# IDOL-Net: An Interactive Dual-Domain Parallel Network for CT Metal Artifact
Reduction

Tao Wang1, Wenjun Xia1, Zexin Lu1, Huaiqiang Sun2, Yan Liu3, Jiliu Zhou1, Yi
Zhang1∗
1College of Computer Science, Sichuan University, Chengdu, China
2Department of Radiology, West China Hospital of Sichuan University, Chengdu,
China
3College of Electrical Engineering, Sichuan University, Chengdu, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Due to the presence of metallic implants, the imaging quality of computed
tomography (CT) would be heavily degraded. With the rapid development of deep
learning, several network models have been proposed for metal artifact
reduction (MAR). Since the dual-domain MAR methods can leverage the hybrid
information from both sinogram and image domains, they have significantly
improved the performance compared to single-domain methods. However, current
dual-domain methods usually operate on both domains in a specific order, which
implicitly imposes a certain priority prior into MAR and may ignore the latent
information interaction between both domains. To address this problem, in this
paper, we propose a novel interactive dual-domain parallel network for CT MAR,
dubbed as IDOL-Net. Different from existing dual-domain methods, the proposed
IDOL-Net is composed of two modules. The disentanglement module is utilized to
generate high-quality prior sinogram and image as the complementary inputs.
The follow-up refinement module consists of two parallel and interactive
branches that simultaneously operate on image and sinogram domain, fully
exploiting the latent information interaction between both domains. The
simulated and clinical results demonstrate that the proposed IDOL-Net
outperforms several state-of-the-art models in both qualitative and
quantitative aspects.

##  1 Introduction

Computed tomography (CT) has been widely applied in clinical, industrial and
other areas [2]. However, when metallic implants exist in scanning regions,
the effects of beam hardening, photon starvation, scattering and nonlinear
partial volume effect become prominent, giving rise to severe radial streak
artifacts in reconstructed CT images, which may seriously affect the
subsequent analysis or diagnosis tasks [18]. As a result, it is of great
importance to develop an efficient method to reduce metal artifacts while
preserving tissue details.

Figure 1: A representative slice processed using different MAR methods. (A)
Metal-corrupted image, (B) CNNMAR, (C) DuDoNet, (D) IDOL-Net

##  2 Method

###  2.1 Overview

During the past decades, many metal artifact reduction (MAR) methods have been
proposed [19]. These methods can be roughly divided into three groups:
sinogram completion, iterative reconstruction and image post-processing. 1)
Sinogram completion methods regard projection data in the metal trace region
as missing, which are estimated using image interpolation or inpainting
methods [36] [3] [9] [7] [16] [17] [10] [28]. This kind of method is difficult
to maintain continuity at the boundaries of metal trace in sinogram domain and
secondary artifacts will be introduced in the reconstructed CT images [30]. 2)
Iterative reconstruction methods usually formulate the objective function
aided by prior information, which can be solved with an iterative manner [26]
[27] [8] [25]. Nevertheless, these approaches are usually time-consuming [3]
and some parameters need to be set manually. 3) Image post-processing methods
directly remove artifacts in reconstructed image [24] [1], but they have to
entail risks of distorting the anatomic structure [34].

Recently, increasing attention has been paid to deep learning (DL). As a
result, many DL-based MAR approaches were proposed, which also can be grouped
into three different categories according to their operation domain. For
sinogram-domain methods, projection data in metal trace region is replaced
with the values generated by neural networks. Image-domain methods dedicate to
suppressing metal artifacts directly without accessing projection data. In the
last few years, dual-domain methods achieve impressive performance, which
simultaneously leverage the information from both image and sinogram domains.
However, current dual-domain methods usually process dual-domain data in a
specific order, which implicitly imposes a certain priority prior into MAR and
may ignore the latent information interaction between both domains. On the
other side, as shown in recent studies [34] [13] [15] [21], it is preferred to
produce a prior image as the auxiliary input to improve the performance of
networks. As a result, how to obtain a high-quality initial estimation becomes
another critical point for DL-based MAR methods.

To tackle these problems mentioned above, in this paper, we propose a novel
Interactive Dual-dOmain paraLlel network for CT MAR, dubbed as IDOL-Net. The
proposed IDOL-Net is composed of two modules as depicted in Fig. 1. The
disentanglement module introduces the idea of disentanglement representation
to generate high-quality prior sinogram and image as the complementary inputs.
The follow-up refinement module consists of two parallel and interactive
branches that simultaneously operate on image and sinogram domain, attempting
to fully exploit the latent information interaction between both domains. The
intermediate results exported from both branches are fused to exchange the
useful features from each other. Finally, the image quality is further
enhanced by the refinement module. A representative slice processed using
different MAR methods is given in Fig. 1.

The main contributions of this work are summarized as follows:

(1) A novel dual-domain network, IDOL-Net, which can simultaneously leverage
the information from both sinogram and image domains, is proposed to estimate
a complex end-to-end mapping from joint metal-affected sinogram and image to
artifact-free image.

(2) Disentanglement representation is introduced to generate a high-quality
hybrid prior estimation from the initial metal-affected sinogram and
reconstructed image.

(3) A refinement module containing two parallel and interactive branches is
designed to simultaneously extract the intermediate features from both
complementary domains, which are delivered and fused during the reconstruction
procedure.

(4) Extensively experimental results on the simulated and clinical datasets
demonstrate the effectiveness compared to other state-of-the-art MAR methods
in both qualitative and quantitative aspects.

##  3 Related works

###  3.1 Single-domain DL-based MAR

Early MAR methods directly applied DL on single-domain data, i.e., sinogram or
image reconstructed by filter backprojection (FBP). For sinogram domain,
neural networks are utilized to synthesize the missing data in the metal trace
region. Park et al. used U-Net [23] to repair inconsistent sinograms by
removing the primary metal-induced beam hardening factors along the metal
trace boundaries [20]. In [32] [29] [4], generative adversarial net (GAN) and
its variant were introduced to improve the inpainting performance. Partial
convolution [14], which is more suitable to sinogram completion, was employed
in [22] and [35]. However, it is hard to guarantee the sinogram continuity at
the boundaries of metal trace and secondary artifacts will appear
consequentially. On the other hand, Gjesteby et al. first applied a simple
plain CNN to suppress the artifacts in image domain [6]. Zhang et al. proposed
to fuse the results obtained using different methods as the prior image and
replace the metal-affected region in the sinogram with the corresponding part
from the prior image [37]. In [5], the authors presented to combine the result
of NMAR [16] and the detail image extracted by guided filter as hybrid inputs
to a dual-stream network. Liao et al. introduced the idea of unsupervised
disentanglement representation to recover the clean image with unpaired
training samples [12]. The main limitation for this method lies in that when
the metallic implants get larger, the artifacts cannot be completely removed
and the anatomic structures are prone to distortion.

###  3.2 Dual-domain DL-based MAR

Recently, dual-domain methods become mainstream. For instance, in [13], two
sub-networks are jointed via a Radon inversion layer to process the linear-
interpolated sinogram and its reconstructed image in sequence. To further
improve the performance, the authors in [15] proposed to insert metal mask
projection into the sinogram network. Yu et al. pointed out that there would
be some tiny anatomical structure changes if image domain enhancement is
adopted at the final stage [34]. Thus, an image domain network is trained at
first to produce a prior image with good quality to guide the subsequent
sinogram domain learning. Peng et al. utilized partial convolution operator to
recover the contaminated values in sinogram and trained an auxiliary sub-
network in image domain to further refine the details [21]. However, since
these methods usually process the sinogram and image in a specific order,
which ignores the latent information interaction, there is still room to
improve the performance by searching for a better network architecture for
dual-domain information fusion.

Figure 2: Overview of the proposed MAR method.

##  4 Method

###  4.1 Overview

As depicted in Fig. 2, the proposed IDOL-Net is composed of two modules, the
disentanglement module and refinement module. In the disentanglement module,
we try to obtain accurate prior estimations, including both initial sinogram
and image. In the refinement module, the preliminarily corrected sinogram and
image from disentanglement module are further refined aided by interactive
feature fusion. The details of both modules are elaborated as follows.

###  4.2 Disentanglement module

Current DL-based methods [34] [13] [15] [37] have demonstrated the
effectiveness in introducing the prior image or sinogram separately as the
auxiliary input for MAR. However, although it is natural to combine both image
and sinogram as the prior inputs, none of these methods have explored this
possibility. To take full advantage of the prior information from both
domains, unlike other DL-based methods, which use linear interpolation or
U-Net to obtain the coarse prior sinogram or image, the proposed
disentanglement module introduces the idea of disentanglement representation
to produce a more accurate hybrid prior estimation.

As illustrated in the left part of Fig. 2, the original metal-affected
sinogram $S_{ma}$ is fed into two networks, $G_{S\\_metal}$ and
$G_{S\\_clean}$, which respectively extract the metal-only sinogram
$S_{metal}$ and the content sinogram $S_{clean}$ from $S_{ma}$:

$S_{metal}=G_{S\\_metal}\ (S_{ma})$ (1) $S_{clean}=G_{S\\_clean}\ (S_{ma})$
(2)

Meanwhile, the artifact-corrupted image $X_{ma}$ reconstructed from $S_{ma}$
using FBP is also processed by two networks, $G_{X\\_art}$ and $G_{X\\_clean}$
, which respectively estimate the artifact image $X_{art}$ and the artifact-
free image $X_{clean}$ :

$X_{art}=G_{X\\_art}\ (X_{ma})$ (3) $X_{clean}=G_{X\\_clean}\ (X_{ma})$ (4)

Unlike the traditional MAR networks, which usually utilize one network to
generate clean sinogram or image, the proposed disentanglement module imposes
additional constraints on the extracted residuals, i.e., the metal-only
sinogram and the artifact image, producing a purer decomposition on the
original sinogram and image. The loss function of the disentanglement module
is defined as:

$\mathcal{L}_{dis}=\mathcal{L}_{clean}+\alpha_{1}\mathcal{L}_{art}+\alpha_{2}\mathcal{L}_{ma}$
(5)

Three components are formulated respectively as

$\mathcal{L}_{clean}=||X_{clean}-X_{clean}^{gt}||_{1}+\theta_{1}||S_{clean}-S_{clean}^{gt}||_{1}$
(6)
$\mathcal{L}_{art}=||X_{art}-X_{art}^{gt}||_{1}+\theta_{2}||S_{metal}-S_{metal}^{gt}||_{1}$
(7) $\displaystyle\mathcal{L}_{ma}$
$\displaystyle=||(X_{clean}+X_{art})-X_{ma}||_{1}$ (8)
$\displaystyle+\theta_{3}||(S_{clean}+S_{metal})-S_{ma}||_{1}$

where $X_{clean}^{gt}$ , $S_{clean}^{gt}$ , $X_{art}^{gt}$ and
$S_{metal}^{gt}$ are the corresponding labels. $\mathcal{L}_{clean}$ denotes
the reconstruction loss to facilitate $X_{clean}\approx X_{clean}^{gt}$ and
$S_{clean}\approx S_{clean}^{gt}$; $\mathcal{L}_{art}$ indicates the
reconstruction loss to impose $X_{art}\approx X_{art}^{gt}$ and
$S_{metal}\approx S_{metal}^{gt}$; $\mathcal{L}_{ma}$ is adopted to ensure
that the decomposition is invertible and without loss of information. For
simplicity, in this paper, $G_{S\\_metal}$ , $G_{S\\_clean}$ , $G_{X\\_art}$
and $G_{X\\_clean}$ share the same structure, which is presented in Fig. 3. A
simple plain CNN with four residual blocks is adopted as the unified network
architecture.

Figure 3: Fig. 2: The details of $G_{S\\_metal}$ , $G_{S\\_clean}$ ,
$G_{X\\_art}$ and $G_{X\\_clean}$ . C: channel numbers, K: kernel, S: stride,
and P: padding sizes.

###  4.3 Refinement module

Figure 4: Illustrations of the $G_{im}$. K: kernel, S: stride, P: padding
sizes and D: dilation. Figure 5: Illustrations of the $F_{sino}$. K: kernel,
S: stride, P: padding sizes and D: dilation.

Existing studies demonstrate that simply processing the corrupted sinogram or
image with one single network has limited performance [34] [13]. Based on this
consideration, a refinement module is proposed to further improve the image
quality. Its structure is shown in the right part of Fig. 2. There are two
branches to separately handle sinogram and image data. Both branches have four
same blocks, i.e., $G_{im}^{i}$ for image branch and $G_{sino}^{i}$ for
sinogram branch. $X_{clean}$ and $S_{clean}$ generated by the disentanglement
module are employed as the inputs of the proposed refinement module. FBP is
applied on $S_{clean}$ to obtain $X_{clean}^{fbp}$, which is concatenated with
$X_{clean}$ as the initial input of the first $G_{im}^{1}$. Similarly, forward
projection (FP) is applied on $X_{clean}$ to obtain $S_{clean}^{fp}$, which is
concatenated with $S_{clean}$ as the initial input of the first
$G_{sino}^{i+1}$ . Similar operations are performed after each $G_{im}^{i}$
and $G_{sino}^{i}$. FBP and FP are applied to the intermediate sinogram and
image results. Then the outputs of both FBP and its corresponding $G_{im}^{i}$
are concatenated and fed into $G_{im}^{i+1}$. The outputs of both FP and its
corresponding $G_{sino}^{i}$ are concatenated and fed into $G_{sino}^{i+1}$.
For simplicity, U-Net is adopted as the backbone of both $G_{sino}^{i}$ and
$G_{im}^{i}$ and we halve the channel numbers to reduce the computational
cost. To suppress the nonlocal artifacts in image domain, nonlocal network
modules [31] are integrated into $G_{im}^{i}$. The architecture of
$G_{im}^{i}$ is illustrated in Fig. 4. For sinogram branch, a sinogram
completion network, denoted as $F_{sino}^{i}$, is adopted to restore
projection data within the metal trace mask and the structure of
$F_{sino}^{i}$ is depicted in Fig. 5. Inspired by the idea of mask pyramid
network (MPN) [11], the metal mask projection is leveraged to alleviate
discontinuity at boundaries of metal trace.

Specifically, in a refinement block, given the metal mask $M$ and metal trace
mask $M_{t}$, there are four outputs, outputs of $G_{sino}^{i+1}$ and
$G_{im}^{i+1}$ recognized as $S_{sino}^{i+1}$ and $X_{im}^{i+1}$, forward
projection of $X_{im}^{i+1}$ denoted as $S_{im}^{i+1}$ and filtered back-
projection of $S_{sino}^{i+1}$ referred to as $X_{sino}^{i+1}$ .

$S_{res}^{i+1}=F_{sino}^{i+1}(S_{im}^{i}\ ,S_{sino}^{i}\ ,M_{t})$ (9)
$\displaystyle S_{sino}^{i+1}$
$\displaystyle=G_{sino}^{i+1}(S_{res}^{i+1},S_{ma},M_{t})$ (10)
$\displaystyle=S_{res}^{i+1}\odot M_{t}+S_{ma}\odot(1-M_{t})$
$X_{im}^{i+1}=G_{im}^{i+1}(X_{im}^{i}\ ,X_{sino}^{i})+X_{im}^{i}$ (11)
$S_{im}^{i+1}=\mathcal{FP}(X_{im}^{i+1})$ (12)
$X_{sino}^{i+1}=\mathcal{FBP}(S_{sino}^{i+1})$ (13)

where i=0,1,2,3. When i=0, $S_{sino}^{1}=S_{clean}$,
$S_{im}^{1}=S_{clean}^{fp}$, $X_{im}^{1}=X_{clean}$ and
$X_{sino}^{1}=X_{clean}^{fbp}$. The first three refinement blocks share the
same structure. The last refine block has similar architecture, but it has no
FP operation on $X_{im}^{4}$. Then, we concatenate $X_{im}^{4}$ and
$X_{sino}^{4}$ of the last block as the inputs of the fusion layer
$\mathcal{F}$, which is a $1\times 1$ convolution layer, to obtain the final
result $X_{corr}$ as

$X_{corr}=\mathcal{F}(X_{im}^{4}\ ,X_{sino}^{4})$ (14)

To optimize the refinement module, $\mathcal{L}_{1}$ loss is adopted to
minimize the differences between $X_{corr}$ and the ground truth:

$\mathcal{L}_{corr}=||(X_{corr}-X_{gt}^{clean})\odot(1-M_{t})||_{1}$ (15)

Thus, the total loss function can be written as:

$\mathcal{L}=\mathcal{L}_{dis}+\gamma\mathcal{L}_{corr}$ (16)

where $\alpha_{1}$, $\alpha_{2}$, $\theta_{1}$,$\theta_{2}$, $\theta_{3}$ and
$\gamma$ are hyper-parameters to balance the weights of different components.
We manually set $\alpha_{1}=\alpha_{2}=1.0$ ,
$\theta_{1}=\theta_{2}=\theta_{3}=0.06$ and $\gamma=10.0$ in this manuscript.

##  5 Experiments

###  5.1 Data set

In our experiments, the manually segmented metal masks dataset [37], which
contains simulated dental fillings, spine crews and hip prostheses, is
utilized to produce the metal masks. We randomly selected 1000 CT images from
DeepLesion dataset [33] and 90 metal masks to build our training set, which
contains totally 90000 images. Another 200 CT images and ten masks were used
to form the testing set with totally 2000 images. The same procedure used in
[37] [12] is adopted to synthesize the metal-affected CT images. Images were
resized to 256×256 to reduce the computational cost. Without loss of
generality, 2D parallel-beam geometry was employed. There were 361 views
evenly distributed from 0° to 180° and 367 bins received photons. In addition,
the pixel values were clipped to [0, 4095], which coincided with the real
scenario.

###  5.2 Implementation details

The ODL library was adopted to implement FBP and FP. The whole network was
trained in an end-to-end manner with the Pytorch deep learning framework and
was optimized using the Adam optimizer with the parameters
$(\beta_{1},\beta_{2})=(0.5,0.999)$. The learning rate was set to 0.0002
initially and halved for every 10 epochs. We trained our network for 100
epochs on a workstation with four Nvidia 1080Ti GPUs.

PSNR/SSIM | 1 | 2 | 3 | 4 | 5 | Average
---|---|---|---|---|---|---
Uncorrected | 15.93/0.7101 | 15.14/0.6617 | 14.86/0.6417 | 15.14/0.6572 | 15.32/0.6564 | 15.33/0.6673
LI | 32.86/0.9545 | 32.60/0.9442 | 30.76/0.9330 | 27.26/0.8934 | 28.94/0.8853 | 30.74/0.9224
NMAR | 32.54/0.9564 | 31.87/0.9394 | 31.20/0.9359 | 27.91/0.9035 | 29.21/0.9000 | 30.83/0.9270
CNNMAR | 32.78/0.9665 | 32.80/0.9641 | 33.24/0.9612 | 32.61/0.9517 | 32.46/0.9455 | 32.72/0.9574
ADN | 30.92/0.9335 | 30.03/0.9162 | 32.08/0.9393 | 26.45/0.9136 | 31.78/0.9407 | 30.58/0.9290
DuDoNet | 38.77/0.9673 | 38.57/0.9648 | 36.44/0.9642 | 33.03/95.33 | 34.26/0.9531 | 36.82/0.9777
IDOL-Net | 42.01/0.9870 | 42.37/0.9878 | 40.67/0.9900 | 35.26/0.9782 | 38.30/0.9842 | 40.20/0.9858
Table 1: Quantitative comparison of different methods on the simulated
dataset. 1-5 represents the sizes of metallic implants in ascending order.

Figure 6: Comparisons with different MAR methods on simulation data with small, middle and big metallic implant. (A1-A3) reference images; (B1-B3) metal corrupted images; (C1-C3) corresponding results of LI; (D1-D3) NMAR; (E1-E3) CNNMAR; (F1-F3) ADN; (G1-G3) DuDoNet and (H1-H3) IDOL-Net. Methods | Refine-Net | Clean-IDOL | Art-IDOL | Prior-Im | Prior-Sino | IDO-Net
---|---|---|---|---|---|---
PSNR | 36.35 | 36.65 | 37.52 | 31.42 | 27.30 | 40.20
SSIM | 0\. 9660 | 0.9716 | 0.9723 | 0.9418 | 0.6005 | 0.9858
Table 2: Quantitative comparison of different variants of our method on the
simulated dataset.

###  5.3 Experimental results on the simulated data set

To validate the effectiveness of the proposed IDOL-Net, several methods,
including LI [9], NMAR [16], CNNMAR [37], ADN [12], DuDoNet [13] were involved
for comparison. LI and NMAR are widely used interpolation-based methods.
CNNMAR takes advantage of the outputs of different MAR methods to generate a
prior image. ADN is an advanced unsupervised image-domain method. DuDoNet is a
state-of-the-art dual-domain approach. All the methods were implemented with
the publicly released code or strictly implemented according to the original
paper. Structural similarity (SSIM) and peak signal to noise ratio (PSNR) were
employed as the quantitative metrics.

1) Quantitative comparisons: Table 1 shows the average quantitative results on
the simulated dataset. It is observed that all the methods can significantly
improve the PSNR and SSIM values and DL-based methods achieve better scores
than traditional MAR methods. DuDoNet and IDOL-Net gain obvious improvements
in terms of both metrics, which quantitatively demonstrated the merits of
dual-domain methods. In addition, our method produces a noticeable advantage
over DuDoNet in terms of both metrics for all the metal sizes, with an average
improvement of 3.3dB in PSNR and 0.81 in SSIM.

2) Qualitative comparisons: Fig. 6 shows the representative results of
different methods on simulated data with different metal sizes. For better
visualization, the metal masks are painted in red. Due to the fact that LI and
NMAR are interpolation-based methods directly discarding the projection data
in metal trace, the metal information is lost and the corrected sinogram
cannot keep the continuity at the boundaries of metal trace, leading to
blurred tissues around the metals and remarkable secondary artifacts. Although
CNNMAR fuses the results of different MAR methods, the improvement is quite
limited. ADN achieves better performance on structure recovery around the
metal implants than LI, NMAR and CNNMAR, and even DuDoNet in some cases, but
it also can be noticed that in the second case (Fig. 6 (F2)), the shadow
artifacts are not well suppressed and the details covered by the artifacts are
hard to identify. This phenomenon probably lies in that ADN is essentially a
post-processing method, which does not leverage the sinogram information.
Compared to other methods, DuDoNet and the proposed IDOL-Net have the best
performance on artifact reduction, but DuDoNet cannot recover the structures
and tissues around the metal well, which can be clearly observed in the
magnified yellow boxes in Fig. 6 (G2) and (G3). Our proposed IDOL-Net not only
effectively removes most artifacts but also recovers the details better than
all the other methods in all cases.

###  5.4 Experimental results on clinical data

To evaluate the robustness of our IDOL-Net for clinical practice, clinical CT
images were tested and Fig. 7 shows the visual comparisons of one
representative slice processed using different methods. The metal was
segmented by the threshold of 2000 HU and the metal mask is painted in red for
better visualization. The test image was normalized to the same range as the
training data and the results were obtained using the model trained with
simulated dataset used in the previous subsection. In Fig. 7 (A), the metal
leads to severe artifacts, which covers some tissues, especially near the
metal. It is observed that LI, NMAR and DuDoNet cannot retrieve the tissues
around the metal, but due to the utilization of dual-domain information,
DuDoNet outperforms most methods in artifact reduction. Since the shadow
artifacts around the metal is very heavy, ADN cannot do much about this, which
is similar to the situation in simulated dataset. It is observed that IDOL-Net
suppresses most of the metal artifacts and effectively preserves the
anatomical structures around the metals, which demonstrates its potential for
real clinical application.

###  5.5 Analytical studies

In this subsection, we investigate the effectiveness of different network
modules and our ablation experiments use the following configurations.

Figure 7: Visual comparison with different MAR methods on a clinical CT image.
(A)-(G) respectively represent uncorrected CT images and corrected results
using LI, NMAR, CNNMAR, ADN, DuDoNet and IDOL-Net.

(a): Refine-Net: network trained without both prior image and sinogram;

(b): Clean-IDOL: network that only learns clean parts of CT images and
sinograms followed by refinement module;

(c): Art-IDOL: network that only learns residual parts of CT images and
sinograms followed by refinement module;

(d): Prior-Im: network with disentanglement module followed by only image
refinement branch;

(e): Prior-Sino: network with disentanglement module followed by only sinogram
refinement branch; and

(f): Our full model with both disentanglement and refinement modules.

1) Effectiveness of prior estimation. Most dual-domain MAR methods usually use
original or LI-interpolated sinogram and corresponding reconstructed CT images
as inputs, or adopted LI’s corrected results as inputs. However, the first
case results in incomplete artifact reduction and maintains some obvious
artifacts in final corrected images. In addition, since LI is just a rough
estimation, which gives rise to details lost around the metal. A possible
solution is to generate both good quality prior sinogram and image. To
investigate the effectiveness of the prior image and sinogram generated by our
proposed disentanglement module, a network without this module, termed Refine-
Net, is trained and we take metal-affected CT image and sinogram as inputs.
Table 2 shows the quantitative results. It is clear to see that aided by prior
image and sinogram, our model achieves much better score in terms of both
metrics. As shown in In Fig. 8 (B), there are obvious artifacts indicated by a
blue arrow.

2) Effectiveness of $\mathcal{L}_{ma}$. There are two ways to obtain high
quality prior image and sinogram. One is just predicting the clean data
directly from contaminated data. The other is learning residual part and get
the clean part by subtracting the learned residual data from original
corrupted data. However, these methods may fail to separate the clean data or
residuals from original data completely. In other words, there could be some
useful information remaining in the restored data or residuals. To tackle this
problem, inspired by disentangling representation, we learn both clean part
and residual part simultaneously. In addition, we adopt $\mathcal{L}_{ma}$ to
produce a lossless decomposition. To investigate the effectiveness of
$\mathcal{L}_{ma}$, we trained two networks, Clean-IDOL and Art-IDOL, without
$\mathcal{L}_{ma}$. It can be seen in Table 2 that both Clean-IDOL and Art-
IDOL achieve lower SSIM and PSNR values than IDOL-Net. In Fig. 8, Clean-IDOL
introduced secondary artifacts indicated by green arrow. Art-IDOL shows slight
anatomic structure distortion marked by yellow dotted box. It is noticeable
that the result of IDOL-Net is most consistent with the referenced image in
all the methods.

3) Effectiveness of parallel branches. To show the effectiveness of our
proposed interactive dual-domain parallel branches, we trained two networks,
dubbed Prior-Im and Prior-Sino, which are only with image domain branch or
sinogram domain branch, respectively. It can be seen in Table 2 that Prior-
Sino gains lower SSIM and PSNR scores than other methods. The possible reason
lies in that Prior-Sino restored projection data within metal trace region
using the original contaminated data and the distortion introduced by
inaccurate result will expand to the whole reconstructed image, which heavily
lower both quantitative scores calculated in image domain. Prior-Im
significantly improves the qualitative and quantitative results compared to
Prior-Sino, but as shown in Table 2 and Fig. 8, our IDOL-Net have superior
performance to Prior-Im, which provides impressive support for our claim that
the proposed interactive dual-domain parallel branches are efficient.

4) Discussion of refinement blocks. To sense the impact of the number of
network blocks in refinement module, different block numbers (from 1 to 5) are
tested. TABLE 3 lists the quantitative results and it can be observed that the
model with four blocks attains the best quantitative score.

Methods | 1 Block | 2 Block | 3 Block | 4 Block | 5 Block
---|---|---|---|---|---
PSNR | 37.70 | 39.40 | 39.34 | 40.20 | 39.37
SSIM | 0\. 9794 | 0.9827 | 0.9831 | 0.9858 | 0.9808

Table 3: The proposed method with different numbers of refinement blocks.
Figure 8: Visual comparison with different MAR methods on a clinical CT image.
(A)-(G) respectively represent ground truth and corrected results using
Refine-Net, Clean-IDOL, Art-IDOL, Prior-Im, Prior-Sino and IDOL-Net.

##  6 Conclusion

In this work, we present a novel interactive dual-domain parallel network for
CT MAR, which contains two main modules. The disentanglement module aims to
generate high quality prior sinogram and image as the initial inputs for the
following refinement module, which is composed of two interactive parallel
branches to exchange the useful information extracted from the intermediate
results of both sinogram and image domains. The whole framework is trained in
a end-to-end manner so that the dual-domain architecture can not only benefit
from each other but also alleviate information loss. Experiment results on
simulated and clinical data demonstrate that our proposed method can
effectively suppress most artifacts and preserve tissues details. In the
future work, we will investigate how will the segmentation result influence
the final MAR result. Integrating the segmentation task into our MAR framework
is also an interesting topic.

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# Counterexamples to maximal regularity
for operators in divergence form

Sebastian Bechtel  Delft Institute of Applied Mathematics, Delft University of
Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
<EMAIL_ADDRESS>,  Connor Mooney  Department of Mathematics, UC Irvine
Rowland Hall 410C, Irvine, CA 92697-3875, USA<EMAIL_ADDRESS>and  Mark
Veraar  Delft Institute of Applied Mathematics, Delft University of
Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we present counterexamples to maximal $L^{p}$-regularity for a
parabolic PDE. The example is a second-order operator in divergence form with
space and time-dependent coefficients. It is well-known from Lions’ theory
that such operators admit maximal $L^{2}$-regularity on $H^{-1}$ under a
coercivity condition on the coefficients, and without any regularity
conditions in time and space. We show that in general one cannot expect
maximal $L^{p}$-regularity on $H^{-1}({\mathbb{R}}^{d})$ or $L^{2}$-regularity
on $L^{2}({\mathbb{R}}^{d})$.

###### Key words and phrases:

Maximal $L^{p}$-regularity, counterexamples, Lions’ problem, divergence form
operators

###### 2010 Mathematics Subject Classification:

Primary: 35K90; Secondary: 35B65

##  1\. Introduction

Let $V$ and $H$ be complex Hilbert spaces such that $V\hookrightarrow H$
densely and continuously. Identifying $H^{*}$ with its dual, we can view $H$
as a subspace of $V^{*}$, which is the dual of $V$. We start with the abstract
problem

(CP) $\left\\{\begin{aligned} u^{\prime}-\mathcal{A}u&=f,\ \ \text{on}\
(0,1),\\\ u(0)&=0.\end{aligned}\right.$

Here $\mathcal{A}:(0,1)\to\mathcal{L}(V,V^{*})$ is strongly measurable and
$f\in L^{2}(0,1;V^{*})$. Moreover, we suppose that there are
$\Lambda,\lambda>0$ such that for all $t\in(0,1)$ and $v\in V$ one has

$\displaystyle\|\mathcal{A}(t)v\|_{V^{*}}\leq\Lambda\|v\|_{V},\qquad\operatorname{Re}\langle\mathcal{A}(t)v,v\rangle\geq\lambda\|v\|_{V}^{2}.$

By Lions’ theory [18] it is known that (CP) has a unique weak solution $u\in
H^{1}(0,1;V^{*})\cap L^{2}(0,1;V)$. More generally (see [8, XVIII.3.5]), if
$f\in L^{2}(0,1;V^{*})+L^{1}(0,1;H)$, then (CP) has a unique weak solution
$u\in L^{2}(0,1;V)$ such that $u^{\prime}\in L^{2}(0,1;V^{*})+L^{1}(0,1;H)$.

In this paper we study two problems concerning the regularity of $u$.

######  Problem 1.

Let $p\in(1,\infty)\setminus\\{2\\}$. Under what condition on $\mathcal{A}$
does the following hold: for all $f\in L^{p}(0,1;V^{*})$ there is a unique
weak solution $u$ which is in $L^{p}(0,1;V)$?

Using the equation this also gives $u\in H^{1,p}(0,1;V^{*})$. Problem 1 can
equivalently be formulated to the question whether there is a constant $C>0$
such that for all step functions $f$ valued in $H$ and with $u$ the unique
solution to (CP) given by Lions’ result for $p=2$ one has the estimate

$\|u\|_{L^{p}(0,1;V)}\leq C\|f\|_{L^{p}(0,1;V^{*})}.$

It is well-known that regularity results fail for the endpoint cases $p=1$ and
$p=\infty$ unless $V=V^{*}$ and thus $\mathcal{A}$ is a family of bounded
operators on $V$ (see [6, 13] and also [15, Thm. 17.4.4 & Cor. 17.4.5]).
Hence, we can safely concentrate on the case $p\in(1,\infty)$ in this paper.

The second problem is a variation of Lions’ problem [18, p. 68], who
originally asked this question for symmetric $\mathcal{A}$.

######  Problem 2.

Under what condition on $\mathcal{A}$ does the following hold: for all $f\in
L^{2}(0,1;H)$ the unique weak solution $u$ satisfies $u^{\prime}\in
L^{2}(0,1;H)$?

Again using the equation one also has $\mathcal{A}u\in L^{2}(0,1;H)$. However,
it is unclear what this tells about the regularity of $u$ since the domain of
the operators $\mathcal{A}(t)$ in $H$ are not easy to describe in general.

Both problems have in common (at least if $p>2$) that they imply regularity of
$u$ such as $u\in C^{\alpha}([0,1];[H,V]_{\lambda})$ for some $\alpha>0$ and
$\lambda\in(0,1)$. Such properties are for instance useful in the study of
non-linear problems. They follow from standard interpolation estimates and
Sobolev embeddings. In Lions’ general $L^{2}$-setting one can only obtain
$C([0,1];H)$ or Hölder regularity of small order, compare with Theorem 2.2.

If $\mathcal{A}$ is autonomous, that is to say, the family $\mathcal{A}(t)$
does not depend on $t$, then in both problems the answer is affirmative.
Indeed, in Problem 1 this can be concluded from Lions’ result and [15, Thm.
17.2.31]. Concerning Problem 2, this follows from a result of de Simon [21].

Otherwise, some conditions are needed. In the case of Problem 1 it is
sufficient that the mapping $t\mapsto\mathcal{A}(t)\in\mathcal{L}(V,V^{*})$ is
(piecewise relatively) continuous [2, 20]. Without any continuity it is only
known that there is some $\varepsilon>0$ depending on $\Lambda$, $\lambda$
such that Problem 1 holds with $p\in(2-\varepsilon,2+\varepsilon)$, see
Theorem 2.2. For Problem 2 the situation is even more delicate. Based on an
abstract counterexample for the Kato square root problem due to McIntosh, Dier
constructed a first nonsymmetric counterexample to Problem 2 in his PhD
thesis, see also [3, Sec. 5]. Moreover, Fackler constructed a counterexample
that is symmetric and $C^{\frac{1}{2}}$-Hölder continuous [11]. To the
contrary, with slightly more regularity (for instance
$C^{\frac{1}{2}+\varepsilon}$-Hölder regularity), many positive results were
given [1, 4, 7, 9, 12, 14, 20]. We also recommend the survey [3] for an
overview of Problem 2.

The counterexamples due to Dier and Fackler are abstract and not differential
operators. As is highlighted by the solution to the Kato square root problem
[5], the extra structure of a differential operator can be beneficial compared
to the general situation. It is hence of interest to find counterexamples to
Problems 1 and 2 that are differential operators. For Problem 2 this was
explicitly pointed out in [12, Problem 6.1] and [3, Prop. 12.1]. To be more
precise with our setting, we work with $H=L^{2}({\mathbb{R}}^{d})$ and
$V=H^{1}({\mathbb{R}}^{d})$. If
$B\colon[0,1]\times{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ is elliptic in
the sense that there exist constants $\Lambda,\lambda>0$ such that for all
$t\in[0,1]$ and $x\in{\mathbb{R}}^{d}$ one has

(1.1)
$\displaystyle|B(t,x)\xi|\leq\Lambda|\xi|,\qquad\operatorname{Re}B(x,t)\xi\cdot\overline{\xi}\geq\lambda|\xi|^{2}\qquad(\xi\in{\mathbb{C}}^{d}),$

then consider the problem

(P) $\left\\{\begin{aligned} u^{\prime}-{\text{div}}(B\nabla
u)&=f,\quad\text{on }(0,1),\\\ u(0)&=0,\end{aligned}\right.$

where $f\in L^{2}(0,1;H^{-1}({\mathbb{R}}^{d}))$. In the notation of (CP) we
have put $\mathcal{A}(t)v=-{\text{div}}(B(t)\nabla v)$. Lions’ theory yields a
unique solution $u$ in the regularity class
$H^{1}(0,1;L^{2}({\mathbb{R}}^{d}))\cap L^{2}(0,1;H^{1}({\mathbb{R}}^{d}))$.
Problems 1 and 2 ask if for all elliptic coefficients one has the regularity
$u\in L^{p}(0,1;H^{1}({\mathbb{R}}^{d}))$ or $u^{\prime}\in
L^{2}(0,1;L^{2}({\mathbb{R}}^{d}))$ when the forcing term $f$ is taken from
$L^{p}(0,1;H^{-1}({\mathbb{R}}^{d}))$ or $L^{2}(0,1;L^{2}({\mathbb{R}}^{d}))$.
The answer is negative in both cases and the respective counterexamples will
be the content of the present article. More precisely, based on a construction
of the second-named author [19], we will obtain equations whose solution fails
to have higher $L^{r}(L^{s})$-integrability (Theorem 2.3). Based on this
result, we produce a counterexample to Problem 1 for every $p\neq 2$ in
Theorem 2.4. In particular, this shows sharpness of Theorem 2.2 which we
mentioned briefly above. Concerning Problem 2, we will show in Theorem 2.6
that it fails in the worst possible way for general elliptic problems in
divergence form.

##  2\. Maximal regularity in Hilbert spaces

###  2.1. Known results

We present the results regarding Problem 1 that were already discussed in the
introduction.

A function $u\in L^{2}(0,1;V)\cap H^{1}(0,1;V^{*})$ is called a solution to
(CP) if for all $t\in[0,1]$,

$u(t)-\int_{0}^{t}\mathcal{A}uds=\int_{0}^{t}fds,$

where the integrals are defined as Bochner integrals in $V^{*}$.

One can check that $u$ is a solution to (CP) if and only if for all $\phi\in
C^{\infty}_{c}((0,1);V)$ one has

(2.1) $\displaystyle-\int_{0}^{1}\langle u(s),\phi^{\prime}(s)\rangle
ds-\int_{0}^{1}\langle\mathcal{A}(s)u(s),\phi(s)\rangle ds=\int_{0}^{1}\langle
f(s),\phi(s)\rangle ds.$

######  Theorem 2.1 (Lions).

In the situation of (CP) there is a unique solution $u\in L^{2}(0,1;V)\cap
H^{1}(0,1;V^{*})$. Constants in the maximal regularity estimate depend only on
$\Lambda$ and $\lambda$.

Based on a perturbation principle for isomorphisms in complex interpolation
scales due to Sneiberg [22], Lions’ result can be extended to $p$ close to
$2$, see [10, Thm. 4.2].

######  Theorem 2.2.

In the situation of Lions’ result (Theorem 2.1) there exists $\varepsilon>0$
depending on $\Lambda$, $\lambda$ and the pair $(V,H)$ such that Problem 1 has
a positive solution for $p\in(2-\varepsilon,2+\varepsilon)$.

It will be clear from Theorem 2.4 that one cannot improve the latter to all
$p\in(1,\infty)$, even if $f\in L^{p}(0,1;H)$.

###  2.2. Failure of $L^{r}(L^{s})$-integrability for variational solutions

The following counterexample is based on a construction due to the second-
named author [19] and is the basis for our subsequent counterexamples to
Problems 1 and 2.

######  Theorem 2.3 (Failure of higher integrability).

Let $d\geq 2$. For every $s,r\in(1,\infty)$ with
$\frac{2}{r}+\frac{d}{s}<\frac{d}{2}$ there exists an elliptic matrix
$B\colon[0,1]\times{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ and $f\in
L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$ such that the unique solution $u$ to
(P) satisfies $u\not\in L^{r}(0,1;L^{s}({\mathbb{R}}^{d}))$.

###### Proof.

Pick $\frac{2}{r}+\frac{d}{s}<\mu<\frac{d}{2}$.

Step 1: general setup. We repeat part of the construction in [19]. There exist
a Lipschitz function $w:{\mathbb{R}}^{d}\to{\mathbb{C}}$ and an elliptic
matrix $a:{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ with the subsequent
properties. The functions $a$ and $w$ are uniformly bounded, smooth away from
zero, Lipschitz in zero, and for all multiindices $\alpha$, $|x|\geq 1$,

(2.2) $\displaystyle|\partial^{\alpha}w(x)|$ $\displaystyle\leq
C_{\alpha}|x|^{-|\alpha|-\mu},$ (2.3) $\displaystyle|\partial^{\alpha}a(x)|$
$\displaystyle\leq C_{\alpha}|x|^{-|\alpha|}.$

These estimates follow from the definitions of $w$ and $a$ given in [19, Sec.
3]. Similar as in [19, Eq. (5), Thm 2.2, and Cor 2.3] define

$\displaystyle B(t,x)$ $\displaystyle\coloneqq a(x/(1-t)^{1/2})$
$\displaystyle\zeta(t,x)$
$\displaystyle\coloneqq(1-t)^{-\mu/2}e^{-\frac{i}{2}\log(1-t)}w(x/(1-t)^{1/2}).$

These were constructed so that $\zeta\in L^{2}_{\rm
loc}([0,1)\times{\mathbb{R}}^{d})$, $\nabla\zeta\in L^{2}_{\rm
loc}([0,1)\times{\mathbb{R}}^{d})$ and the equation

(2.4)
$\displaystyle\partial_{t}\zeta(t,x)={\text{div}}(B(t,x)\nabla\zeta(t,x))$

is satisfied pointwise for $t\in(0,1)$ and
$x\in{\mathbb{R}}^{d}\setminus\\{0\\}$. The latter means that

$(\zeta,\partial_{t}\phi)_{L^{2}((0,1)\times{\mathbb{R}}^{d})}=(B\nabla\zeta,\nabla\phi)_{L^{2}((0,1)\times{\mathbb{R}}^{d})}$

holds for all $\phi\in C^{\infty}_{c}((0,1)\times{\mathbb{R}}^{d})$. It is
clear from the constructions that the weak derivative $\partial_{t}\zeta$
exists on $(0,T)$ as an $L^{2}(B_{R})$-valued function, for any $T<1$ and
$R>0$. Now let

$\displaystyle u(t,x)$ $\displaystyle=t\zeta(t,x)\eta(x),$ $\displaystyle
f(t,x)$
$\displaystyle=\eta(x)\zeta(t,x)-t\sum_{k,\ell=1}^{d}[B_{k,\ell}(t,x)+B_{\ell,k}(t,x)]\partial_{k}\zeta(t,x)\partial_{\ell}\eta(x)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-t\zeta(t,x){\text{div}}(B(t,x)\nabla\eta(x)),$

where the cut-off function $\eta\in C^{\infty}_{c}({\mathbb{R}}^{d})$ is such
that $\eta(x)=1$ for $|x|\leq 1$, $\eta(x)=0$ for $|x|\geq 2$ and
$0\leq\eta\leq 1$ on ${\mathbb{R}}^{d}$.

We claim that $u\in L^{2}(0,1;H^{1}({\mathbb{R}}^{d}))\cap
H^{1}(0,1;H^{-1}({\mathbb{R}}^{d}))$, $f\in
L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$ and $u$ is the unique solution to
(P). Then, using (2.4) it is elementary to check that for $t\in(0,1)$ and
$x\in{\mathbb{R}}^{d}\setminus\\{0\\}$ one pointwise has

$\left\\{\begin{aligned} \partial_{t}u(t,x)-{\text{div}}(B(t,x)\nabla
u(t,x))&=f(t,x),\\\ u(t,0)&=0.\end{aligned}\right.$

Moreover, it also holds in distributional sense on
$(0,1)\times{\mathbb{R}}^{d}$. Hence, by density and (2.1), it is a solution
to (P).

Therefore, it remains to check $u\in L^{2}(0,1;H^{1}({\mathbb{R}}^{d}))$ and
$f\in L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$ in order to find that $u$ is
the unique solution to (P) provided by Theorem 2.1.

Step 2: $u\in L^{2}(0,1;H^{1}({\mathbb{R}}^{d}))$. Fix $t\in(0,1)$ and let
$\alpha$ be a multiindex with $|\alpha|\leq 1$. By definition of $\zeta$ and
using (2.2) one has

(2.5)
$\displaystyle\begin{split}\|\partial^{\alpha}\zeta(t,\cdot)\|_{L^{2}(B_{2})}&\leq(1-t)^{-\frac{\mu+|\alpha|}{2}+\frac{d}{4}}\|\partial^{\alpha}w\|_{L^{2}(B_{{2}/{(1-t)^{1/2}}})}\\\
&\leq(1-t)^{-\frac{\mu+|\alpha|}{2}+\frac{d}{4}}\Big{[}C_{d,w}+C_{0}\||\cdot|^{-\mu-|\alpha|}\|_{L^{2}(B_{{2}/{(1-t)^{1/2}}}\setminus
B_{1})}\Big{]}\\\
&\leq(1-t)^{-\frac{\mu+|\alpha|}{2}+\frac{d}{4}}\Big{[}C_{d,w}+C_{0}C_{d}(1-t)^{\frac{\mu+|\alpha|}{2}-\frac{d}{4}}\Big{]}\\\
&\leq
C_{d,w}(1-t)^{-\frac{\mu+|\alpha|}{2}+\frac{d}{4}}+C_{d,w}^{\prime}\end{split}$

for some constant $C_{d,w}^{\prime}$ only depending on $d$ and $w$ and with

$C_{d,w}=|B_{1}|^{1/2}\sup_{x\in B_{1}}(|w(x)|+|\nabla w(x)|).$

Note that $C_{d,w}$ is finite since $w$ is Lipschitz. Recall that
$\frac{d}{4}-\frac{\mu}{2}$ is positive. Thus, by definition of $u$ and using
(2.5) we find

$\displaystyle\|u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}\leq
t\|\zeta(t,\cdot)\|_{L^{2}(B_{2})}\leq C_{d,w}+C_{d,w}^{\prime}.$

Thus, $u$ is bounded as an $L^{2}({\mathbb{R}}^{d})$-valued function and
therefore in particular $u\in L^{2}(0,1;L^{2}({\mathbb{R}}^{d}))$. Similarly,

$\displaystyle\|\nabla u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}$
$\displaystyle\leq\|\nabla\zeta(t,\cdot)\|_{L^{2}(B_{2})}+\|\nabla\eta\|_{\infty}\|\zeta(t,\cdot)\|_{L^{2}(B_{2}\setminus
B_{1})}$ $\displaystyle\leq
C_{d,w}(1-t)^{-\frac{\mu+1}{2}+\frac{d}{4}}+C_{d,w}^{\prime}+C_{d,w}^{\prime}\|\eta\|_{\infty}.$

By choice of $\mu$ we have $-\frac{\mu+1}{2}+\frac{d}{4}>-\frac{1}{2}$,
therefore also $\nabla u\in L^{2}(0,1;L^{2}({\mathbb{R}}^{d}))$.

Step 3: $f\in L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$. To estimate the norm
of $f$ we consider all parts separately. The fact that $\eta\zeta\in
L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$ follows from (2.5). Due to the
support properties of $\nabla\eta$ and the boundedness of $B$ it suffices to
prove a uniform estimate for
$\|\partial_{m}\zeta(t,\cdot)\|_{L^{2}(B_{2}\setminus B_{1})}$ to estimate
$B_{k,\ell}\partial_{m}\zeta\partial_{n}\eta$. For this calculate with (2.2)
that

(2.6)
$\displaystyle\begin{split}\|\partial_{m}\zeta(t,\cdot)\|_{L^{2}(B_{2}\setminus
B_{1})}&=(1-t)^{-\frac{\mu}{2}-\frac{1}{2}}\|\partial_{m}w(\cdot/(1-t)^{1/2}\|_{L^{2}(B_{2}\setminus
B_{1})}\\\ &\leq C_{1}\||\cdot|^{-1-\mu}\|_{L^{2}(B_{2}\setminus
B_{1})}.\end{split}$

It remains to estimate $\zeta{\text{div}}(B\nabla\eta)$. Again, due to the
support properties of $\nabla\eta$, it suffices to consider $x\in
B_{2}\setminus B_{1}$. First, by (2.2) and definition of $\zeta$,

$|\zeta(t,x)|\leq(1-t)^{-\mu/2}|w(x/(1-t)^{1/2})|\leq C_{1,0}.$

Thus it remains to estimate
$\|\partial^{\alpha}B_{k,\ell}(t,\cdot)\partial^{\beta}\eta\|_{L^{2}(B_{2}\setminus
B_{1})}$ for $|\alpha|\leq 1$ and $1\leq|\beta|\leq 2$. Since $B_{k,\ell}$ and
$\partial^{\beta}\eta$ are uniformly bounded, it is enough to show that
$\|\partial^{\alpha}B_{k,\ell}(t,\cdot)\|_{L^{2}(B_{2}\setminus B_{1})}$ is
uniformly bounded for $|\alpha|=1$. Indeed, this follows from the analogous
calculation to (2.6) using (2.3) instead of (2.2).

Step 4: $u\not\in L^{r}(0,1;L^{s}({\mathbb{R}}^{d}))$. Similar to (2.5) but
using $B_{1}\subseteq B_{{1}/{(1-t)^{1/2}}}$,

$\displaystyle\|\zeta(t,\cdot)\|_{L^{s}(B_{1})}=(1-t)^{-\frac{\mu}{2}+\frac{d}{2s}}\|w\|_{L^{s}(B_{1/(1-t)^{1/2}})}\geq(1-t)^{-\frac{\mu}{2}+\frac{d}{2s}}\|w\|_{L^{s}(B_{1})}.$

Therefore, with $t\in(\frac{1}{2},1)$,

$\displaystyle\|u(t,\cdot)\|_{L^{s}({\mathbb{R}}^{d})}\geq
t\|\zeta(t,\cdot)\|_{L^{s}(B_{1})}\geq\frac{1}{2}(1-t)^{-\frac{\mu}{2}+\frac{d}{2s}}\|w\|_{L^{s}(B_{1})}.$

By choice of $\mu$, $r$ and $s$ we have
$-\frac{\mu}{2}+\frac{d}{2s}<-\frac{1}{r}$, therefore $u\not\in
L^{r}(0,1;L^{s}({\mathbb{R}}^{d}))$. ∎

###  2.3. Negative result concerning problem 1

The following result shows that for time-dependent operators in the
variational setting, maximal $L^{2}$-regularity cannot be extrapolated to
maximal $L^{p}$-regularity for $p\neq 2$ besides the small interval given in
Theorem 2.2. It answers Problem 1 in a negative way in the setting of elliptic
differential operators.

######  Theorem 2.4 (Failure of extrapolation of maximal $L^{p}$-regularity).

Let $d\geq 2$. For every $p\in(1,\infty)\setminus\\{2\\}$ there exists
$B:[0,1]\times{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ elliptic and $f\in
L^{p}(0,1;H^{-1}({\mathbb{R}}^{d}))$ such that the unique solution $u$ to (P)
satisfies $u\notin L^{p}(0,1;H^{1}({\mathbb{R}}^{d}))$.

###### Proof.

We divide the proof into two cases.

Case 1: $p>2$. We appeal to Theorem 2.3. If $d\geq 3$ put $r=p$ and
$s=2^{*}\coloneqq\frac{2d}{d-2}$ and if $d=2$ put $r=p$ and
$s>\frac{2p}{p-2}$. In both cases the condition
$\frac{2}{r}+\frac{d}{s}<\frac{d}{2}$ is satisfied, so that Theorem 2.3 yields
$B:[0,1]\times{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ elliptic and $f\in
L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))\subseteq
L^{p}(0,1;L^{2}({\mathbb{R}}^{d}))$ such that the unique solution $u$ to (2.3)
satisfies $u\not\in L^{r}(0,1;L^{s}({\mathbb{R}}^{d}))$. This implies
$u\not\in L^{p}(0,1;H^{1}({\mathbb{R}}^{d}))$ since otherwise the Sobolev
embedding yields a contradiction to the previous assertion.

Case 2: $p<2$. We use a duality argument that resembles [16, Thm. 6.2]. By the
first part there are $f\in L^{p^{\prime}}(0,1;H^{-1}({\mathbb{R}}^{d}))$,
elliptic coefficients
$B\colon[0,1]\times{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ and a solution
$u\in L^{2}(0,1;H^{1}({\mathbb{R}}^{d}))\cap
H^{1}(0,1;H^{-1}({\mathbb{R}}^{d}))$ to the equation

(2.7) $\left\\{\begin{aligned} \partial_{t}u-{\text{div}}(B\nabla u)&=f,\\\
u(0)&=0,\end{aligned}\right.$

such that $u\not\in L^{p^{\prime}}(0,1;H^{1}({\mathbb{R}}^{d}))$. Assume for
the sake of contradiction that for every
$A\colon[-1,0]\times{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ and
$\widetilde{g}\in L^{2}(-1,0;H^{-1}({\mathbb{R}}^{d}))$ there is a unique
solution $\widetilde{v}\in L^{2}(-1,0;H^{1}({\mathbb{R}}^{d}))\cap
H^{1}(-1,0;H^{-1}({\mathbb{R}}^{d}))$ to the problem

(2.8) $\left\\{\begin{aligned}
\partial_{t}\widetilde{v}-{\text{div}}(A\nabla\widetilde{v})&=\widetilde{g},\\\
\widetilde{v}(-1)&=0,\end{aligned}\right.$

satisfying the estimate

(2.9)
$\displaystyle\|\partial_{t}\widetilde{v}\|_{L^{p}(-1,0;H^{-1}({\mathbb{R}}^{d}))}+\|\widetilde{v}\|_{L^{p}(-1,0;H^{1}({\mathbb{R}}^{d}))}\leq
C\|\widetilde{g}\|_{L^{p}(-1,0;H^{-1}({\mathbb{R}}^{d}))},$

where the constant $C>0$ does not depend on $\widetilde{v}$ and
$\widetilde{g}$. By translation the question on $(-1,0)$ is equivalent to that
on $(0,1)$. Both intervals are related by the transformation $v\to-v$. We
write for instance $\widetilde{u}(t)=u(-t)$ to translate $u$ to a function on
$(-1,0)$ and vice versa. Specialize $A=(\widetilde{B})^{*}$ in (2.8). Since
$u(0)=0=v(1)$ we can use integration by parts to obtain

$\displaystyle\int_{0}^{1}\langle g,u\rangle
dt=\int_{-1}^{0}\langle\widetilde{g},\widetilde{u}\rangle dt$
$\displaystyle=\int_{-1}^{0}\langle(\widetilde{v})^{\prime},\widetilde{u}\rangle+((\widetilde{B})^{*}\nabla\widetilde{v}|\nabla\widetilde{u})_{2}dt$
$\displaystyle=\int_{0}^{1}-\langle v^{\prime},u\rangle+(\nabla v|B\nabla
u)_{2}dt$ $\displaystyle=\int_{0}^{1}\langle v,u^{\prime}\rangle+(\nabla
v|B\nabla u)_{2}dt.$

Plug in (2.7) to deduce

$\displaystyle\int_{0}^{1}\langle g,u\rangle dt=\int_{0}^{1}\langle v,f\rangle
dt.$

Hence, using (2.9) we can estimate

$\displaystyle\bigl{|}\int_{0}^{1}\langle g,u\rangle dt\bigr{|}$
$\displaystyle\leq\|v\|_{L^{p}(0,1;H^{1}({\mathbb{R}}^{d}))}\|f\|_{L^{p^{\prime}}(0,1;H^{-1}({\mathbb{R}}^{d}))}$
$\displaystyle\leq
C\|g\|_{L^{p}(0,1;H^{-1}({\mathbb{R}}^{d}))}\|f\|_{L^{p^{\prime}}(0,1;H^{-1}({\mathbb{R}}^{d}))}.$

Since $g$ was arbitrary, deduce by duality the contradiction $u\in
L^{p^{\prime}}(0,1;H^{1}({\mathbb{R}}^{d}))$. ∎

######  Remark 2.5.

For the case $p>2$, we saw that although $f\in
L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$, the unique solution $u$ to (P)
satisfies $u\notin L^{p}(0,1;H^{1}({\mathbb{R}}^{d}))$.

###  2.4. Negative result concerning problem 2

We show that Problem 2 fails in the worst possible way in the case of elliptic
operators in divergence form: for every $\nu\in(1/2,1]$ there exist
coefficients $B$ and a forcing term $f\in
L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$ such that the unique solution $u$ to
(P) satisfies $u\notin H^{\nu}(0,1;L^{2}({\mathbb{R}}^{d}))$. This solves
Lions’ problem in a negative way also for elliptic operators. It would still
be interesting to find a counterexample for $\nu=1$ where the coefficients are
Hölder continuous of order $\alpha$ for some $\alpha\leq 1/2$, and this would
then be an optimal counterexample due to the positive results for which we
refer to the survey [3]. The present coefficient function cannot even be
continuous, see Remark 2.7. There seems to be some room for improvement in the
regularity of $B$ as the regularity is much worse than $H^{1}$ in time.

######  Theorem 2.6 (Failure of maximal $L^{2}$-regularity on
$L^{2}({\mathbb{R}}^{d})$).

Let $d\geq 2$. For every $\nu\in(1/2,1]$ there exists
$B:[0,1]\times{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ elliptic and $f\in
L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$ such that the unique solution $u$ to
(P) satisfies $u\notin H^{\nu}(0,1;L^{2})$.

###### Proof.

The strategy is as for the case $p>2$ in Theorem 2.4. Let $\nu\in(1/2,1]$ and
let $0<\theta<\frac{1}{2\nu}$. Then define parameters $r$ and $s$ through
$\frac{1}{r}=\frac{1}{2}-\nu\theta$ and
$\frac{1}{s}=\frac{1}{2}-\frac{1-\theta}{d}$. They satisfy the condition in
Theorem 2.4, consequently there is
$B:[0,1]\times{\mathbb{R}}^{d}\to{\mathbb{C}}^{d\times d}$ elliptic and $f\in
L^{\infty}(0,1;L^{2}({\mathbb{R}}^{d}))$ such that the unique solution $u$ to
(P) satisfies $u\notin L^{r}(0,1;L^{s}({\mathbb{R}}^{d}))$. Suppose that $u\in
H^{\nu}(0,1;L^{2}({\mathbb{R}}^{d}))$. Since $u\in
L^{2}(0,1;H^{1}({\mathbb{R}}^{d})$ by Theorem 2.1, complex interpolation (see
[15, Theorem 14.7.12]) yields $u\in
H^{\nu\theta}(0,1;H^{1-\theta}({\mathbb{R}}^{d}))$. By choice of $\theta$, the
Sobolev embedding is applicable and gives $u\in
L^{r}(0,1;L^{s}({\mathbb{R}}^{d}))$, a contradiction. ∎

######  Remark 2.7.

By construction, the given counterexample is at the same time a counterexample
for Problem 1. We have discussed in the introduction that if $t\mapsto B(t)\in
L^{\infty}({\mathbb{R}}^{d})$ is (piecewise relatively) continuous, then
Problem 1 is automatically true for all $p\in(1,\infty)$. Therefore, the
function $B$ in Theorem 2.6 is not continuous as a map into
$L^{\infty}({\mathbb{R}}^{d})$.

The examples show some limitations of what regularity estimates can hold for
elliptic operators in divergence form. However, several issues remain for
Problems 1 and 2. For instance, the operator $\mathcal{A}$ used in the above
counterexamples is not symmetric/hermitian. We do not know what can be said
for the case $d=1$. The only reference we found on the one-dimensional setting
is [17], where counterexamples are given in case of $L^{p}$-integrability in
time and space for both the divergence and non-divergence setting for the
range $p\in(1,3/2)\cup(3,\infty)$. Finally, it would be interesting to see
what can be said about Problem 2 if $\mathcal{A}$ is more regular in time
(i.e. continuous or Hölder continuous).

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