approach similar to before – defining the higher moments as functions on the
secular manifold, and then computing the corresponding integral. Yet, it turns
out that the higher moments cannot be expressed as well defined functions on
the secular manifold.
Since this approach fails for the higher moments, this problem holds an
additional challenge of finding a different way to perform the computation.

  2. (2)

The RNG was defined using the $\delta$ vertex condition, which is a special
case of the $\delta_{s}$ family of vertex conditions. The results for the RNG
given in this work immediately raise the question whether similar results can
be obtained for the $\delta_{s}$ family (and whether further geometric
information can be obtained from studying the corresponding gaps).
For instance, for the $\delta_{0}$ type condition, it seems (at least
numerically) that the mean value of the corresponding gap converges as well
(see Figure 8.3) Nevertheless, it is not clear what it converges to, and what
additional properties this gap possesses.

Figure 8.3. The first three-hundred values of the spectral gap for the
$\delta_{0}$ condition on a star graph (red points), and the running mean
value of the sequence (blue points). Similar to the RNG, the sequence seems to
be bounded, and the mean value seems to converge.

When trying to repeat the computation shown for the RNG with different vertex
conditions from the $\delta_{s}$ family, one comes across a problem similar to
the one described in the previous bullet. It turns out that for $s\neq\infty$,
it is not simple to define the corresponding spectral gap as a function on the
secular manifold, which makes it difficult to apply the approach presented in
this work.
So once again, an interesting challenge could be to find an alternative way to
perform this computation, and see if any additional geometric information can
be derived from the statistics of the corresponding gap.

  3. (3)

In this work, we have assumed $\Gamma$ to be a compact metric graph. In the
case where $\Gamma$ is not compact, but periodic, then the spectrum is no
longer discrete, and consists of bands and gaps.
If we periodically place a Robin parameter $\sigma$ along the graph (as done
in the famous Kronig-Penney model, see [36]), then as $\sigma$ changes, these
spectral bands change as well. What one obtains are not exactly spectral
curves (since they have ‘width’), but many of the usual notions from compact
graphs still apply (including the secular manifold).
It could be interesting to try and define a notion of a RNG for such systems,
which will quantify how the structure of the spectral bands and gaps changes
with respect to the Robin parameter. Then, one could study its properties and
see if any analogies to the compact case exist.

  4. (4)

Fix a certain graph eigenfunction $f_{n}$ (as in Figure 8.4). If we follow the
value of the quantity
$s\left(x\right):=\frac{f_{n}^{\prime}\left(x\right)}{f_{n}\left(x\right)}$
near a vertex, we see that its behavior changes suddenly when we reach the
vertex. Denote by $s_{e}\left(v\right)$ the limit of the quantity
$s\left(x\right)$ as we approach the vertex $v$ from the edge $e$. Then we see
that for $s$ value slightly smaller than $s_{e}\left(v\right)$, the number of
$s$ points of $f_{n}$ is usually different than that for $s$ value slightly
larger than $s_{e}\left(v\right)$. These so called critical $s$ values are
exactly the $s$ values for which the $s$ count of $f_{n}$ changes (see Figure
8.5), and they hold a particular interest in studying the $s$ points of
$f_{n}$.

Figure 8.4. When approaching the central vertex through each individual edge,
the limit of the quantity $s\left(x\right)$ is different. These three critical
$s$ values result in $f_{n}$ having a different number of $s$ points for
different values of $s$.

Theorem 3.8 shows that the number of $s$ points of a given eigenfunction is
equal to the spectral flow of the corresponding $\delta_{s}$ family. Thus, at
these critical $s$ values, the spectral flow for the $\delta_{s}$ family
changes suddenly. This can be seen by considering not the spectral curves, but
in fact the spectral surfaces of the two-parameter family
$\delta_{s}\left(t\right)$.

Figure 8.5. The spectral surfaces for the $\delta_{s}\left(t\right)$ family
for a fixed eigenfunction $f$ of a star graph. For each value of $s$, we place
the $\delta_{s}$ points on the $s$ points of $f$. At $s=0.3$, the number of
$s$ points of $f$ jumps from two to one, which results in branching of the
spectral surfaces (red dashed lines).

Whenever we reach a critical $s$ value, the spectral flow changes (either
increases by one or decreases by one). This means that at the critical $s$
values, a single branch of the spectral surfaces is created/destroyed, as seen
in Figure 8.5.
It could be interesting to study these spectral surfaces in order to gain
insight on the possible number of $s$ points an eigenfunction could have.

  5. (5)

The study of the statistical behavior of the nodal count on quantum graphs
(such as the mean nodal deficiency) was the subject of several works in recent
years, such as [3, 4, 1, 7]. The newly introduced definition of $s$ domains
opens a door for studying the statistics of $s$ domains, rather than just
nodal domains.
The study of Neumann domains ([2, 5]) shows that different $s$ values can
display different statistics. It could thus be interesting to try and study
the collective statistical behavior of $s$ points, in order to gain a better
understanding of the behavior of the graph eigenfunctions.

##  Appendix A  Derivation of the sesquilinear form for the $\delta_{s}$
condition

We wish to compute the sesquilinear form $L_{t}^{s}$ which corresponds to the
$\delta_{s}\left(t\right)$ condition. To do this, we apply the method
presented in Theorem $1.4.11$ in [18].

We perform the computation for $s\neq\infty$. The computation for $s=\infty$
(which is just the $\delta$ condition) is standard and appears in the
reference above.

Let us start with the case $t\neq 0,\infty$. Working in the standard
coordinates on the graph (where the derivative of a function $u$ is taken into
the edge), the $\delta_{s}$ vertex conditions presented in formulas
(2.24,2.25) can be written at each individual vertex $v$ in the following
form:

(A.5) $\displaystyle A_{v}\left(\begin{array}[]{c}u_{1}\left(v\right)\\\
u_{2}\left(v\right)\end{array}\right)+B_{v}\left(\begin{array}[]{c}u_{1}^{\prime}\left(v\right)\\\
u_{2}^{\prime}\left(v\right)\end{array}\right)=0,$ (A.8) $\displaystyle
A_{v}=\left(\begin{array}[]{cc}\cos\left(\alpha\right)&-\cos\left(\alpha\right)\\\
\frac{1}{2}\cos\left(\alpha\right)t+\sin\left(\alpha\right)&\frac{1}{2}\cos\left(\alpha\right)t-\sin\left(\alpha\right)\end{array}\right),$
(A.11) $\displaystyle
B_{v}=\left(\begin{array}[]{cc}\sin\left(\alpha\right)&\sin\left(\alpha\right)\\\
\frac{1}{2}\sin\left(\alpha\right)t-\cos\left(\alpha\right)&-\frac{1}{2}\sin\left(\alpha\right)t-\cos\left(\alpha\right)\end{array}\right),$

where $\alpha$ is the Pr fer angle, $s=\cot\left(\alpha\right)$.

By the result presented in [18], the sesquilinear form is given by

(A.12)
$L_{t}^{s}\left(f,g\right)=\int_{\Gamma}\frac{df}{dx}\overline{\frac{dg}{dx}}dx-\sum_{v\in
B}\left\langle\Lambda_{v}P_{R,v}F,P_{R,v}G\right\rangle,$

where:

(A.13) $\displaystyle P_{R,v}=I-P_{D,v}-P_{N,v},$ (A.14)
$\displaystyle\Lambda_{v}=B_{v}^{-1}A_{v}P_{R,v},$

and $P_{D,v},P_{N,v}$ are the orthogonal projections onto
$\ker\left(B_{v}\right)$ and $\ker\left(A_{v}\right)$ correspondingly.

A straightforward computation gives that for $t\neq 0$

(A.15) $\displaystyle P_{R,v}=I,$ (A.18)
$\displaystyle\Lambda_{v}=\frac{1}{\sin^{2}\left(\alpha\right)t}\left(\begin{array}[]{cc}1+\frac{1}{2}\sin\left(2\alpha\right)t&-1\\\
-1&1-\frac{1}{2}\sin\left(2\alpha\right)t\end{array}\right).$

Plugging this into formula A.12 we get

(A.19)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,L_{t}^{s}\left(f,g\right)=\int_{\Gamma}\frac{df}{dx}\overline{\frac{dg}{dx}}dx$
(A.26) $\displaystyle-\frac{1}{\sin^{2}\left(\alpha\right)t}\sum_{v\in
B}\left(\begin{array}[]{c}f_{1}\left(v\right)\\\
f_{2}\left(v\right)\end{array}\right)^{*}\left(\begin{array}[]{cc}1+\frac{1}{2}\sin\left(2\alpha\right)t&-1\\\
-1&1-\frac{1}{2}\sin\left(2\alpha\right)t\end{array}\right)\left(\begin{array}[]{c}g_{1}\left(v\right)\\\
g_{2}\left(v\right)\end{array}\right).$

Note that while the computation was done in the non-oriented coordinates,
since the sesquilinear form only depends on the value of the function (and not
the orientation of the derivative), the same formula holds in the oriented
coordinates we use in this work.

By [18], the domains of the corresponding operators consist of all functions
in $H^{1}\left(\Gamma\right)$ which satisfy $P_{D,v}=0$ at each vertex $v$.
Since $P_{D,v}=0$, we conclude that the domain is $H^{1}\left(\Gamma\right)$.

Now for the case $t=\infty$. This time, the corresponding matrices are given
by

(A.29) $\displaystyle
A_{v}=\left(\begin{array}[]{cc}\cos\left(\alpha\right)&-\cos\left(\alpha\right)\\\
\cos\left(\alpha\right)&\cos\left(\alpha\right)\end{array}\right),$ (A.32)
$\displaystyle
B_{v}=\left(\begin{array}[]{cc}\sin\left(\alpha\right)&\sin\left(\alpha\right)\\\
\sin\left(\alpha\right)&-\sin\left(\alpha\right)\end{array}\right).$

The same computation as before gives

(A.33) $\displaystyle P_{R,v}=I,$ (A.36)
$\displaystyle\Lambda_{v}=\frac{1}{2\sin^{2}\left(\alpha\right)}\left(\begin{array}[]{cc}\sin\left(2\alpha\right)&0\\\
0&-\sin\left(2\alpha\right)\end{array}\right),$

which overall gives

(A.37)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,L_{\infty}^{s}\left(f,g\right)=\int_{\Gamma}\frac{df}{dx}\overline{\frac{dg}{dx}}dx$
(A.44) $\displaystyle-\frac{1}{\sin^{2}\left(\alpha\right)}\sum_{v\in
B}\left(\begin{array}[]{c}f_{1}\left(v\right)\\\
f_{2}\left(v\right)\end{array}\right)^{*}\left(\begin{array}[]{cc}\frac{1}{2}\sin\left(2\alpha\right)t&0\\\
0&-\frac{1}{2}\sin\left(2\alpha\right)t\end{array}\right)\left(\begin{array}[]{c}g_{1}\left(v\right)\\\
g_{2}\left(v\right)\end{array}\right).$

Once again, $P_{D,v}=0$ and so the domain is given by
$H^{1}\left(\Gamma\right)$.

Lastly, in the case $t=0$, we have that

(A.47) $\displaystyle
A_{v}=\left(\begin{array}[]{cc}\cos\left(\alpha\right)&-\cos\left(\alpha\right)\\\
\sin\left(\alpha\right)&-\sin\left(\alpha\right)\end{array}\right),$ (A.50)
$\displaystyle
B_{v}=\left(\begin{array}[]{cc}\sin\left(\alpha\right)&\sin\left(\alpha\right)\\\
-\cos\left(\alpha\right)&-\cos\left(\alpha\right)\end{array}\right).$

This time, the Dirichlet and Neumann projections are non-trivial:

(A.51)
$\displaystyle\ker\left(B_{v}\right)=sp\left\\{\left(1,-1\right)\right\\},$
(A.54) $\displaystyle\Rightarrow$ $\displaystyle
P_{D,v}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&-1\\\
-1&1\end{array}\right),$ (A.55)
$\displaystyle\ker\left(A_{v}\right)=sp\left\\{\left(1,1\right)\right\\},$
(A.58) $\displaystyle\Rightarrow$ $\displaystyle
P_{N,v}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right),$

Which gives $\Lambda=0$ and the following sesquilinear form:

(A.59) $\displaystyle
L_{0}^{s}\left(f,g\right)=\int_{\Gamma}\frac{df}{dx}\overline{\frac{dg}{dx}}dx.$

By the expression above for $P_{D,v}$, we also see that the domain of
$L_{0}^{s}$ consists of all functions in $H^{1}\left(\Gamma\right)$ which are
continuous at the selected set of vertices.

##  Appendix B  Explicit computation of $\Lambda_{\infty}\left(c\right)$ for
specific graph

We give an example of a computation for the Robin map
$\Lambda_{\infty}\left(c\right)$ for the graph displayed in Figure B.1. For
simplicity of the computation, we take all edge lengths to be equal to $L$.

Figure B.1. Computation of the Robin map $\Lambda_{\infty}\left(c\right)$ for
the graph above, where the vertex set $B$ is marked in red. The graph is
decomposed into to star graphs, $D_{1}$ and $D_{2}$, and the computation of
$\Lambda_{\infty}\left(c\right)$ is done by computing the matrices for
$\Lambda_{\infty}^{D_{1}}\left(c\right)$ and
$\Lambda_{\infty}^{D_{1}}\left(c\right)$ and taking their block sum.

To compute the Robin map, we first compute the map
$\Lambda_{\infty}^{D}\left(c\right)$ for a star graph with three edges, where
the set $B$ is taken to be its exterior vertices (and the Neumann-Kirchhoff
condition is imposed at the center). We will then construct the matrix for
$\Lambda_{\infty}\left(c\right)$ by using the decomposition of the given graph
into two stars. For convenience, we parameterize our star graph so that the
interior vertices are located at $x=0$ and the center is located at $x=L$.

By using the definition of our trace maps, we see that for the case
$s=\infty$, the map $\Lambda_{\infty}^{D}\left(c\right)$ is simply the
Dirichlet to Neumann map, which sends the Dirichlet trace data to the Neumann
trace data.

Let
$k^{2}\notin\textrm{\text{Spec}}\left(H^{\infty}\left(\infty\right)\right)$.
For simplicity, we choose $c>0$ and write $c=k^{2}$. Let $w\in\mathbb{R}^{3}$
be given. We are interested in solving the following boundary value problem:

(B.1) $\displaystyle-\frac{d^{2}f}{dx^{2}}=k^{2}f,$ (B.2) $\displaystyle
f_{e}\left(0\right)=w_{e},e\in\left\\{1,2,3\right\\},$ (B.3) $\displaystyle
f_{e}\left(L\right)=f_{e^{\prime}}\left(L\right),\forall
e,e^{\prime}\in\left\\{1,2,3\right\\},$ (B.4)
$\displaystyle\sum_{e=1}^{3}f_{e}^{\prime}\left(L\right)=0.$

For each edge $e\in\left\\{1,2,3\right\\}$, we suggest a solution of the form

(B.5) $f_{e}\left(x\right)=A_{e}\sin\left(kx\right)+B_{e}\cos\left(kx\right).$

Then the condition at the outer vertices gives

(B.6) $B_{e}=w_{e}.$

The continuity condition at the center gives that for every
$e\in\left\\{1,2,3\right\\}$:

(B.7) $\displaystyle
A_{1}\sin\left(kL\right)+w_{1}\cos\left(kL\right)=A_{e}\sin\left(kL\right)+w_{e}\cos\left(kL\right)$
(B.8) $\displaystyle\Rightarrow$ $\displaystyle
A_{e}=\frac{1}{\sin\left(kL\right)}\left(A_{1}\sin\left(kL\right)+w_{1}\cos\left(kL\right)-u_{e}\cos\left(kL\right)\right)$
(B.9) $\displaystyle\Rightarrow$ $\displaystyle
f_{e}\left(x\right)=\left(A_{1}+w_{1}\cot\left(kL\right)-u_{e}\cot\left(kL\right)\right)\sin\left(kx\right)+w_{e}\cos\left(kx\right).$

Plugging this into the condition for the sum of derivatives at the center, a
straightforward computation yields that for each edge $e$

(B.10)
$A_{e}=\frac{\sum_{j=1}^{3}\left(\frac{1}{\sin\left(kL\right)}\left(w_{j}\cos\left(kL\right)-w_{e}\cos\left(kL\right)\right)\cos\left(kL\right)+w_{j}\sin\left(kL\right)\right)}{3\cos\left(kL\right)}.$

This overall gives

(B.11) $\displaystyle
f_{e}\left(x\right)=\left[\frac{1}{\sin\left(kL\right)\cos\left(kL\right)}w_{j}-w_{e}\sin\left(kL\right)\right]\sin\left(kx\right)+w_{e}\cos\left(\lambda
x\right),$

and the corresponding Neumann trace is thus

(B.12)
$f_{e}^{\prime}\left(0\right)=\frac{k}{\sin\left(kL\right)\cos\left(kL\right)}w_{j}-kw_{e}\sin\left(kL\right).$

We thus see that $\Lambda_{\infty}^{D}\left(k^{2}\right)$ can be written in
the following matrix form:

(B.13)
$\left(\Lambda_{\infty}^{D}\left(k^{2}\right)\right)_{ee^{\prime}}=\begin{cases}\frac{k}{\sin\left(kL\right)\cos\left(kL\right)}&e\neq
e^{\prime}\\\ \frac{k}{\cos\left(kL\right)}&e=e^{\prime}\end{cases}$

We now construct the complete matrix $\Lambda_{\infty}\left(k^{2}\right)$,
which is a square matrix of size four acting on
$\ell^{2}\left(\left\\{v_{1},...,v_{4}\right\\}\right)$. From the orientation
on the edges (and definition of our weight functions $\chi_{D}$), we see that
$\Lambda_{\infty}^{D_{1}}\left(k^{2}\right)$ comes with a minus sign while
$\Lambda_{\infty}^{D_{2}}\left(k^{2}\right)$ comes with a plus sign, which
means that most terms cancel. Taking the block sum (and noting that $D_{1}\cap
D_{2}=\left\\{v_{2},v_{3}\right\\}$) we finally get

(B.14)
$\Lambda_{\infty}\left(k^{2}\right)=\left(\begin{array}[]{cccc}-\frac{k}{\cos\left(kL\right)}&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&\frac{k}{\cos\left(kL\right)}\end{array}\right).$

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60.4 $\pm$ 1.0 53.4 $\pm$ 0.5 74.9 $\pm$ 0.1 76.1 $\pm$ 0.7 66.2 DANN 60.6
$\pm$ 1.4 51.8 $\pm$ 0.7 73.4 $\pm$ 0.5 75.5 $\pm$ 0.9 65.3 CDANN 57.9 $\pm$
0.2 52.1 $\pm$ 1.2 74.9 $\pm$ 0.7 76.2 $\pm$ 0.2 65.3 MTL 60.7 $\pm$ 0.8 53.5
$\pm$ 1.3 75.2 $\pm$ 0.6 76.6 $\pm$ 0.6 66.5 SagNet 62.7 $\pm$ 0.5 53.6 $\pm$
0.5 76.0 $\pm$ 0.3 77.8 $\pm$ 0.1 67.5 ARM 58.8 $\pm$ 0.5 51.8 $\pm$ 0.7 74.0
$\pm$ 0.1 74.4 $\pm$ 0.2 64.8 V-REx 59.6 $\pm$ 1.0 53.3 $\pm$ 0.3 73.2 $\pm$
0.5 76.6 $\pm$ 0.4 65.7 RSC 61.7 $\pm$ 0.8 53.0 $\pm$ 0.9 74.8 $\pm$ 0.8 76.3
$\pm$ 0.5 66.5 AND-mask 60.3 $\pm$ 0.5 52.3 $\pm$ 0.6 75.1 $\pm$ 0.2 76.6
$\pm$ 0.3 66.1 SAND-mask 59.9 $\pm$ 0.7 53.6 $\pm$ 0.8 74.3 $\pm$ 0.4 75.8
$\pm$ 0.5 65.9 Fishr 63.4 $\pm$ 0.8 54.2 $\pm$ 0.3 76.4 $\pm$ 0.3 78.5 $\pm$
0.2 68.2 SelfReg 64.2 $\pm$ 0.6 53.6 $\pm$ 0.7 76.7 $\pm$ 0.3 77.9 $\pm$ 0.5
68.1 IDM 64.4 $\pm$ 0.3 54.4 $\pm$ 0.6 76.5 $\pm$ 0.3 78.0 $\pm$ 0.4 68.3

TABLE XVII: Detailed results on TerraIncognita in DomainBed.

Algorithm L100 L38 L43 L46 Average ERM 59.4 $\pm$ 0.9 49.3 $\pm$ 0.6 60.1
$\pm$ 1.1 43.2 $\pm$ 0.5 53.0 IRM 56.5 $\pm$ 2.5 49.8 $\pm$ 1.5 57.1 $\pm$ 2.2
38.6 $\pm$ 1.0 50.5 GroupDRO 60.4 $\pm$ 1.5 48.3 $\pm$ 0.4 58.6 $\pm$ 0.8 42.2
$\pm$ 0.8 52.4 Mixup 67.6 $\pm$ 1.8 51.0 $\pm$ 1.3 59.0 $\pm$ 0.0 40.0 $\pm$
1.1 54.4 MLDG 59.2 $\pm$ 0.1 49.0 $\pm$ 0.9 58.4 $\pm$ 0.9 41.4 $\pm$ 1.0 52.0
CORAL 60.4 $\pm$ 0.9 47.2 $\pm$ 0.5 59.3 $\pm$ 0.4 44.4 $\pm$ 0.4 52.8 MMD
60.6 $\pm$ 1.1 45.9 $\pm$ 0.3 57.8 $\pm$ 0.5 43.8 $\pm$ 1.2 52.0 DANN 55.2
$\pm$ 1.9 47.0 $\pm$ 0.7 57.2 $\pm$ 0.9 42.9 $\pm$ 0.9 50.6 CDANN 56.3 $\pm$
2.0 47.1 $\pm$ 0.9 57.2 $\pm$ 1.1 42.4 $\pm$ 0.8 50.8 MTL 58.4 $\pm$ 2.1 48.4
$\pm$ 0.8 58.9 $\pm$ 0.6 43.0 $\pm$ 1.3 52.2 SagNet 56.4 $\pm$ 1.9 50.5 $\pm$
2.3 59.1 $\pm$ 0.5 44.1 $\pm$ 0.6 52.5 ARM 60.1 $\pm$ 1.5 48.3 $\pm$ 1.6 55.3
$\pm$ 0.6 40.9 $\pm$ 1.1 51.2 V-REx 56.8 $\pm$ 1.7 46.5 $\pm$ 0.5 58.4 $\pm$
0.3 43.8 $\pm$ 0.3 51.4 RSC 59.9 $\pm$ 1.4 46.7 $\pm$ 0.4 57.8 $\pm$ 0.5 44.3
$\pm$ 0.6 52.1 AND-mask 54.7 $\pm$ 1.8 48.4 $\pm$ 0.5 55.1 $\pm$ 0.5 41.3
$\pm$ 0.6 49.8 SAND-mask 56.2 $\pm$ 1.8 46.3 $\pm$ 0.3 55.8 $\pm$ 0.4 42.6
$\pm$ 1.2 50.2 Fishr 60.4 $\pm$ 0.9 50.3 $\pm$ 0.3 58.8 $\pm$ 0.5 44.9 $\pm$
0.5 53.6 SelfReg 60.0 $\pm$ 2.3 48.8 $\pm$ 1.0 58.6 $\pm$ 0.8 44.0 $\pm$ 0.6
52.8 IDM 60.1 $\pm$ 1.4 48.8 $\pm$ 1.9 57.9 $\pm$ 0.2 44.3 $\pm$ 1.2 52.8

TABLE XVIII: Detailed results on DomainNet in DomainBed.

Algorithm clip info paint quick real sketch Average ERM 58.6 $\pm$ 0.3 19.2
$\pm$ 0.2 47.0 $\pm$ 0.3 13.2 $\pm$ 0.2 59.9 $\pm$ 0.3 49.8 $\pm$ 0.4 41.3 IRM
40.4 $\pm$ 6.6 12.1 $\pm$ 2.7 31.4 $\pm$ 5.7 9.8 $\pm$ 1.2 37.7 $\pm$ 9.0 36.7
$\pm$ 5.3 28.0 GroupDRO 47.2 $\pm$ 0.5 17.5 $\pm$ 0.4 34.2 $\pm$ 0.3 9.2 $\pm$
0.4 51.9 $\pm$ 0.5 40.1 $\pm$ 0.6 33.4 Mixup 55.6 $\pm$ 0.1 18.7 $\pm$ 0.4
45.1 $\pm$ 0.5 12.8 $\pm$ 0.3 57.6 $\pm$ 0.5 48.2 $\pm$ 0.4 39.6 MLDG 59.3
$\pm$ 0.1 19.6 $\pm$ 0.2 46.8 $\pm$ 0.2 13.4 $\pm$ 0.2 60.1 $\pm$ 0.4 50.4
$\pm$ 0.3 41.6 CORAL 59.2 $\pm$ 0.1 19.9 $\pm$ 0.2 47.4 $\pm$ 0.2 14.0 $\pm$
0.4 59.8 $\pm$ 0.2 50.4 $\pm$ 0.4 41.8 MMD 32.2 $\pm$ 13.3 11.2 $\pm$ 4.5 26.8
$\pm$ 11.3 8.8 $\pm$ 2.2 32.7 $\pm$ 13.8 29.0 $\pm$ 11.8 23.5 DANN 53.1 $\pm$
0.2 18.3 $\pm$ 0.1 44.2 $\pm$ 0.7 11.9 $\pm$ 0.1 55.5 $\pm$ 0.4 46.8 $\pm$ 0.6
38.3 CDANN 54.6 $\pm$ 0.4 17.3 $\pm$ 0.1 44.2 $\pm$ 0.7 12.8 $\pm$ 0.2 56.2
$\pm$ 0.4 45.9 $\pm$ 0.5 38.5 MTL 58.0 $\pm$ 0.4 19.2 $\pm$ 0.2 46.2 $\pm$ 0.1
12.7 $\pm$ 0.2 59.9 $\pm$ 0.1 49.0 $\pm$ 0.0 40.8 SagNet 57.7 $\pm$ 0.3 19.1
$\pm$ 0.1 46.3 $\pm$ 0.5 13.5 $\pm$ 0.4 58.9 $\pm$ 0.4 49.5 $\pm$ 0.2 40.8 ARM
49.6 $\pm$ 0.4 16.5 $\pm$ 0.3 41.5 $\pm$ 0.8 10.8 $\pm$ 0.1 53.5 $\pm$ 0.3
43.9 $\pm$ 0.4 36.0 V-REx 43.3 $\pm$ 4.5 14.1 $\pm$ 1.8 32.5 $\pm$ 5.0 9.8
$\pm$ 1.1 43.5 $\pm$ 5.6 37.7 $\pm$ 4.5 30.1 RSC 55.0 $\pm$ 1.2 18.3 $\pm$ 0.5
44.4 $\pm$ 0.6 12.5 $\pm$ 0.1 55.7 $\pm$ 0.7 47.8 $\pm$ 0.9 38.9 AND-mask 52.3
$\pm$ 0.8 17.3 $\pm$ 0.5 43.7 $\pm$ 1.1 12.3 $\pm$ 0.4 55.8 $\pm$ 0.4 46.1
$\pm$ 0.8 37.9 SAND-mask 43.8 $\pm$ 1.3 15.2 $\pm$ 0.2 38.2 $\pm$ 0.6 9.0
$\pm$ 0.2 47.1 $\pm$ 1.1 39.9 $\pm$ 0.6 32.2 Fishr 58.3 $\pm$ 0.5 20.2 $\pm$
0.2 47.9 $\pm$ 0.2 13.6 $\pm$ 0.3 60.5 $\pm$ 0.3 50.5 $\pm$ 0.3 41.8 SelfReg
60.7 $\pm$ 0.1 21.6 $\pm$ 0.1 49.5 $\pm$ 0.1 14.2 $\pm$ 0.3 60.7 $\pm$ 0.1
51.7 $\pm$ 0.1 43.1 IDM 58.8 $\pm$ 0.3 20.7 $\pm$ 0.2 48.3 $\pm$ 0.1 13.7
$\pm$ 0.4 59.1 $\pm$ 0.1 50.2 $\pm$ 0.3 41.8

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# Improved polynomial decay for unbounded semigroups

Chenxi Deng  School of Mathematics and Statistics
Beijing Institute of Technology
Beijing 100081
China<EMAIL_ADDRESS>,  Jan Rozendaal  Institute of Mathematics,
Polish Academy of Sciences
Śniadeckich 8
00-656 Warsaw
Poland<EMAIL_ADDRESS>and  Mark Veraar  Delft Institute of Applied
Mathematics
Delft University of Technology
P.O. Box 5031
2628 CD Delft
The Netherlands<EMAIL_ADDRESS>
###### Abstract.

We obtain polynomial decay rates for $\displaystyle C_{0}$-semigroups,
assuming that the resolvent grows polynomially at infinity in the complex
right half-plane. Our results do not require the semigroup to be uniformly
bounded, and for unbounded semigroups we improve upon previous results by, for
example, removing a logarithmic loss on non-Hilbertian Banach spaces.

###### Key words and phrases:

$\displaystyle C_{0}$-semigroup, polynomial stability, Fourier multiplier

###### 2020 Mathematics Subject Classification:

Primary 47D06. Secondary 35B40, 42B37, 46B20

The first author is supported by China Scholarship Council (CSC). This
research was funded in part by the National Science Center, Poland, grant
2021/43/D/ST1/00667. The second author is partially supported by NCN grant
UMO-2023/49/B/ST1/01961. The third author is supported by the VICI subsidy
VI.C.212.027 of the Netherlands Organization for Scientific Research (NWO)

##  1\. Introduction

###  1.1. Setting

We study the asymptotic behavior of solutions to the abstract Cauchy problem

(1.1) $\displaystyle\dot{u}(t)$ $\displaystyle=Au(t)\quad(t\geq 0),$
$\displaystyle u(0)$ $\displaystyle=x,$

on a Banach space $\displaystyle X$. We assume that (1.1) is well posed, so
that the solution operators form a $\displaystyle C_{0}$-semigroup
$\displaystyle(T(t))_{t\geq 0}\subseteq\mathcal{L}(X)$ of bounded operators,
with generator $\displaystyle A$. Throughout, we will consider $\displaystyle
A$ satisfying $\displaystyle\overline{{\mathbb{C}}_{+}}\subseteq\rho(A)$,
where
$\displaystyle{\mathbb{C}}_{+}:=\\{z\in{\mathbb{C}}\mid\operatorname{Re}(z)>0\\}$
and $\displaystyle\rho(A):={\mathbb{C}}\setminus\sigma(A)$ is the resolvent
set of $\displaystyle A$. Under these assumptions, there are two well-known
flavors of results that relate information about the resolvent operators
$\displaystyle R(\lambda,A):=(\lambda-A)^{-1}$,
$\displaystyle\lambda\in\rho(A)$, to asymptotic behavior of the semigroup
orbits.

Firstly, the classical Gearhart–Huang–Prüss–Greiner theorem [13, 17, 23] says
that, if $\displaystyle X$ is a Hilbert space, then the semigroup
$\displaystyle(T(t))_{t\geq 0}$ is uniformly stable, and all orbits decay
exponentially to zero, if and only if

(1.2)
$\sup_{\lambda\in{\mathbb{C}}_{+}}\|R(\lambda,A)\|_{\mathcal{L}(X)}<\infty.$

Versions of this theorem on non-Hilbertian Banach spaces were discovered later
[36, 37, 35]. Here an assumption such as (1.2) typically guarantees
exponential decay only for sufficiently smooth initial data, with the degree
of smoothness depending on the geometry of the underlying Banach space. It is
relevant to note that all these results make no a priori assumptions on the
growth of the semigroup; only spectral information is required.

On the other hand, a more recent line of research considers the setting where
the resolvent is not bounded on the right half-plane, but instead blows up
along the imaginary axis at a specified rate. In this case the semigroup is
not uniformly stable, and one can at best hope to obtain uniform decay rates
for sufficiently smooth initial data. Semigroups with these properties arise
naturally in the study of the damped wave equation

(1.3)
$\partial_{t}^{2}u(t,x)=\Delta_{g}u(t,x)-a(x)\partial_{t}u(t,x)\quad((t,x)\in\mathbb{R}\times
M),$

on a Riemannian manifold $\displaystyle(M,g)$, where $\displaystyle a\in C(M)$
[24, 25, 21, 7, 1, 8]. A succession of results in semigroup theory [3, 5, 6,
4, 29] has elucidated the relationship between the rate of resolvent blowup
and the rate of decay of classical solutions to (1.1), in the case where the
semigroup is a priori assumed to be uniformly bounded. The latter assumption
is in turn satisfied if the damping function $\displaystyle a$ in (1.3) is
non-negative.

However, when considering functions $\displaystyle a$ in (1.3) that change
sign, the associated semigroup need not be uniformly bounded and one may
encounter unexpected spectral behavior (see e.g. [26, 30]). Moreover,
polynomially growing semigroups appear naturally in the analysis of
Schrödinger operators with unbounded potentials [10, 15], perturbed wave
equations [14, 22], delay differential equations [33] and hyperbolic equations
on non-Hilbertian Banach spaces [9, 28].

Hence it is natural to wonder what can be said when one combines some of the
difficulties of both the lines of research mentioned above, that is, if the
semigroup $\displaystyle(T(t))_{t\geq 0}$ is not uniformly bounded and the
resolvent is not uniformly bounded on the right half-plane. This is the
setting that will be considered in this article.

###  1.2. Previous work

Throughout, we suppose that $\displaystyle{\mathbb{C}}_{+}\subseteq\rho(A)$
and that there exist $\displaystyle\beta,C\geq 0$ such that

$\displaystyle\|R(\lambda,A)\|_{\mathcal{L}(X)}\leq
C(1+|\lambda|)^{\beta}\quad(\lambda\in{\mathbb{C}}_{+}).$

It then follows that
$\displaystyle\overline{{\mathbb{C}}_{+}}\subseteq\rho(A)$, but unless
$\displaystyle\beta=0$, i.e. unless (1.2) holds, the resolvent might blow up
along the imaginary axis, with polynomial rate at most $\displaystyle
O(|\lambda|^{\beta})$. As in the work for uniformly bounded semigroups
mentioned above, one hopes to derive polynomial rates of decay for semigroup
orbits with sufficiently smooth initial data.

In this regard, it was first shown in [3] that, on general Banach spaces, for
each $\displaystyle\rho\geq 0$ and $\displaystyle\tau>(\rho+1)\beta+1$ one has

(1.4) $\|T(t)x\|_{X}\lesssim t^{-\rho}\|x\|_{D((-A)^{\tau})}$

for all $\displaystyle t\geq 1$ and $\displaystyle x\in D((-A)^{\tau})$.
Later, [31] improved this estimate under additional geometric assumptions on
the underlying Banach space. Namely, if $\displaystyle X$ has Fourier type
$\displaystyle p\in[1,2]$ (see Section 2.1), then (1.4) holds for each
$\displaystyle\tau>(\rho+1)\beta+\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$.
Moreover, if $\displaystyle p=2$, i.e. if $\displaystyle X$ is a Hilbert
space, then one may let $\displaystyle\tau=(\rho+1)\beta$. However, it was
left as an open question whether one may also let
$\displaystyle\tau=(\rho+1)\beta+\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$ for
Banach spaces with Fourier type $\displaystyle p\in[1,2)$ (see also [27,
Appendix B]).

Recently, it was shown in [32] that the results from [31] regarding (1.4) can
in fact be improved. More precisely, for each $\displaystyle\rho>0$ and
$\displaystyle\sigma>\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$ one has

(1.5) $\|T(t)x\|_{X}\lesssim t^{-\rho}\log(t)^{\sigma}\|x\|_{D((-A)^{\tau})}$

for $\displaystyle t\geq 2$ and $\displaystyle x\in D((-A)^{\tau})$, where
$\displaystyle\tau=(\rho+1)\beta+\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$. That is,
for $\displaystyle\rho>0$ and $\displaystyle p\in[1,2)$, (1.5) attains the
missing endpoint exponent from [31], up to a logarithmic loss. In fact, [32]
combined methods from [31] with ones from the theory for bounded semigroups in
[4] and considered resolvents with more general growth behavior, but
specializing to polynomially growing resolvents leads to (1.5).

Finally, it is important to emphasize that the results from [31] and [32] are
far from optimal if the semigroup $\displaystyle(T(t))_{t\geq 0}$ is uniformly
bounded. Indeed, in this case [5] yields, on general Banach spaces and for all
$\displaystyle\rho\geq 0$,

(1.6) $\|T(t)x\|_{X}\lesssim t^{-\rho}\log(t)^{\rho}\|x\|_{D((-A)^{\tau})}$

for $\displaystyle t\geq 2$ and $\displaystyle x\in D((-A)^{\tau})$, where
$\displaystyle\tau=\rho\beta$. Moreover, by [6], if $\displaystyle X$ is a
Hilbert space then the logarithmic factor in (1.6) can be removed, yielding
(1.4) for $\displaystyle\tau=\rho\beta$. On the other hand, for unbounded
semigroups on Hilbert spaces and for $\displaystyle\rho=0$ one cannot in
general expect to obtain (1.4) for $\displaystyle\tau<(\rho+1)\beta$, as
follows from an example of Wrobel (see [38, Example 4.1] and [31, Example
4.20]).

###  1.3. Main result

For $\displaystyle\tau>0$ and $\displaystyle q\in[1,\infty]$, we will work
with the real interpolation space $\displaystyle
D_{A}(\tau,q):=(X,D(A^{m}))_{\tau/m,q}$, where $\displaystyle m\in\mathbb{N}$
with $\displaystyle m>\tau$ is arbitrary (see also (2.7)). Moreover, we refer
to (2.1) and (2.2) for the definitions of Hardy–Littlewood type and
Hardy–Littlewood cotype, respectively. The following is our main result.

######  Theorem 1.1.

Let $\displaystyle A$ be the generator of a $\displaystyle C_{0}$-semigroup
$\displaystyle(T(t))_{t\geq 0}$ on a Banach space $\displaystyle X$ with
Fourier type $\displaystyle p\in[1,2]$. Suppose that
$\displaystyle{\mathbb{C}}_{+}\subseteq\rho(A)$, and that there exist
$\displaystyle\beta>0$ and $\displaystyle C\geq 0$ such that

(1.7) $\|R(\lambda,A)\|_{\mathcal{L}(X)}\leq
C(1+|\lambda|)^{\beta}\quad(\lambda\in{\mathbb{C}}_{+}).$

Let $\displaystyle\rho\geq 0$ and set
$\displaystyle\tau:=(\rho+1)\beta+\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$. Then
there exists a $\displaystyle C_{\rho}\geq 0$ such that

(1.8) $\|T(t)x\|_{X}\leq C_{\rho}t^{-\rho}\|x\|_{D_{A}(\tau,p)}$

for all $\displaystyle t\geq 1$ and $\displaystyle x\in D_{A}(\tau,p)$. If
$\displaystyle\rho>0$, then (1.8) also holds with $\displaystyle
D_{A}(\tau,p)$ replaced by $\displaystyle D_{A}(\tau,q)$ for any
$\displaystyle q\in[1,\infty]$, or by $\displaystyle D((-A)^{\tau})$.

Suppose, additionally, that $\displaystyle p>1$ and that $\displaystyle X$ has
Hardy–Littlewood type $\displaystyle p$ or Hardy–Littlewood cotype
$\displaystyle p^{\prime}$. Then, for $\displaystyle\rho=0$, (1.8) also holds
with $\displaystyle D_{A}(\tau,p)$ replaced by $\displaystyle D((-A)^{\tau})$.

The first two statements of Theorem 1.1 are contained in the main text as
Theorem 3.3, while the last statement is Theorem 4.2.

Given that any Banach space has Fourier type $\displaystyle p=1$, the first
part of Theorem 1.1 applies to general Banach spaces. For $\displaystyle
p\in(1,2]$, the assumptions on $\displaystyle X$ in Theorem 1.1 are satisfied
in particular if $\displaystyle X$ is isomorphic to a closed subspace of
$\displaystyle L^{r}(\Omega)$, for $\displaystyle\Omega$ a measure space and
$\displaystyle r=p$ or $\displaystyle r=p^{\prime}$ (see Section 2.1).

For $\displaystyle p\in[1,2)$, the first part of Theorem 1.1 improves (1.5) by
removing the logarithmic factor for $\displaystyle\rho>0$, and it yields an
endpoint result for $\displaystyle\rho=0$. The second part of Theorem 1.1 in
turn fully extends (1.4) to $\displaystyle\rho=0$ and
$\displaystyle\tau=\beta+\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$, under additional
geometric assumptions. Also note that, for all $\displaystyle p\in[1,2]$ and
$\displaystyle\rho>0$, (1.8) involves a larger space of initial data than
considered in [31] and [32], since $\displaystyle D((-A)^{\tau})\subseteq
D_{A}(\tau,\infty)$. On the other hand, for $\displaystyle\rho=0$, (1.8)
complements the main result of [31] on Hilbert spaces, since in general one
neither has $\displaystyle D((-A)^{\tau})\subseteq D_{A}(\tau,2)$ nor
$\displaystyle D_{A}(\tau,2)\subseteq D((-A)^{\tau})$.

The exponent $\displaystyle\tau$ in Theorem 1.1 is sharp for $\displaystyle
p=2$ and $\displaystyle\rho=0$, as noted above, and for general $\displaystyle
p\in[1,2]$ as $\displaystyle\beta\to 0$, as follows from a modification of an
example of Arendt concerning exponential stability (see [2, Example 5.1.11]
and [37, Section 4]). We do not know whether, for a general Banach space with
Fourier type $\displaystyle p\in[1,2)$ and for $\displaystyle\rho=0$, (1.8)
also holds with $\displaystyle D_{A}(\tau,p)$ replaced by $\displaystyle
D((-A)^{\tau})$.

For any $\displaystyle C_{0}$-semigroup $\displaystyle(T(t))_{t\geq 0}$ there
exists an $\displaystyle\omega\in\mathbb{R}$ such that

(1.9) $\|T(t)\|_{\mathcal{L}(X)}\lesssim e^{\omega t}\quad(t\geq 0).$

As already noted, only the case $\displaystyle\omega>0$ will be of interest in
this article. However, it follows from (1.9) that (1.7) holds whenever
$\displaystyle\operatorname{Re}(\lambda)\geq\omega_{0}$, for
$\displaystyle\omega_{0}>\omega$. Moreover, (1.7) directly extends to
$\displaystyle\lambda\in i\mathbb{R}$ as well. Hence (1.7) is in fact an
assumption on the growth of the resolvent as $\displaystyle\lambda$ tends to
infinity in the strip $\displaystyle\\{\lambda\in{\mathbb{C}}\mid
0\leq\operatorname{Re}(\lambda)\leq\omega_{0}\\}$.

One may weaken assumption (1.7) somewhat, by requiring instead that

(1.10)
$\|R(\lambda,A)\|_{\mathcal{L}(X)}\lesssim(1+|\lambda|)^{\beta_{0}}\quad(\lambda\in{\mathbb{C}}_{+})$

for _some_ $\displaystyle\beta_{0}>0$, and that

$\displaystyle\|R(i\xi,A)\|_{\mathcal{L}(X)}\lesssim(1+|\xi|)^{\beta}\quad(\xi\in\mathbb{R}).$

Then the conclusion of Theorem 1.1 still holds, and the specific value of
$\displaystyle\beta_{0}$ in (1.10) plays no role. Indeed, the place in the
proof of Theorem 1.1 where one genuinely uses polynomial resolvent bounds for
$\displaystyle\lambda\in{\mathbb{C}}_{+}$ is in the proofs of Theorems 3.2 and
4.2, to obtain a dense subset of initial values for which the semigroup orbits
are integrable, and there the value of $\displaystyle\beta_{0}$ is irrelevant.
Instead, as in the theory for uniformly bounded semigroups, to obtain concrete
rates of decay we work with the behavior of the resolvent on the imaginary
axis.

As in [31, 32], our techniques in principle also allow for $\displaystyle A$
to have a singularity at zero. More precisely, one could suppose that (1.7)
holds for $\displaystyle|\lambda|\geq 1$, and that there exists an
$\displaystyle\alpha>0$ such that
$\displaystyle\|R(\lambda,A)\|_{\mathcal{L}(X)}\lesssim|\lambda|^{-\alpha}$
for $\displaystyle|\lambda|<1$. In this case one has to assume additionally
that $\displaystyle-A$ is an injective sectorial operator (see Remark 2.3),
and the initial values have to be restricted to the range of a suitable
fractional power of $\displaystyle-A$. For simplicity, we will not consider
such a setting in this article.

###  1.4. The strategy of the proof

Our approach is similar to that in [31] (see also [27]), applying Fourier
multiplier theory to the resolvent on the imaginary axis. However, whereas
[31] mostly involved Fourier multipliers from $\displaystyle
L^{p}(\mathbb{R};Y)$ to $\displaystyle L^{q}(\mathbb{R};X)$ for
suitable$\displaystyle 1\leq p\leq q\leq\infty$ and $\displaystyle Y\subseteq
X$, in the present article we proceed differently.

Namely, the first part of Theorem 1.1 is proved using Proposition 2.2, which
considers multipliers between the Besov space $\displaystyle
B^{s}_{p,p}(\mathbb{R};Y)$ and $\displaystyle L^{p^{\prime}}(\mathbb{R};X)$,
for suitable values of $\displaystyle p$ and $\displaystyle s$. Working with
such multipliers allows us to obtain endpoint estimates. In turn, Besov spaces
are intimately connected to the real interpolation method, and in Proposition
2.5 we show that real interpolation spaces can also be used effectively to
cancel out resolvent growth, as is required to satisfy the conditions of our
Fourier multiplier theorems. This somewhat different approach also
necessitates other changes to the setup from [31].

On the other hand, for the second part of Theorem 1.1 we consider Fourier
multipliers between weighted spaces $\displaystyle L^{p}(\mathbb{R},w;Y)$ and
$\displaystyle L^{q}(\mathbb{R},v;X)$, for suitable weights $\displaystyle w$
and $\displaystyle v$. This setting allows us to obtain endpoint results
involving fractional domains, at the cost of having to make a priori
assumptions about the mapping properties of the Fourier transform between such
weighted spaces.

###  1.5. Organization

In Section 2 we collect some preliminaries on the vector-valued Fourier
transform, vector-valued Besov spaces, and interpolation spaces associated
with semigroup generators. In Section 3 we then prove the first part of
Theorem 1.1, and in Section 4 we prove the final statement in Theorem 1.1.

###  1.6. Notation and terminology

The natural numbers are $\displaystyle\mathbb{N}=\\{1,2,\ldots\\}$, and
$\displaystyle\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. We write
$\displaystyle{\mathbb{C}}_{+}:=\\{z\in{\mathbb{C}}\mid\operatorname{Re}(z)>0\\}$
for the open complex right half-plane.

For $\displaystyle p\in[1,\infty]$ and $\displaystyle
w:\mathbb{R}\to[0,\infty)$ measurable, we denote by $\displaystyle
L^{p}(\mathbb{R},w;X)$ the Bochner space of equivalence classes of strongly
measurable, $\displaystyle p$-integrable, $\displaystyle X$-valued functions
on $\displaystyle\mathbb{R}$ with respect to the weight $\displaystyle w$,
endowed with the norm

$\displaystyle\|f\|_{L^{p}(\mathbb{R},w;X)}:=\Big{(}\int_{\mathbb{R}}\|f(x)\|_{X}^{p}w(x)\mathrm{d}x\Big{)}^{1/p}$

for $\displaystyle f\in L^{p}(\mathbb{R},w;X)$. We simply denote this space by
$\displaystyle L^{p}(\mathbb{R};X)$ when $\displaystyle w\equiv 1$. For
$\displaystyle\gamma\in\mathbb{R}$, the weight $\displaystyle
w_{\gamma}:\mathbb{R}\to[0,\infty)$ is given by

(1.11) $w_{\gamma}(x):=|x|^{\gamma}\quad(x\in\mathbb{R}).$

The Hölder conjugate $\displaystyle p^{\prime}\in[1,\infty]$ of $\displaystyle
p\in[1,\infty]$ is defined by $\displaystyle
1=\tfrac{1}{p}+\tfrac{1}{p^{\prime}}$. We write
$\displaystyle{\mathbf{1}}_{S}$ for the indicator function of a set
$\displaystyle S$.

The space of bounded operators between complex Banach spaces $\displaystyle X$
and $\displaystyle Y$ is $\displaystyle\mathcal{L}(X,Y)$, and
$\displaystyle\mathcal{L}(X):=\mathcal{L}(X,X)$. The domain of a closed
operator $\displaystyle A$ on $\displaystyle X$ is $\displaystyle D(A)$.

We use the notation $\displaystyle f(s)\lesssim g(s)$ to indicate that
$\displaystyle f(s)\leq Cg(s)$ for all $\displaystyle s$ and a constant
$\displaystyle C\geq 0$ independent of $\displaystyle s$, and similarly for
$\displaystyle f(s)\gtrsim g(s)$ and $\displaystyle f(s)\eqsim g(s)$.

##  2\. Preliminaries

In this section we first collect some basic definitions involving the vector-
valued Fourier transform, and then we introduce Besov spaces and state two
results which will be needed to prove the first half of Theorem 1.1. Finally,
we collect background on interpolation spaces and we prove two key results
about them.

###  2.1. The Fourier transform

Let $\displaystyle X$ be a Banach space. The class of $\displaystyle X$-valued
Schwartz functions on $\displaystyle\mathbb{R}$ is denoted by
$\displaystyle\mathcal{S}(\mathbb{R};X)$, and the space of $\displaystyle
X$-valued tempered distributions by
$\displaystyle\mathcal{S}^{\prime}(\mathbb{R};X)$. The Fourier transform of
$\displaystyle f\in\mathcal{S}^{\prime}(\mathbb{R};X)$ is denoted by
$\displaystyle{\mathcal{F}}f$ or $\displaystyle\widehat{f}$. If $\displaystyle
f\in L^{1}(\mathbb{R};X)$ then

$\displaystyle{\mathcal{F}}f(\xi)=\int_{\mathbb{R}}e^{-i\xi
t}f(t)\,\mathrm{d}t\quad(\xi\in\mathbb{R}).$

One says that $\displaystyle X$ has _Fourier type_ $\displaystyle p\in[1,2]$
if $\displaystyle{\mathcal{F}}:L^{p}(\mathbb{R};X)\to
L^{p^{\prime}}(\mathbb{R};X)$ is bounded. Every Banach space $\displaystyle X$
has Fourier type $\displaystyle 1$, and $\displaystyle X$ has Fourier type
$\displaystyle 2$ if and only if $\displaystyle X$ is isomorphic to a Hilbert
space (see [20]).

We say that $\displaystyle X$ has _Hardy–Littlewood type_ $\displaystyle
p\in(1,2]$ if

(2.1) ${\mathcal{F}}:L^{p}(\mathbb{R};X)\to L^{p}(\mathbb{R},w_{p-2};X)$

is bounded, where $\displaystyle w_{\gamma}$ is defined in (1.11) for
$\displaystyle\gamma\in\mathbb{R}$. Moreover, $\displaystyle X$ has
_Hardy–Littlewood cotype_ $\displaystyle q\in[2,\infty)$ if

(2.2) ${\mathcal{F}}:L^{q}(\mathbb{R},w_{q-2};X)\to L^{q}(\mathbb{R};X)$

is bounded. Note that, if $\displaystyle X={\mathbb{C}}$, then (2.1) is the
Hardy–Littlewood inequality. In the latter case, and in fact for any Hilbert
space $\displaystyle X$, (2.1) holds for all $\displaystyle p\in(1,2]$, and
(2.2) for all $\displaystyle q\in[2,\infty)$.

Any Banach space $\displaystyle X$ trivially has Fourier type $\displaystyle
1$. If $\displaystyle X$ has Fourier type $\displaystyle p_{0}\in(1,2]$, then
$\displaystyle X$ has Hardy–Littlewood type $\displaystyle p$ for all
$\displaystyle p\in(1,p_{0})$, and Hardy–Littlewood cotype for all
$\displaystyle q\in(p_{0}^{\prime},\infty)$ (see [11, Proposition 3.5]). Also,
if $\displaystyle X$ is a Banach lattice which is $\displaystyle p$-convex and
$\displaystyle p$-concave with $\displaystyle p\in(1,\infty)$, then
$\displaystyle X$ has Fourier type $\displaystyle p$ and Hardy–Littlewood type
$\displaystyle p$ if $\displaystyle p\leq 2$, and Fourier type $\displaystyle
p^{\prime}$ and Hardy–Littlewood cotype $\displaystyle p$ if $\displaystyle
p\geq 2$ (see [12, Proposition 2.2] and [11, Proposition 6.9]). This holds in
particular if $\displaystyle X$ is isomorphic to a closed subspace of
$\displaystyle L^{p}(\Omega)$, for $\displaystyle\Omega$ any measure space.
For more on the relation between the notions of Fourier type, Hardy–Littlewood
(co)type, and convexity and concavity in Banach lattices, we refer to [11].

Let $\displaystyle Y$ be a Banach space and $\displaystyle
m:\mathbb{R}\to\mathcal{L}(Y,X)$. We say that $\displaystyle m$ is
_$\displaystyle X$ -strongly measurable_ if $\displaystyle\xi\mapsto m(\xi)y$
is a strongly measurable $\displaystyle X$-valued map for every $\displaystyle
y\in Y$. In this article we will consider $\displaystyle m$ which have the
additional property that there exist $\displaystyle\alpha,C_{\alpha}\geq 0$
such that $\displaystyle\|m(\xi)\|_{\mathcal{L}(Y,X)}\leq
C_{\alpha}(1+|\xi|)^{\alpha}$ for all $\displaystyle\xi\in\mathbb{R}$. In this
case, we may set

$\displaystyle
T_{m}f:=\mathcal{F}^{-1}(m\widehat{f}\,)\quad(f\in\mathcal{S}(\mathbb{R};X)).$

Then $\displaystyle
T_{m}:\mathcal{S}(\mathbb{R};X)\to\mathcal{S}^{\prime}(\mathbb{R};Y)$ is the
_Fourier multiplier operator_ with _symbol_ $\displaystyle m$.

###  2.2. Besov spaces

Throughout this article, fix an inhomogeneous Littlewood–Paley sequence
$\displaystyle(\phi_{k})_{k\in\mathbb{N}_{0}}\subseteq
C^{\infty}_{c}(\mathbb{R})$. That is, one has $\displaystyle\phi_{1}(\xi)=0$
if $\displaystyle|\xi|\notin[1/2,2]$,
$\displaystyle\phi_{k}(\xi)=\phi_{1}(2^{-k+1}\xi)$ for each $\displaystyle
k>1$ and $\displaystyle\xi\in\mathbb{R}$, and

$\displaystyle\sum_{k=0}^{\infty}\phi_{k}(\xi)=1\quad(\xi\in\mathbb{R}).$

Let $\displaystyle p,q\in[1,\infty]$ and $\displaystyle s\in\mathbb{R}$. Then
the Besov space $\displaystyle B^{s}_{p,q}(\mathbb{R};X)$ consists of all
$\displaystyle f\in\mathcal{S}^{\prime}(\mathbb{R};X)$ such that
$\displaystyle{\mathcal{F}}^{-1}(\phi_{k})*f\in L^{p}(\mathbb{R};X)$ for each
$\displaystyle k\geq 0$, and such that

$\displaystyle\|f\|_{B^{s}_{p,q}(\mathbb{R};X)}:=\|(2^{ks}{\mathcal{F}}^{-1}(\phi_{k})*f)_{k\geq
0}\|_{\ell^{q}(L^{p}(\mathbb{R};X))}<\infty.$

Then $\displaystyle\mathcal{S}(\mathbb{R};X)\subseteq
B^{s}_{p,q}(\mathbb{R};X)$, by [19, Proposition 14.4.3]. Moreover, if
$\displaystyle p,q<\infty$, then
$\displaystyle\mathcal{S}(\mathbb{R};X)\subseteq B^{s}_{p,q}(\mathbb{R};X)$
lies dense. Finally, we will use the simple observation that

(2.3) $B^{s}_{p,q}(\mathbb{R};X)\subseteq B^{r}_{p,1}(\mathbb{R};X)$

for all $\displaystyle p,q\in[1,\infty]$ and $\displaystyle s,r\in\mathbb{R}$
with $\displaystyle s>r$, and that

(2.4) $B^{0}_{p,1}(\mathbb{R};X)\subseteq L^{p}(\mathbb{R};X)\subseteq
B^{0}_{p,\infty}(\mathbb{R};X)$

for all $\displaystyle p\in[1,\infty]$ (see e.g. [19, Proposition 14.4.18]).

The following lemma will be used in the proof of Proposition 2.4.

######  Lemma 2.1.

Let $\displaystyle p\in[1,\infty)$, $\displaystyle q\in[1,\infty]$ and
$\displaystyle s\in(0,1/p)$. Then there exists a $\displaystyle C\geq 0$ such
that $\displaystyle{\mathbf{1}}_{(0,\infty)}f\in B^{s}_{p,q}(\mathbb{R};X)$
for all $\displaystyle f\in B^{s}_{p,q}(\mathbb{R};X)$, and

$\displaystyle\|{\mathbf{1}}_{(0,\infty)}f\|_{B^{s}_{p,q}(\mathbb{R};X)}\leq
C\|f\|_{B^{s}_{p,q}(\mathbb{R};X)}.$

###### Proof.

For $\displaystyle p>1$, the statement in fact holds for $\displaystyle
s\in(-1/p,1/p)$, as is shown in [19, Corollary 14.6.35]. In the proof of the
latter result, one can see that for $\displaystyle s\in(0,1/p)$ one may also
allow $\displaystyle p=1$. ∎

Finally, the following Fourier multiplier result, [19, Proposition 14.5.7], is
one of the key ingredients in the proof of the first half of Theorem 1.1.

######  Proposition 2.2.

Let $\displaystyle X$ and $\displaystyle Y$ be Banach spaces with Fourier type
$\displaystyle p\in[1,2]$, and let $\displaystyle
m:\mathbb{R}\to\mathcal{L}(Y,X)$ be $\displaystyle X$-strongly measurable,
with
$\displaystyle\sup_{\xi\in\mathbb{R}}\|m(\xi)\|_{\mathcal{L}(Y,X)}<\infty$.
Then

$\displaystyle T_{m}:B^{1/p-1/p^{\prime}}_{p,p}(\mathbb{R};Y)\to
L^{p^{\prime}}(\mathbb{R};X)$

is bounded.

###  2.3. Interpolation spaces

Let $\displaystyle A$ be a linear operator on a Banach space $\displaystyle
X$. For $\displaystyle\omega\in(0,\pi)$, set $\displaystyle
S_{\omega}:=\\{z\in{\mathbb{C}}\setminus\\{0\\}\mid|\arg(z)|<\omega\\}$. Then
$\displaystyle-A$ is a _sectorial operator_ if there exists an
$\displaystyle\omega\in(0,\pi)$ such that
$\displaystyle\sigma(-A)\subseteq\overline{S_{\omega}}$, and

(2.5)
$\sup\\{\|\lambda(\lambda+A)^{-1}\|_{\mathcal{L}(X)}\,|\,\lambda\in{\mathbb{C}}\backslash\overline{S_{\omega^{\prime}}}\\}<\infty$

for each $\displaystyle\omega^{\prime}\in(\omega,\pi)$. If $\displaystyle-A$
is a sectorial operator, then the fractional power
$\displaystyle(-A)^{\alpha}$ is well defined for each
$\displaystyle\alpha\in{\mathbb{C}}_{+}$, cf. [16, Chapter 3]. If,
additionally, $\displaystyle A$ is injective, then
$\displaystyle(-A)^{\alpha}$ is well defined for all
$\displaystyle\alpha\in{\mathbb{C}}$. Note that $\displaystyle
D((-A)^{\beta})\subseteq D((-A)^{\alpha})$ whenever
$\displaystyle\beta\in{\mathbb{C}}$ satisfies
$\displaystyle\operatorname{Re}(\beta)>\operatorname{Re}(\alpha)$.

######  Remark 2.3.

Throughout this article, as in Theorem 1.1, we will consider $\displaystyle
C_{0}$-semigroups $\displaystyle(T(t))_{t\geq 0}$ with generator
$\displaystyle A$ such that $\displaystyle{\mathbb{C}}_{+}\subseteq\rho(A)$
and

(2.6) $\|R(\lambda,A)\|_{\mathcal{L}(X)}\leq
C(1+|\lambda|)^{\beta}\quad(\lambda\in{\mathbb{C}}_{+}),$

where $\displaystyle\beta,C\geq 0$ are independent of $\displaystyle\lambda$.
Under these assumptions, $\displaystyle-A$ is a sectorial operator of angle
$\displaystyle\pi/2$. Indeed, the semigroup generation property implies that
$\displaystyle\|R(\lambda,A)\|_{\mathcal{L}(X)}\lesssim
1/\operatorname{Re}(\lambda)$ for $\displaystyle\operatorname{Re}(\lambda)$
large (as follows from (1.9)), which gives a uniform bound in (2.5) for
$\displaystyle|\lambda|$ large if $\displaystyle\omega^{\prime}>\pi/2$. On the
other hand, (2.6) implies that
$\displaystyle\overline{{\mathbb{C}}_{+}}\subseteq\rho(A)$, which in turn
yields the required bound in (2.5) for $\displaystyle|\lambda|$ small.

Let $\displaystyle-A$ be a sectorial operator on a Banach space $\displaystyle
X$, and let $\displaystyle\tau\in(0,\infty)$ and $\displaystyle
q\in[1,\infty]$. Then the real interpolation space associated with
$\displaystyle A$, $\displaystyle\tau$ and $\displaystyle q$ is

(2.7) $D_{A}(\tau,q):=(X,D((-A)^{\alpha}))_{\tau/\alpha,q},$

where $\displaystyle\alpha\in(\tau,\infty)$ is arbitrary. It follows from
reiteration that $\displaystyle D_{A}(\tau,q)$ is independent of the choice of
$\displaystyle\alpha$. In particular, one has

$\displaystyle
D_{A}(\tau,q)=(X,D((-A)^{m}))_{\tau/m,q}=(X,D(A^{m}))_{\tau/m,q}$

whenever $\displaystyle m\in\mathbb{N}$ satisfies $\displaystyle m>\tau$. By
basic properties of interpolation spaces, $\displaystyle
D_{A}(\tau,q)\subseteq D_{A}(\sigma,r)$ if $\displaystyle\sigma<\tau$, or if
$\displaystyle\sigma=\tau$ and $\displaystyle r\geq q$. By [16, Corollary
6.6.3],

(2.8) $D_{A}(\tau,1)\subseteq D((-A)^{\tau})\subseteq D_{A}(\tau,\infty)$

for all $\displaystyle\tau>0$. Also, $\displaystyle D((-A)^{\alpha})\subseteq
D_{A}(\tau,q)$ lies dense for all $\displaystyle\alpha>\tau$ and
$\displaystyle q<\infty$, by [16, Theorem 6.6.1].

Finally, if $\displaystyle X$ has Fourier type $\displaystyle p\in[1,2]$ and
$\displaystyle A$ is injective, then both $\displaystyle D((-A)^{\tau})$ and
$\displaystyle D_{A}(\tau,q)$ also have Fourier type $\displaystyle p$, for
all $\displaystyle\tau>0$ and $\displaystyle q\in[1,\infty]$. Indeed,
$\displaystyle(-A)^{\tau}:D((-A)^{\tau})\to X$ is an isomorphism, while for
$\displaystyle D_{A}(\tau,q)$ the statement follows from (2.7) by
interpolation (see also [18, Proposition 2.4.17]).

The following proposition, connecting interpolation spaces to the Besov spaces
from the previous subsection, will play a key role in the proof of part of
Theorem 3.2.

######  Proposition 2.4.

Let $\displaystyle A$ be the generator of a $\displaystyle C_{0}$-semigroup
$\displaystyle(T(t))_{t\geq 0}$ on a Banach space $\displaystyle X$, and
suppose that $\displaystyle-A$ is a sectorial operator. Let $\displaystyle
M\geq 0$ and $\displaystyle\omega\in\mathbb{R}$ be such that
$\displaystyle\|T(t)\|_{\mathcal{L}(X)}\leq Me^{(\omega-1)t}$ for all
$\displaystyle t\geq 0$, and let $\displaystyle p\in[1,\infty)$,
$\displaystyle q\in[1,\infty]$ and $\displaystyle s\in(0,1/p)$. Then there
exists a constant $\displaystyle C\geq 0$ such that
$\displaystyle[t\mapsto{\mathbf{1}}_{(0,\infty)}(t)e^{-\omega t}T(t)x]\in
B^{s}_{p,q}(\mathbb{R};X)$ for all $\displaystyle x\in D_{A}(s,q)$, with

$\displaystyle\|[t\mapsto{\mathbf{1}}_{(0,\infty)}(t)e^{-\omega
t}T(t)x]\|_{B^{s}_{p,q}(\mathbb{R};X)}\leq C\|x\|_{D_{A}(s,q)}.$

###### Proof.

Let $\displaystyle J:X\to L^{p}(\mathbb{R};X)$ be the bounded linear operator
given by $\displaystyle Jx(t):=e^{-\omega|t|}T(|t|)x$, for $\displaystyle x\in
X$ and $\displaystyle t\in\mathbb{R}$. Since
$\displaystyle(Jx)^{\prime}(t)=-\text{sign}(t)e^{-\omega|t|}T(|t|)(\omega-A)x$
for $\displaystyle x\in D(A)$ and $\displaystyle t\neq 0$, the restricted
operator $\displaystyle J:D(A)\to W^{1,p}(\mathbb{R};X)$ is bounded. Real
interpolation (see [19, Theorem 14.4.31]) shows that $\displaystyle
J:D_{A}(s,q)\to B^{s}_{p,q}(\mathbb{R};X)$ is bounded as well. Now the proof
is concluded by applying Lemma 2.1. ∎

In turn, the following proposition will be crucial for the proof of Theorem
1.1.

######  Proposition 2.5.

Let $\displaystyle A$ be be the generator of a $\displaystyle C_{0}$-semigroup
$\displaystyle(T(t))_{t\geq 0}$ on a Banach space $\displaystyle X$. Suppose
that $\displaystyle{\mathbb{C}}_{+}\subseteq\rho(A)$, and that there exist
$\displaystyle\beta>0$ and $\displaystyle C\geq 0$ such that

(2.9) $\|R(\lambda,A)\|_{\mathcal{L}(X)}\leq
C(1+|\lambda|)^{\beta}\quad(\lambda\in{\mathbb{C}}_{+}).$

Then $\displaystyle i\mathbb{R}\subseteq\rho(A)$, and

$\displaystyle\sup\\{\|R(i\xi,A)^{k}\|_{\mathcal{L}(D_{A}((n+1)\beta,q),X)}\mid\xi\in\mathbb{R},k\in\\{0,\ldots,n+1\\}\\}<\infty$

for all $\displaystyle n\in\mathbb{N}_{0}$ and $\displaystyle q\in[1,\infty]$.

###### Proof.

The required statement is trivial for $\displaystyle k=0$ and
$\displaystyle|\xi|<1$, so henceforth we will consider $\displaystyle k>0$ and
$\displaystyle\xi\in\mathbb{R}$ with $\displaystyle|\xi|\geq 1$.

By basic properties of resolvents, it follows from (2.9) that $\displaystyle
i\mathbb{R}\subseteq\rho(A)$ and

(2.10) $\|R(i\xi,A)\|_{\mathcal{L}(X)}\leq
C(1+|\xi|)^{\beta}\quad(\xi\in\mathbb{R}).$

As noted in Remark 2.3, $\displaystyle-A$ is a sectorial operator. Moreover,
$\displaystyle D((-A)^{\alpha})=D((1-A)^{\alpha})$ for any
$\displaystyle\alpha>0$, by [19, Proposition 15.2.12]. Hence, combining (2.10)
and [31, Proposition 3.4] yields

(2.11) $\sup_{|\xi|\geq 1}\|R(i\xi,A)(-A)^{-\beta}\|_{\mathcal{L}(X)}<\infty.$

Then, for $\displaystyle k\in\\{1,\ldots,n\\}$ and $\displaystyle|\xi|\geq 1$,

$\displaystyle\displaystyle\|R(i\xi,A)^{k}\|_{\mathcal{L}(D((-A)^{n\beta}),X)}$
$\displaystyle\displaystyle\eqsim\|R(i\xi,A)^{k}(-A)^{-n\beta}\|_{\mathcal{L}(X)}$
$\displaystyle\displaystyle\leq\|R(i\xi,A)^{k}(-A)^{-k\beta}\|_{\mathcal{L}(X)}\|(-A)^{-(n-k)\beta}\|_{\mathcal{L}(X)}\lesssim
1.$

Together with (2.10), this implies

(2.12)
$\|R(i\xi,A)^{k}\|_{\mathcal{L}(D((-A)^{n\beta}),X)}\lesssim(1+|\xi|)^{\beta}\quad(k\in\\{1,\ldots,n+1\\}).$

On the other hand, another application of [31, Proposition 3.4] shows that

$\displaystyle\|R(i\xi,A)(-A)^{-\beta-1}\|_{\mathcal{L}(X)}\lesssim(1+|\xi|)^{-1}.$

This, combined with (2.11), yields

(2.13)
$\displaystyle\|R(i\xi,A)^{k}\|_{\mathcal{L}(D((-A)^{(n+1)\beta+1}),X)}\eqsim\|R(i\xi,A)^{k}(-A)^{-(n+1)\beta-1}\|_{\mathcal{L}(X)}$
$\displaystyle\leq\|R(i\xi,A)(-A)^{-\beta}\|^{k-1}_{\mathcal{L}(X)}\|R(i\xi,A)(-A)^{-\beta-1}\|_{\mathcal{L}(X)}\|(-A)^{-\beta}\|_{\mathcal{L}(X)}^{n-k+1}$
$\displaystyle\lesssim(1+|\xi|)^{-1},$

for all $\displaystyle k\in\\{1,\ldots,n+1\\}$ and $\displaystyle|\xi|\geq 1$.

Now, by (2.8), (2.12) and (2.13), we have

$\displaystyle\displaystyle\|R(i\xi,A)^{k}\|_{\mathcal{L}(D_{A}(n\beta,1),X)}$
$\displaystyle\displaystyle\lesssim(1+|\xi|)^{\beta},$
$\displaystyle\displaystyle\|R(i\xi,A)^{k}\|_{\mathcal{L}(D_{A}((n+1)\beta+1,1),X)}$
$\displaystyle\displaystyle\lesssim(1+|\xi|)^{-1},$

for $\displaystyle|\xi|\geq 1$ and $\displaystyle k\in\\{1,\ldots,n+1\\}$.
Finally, since

$\displaystyle
D_{A}((n+1)\beta,q)=(D_{A}(n\beta,1),D_{A}((n+1)\beta+1,1))_{\frac{\beta}{1+\beta},q},$

interpolating these estimates yields $\displaystyle\sup_{|\xi|\geq
1}\|R(i\xi,A)^{k}\|_{\mathcal{L}(D_{A}((n+1)\beta,q),X)}<\infty$. ∎

##  3\. Polynomial stability on real interpolation spaces

This section is devoted to the proof of the first half of Theorem 1.1. To this
end, we need two preliminary results.

We first require the following extension of [31, Proposition 3.2] to the mixed
Besov-Lebesgue setting.

######  Proposition 3.1.

Let $\displaystyle A$ be the generator of a $\displaystyle C_{0}$-semigroup
$\displaystyle(T(t))_{t\geq 0}$ on a Banach space $\displaystyle X$, and let
$\displaystyle Y$ be a Banach space that is continuously embedded in
$\displaystyle X$. Suppose that $\displaystyle i\mathbb{R}\subseteq\rho(A)$,
and that there exist $\displaystyle\beta>0$ and $\displaystyle C\geq 0$ such
that

(3.1) $\|R(i\xi,A)\|_{\mathcal{L}(Y,X)}\leq
C(1+|\xi|)^{\beta}\quad(\xi\in\mathbb{R}).$

Let $\displaystyle p\in[1,\infty)$ and $\displaystyle s\in[0,\infty)$ be such
that either $\displaystyle s>0$, or $\displaystyle s=0$ and $\displaystyle
p=1$, and suppose that there exist $\displaystyle q\in[1,\infty]$ and
$\displaystyle n\in\mathbb{N}$ such that

(3.2) $T_{R(i\cdot,A)^{j}}:B^{s}_{p,p}(\mathbb{R};Y)\to L^{q}(\mathbb{R};X)$

is bounded for each $\displaystyle j\in\\{n-1,n\\}\cap\mathbb{N}$. Then

$\displaystyle T_{R(i\cdot,A)^{n}}:B^{s}_{p,p}(\mathbb{R};Y)\to
L^{\infty}(\mathbb{R};X)$

is bounded.

We only assume (3.1) to guarantee that the Fourier multiplier operator in
(3.2) is well defined; the specific value of $\displaystyle\beta$ plays no
role here.

###### Proof.

The proof is analogous to that of [31, Proposition 3.2]. For the convenience
of the reader, we provide the argument. It suffices to show that there exists
a $\displaystyle C\geq 0$ such that

(3.3) $\sup_{k\leq\sigma\leq k+1}\|T_{R(i\cdot,A)^{n}}f(\sigma)\|_{X}\leq
C\|f\|_{B^{s}_{p,p}(\mathbb{R};Y)}$

for every $\displaystyle f\in\mathcal{S}(\mathbb{R};Y)$ and $\displaystyle
k\in\mathbb{Z}$, since $\displaystyle\mathcal{S}(\mathbb{R};Y)\subseteq
B^{s}_{p,p}(\mathbb{R};Y)$ lies dense.

For each $\displaystyle j\in\\{n-1,n\\}\cap\mathbb{N}$ there exists a
$\displaystyle K_{j}\geq 0$ independent of $\displaystyle f$ such that

(3.4) $\|T_{R(i\cdot,A)^{j}}f\|_{L^{q}(\mathbb{R};X)}\leq
K_{j}\|f\|_{B^{s}_{p,p}(\mathbb{R};Y)}.$

Hence there exists a $\displaystyle t\in[k-1,k]$ such that

(3.5) $\|T_{R(i\cdot,A)^{j}}f(t)\|_{X}\leq
K_{j}\|f\|_{B^{s}_{p,p}(\mathbb{R};Y)}.$

Now let $\displaystyle\tau\in[0,2]$. One can check that

(3.6)
$T(\tau)T_{R(i\cdot,A)^{n}}f(t)=T_{R(i\cdot,A)^{n}}f(t+\tau)-\int_{0}^{\tau}T(r)T_{R(i\cdot,A)^{n-1}}f(t+\tau-r)\mathrm{d}r.$

Hence, by (3.5), Hölder’s inequality and (3.4), for $\displaystyle n>1$ one
has

$\displaystyle\displaystyle\|T_{R(i\cdot,A)^{n}}f(t+\tau)\|_{X}$
$\displaystyle\displaystyle\lesssim\|T_{R(i\cdot,A)^{n}}f(t)\|_{X}+\int_{0}^{\tau}\|T_{R(i\cdot,A)^{n-1}}f(t+\tau-r)\|_{X}\mathrm{d}r$
$\displaystyle\displaystyle\leq
K_{n}\|f\|_{B^{s}_{p,p}(\mathbb{R};Y)}+\tau^{1/{q^{\prime}}}\|T_{R(i\cdot,A)^{n-1}}f\|_{L^{q}(\mathbb{R};X)}$
$\displaystyle\displaystyle\lesssim\|f\|_{B^{s}_{p,p}(\mathbb{R};Y)}.$

This implies (3.3) for $\displaystyle n>1$.

Finally, for $\displaystyle n=1$, by the assumptions on $\displaystyle p$ and
$\displaystyle s$ as well as (2.3) and (2.4), one has $\displaystyle
B^{s}_{p,p}(\mathbb{R};Y)\subseteq B^{0}_{p,1}(\mathbb{R};Y)\subseteq
L^{p}(\mathbb{R};Y)$. Hence Hölder’s inequality again gives

$\displaystyle\int_{0}^{\tau}\|f(t+\tau-r)\|_{X}\mathrm{d}r\lesssim\int_{0}^{\tau}\|f(t+\tau-r)\|_{Y}\mathrm{d}r\leq\tau^{1/p^{\prime}}\|f\|_{L^{p}(\mathbb{R};Y)}\lesssim\|f\|_{B^{s}_{p,p}(\mathbb{R};Y)}.$

By combining this with (3.6) in the same manner as before, one obtains (3.3).
∎

We will also rely on the following version of [31, Theorem 4.6] in the mixed
Besov-Lebesgue setting.

######  Theorem 3.2.

Let $\displaystyle A$ be the generator of a $\displaystyle C_{0}$-semigroup
$\displaystyle(T(t))_{t\geq 0}$ on a Banach space $\displaystyle X$. Suppose
that $\displaystyle{\mathbb{C}}_{+}\subseteq\rho(A)$, and that there exist
$\displaystyle\beta>0$ and $\displaystyle C\geq 0$ such that

(3.7) $\|R(\lambda,A)\|_{\mathcal{L}(X)}\leq
C(1+|\lambda|)^{\beta}\quad(\lambda\in{\mathbb{C}}_{+}).$

Let $\displaystyle\gamma>0$, $\displaystyle p\in[1,\infty)$ and $\displaystyle
s\in(0,1/p)$, and suppose that there exist $\displaystyle n\in\mathbb{N}_{0}$
and $\displaystyle q\in[1,\infty]$ such that

(3.8) $T_{R(i\cdot,A)^{k}}:B^{s}_{p,p}(\mathbb{R};D_{A}(\gamma,p))\to
L^{q}(\mathbb{R};X)$

is bounded for each $\displaystyle k\in\\{n-1,n,n+1\\}\cap\mathbb{N}$. Then
there exists a $\displaystyle C_{n}\geq 0$ such that

(3.9) $\|T(t)x\|_{X}\leq C_{n}t^{-n}\|x\|_{D_{A}(\gamma+s,p)}$

for all $\displaystyle t\geq 1$ and $\displaystyle x\in D_{A}(\gamma+s,p)$.

Note that (3.7) automatically extends to all
$\displaystyle\lambda\in\overline{{\mathbb{C}}_{+}}$, so (3.8) is well
defined. Also, as in Proposition 3.1, the specific value of
$\displaystyle\beta$ in (3.7) plays no role.

###### Proof.

We want to show that $\displaystyle\|T(t)x\|_{X}\leq
C_{n}t^{-n}\|x\|_{D_{A}(\gamma+s,p)}$ for all $\displaystyle t\geq 1$ and
$\displaystyle x\in D_{A}(\gamma+s,p)$. Since $\displaystyle D(A^{l})$ is
dense in $\displaystyle D_{A}(\gamma+s,p)$ whenever $\displaystyle
l\in\mathbb{N}$ satisfies $\displaystyle l>\gamma+s$, we may suppose
throughout that $\displaystyle x\in D(A^{l})$ for some large $\displaystyle
l$. Hence, setting $\displaystyle
g(t):=t^{n}{\mathbf{1}}_{(0,\infty)}(t)T(|t|)x$ for $\displaystyle
t\in\mathbb{R}$, by [31, Proposition 4.3] we may suppose that $\displaystyle
g\in L^{1}(\mathbb{R};X)$. In turn, [31, Lemma 3.1] then implies that
$\displaystyle\widehat{g}(\xi)=n!R(i\xi,A)^{n+1}x$ for all
$\displaystyle\xi\in\mathbb{R}$.

Next, note that $\displaystyle(T(t))_{t\geq 0}$ restricts to a $\displaystyle
C_{0}$-semigroup on $\displaystyle D_{A}(\gamma,p)$, the generator of which is
the part of $\displaystyle A$ in $\displaystyle D_{A}(\gamma,p)$, which has
domain $\displaystyle D_{A}(\gamma+1,p)$. In particular, we may fix
$\displaystyle M\geq 1$ and $\displaystyle\omega\in\mathbb{R}$ such that
$\displaystyle\|T(t)\|_{\mathcal{L}(D_{A}(\gamma,p))}\leq Me^{(\omega-1)t}$
for all $\displaystyle t\geq 0$. Set $\displaystyle
f(t):={\mathbf{1}}_{(0,\infty)}(t)e^{-\omega t}T(|t|)x$. Then

(3.10) $\|f\|_{L^{\infty}(\mathbb{R};D_{A}(\gamma,p))}\leq
M\|x\|_{D_{A}(\gamma,p)}\lesssim\|x\|_{D_{A}(\gamma+s,p)}.$

Moreover,

(3.11)
$\|f\|_{B^{s}_{p,p}(\mathbb{R};D_{A}(\gamma,p))}\lesssim\|x\|_{(D_{A}(\gamma,p),D_{A}(\gamma+1,p))_{s,p}}\lesssim\|x\|_{D_{A}(\gamma+s,p)},$

as follows from Proposition 2.4 and [34, Theorem 1.10.2]. Also, again by [31,
Lemma 3.1], $\displaystyle\widehat{f}(\xi)=R(\omega+i\xi,A)x$ for all
$\displaystyle\xi\in\mathbb{R}$. In particular, if we set

$\displaystyle m(\xi):=n!(R(i\xi,A)^{n}+\omega R(i\xi,A)^{n+1}),$

then $\displaystyle m(\xi)\widehat{f}(\xi)=\widehat{g}(\xi)$.

By combining all this, we see that

(3.12) $\displaystyle\sup_{t\geq 0}\|t^{n}T(t)x\|_{X}$
$\displaystyle=\|T_{m}f\|_{L^{\infty}(\mathbb{R};X)}$ $\displaystyle\leq
n!\big{(}\|T_{R(i\cdot,A)^{n}}f\|_{L^{\infty}(\mathbb{R};X)}+\omega\|T_{R(i\cdot,A)^{n+1}}f\|_{L^{\infty}(\mathbb{R};X)}\big{)}.$

For $\displaystyle n>0$, one can apply (3.8) and Proposition 3.1 to the final
line, and then use (3.11) as well, to obtain

$\displaystyle\sup_{t\geq
0}\|t^{n}T(t)x\|_{X}\lesssim\|f\|_{B^{s}_{p,p}(\mathbb{R};D_{A}(\gamma,p))}\lesssim\|x\|_{D_{A}(\gamma+s,p)}.$

For $\displaystyle n=0$, the same reasoning can be used for the second term in
brackets in (3.12), while for the first term one can directly rely on (3.10),
since $\displaystyle T_{R(i\cdot,A)^{0}}f=f$. ∎

We are now ready to prove the first part of Theorem 1.1.

######  Theorem 3.3.

Let $\displaystyle A$ be the generator of a $\displaystyle C_{0}$-semigroup
$\displaystyle(T(t))_{t\geq 0}$ on a Banach space $\displaystyle X$ with
Fourier type $\displaystyle p\in[1,2]$. Suppose that
$\displaystyle{\mathbb{C}}_{+}\subseteq\rho(A)$, and that there exist
$\displaystyle\beta>0$ and $\displaystyle C\geq 0$ such that

$\displaystyle\|R(\lambda,A)\|_{\mathcal{L}(X)}\leq
C(1+|\lambda|)^{\beta}\quad(\lambda\in{\mathbb{C}}_{+}).\\\ $

Let $\displaystyle\rho\geq 0$ and set
$\displaystyle\tau:=(\rho+1)\beta+\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$. Then
there exists a $\displaystyle C_{\rho}\geq 0$ such that

(3.13) $\|T(t)x\|_{X}\leq C_{\rho}t^{-\rho}\|x\|_{D_{A}(\tau,p)}$

for all $\displaystyle t\geq 1$ and $\displaystyle x\in D_{A}(\tau,p)$.
Moreover, if $\displaystyle\rho>0$, then (3.13) also holds with $\displaystyle
D_{A}(\tau,p)$ replaced by $\displaystyle D_{A}(\tau,q)$ for any
$\displaystyle q\in[1,\infty]$, or by $\displaystyle D((-A)^{\tau})$.

###### Proof.

We first consider the case where $\displaystyle\rho\in\mathbb{N}_{0}$. Recall
that, since $\displaystyle X$ has Fourier type $\displaystyle p$, so does
$\displaystyle D_{A}((\rho+1)\beta,p)$. Moreover, by Proposition 2.5,

$\displaystyle\sup_{\xi\in\mathbb{R}}\|R(i\xi,A)^{k}\|_{\mathcal{L}(D_{A}((\rho+1)\beta,p),X)}<\infty$

for all $\displaystyle k\in\\{0,\ldots,\rho+1\\}$. Hence Proposition 2.2
implies that

$\displaystyle
T_{R(i\cdot,A)^{k}}:B^{1/p-1/p^{\prime}}_{p,p}(\mathbb{R};D_{A}((\rho+1)\beta,p))\to
L^{p^{\prime}}(\mathbb{R};X)$

is bounded for every $\displaystyle k\in\\{0,\ldots,\rho+1\\}$. Finally,
Theorem 3.2 yields (3.13).

To extend (3.13) to general $\displaystyle\rho\geq 0$ we proceed as follows.
Fix $\displaystyle t\geq 1$ and $\displaystyle\rho>0$. Let
$\displaystyle\rho_{0},\rho_{1}\in\mathbb{N}_{0}$ be such that
$\displaystyle\rho_{0}<\rho<\rho_{1}$, and let $\displaystyle\theta\in(0,1)$
be such that $\displaystyle\rho=(1-\theta)\rho_{0}+\theta\rho_{1}$. Set
$\displaystyle\tau_{i}:=(\rho_{i}+1)\beta+\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$
for $\displaystyle i\in\\{0,1\\}$. Then, by what we have already shown,

$\displaystyle\|T(t)\|_{\mathcal{L}(D_{A}(\tau_{i},p),X)}\leq
C_{\rho_{i}}t^{-\rho_{i}}$

for some constant $\displaystyle C_{\rho_{i}}\geq 0$ independent of
$\displaystyle t$. Now, due to reiteration, real interpolation with parameters
$\displaystyle\theta$ and $\displaystyle q\in[1,\infty]$ gives

$\displaystyle\|T(t)\|_{\mathcal{L}(D_{A}(\tau,q),X)}\leq C_{\rho}t^{-\rho}$

for some $\displaystyle C_{\rho}\geq 0$ independent of $\displaystyle t$. This
proves both (3.13) and the final statement of the theorem, since
$\displaystyle D((-A)^{\tau})\subseteq D_{A}(\tau,\infty)$. ∎

##  4\. Polynomial stability on fractional domains

This section is devoted to the proof of the final statement in Theorem 1.1.

The following proposition will play the same role in this section that
Proposition 2.2 did in the previous section. For
$\displaystyle\gamma\in\mathbb{R}$, recall the definition of the weight
$\displaystyle w_{\gamma}:\mathbb{R}\to[0,\infty)$ from (1.11).

######  Proposition 4.1.

Let $\displaystyle p,q\in[1,\infty]$, $\displaystyle r\in[1,\infty)$,
$\displaystyle\gamma\in\mathbb{R}$ and
$\displaystyle\delta\in(-\infty,1/r^{\prime})$. Let $\displaystyle Y$ be a
Banach space such that

(4.1) ${\mathcal{F}}:L^{p}(\mathbb{R};Y)\to L^{r}(\mathbb{R},w_{\gamma r};Y)$

is bounded, and let $\displaystyle X$ be a Banach space such that

(4.2) ${\mathcal{F}}:L^{r}(\mathbb{R},w_{\delta r};X)\to L^{q}(\mathbb{R};X)$

is bounded. Let $\displaystyle
m:\mathbb{R}\setminus\\{0\\}\to\mathcal{L}(Y,X)$ be an $\displaystyle
X$-strongly measurable map for which there exists a $\displaystyle C\geq 0$
such that $\displaystyle\left\|m(\xi)\right\|_{\mathcal{L}(Y,X)}\leq
C|\xi|^{\gamma-\delta}$ for all
$\displaystyle\xi\in\mathbb{R}\setminus\\{0\\}$. Then $\displaystyle
T_{m}:L^{p}(\mathbb{R};Y)\to L^{q}(\mathbb{R};X)$ is bounded.

###### Proof.

Simply combine the assumptions on $\displaystyle X$, $\displaystyle m$ and
$\displaystyle Y$:

$\displaystyle\displaystyle\|T_{m}f\|_{L^{q}(\mathbb{R};X)}\lesssim\|m\widehat{f}\,\|_{L^{r}(\mathbb{R},w_{\delta
r};X)}\lesssim\|w_{\delta}w_{\gamma-\delta}\widehat{f}\,\|_{L^{r}(\mathbb{R};Y)}\lesssim\|f\|_{L^{p}(\mathbb{R};Y)}$

for all $\displaystyle f\in L^{p}(\mathbb{R};Y)$. Note that $\displaystyle
L^{r}(\mathbb{R},w_{\delta r};X)\subseteq\mathcal{S}^{\prime}(\mathbb{R};X)$,
since $\displaystyle\delta<1/r^{\prime}$. ∎

We are now ready to prove the last statement in Theorem 1.1, as a special case
of the following result.

######  Theorem 4.2.

Let $\displaystyle A$ be the generator of a $\displaystyle C_{0}$-semigroup
$\displaystyle(T(t))_{t\geq 0}$ on a Banach space $\displaystyle X$ with
Fourier type $\displaystyle p\in(1,2]$, and suppose that $\displaystyle X$ has
Hardy–Littlewood type $\displaystyle p$ or Hardy–Littlewood cotype
$\displaystyle p^{\prime}$. Suppose that
$\displaystyle{\mathbb{C}}_{+}\subseteq\rho(A)$, and that there exist
$\displaystyle\beta>0$ and $\displaystyle C\geq 0$ such that

$\displaystyle\|R(\lambda,A)\|_{\mathcal{L}(X)}\leq
C(1+|\lambda|)^{\beta}\quad(\lambda\in{\mathbb{C}}_{+}).$

Let $\displaystyle\rho\geq 0$ and set
$\displaystyle\tau:=(\rho+1)\beta+\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$. Then
there exists a $\displaystyle C_{\rho}\geq 0$ such that

$\displaystyle\|T(t)x\|_{X}\leq C_{\rho}t^{-\rho}\|x\|_{D((-A)^{\tau})}$

for all $\displaystyle t\geq 1$ and $\displaystyle x\in D((-A)^{\tau})$.

Note that Theorem 4.2 is independent of Theorem 3.3 in the special case where
$\displaystyle\rho=0$. For $\displaystyle\rho>0$ the conclusion of Theorem 4.2
already follows from Theorem 3.3.

###### Proof.

The proof is analogous to that of [31, Theorem 4.9], and as such also similar
to the proof of Theorem 3.3. We will indicate what the key steps are.

By interpolation, cf. [31, Lemma 4.2], it suffices to consider
$\displaystyle\rho\in\mathbb{N}_{0}$. Then, using arguments as before but
relying on [31, Proposition 3.4] instead of Proposition 2.5, one can show that

(4.3)
$\|R(i\xi,A)^{k}\|_{\mathcal{L}(D((-A)^{\tau}),X)}\lesssim(1+|\xi|)^{-(\frac{1}{p}-\frac{1}{p^{\prime}})}$

for all $\displaystyle k\in\\{1,\ldots,\rho+1\\}$ and an implicit constant
independent of $\displaystyle\xi\in\mathbb{R}$.

Next, note that $\displaystyle D((-A)^{\tau})$ has the same Fourier type and
Hardy–Littlewood type and cotype as $\displaystyle X$, because $\displaystyle
X$ and $\displaystyle D((-A)^{\tau})$ are isomorphic. In particular, if
$\displaystyle X$ has Hardy–Littlewood type $\displaystyle p$, then one may
apply Proposition 4.1 with $\displaystyle r=p$, $\displaystyle q=p^{\prime}$,
$\displaystyle\gamma=\tfrac{1}{p^{\prime}}-\tfrac{1}{p}$ and
$\displaystyle\delta=0$. On the other hand, if $\displaystyle X$ has
Hardy–Littlewood cotype $\displaystyle p^{\prime}$, then one can apply
Proposition 4.1 with $\displaystyle q=r=p^{\prime}$, $\displaystyle\gamma=0$
and $\displaystyle\delta=\tfrac{1}{p}-\tfrac{1}{p^{\prime}}$. In both cases,
it follows from (4.3) that

$\displaystyle T_{R(i\cdot,A)^{k}}:L^{p}(\mathbb{R};D((-A)^{\tau}))\to
L^{p^{\prime}}(\mathbb{R};X)$

is bounded for all $\displaystyle k\in\\{1,\ldots,\rho+1\\}$. Now [31, Theorem
4.6] concludes the proof. ∎

## Acknowledgments

The second author is grateful to Yuri Tomilov for helpful conversations about
the topic of this article. The third author would like to thank Oscar
Dominguez for pointing out reference [12] and for helpful discussions.

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