystyle\lesssim\delta^{2}\int_{0}^{\infty}v_{t,t^{\prime},\delta}(1\wedge(t+2t^{\prime}))^{-3(1-\varepsilon)}\sup_{y\in\operatorname{supp}K}\left|(e^{(t+2t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}\Delta
K_{k,l})(y)\right|^{\varepsilon}dt^{\prime}$
$\displaystyle\lesssim\delta^{3}(\delta|\vartheta_{3}|+\vartheta_{2})e^{-\varepsilon
c^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}\int_{0}^{\infty}(1\wedge(t^{\prime})^{-1-\varepsilon
d/2})dt^{\prime}\lesssim\delta^{3}(\delta|\vartheta_{3}|+\vartheta_{2})e^{-\varepsilon
c^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}.$

Next, using $\vartheta_{2}\leq 1$, we have $\lVert\Delta
V_{t,t^{\prime},\delta}\rVert_{L^{\infty}(\Lambda_{\delta,x_{k}})}+\lVert|\nabla
V_{t,t^{\prime},\delta}|\rVert_{L^{\infty}(\Lambda_{\delta,x_{k}})}\lesssim\delta\vartheta_{2}$,
and so analogously to the computations in the last display

$\displaystyle\delta^{2}\left|\int_{0}^{\infty}\langle(\Delta
V_{t,t^{\prime},\delta})\bar{K}+2\nabla
V_{t,t^{\prime},\delta}\cdot\nabla\bar{K},e^{(t+2t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}\right|$
$\displaystyle\lesssim\delta^{3}\vartheta_{2}\int_{0}^{\infty}(\lVert\bar{K}\rVert_{L^{1}(\operatorname{{\mathbb{R}}}^{d})}+\lVert|\nabla\bar{K}|\rVert_{L^{1}(\operatorname{{\mathbb{R}}}^{d})})\sup_{y\in\operatorname{supp}K}\left|(e^{(t+2t^{\prime})\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l})(y)\right|dt^{\prime}$
$\displaystyle\lesssim\delta^{3}\vartheta_{2}e^{-\varepsilon
c^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}.$

In all, this means
$|\kappa_{k,l}^{(3)}(t\delta^{2})|\lesssim\delta^{3}(\delta|\vartheta_{3}|+\vartheta_{2})e^{-\varepsilon
c^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}$. Arguing as for (A.16) and
(A.17), as well as using (A.19) we conclude that
$|\kappa^{(3)}_{k,l}(t\delta^{2})|\lesssim\delta^{3}(\delta|\vartheta_{3}|+\vartheta_{2})t^{-1/2-d/2}e^{-\varepsilon^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}$
for some $\varepsilon^{\prime}>0$ and

$\int_{0}^{T}\kappa_{k,l}^{(3)}(t)^{2}dt\lesssim\frac{\delta^{8+2d}(\delta|\vartheta_{3}|+\vartheta_{2})^{2}}{|x_{k}-x_{l}|^{2d}}.$

We thus get for diagonal and off-diagonal terms that $\sum_{1\leq k,l\leq
M}\lVert\kappa_{k,l}^{(3)}\rVert^{2}_{L^{2}([0,T])}\leq
c\delta^{8}(\delta|\vartheta_{3}|+\vartheta_{2})^{2}M$ for a constant $c$
depending only on $K$.

##### Case $j=4$

As in the previous cases we have

$\displaystyle\kappa_{k,l}^{(4)}(t\delta^{2})=\delta^{2}\int_{0}^{\infty}\langle(e^{-\delta(\vartheta_{2}/\vartheta_{1})b\cdot
x}-1)S^{*}_{\vartheta^{1},\delta,x_{k}}(t+t^{\prime})K,e^{t^{\prime}\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}dt^{\prime}.$

Using the Cauchy-Schwarz inequality, Proposition 6.2(i) and Lemma 6.4 with
$K=\Delta^{2}\tilde{K}$ we get for any $\varepsilon>0$, and recalling that
$\vartheta_{1}\geq 1$,

$\displaystyle\langle(e^{-\delta(\vartheta_{2}/\vartheta_{1})b\cdot
x}-1)S^{*}_{\vartheta^{1},\delta,x_{k}}(t+t^{\prime})K,e^{t^{\prime}\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}$
$\displaystyle\lesssim\delta\vartheta_{2}(1\wedge(t+t^{\prime})^{-2-d/4+\varepsilon})\lVert|x|e^{t^{\prime}\vartheta_{1}\Delta_{0}}|K_{k,l}|\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}.$
(A.20)

Note that $K_{k,l}\in C^{1}_{c}(\operatorname{{\mathbb{R}}}^{d})$ such that
$|K_{k,l}|\in H^{1,\infty}(\operatorname{{\mathbb{R}}}^{d})$ and
$\nabla|K_{k,l}|\in L^{\infty}(\operatorname{{\mathbb{R}}}^{d})$ with compact
support. Using now [4, Lemma A.2(ii)] to the extent that

$\displaystyle
x(e^{t^{\prime}\vartheta_{1}\Delta_{0}}|K_{k,l}|)(x)=(e^{t^{\prime}\vartheta_{1}\Delta_{0}}(-2t^{\prime}\vartheta_{1}\nabla|K_{k,l}|+x|K_{k,l}|))(x),$
(A.21)

we find that the $L^{2}(\operatorname{{\mathbb{R}}}^{d})$-norm in (A.20) is
uniformly bounded in $t^{\prime}>0$. Hence,
$|\kappa_{k,l}^{(4)}(t\delta^{2})|\lesssim\delta^{3}\vartheta_{2}(1\wedge
t^{-1/2-d/4-\varepsilon})$ and changing variables shows for the sum of
diagonal terms $\sum_{1\leq k\leq
M}\lVert\kappa_{k,k}^{(4)}\rVert^{2}_{L^{2}([0,T])}\lesssim\delta^{8}\vartheta_{2}^{2}M$.
Regarding the off-diagonal terms we have similarly for some $\bar{K}\in
L^{\infty}(\operatorname{{\mathbb{R}}}^{d})$ having compact support

$\displaystyle\left|\langle(e^{-\delta(\vartheta_{2}/\vartheta_{1})b\cdot
x}-1)S^{*}_{\vartheta^{1},\delta,x_{k}}(t+t^{\prime})K,e^{t^{\prime}\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}\right|$
$\displaystyle=\left|\langle
K,S_{\vartheta^{1},\delta,x_{k}}(t+t^{\prime})(e^{-\delta(\vartheta_{2}/\vartheta_{1})b\cdot
x}-1)e^{t^{\prime}\vartheta_{1}\Delta_{\delta,x_{k}}}K_{k,l}\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}\right|$
$\displaystyle\lesssim\delta\vartheta_{2}\lVert
K\rVert_{L^{1}(\operatorname{{\mathbb{R}}}^{d})}\sup_{y\in\operatorname{supp}K}\left|\left(e^{(t+t^{\prime})\vartheta_{1}\Delta_{0}}|x|e^{t^{\prime}\vartheta_{1}\Delta_{0}}|K_{k,l}|\right)(y)\right|$
$\displaystyle\lesssim\delta\vartheta_{2}(1\vee
t^{\prime})\sup_{y\in\operatorname{supp}K}\left|\left(e^{(t+2t^{\prime})\vartheta_{1}\Delta_{0}}|\bar{K}_{k,l}|\right)(y)\right|$
$\displaystyle\lesssim\delta\vartheta_{2}(1\vee
t^{\prime})t^{-d/2}e^{-c^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}},$

using (A.15). Arguing as for (A.16) and (A.17) we then find from combining the
last display with (A.20) that
$|\kappa^{(4)}_{k,l}(t\delta^{2})|\lesssim\delta^{3}\vartheta_{2}t^{-1/2-d/4-\varepsilon^{\prime}}e^{-\varepsilon^{\prime}\frac{|x_{k}-x_{l}|^{2}}{\delta^{2}t}}$
for some $\varepsilon^{\prime}>0$ and

$\int_{0}^{T}\kappa_{k,l}^{(4)}(t)^{2}dt\lesssim\frac{\delta^{8+d+4\varepsilon^{\prime}}\vartheta_{2}^{2}}{|x_{k}-x_{l}|^{4\varepsilon^{\prime}+d}},$

and so in all, for diagonal and off-diagonal terms, $\sum_{1\leq k,l\leq
M}\lVert\kappa_{k,l}^{(4)}\rVert^{2}_{L^{2}([0,T])}\leq
c\delta^{8}\vartheta_{2}^{2}M$ for a constant $c$ depending only on $K$. ∎

######  Lemma A.3.

Let $x_{1},\dots,x_{M}$ be $\delta$-separated points in
$\operatorname{{\mathbb{R}}}^{d}$, and let $p>d$. Then we have for a constant
$C=C(d,p)$

$\displaystyle\sum_{k=2}^{M}\frac{1}{|x_{1}-x_{k}|^{p}}\leq C\delta^{-p}.$

###### Proof.

Since $x_{1},\dots,x_{M}$ are $\delta$-separated, the Euclidean balls
$B(x_{k},\delta/2)=\\{y\in\operatorname{{\mathbb{R}}}^{d}:|y-x_{k}|\leq\delta/2\\}$
around the $x_{k}$ of radius $\delta/2$ are disjoint. Moreover, for $y\in
B(x_{k},\delta/2)$ and $k>1$, we have

$\displaystyle|y-x_{1}|\leq|y-x_{k}|+|x_{k}-x_{1}|\leq\frac{\delta}{2}+|x_{k}-x_{1}|\leq\frac{3}{2}|x_{k}-x_{1}|,$

implying that

$\displaystyle\frac{1}{|x_{k}-x_{1}|}\leq\frac{3}{2}\frac{1}{|y-x_{1}|}.$

We conclude that

$\displaystyle\sum_{k=2}^{M}\frac{1}{|x_{1}-x_{k}|^{p}}$
$\displaystyle=\sum_{k=2}^{M}\frac{1}{\operatorname{vol}(B(x_{k},\delta/2))}\int_{B(x_{k},\delta/2)}\limits\frac{1}{|x_{1}-x_{k}|^{p}}dy$
$\displaystyle\leq\frac{(3/2)^{p}}{(\delta/2)^{d}\operatorname{vol}(B(0,1))}\sum_{k=2}^{M}\int_{B(x_{k},\delta/2)}\limits\frac{1}{|y-x_{1}|^{p}}dy$
$\displaystyle\leq\frac{(3/2)^{p}}{(\delta/2)^{d}\operatorname{vol}(B(0,1))}\int_{B(x_{1},\delta/2)^{c}}\limits\frac{1}{|y-x_{1}|^{p}}dy$
$\displaystyle=\frac{(3/2)^{p}}{(\delta/2)^{d}\operatorname{vol}(B(0,1))}\int_{B(0,\delta/2)^{c}}\limits\frac{1}{|y|^{p}}dy.$

Changing to polar coordinates, we arrive at

$\displaystyle\sum_{k=2}^{M}\frac{1}{|x_{1}-x_{k}|^{p}}$
$\displaystyle\leq\frac{d(3/2)^{p}}{(\delta/2)^{d}\operatorname{vol}(B(0,1))}\int_{\delta/2}^{\infty}\frac{1}{t^{p}}t^{d-1}dt$
$\displaystyle=\frac{d(3/2)^{p}}{(\delta/2)^{p}\operatorname{vol}(B(0,1))}\int_{1}^{\infty}s^{d-p-1}ds.$

Since $p>d$, the latter integral is finite, and the claim follows. ∎

###  A.6 Proof of Theorem 3.3

Argue as in the proof of Theorem 3.1, using slight modifications of Lemmas 6.9
and 6.10. The key additional ingredient is an appropriate extension of
Corollary 4.3. For this, let $A_{1},\dots,A_{p}$ be differential operators of
the form (2.1) with $\bar{A}_{i}=A_{i}$ (meaning that each $A_{i}$ includes
only summands of the same differential order). We assume that

$\displaystyle(A_{i}^{*}K)_{i=1}^{p}\text{ are independent in
}L^{2}(\Lambda).$ (A.22)

Define

$\displaystyle H=\Big{(}\Big{\langle}\frac{A_{i}^{*}K}{\lVert
A_{i}^{*}K\rVert_{L^{2}(\Lambda)}},\frac{A_{j}^{*}K}{\lVert
A_{j}^{*}K\rVert_{L^{2}(\Lambda)}}\Big{\rangle}\Big{)}_{i,j=1}^{p},$

and let $\lambda_{\min}=\lambda_{\min}(H)$ be the smallest eigenvalue of $H$.
By (A.22), $H$ is non-singular, meaning that $\lambda_{\min}(H)>0$. Finally,
let

$\displaystyle v=\sum_{i=1}^{p}\frac{\lVert\Delta
A_{i}^{*}K\rVert_{L^{2}(\Lambda)}^{2}}{\lVert
A_{i}^{*}K\rVert_{L^{2}(\Lambda)}^{2}}.$

######  Corollary A.4.

Let $(H_{X_{\delta}},\|\cdot\|_{X_{\delta}})$ be the RKHS of the measurements
$X_{\delta}$ with differential operator $A_{\vartheta}=\Delta$, where
$X_{\delta}(t)=(\langle X(t),K_{1k}\rangle,\dots,\langle
X(t),K_{pk}\rangle)_{k=1}^{M}$ and $K_{ik}=A_{i}^{*}K_{\delta,x_{k}}/\lVert
A_{i}^{*}K_{\delta,x_{k}}\rVert_{L^{2}(\Lambda)}$. Then we have
$H_{X_{\delta}}=(H^{p})^{M}$ and

$\displaystyle\|((h_{ip})_{i=1}^{p})_{k=1}^{M}\|_{X_{\delta}}^{2}$
$\displaystyle\leq\frac{4vp}{\delta^{4}\lambda^{2}_{\min}}\sum_{k=1}^{M}\sum_{i=1}^{p}\|h_{ik}\|_{L^{2}([0,T])}^{2}+\frac{2}{\lambda^{2}_{\min}}\sum_{k=1}^{M}\sum_{i=1}^{p}\|h_{ik}^{\prime}\|_{L^{2}([0,T])}^{2}$

for all $((h_{ip})_{i=1}^{p})_{k=1}^{M}\in(H^{p})^{M}$,
$\delta^{2}\leq\sqrt{v}$ and $T\geq 1$.

###### Proof of Corollary A.4.

First, let $M=1$. Additionally to $H$, define

$\displaystyle H_{\Delta}$ $\displaystyle=\Big{(}\Big{\langle}\frac{\Delta
A_{i}^{*}K}{\lVert A_{i}^{*}K\rVert_{L^{2}(\Lambda)}},\frac{\Delta
A_{j}^{*}K}{\lVert
A_{j}^{*}K\rVert_{L^{2}(\Lambda)}}\Big{\rangle}\Big{)}_{i,j=1}^{p}.$

By the Cauchy-Schwarz inequality, we have $\lVert
H_{\Delta}\rVert_{\operatorname{op}}\leq v$. Moreover, using also that
$\bar{A}_{i}=A_{i}$ for all $i=1,\dots,p$, we have $G=H$ and
$G_{\Delta}=\delta^{-4}H_{\Delta}$. Inserting these bounds into Theorem 4.2,
we obtain that

$\displaystyle\|(h_{i})_{i=1}^{p}\|_{X_{K}}^{2}$
$\displaystyle\leq\Big{(}\frac{3v}{\delta^{4}\lambda_{\min}^{2}}+\frac{1}{\lambda_{\min}}\Big{)}\sum_{i=1}^{p}\lVert
h_{i}\rVert^{2}_{L^{2}([0,T])}+\frac{2}{\lambda_{\min}}\sum_{i=1}^{p}\lVert
h_{i}^{\prime}\rVert^{2}_{L^{2}([0,T])}.$

Next, let $M>1$. Then $G$ and $G_{\Delta}$ are block-diagonal with $M$ equal
$p\times p$-blocks all of the above form and we get

$\displaystyle\|((h_{i})_{i=1}^{p})_{k=1}^{M}\|_{X_{K}}^{2}$
$\displaystyle\leq\frac{4v}{\delta^{4}\lambda_{\min}^{2}}\sum_{i=1}^{p}\sum_{k=1}^{M}\lVert
h_{ik}\rVert^{2}_{L^{2}([0,T])}+\frac{2}{\lambda_{\min}^{2}}\sum_{i=1}^{p}\sum_{k=1}^{M}\lVert
h_{ik}^{\prime}\rVert^{2}_{L^{2}([0,T])},$

where we also used that $\lambda_{\min}\in(0,1]$ and $\delta^{2}\leq\sqrt{v}$.
∎

####  A.6.1 Second proof of Theorem 4.2

In this Appendix, we prove Theorem 4.2 under the weaker assumption
$K_{1},\dots,K_{M}\in\mathcal{D}(A_{\vartheta})$. This is achieved by an
additional approximation argument. Let $X_{m}(t)=\sum_{j\leq m}Y_{j}(t)e_{j}$,
$0\leq t\leq T$, be the projection of $X(t)$ onto
$V_{m}=\operatorname{span}\\{e_{1},\dots,e_{m}\\}$, and taking values in
$L^{2}([0,T];V_{m})$. We start with the following consequence of Lemma 6.11.

######  Lemma A.5.

For every $m\geq 1$, the RKHS $(H_{X_{m}},\|\cdot\|_{X_{m}})$ of $X_{m}$
satisfies

$H_{X_{m}}=\Big{\\{}h=\sum_{j=1}^{m}h_{j}e_{j}:h_{j}\in H,1\leq j\leq
m\Big{\\}}\quad\text{and}\quad\|h\|_{X_{m}}^{2}=\sum_{j=1}^{m}\lVert
h_{j}\rVert_{Y_{j}}^{2}.$

Moreover, we have $\|h\|_{X_{m}}=\|h\|_{X}$ with the latter norm defined in
Theorem 4.1.

###### Proof.

Since $L^{2}([0,T];V_{m})$ is isomorphic to $L^{2}([0,T])^{m}$, it suffices to
compute the RKHS of the coefficient vector $Y=(Y_{1},\dots,Y_{m})$. Using that
$Y_{1},\dots,Y_{m}$ are independent stationary Ornstein-Uhlenbeck processes,
the vector $Y$ is a Gaussian process in $L^{2}([0,T])^{m}$ with expectation
zero and covariance operator $\bigoplus_{j=1}^{m}C_{Y_{j}}$ with
$C_{Y_{j}}:L^{2}([0,T])\rightarrow L^{2}([0,T])$ being the covariance operator
of $Y_{j}$. Combining this with (6.10) and Lemma 6.11, we conclude that the
RKHS of $Y$ is equal to $H^{m}$ with norm $\sum_{j=1}^{m}\lVert
h_{j}\rVert_{Y_{j}}^{2}$. Translating this back to $X_{m}$, the first claim
follows. The second one follows from $(\lambda_{j},e_{j})_{j=1}^{\infty}$
being an eigensystem of $-A_{\vartheta}$. ∎

###### Proof of Theorem 4.2.

The first step will be to compute the RKHS $(H_{X_{K,m}},\|\cdot\|_{X_{K,m}})$
of $X_{K,m}=(\langle X_{m},K_{k}\rangle_{\mathcal{H}})_{k=1}^{M}$. To this end
define the bounded linear map

$\displaystyle L:L^{2}([0,T];V_{m})\rightarrow
L^{2}([0,T])^{M},f\mapsto(\langle f,K_{k}\rangle_{\mathcal{H}})_{k=1}^{M}.$

Combining the fact that $LX_{m}=X_{K,m}$ in distribution with Proposition 4.1
in [32] and Lemma A.5, we obtain that
$H_{X_{K,m}}=L(\\{h:h=\sum_{j=1}^{m}h_{j}e_{j}:h_{j}\in H\\})$. This implies
$H_{X_{K,m}}\subseteq H^{M}$. To see the reverse inclusion, let $P_{m}$ be the
orthogonal projection of $L^{2}(\Lambda)$ onto
$V_{m}=\operatorname{span}\\{e_{1},\dots,e_{m}\\}$, and let $G_{m}=(\langle
P_{m}K_{k},P_{m}K_{l}\rangle_{\mathcal{H}})_{k,l=1}^{M}$. Since
$(e_{j})_{j\geq 1}$ is an orthonormal basis of $\mathcal{H}$, $G_{m}$ tends
(e.g., in operator norm) to $G$ as $m\rightarrow\infty$. Since $G$ is non-
singular, we deduce that $G_{m}$ is non-singular for all $m$ large enough
(which we assume from now on). Hence, for $(h_{k})_{k=1}^{M}\in H^{M}$, we
have that

$\displaystyle f=\sum_{k,l=1}^{M}(G_{m})^{-1}_{k,l}P_{m}K_{k}h_{l}\in
H_{X_{m}}\quad\text{satisfies}\quad Lf=(h_{k})_{k=1}^{M},$ (A.23)

where we also used that $P_{m}K_{k}\in V_{m}$ for all $1\leq k\leq M$. Hence,
$H^{M}\subseteq H_{X_{K,m}}$ and therefore $H_{X_{K,m}}=H^{M}$. Moreover,
combining (A.23) with Proposition 4.1 in [32] and the fact that
$A_{\vartheta}P_{m}=P_{m}A_{\vartheta}$, we get from Lemma A.5

$\displaystyle\lVert(h_{k})_{k=1}^{M}\rVert_{X_{K,m}}^{2}\leq\|\sum_{k,l=1}^{M}(G_{m})^{-1}_{k,l}P_{m}K_{k}h_{l}\|_{X}^{2}$
$\displaystyle\leq
3\|P_{m}\sum_{k,l=1}^{M}(G_{m})^{-1}_{k,l}A_{\vartheta}K_{k}h_{l}\|_{L^{2}([0,T];\mathcal{H})}^{2}+\|P_{m}\sum_{k,l=1}^{M}(G_{m})^{-1}_{k,l}K_{k}h_{l}\|_{L^{2}([0,T];\mathcal{H})}^{2}$
$\displaystyle+2\|P_{m}\sum_{k,l=1}^{M}(G_{m})^{-1}_{k,l}K_{k}h_{l}^{\prime}\|_{L^{2}([0,T];\mathcal{H})}^{2}.$

Letting $m$ go to infinity, in which case $(G_{m})^{-1}$ converges to
$G^{-1}$, and so by definition of $G_{A_{\vartheta}}$, the last display
becomes

$\displaystyle\limsup_{m\rightarrow\infty}\lVert(h_{k})_{k=1}^{M}\rVert_{X_{K,m}}^{2}$
$\displaystyle\leq
3\int_{0}^{T}\sum_{k,l=1}^{M}(G^{-1}G_{A_{\vartheta}}G^{-1})_{kl}h_{k}(t)h_{l}(t)\,dt+\int_{0}^{T}\sum_{k,l=1}^{M}(G^{-1})_{kl}h_{k}(t)h_{l}(t)\,dt$
$\displaystyle+2\int_{0}^{T}\sum_{k,l=1}^{M}(G^{-1})_{kl}h_{k}^{\prime}(t)h_{l}^{\prime}(t)\,dt.$

Using standard results for the operator norm of symmetric matrices yields thus
for $\limsup_{m\rightarrow\infty}\lVert(h_{k})_{k=1}^{M}\rVert_{X_{K,m}}^{2}$
the upper bound claimed in the statement of the theorem to hold for $\lVert
h\rVert_{X_{K}}^{2}$.

Next, we use the above results to compute the RKHS of $X_{K}=(\langle
X,K_{k}\rangle_{\mathcal{H}})_{k=1}^{M}$. First, let us argue that the RKHS of
a single measurement $\langle X,K_{k}\rangle_{\mathcal{H}}$ (as a set) equals
$H$. Combining Girsanov’s theorem for the Itô process $\langle
X,K_{k}\rangle_{\mathcal{H}}$ in (2.3) with Feldman-Hájek’s theorem, the RKHS
of $\langle X,K_{k}\rangle_{\mathcal{H}}$ starting in zero is $H_{\beta}$.
Adding an independent Gaussian random variable with variance greater zero, we
obtain that in the stationary case $\langle X,K_{k}\rangle_{\mathcal{H}}$ has
RKHS $H=H_{\bar{\beta}}$ (see also the proof of Lemma 6.11). Now, consider the
case $M>1$. By Proposition 4.1 in [32], each coordinate projection maps the
RKHS of $X_{K}$ to the RKHS of a single measurement, thus to $H$ by the first
step. Hence, we have $H_{X_{K}}\subseteq H^{M}$. It remains to show the
reverse inclusion $H^{M}=H_{X_{K,m}}\subseteq H_{X_{K}}$. To see this, note
that

$\langle X_{m},K_{k}\rangle_{\mathcal{H}}=\sum_{j=1}^{m}\langle
K_{k},e_{j}\rangle_{\mathcal{H}}Y_{j}\quad\text{and}\quad\langle
X,K_{k}\rangle_{\mathcal{H}}=\sum_{j=1}^{\infty}\langle
K_{k},e_{j}\rangle_{\mathcal{H}}Y_{j},$

so that $X_{K}=X_{K,m}+(X_{K}-X_{K,m})$ can be written as a sum of two
independent processes taking values in the Hilbert space $L^{2}([0,T])^{M}$.
Letting $C_{K}$ and $C_{K,m}$ be the covariance operators of $X_{K}$ and
$X_{K,m}$, respectively, this implies that $C_{K}=C_{K,m}+\tilde{C}$ with
$\tilde{C}$ self-adjoint and positive. Combining this with the
characterisation of the RKHS norm in Proposition 2.6.8 of [19], we get
$\lVert(h_{k})_{k=1}^{M}\rVert^{2}_{X_{K}}\leq\lVert(h_{k})_{k=1}^{M}\rVert^{2}_{X_{K,m}}$
and $H_{X_{K,m}}\subseteq H_{X_{K}}$ for all $m\geq 1$. Finally, inserting the
upper bound on $\lVert(h_{k})_{k=1}^{M}\rVert^{2}_{X_{K,m}}$ derived above,
the proof is complete. ∎

##### Acknowledgement.

The research of AT and MW has been partially funded by the Deutsche
Forschungsgemeinschaft (DFG)- Project-ID 318763901 - SFB1294. AT further
acknowledges financial support of Carlsberg Foundation Young Researcher
Fellowship grant CF20-0604. RA gratefully acknowledges support by the European
Research Council, ERC grant agreement 647812 (UQMSI).

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