# unique continuation problem on RCD Spaces. I

Qin Deng  and  Xinrui Zhao  MIT, Dept. of Math.
77 Massachusetts Avenue, Cambridge, MA 02139-4307.

###### Abstract.

In this note we establish the weak unique continuation theorem for caloric
functions on compact $RCD(K,2)$ spaces and show that there exists an
$RCD(K,4)$ space on which there exist non-trivial eigenfunctions of the
Laplacian and non-stationary solutions of the heat equation which vanish up to
infinite order at one point. We also establish frequency estimates for
eigenfunctions and caloric functions on the metric horn. In particular, this
gives a strong unique continuation type result on the metric horn for harmonic
functions with a high rate of decay at the horn tip, where it is known that
the standard strong unique continuation property fails.

##  0\. introduction

In this note we study the weak and strong unique continuation properties for
the heat equation on $\operatorname{RCD}(K,N)$ spaces. Consider the equation

(0.1) $\displaystyle\Delta u-\partial_{t}u=0.$

Let $T>0$. On an $\operatorname{RCD}(K,N)$ space
$(X,\mathop{}\\!\mathrm{d},\mathfrak{m})$, a solution to (0.1) on $[0,T]$ is
defined as follows.

######  Definition .

A function $u:X\to\mathbb{R}$ is a solution (resp. sub-solution, super-
solution) of (0.1) if $u\in W^{1,2}_{loc}(X\times[0,T])$ and that, for all
$\phi\in{\text{Lip}}_{0}(X\times[0,T])$ with $\phi\geq 0$, we have

(0.2) $\displaystyle\int_{X\times[0,T]}-\langle\nabla
u,\nabla\phi\rangle-\phi\,\partial_{t}u\,\,d\mathfrak{m}dt=0\,(\text{resp.
}\leq 0,\geq 0).$

This paper is concerned with the validity of the strong and weak unique
continuation property for the heat equation (0.1). To be precise, we say that

  * •

The heat equation satisfies the weak unique continuation property on an
$\operatorname{RCD}(K,N)$ space $(X,\mathop{}\\!\mathrm{d},\mathfrak{m})$ if,
for any solution $u\in W^{1,2}(X\times[0,T])$ of (0.1), if $u$ vanishes on any
non-empty open subset of $X\times[0,T]$, then $u\equiv 0$.

  * •

A function $u\in L^{2}(X\times[0,T])$ vanishes up to infinite order at some
$(x_{0},t_{0})\in X\times[0,T]$ if there exists some $R>0$ so that for any
integer $N>0$, there exists $C(N)>0$ so that

$\int_{B_{r}(x_{0})\times((t_{0}-r^{2},t_{0}+r^{2})\cap[0,T])}|u|^{2}\,\,d\mathfrak{m}dt\leq
Cr^{N}.$

  * •

The heat equation satisfies the strong unique continuation property on an
$\operatorname{RCD}(K,N)$ space $(X,\mathop{}\\!\mathrm{d},\mathfrak{m})$ if,
for any solution $u\in W^{1,2}(X\times[0,T])$ of (0.1), if $u$ vanishes up to
infinite order at any $(x_{0},t_{0})\in X\times(0,T]$, then $u\equiv 0$.

Note that we restrict $t_{0}$ to be away from $0$ in the definition of strong
unique continuation as it is always possible to solve the heat equation
starting from some initial data $u(\cdot,0)$, which vanishes up to infinite
order (spatially) at some $x_{0}\in X$. This would easily imply that the heat
flow also vanishes up to infinite order at $(x_{0},0)$ in the smooth case. We
mention that unique continuation type results are related to giving an upper
bound for the measure of the nodal set of non-trivial solutions and that,
recently, there has also been work to establish lower bounds in the nonsmooth
setting, see [CF21, DPF21].

Our first result in Section 1 gives the validity of the weak unique
continuation property for compact $\operatorname{RCD}(K,2)$ spaces:

######  Theorem .

Let $(X,\mathop{}\\!\mathrm{d},\mathfrak{m})$ be a compact
$\operatorname{RCD}(K,2)$ space. The heat equation on $X$ satisfies the weak
unique continuation property.

As in [DZ21], where the same theorem was shown for harmonic functions, the
main idea to handle the spatial direction is to leverage the
$C^{0}$-Riemannian structure of non-collapsed $\operatorname{RCD}(K,2)$ spaces
from [LS18, OS94]. To handle the time direction, we also show the time
analyticity of solutions of (0.1) following the recent [Z21].

In [DZ21], an $\operatorname{RCD}(K,4)$ space was constructed on which there
exists non-trivial harmonic functions which vanish up to infinite order at
some point. The heat flow of any such harmonic function would immediately give
a counterexample to strong unique continuation for the heat equation as well.
As such, we will be primarily interested in the strong unique continuation
property for non-stationary solutions of the heat equation in this paper. We
extend our result from [DZ21] as follows in Sections 3 and 5 respectively:

######  Theorem .

There exists an $\operatorname{RCD}(K,4)$ space and a non-trivial
eigenfunction on it with eigenvalue $\mu\neq 0$ which vanishes up to infinite
order at one point.

######  Theorem .

There exists an $\operatorname{RCD}(K,4)$ space and a non-stationary solution
of the heat equation on it which vanishes up to infinite order at one point.

Besides these results, we also give frequency estimate on metric horns in
Sections 2 and 4 and establish a strong unique continuation type result for
eigenfunctions and caloric functions on metric horns, where the classical
strong unique continuation property fails by the results of [DZ21].

We have to use the geometry setting of this equation to deal with the
difficulty of coefficients which are non-Lipschitz. For discussions on smooth
manifolds with Ricci curvarure lower bound and some related questions, see for
example [CM97a, CC97, CM97b, CC00a, CC00b, CM11, CN12, CM21b, CM22]. For
references on the theory of $\operatorname{RCD}$ spaces, see [S06a, S06b,
LV09, AGS14, G15, AGMR15, AGS15, EKS15, DPG18, MN19, BS20, CM20a, CM20b,
CM21a, KM21, BNS22].

## Acknowledgement

We are very grateful to Prof. Tobias Colding for his interest on this unique
continuation problem and his constant encouragements. Xinrui Zhao is supported
by NSF Grant DMS 1812142 and NSF Grant DMS 1811267.

##  1\. weak unique continuation of caloric functions on
$\operatorname{RCD}(K,2)$ spaces

In this section we will consider the case of $\operatorname{RCD}(K,2)$ spaces.
In the smooth setting, the Ricci curvature in $2$ dimensions completely
determines sectional curvature and hence a Ricci lower bound translates to a
sectional curvature lower bound. This is also true for any non-collapsed
$\operatorname{RCD}(K,2)$ space $(X,\mathop{}\\!\mathrm{d},\mathfrak{m})$.

######  Theorem .

([LS18]) If $\mathfrak{m}=c\mathcal{H}^{2}$ with $c\geq 0$ (i.e.
$(X,\mathop{}\\!\mathrm{d},\mathfrak{m})$ is non-collapsed), then
$(X,\mathop{}\\!\mathrm{d})$ is an Alexandrov space with curvature bounded
below by $K$.

Therefore, it suffices to consider $2$-dimensional Alexandrov spaces of
curvature at least $K$ and collapsed $\operatorname{RCD}(K,2)$ spaces.
Alexandrov spaces are known to have a generalized Riemannian structure by
[OS94]. We refer to [OS94] for relevant definitions.

######  Theorem .

([OS94]) Let $(X,\mathop{}\\!\mathrm{d})$ be an $n$-dimensional Alexandrov
space and denote $S_{X}$ as the set of singular points. Then there exists a
$C^{0}$-Riemannian structure on $X\setminus S_{X}\subset X$ satisfying the
following:

  * (1)

There exists an $X_{0}\subset X\setminus S_{X}$ such that $X\setminus X_{0}$
is of $n$-dimensional Hausdorff measure zero and that the Riemannian structure
is $C^{\frac{1}{2}}$-continuous on $X_{0}\subset X$;

  * (2)

The metric structure on $X\setminus S_{X}$ induced from the Riemannian
structure coincides with the original metric of $X$.

In a coordinate neighborhood $U$ given by the $C^{0}$-Riemannian structure of
$X$, with corresponding metric $g_{ij}$, a solution $u$ of (0.1) satisfies

(1.1)
$\displaystyle\int_{U\times[0,T]}-g^{ij}\,u_{i}\,\phi_{j}-\phi\,\partial_{t}u\,\,d\mathfrak{m}\,dt=0$

for all $\phi\in Lip_{0}(X\times[0,T])$ supported in $U\times[0,T]$ (Cf.
[KMS01]). Note that since $g^{ij}$ may not be Lipschitz, the results of [L90]
do not apply. We will instead use techniques which are special in the
2-dimensional case. For more details, see for example [CH62].

We proceed by taking local isothermal coordinates. To be precise, consider
functions $\sigma,\rho:U\to\mathbb{R}^{2}$ satisfying

(1.2)
$\displaystyle\sigma_{x}=\frac{g^{xy}\rho_{x}+g^{yy}\rho_{y}}{\sqrt{g^{xx}g^{yy}-(g^{xy})^{2}}}$
(1.3)
$\displaystyle-\sigma_{y}=\frac{g^{xx}\rho_{x}+g^{xy}\rho_{y}}{\sqrt{g^{xx}g^{yy}-(g^{xy})^{2}}},$

where derivatives are taken with respect to the coordinate chart for $U$. The
existence of such functions is given by [M38], see also [BN54, CH62]. This is
equivalent to solving the complex equation

(1.4) $\displaystyle w_{\bar{z}}=\mu w_{z},$

where
$\mu=\frac{g^{yy}-g^{xx}-2ig^{xy}}{g^{xx}+g^{yy}+2\sqrt{g^{xx}g^{yy}-(g^{xy})^{2}}}$.
Note that

(1.5)
$\displaystyle|w_{z}|^{2}=\frac{(\sigma_{x}\rho_{y}-\sigma_{y}\rho_{x})}{4\sqrt{g^{xx}g^{yy}-(g^{xy})^{2}}}(g^{xx}+g^{yy}+2\sqrt{g^{xx}g^{yy}-(g^{xy})^{2}})$

We have

(1.6) $\displaystyle u_{x}=u_{\rho}\rho_{x}+u_{\sigma}\sigma_{x},$ (1.7)
$\displaystyle u_{y}=u_{\rho}\rho_{y}+u_{\sigma}\sigma_{y},$

and so

$\displaystyle\int_{U\times[0,T]}$
$\displaystyle-(g^{xx}\,(u_{\rho}\rho_{x}+u_{\sigma}\sigma_{x})\,(\phi_{\rho}\rho_{x}+\phi_{\sigma}\sigma_{x}))-(g^{xy}\,(u_{\rho}\rho_{x}+u_{\sigma}\sigma_{x})\,(\phi_{\rho}\rho_{y}+\phi_{\sigma}\sigma_{y}))$
(1.8)
$\displaystyle-(g^{xy}\,(u_{\rho}\rho_{y}+u_{\sigma}\sigma_{y})\,(\phi_{\rho}\rho_{x}+\phi_{\sigma}\sigma_{x}))-(g^{yy}\,(u_{\rho}\rho_{y}+u_{\sigma}\sigma_{y})\,(\phi_{\rho}\rho_{y}+\phi_{\sigma}\sigma_{y}))-\phi\,\partial_{t}u\,\,d\mathfrak{m}\,dt=0.$

After rearranging the terms,

(1.9)
$\displaystyle\int_{U\times[0,T]}\sqrt{g^{xx}g^{yy}-(g^{xy})^{2}}(\sigma_{x}\rho_{y}-\sigma_{y}\rho_{x})(u_{\rho}\phi_{\rho}+u_{\sigma}\phi_{\sigma})-\phi\,\partial_{t}u\,\,d\mathfrak{m}\,dt=0.$

and so in the new coordinates, we have

(1.10)
$\displaystyle\int_{\mathbb{R}^{2}\times[0,T]}(u_{\rho}\phi_{\rho}+u_{\sigma}\phi_{\sigma})-\frac{\phi\,\partial_{t}u}{\sqrt{g^{xx}g^{yy}-(g^{xy})^{2}}(\sigma_{x}\rho_{y}-\sigma_{y}\rho_{x})}\,\,dx\,dy\,dt=0.$

Thus in the coordinate $(\rho,\sigma)$ the equation becomes

(1.11) $\displaystyle\Delta u-a\partial_{t}u=0,$

where $a$ is measurable and Hölder on a full measure subset. Note that as now
we have a possibly discontinuous coefficient in front of $\partial_{t}$, the
result in [L90] cannot be used. We will instead use a geometrical argument to
deal with this difficulty.

Now we are ready to state the main result of this section.

######  Theorem .

Let $u\in W^{1,2}(X\times[0,T])$ be a solution of (0.1) on an
$\operatorname{RCD}(K,2)$ space. If $u$ vanishes on a non-empty open set
$\Gamma\subset X\times(0,T]$, then $u\equiv 0$ on
$\text{proj}_{X}(\Gamma)\times[0,T]$. Moreover if $X$ is compact, then
$u\equiv 0$ on $X\times[0,T]$.

###### Proof.

By a similar discussion as in [DZ21], we can only consider the non-collapsed
case. We assume that u vanishes on a non-empty open set
$V\times(t_{1},t_{2})$.

Since $u$ satisfies the heat equation, we have the following a priori
estimates (cf.[G20, Remark 5.2.11]):

(1.12) $\displaystyle\|u(\cdot,t)\|_{L^{2}}\leq\|u(\cdot,0)\|_{L^{2}},$ (1.13)
$\displaystyle\||\nabla
u(\cdot,t)|\|_{L^{2}}\leq\frac{\|u(\cdot,0)\|_{L^{2}}}{\sqrt{t}},$ (1.14)
$\displaystyle\|\partial_{t}u(\cdot,t)\|_{L^{2}}=\|\Delta
u(\cdot,t)\|_{L^{2}}\leq\frac{\|u(\cdot,0)\|_{L^{2}}}{t},$

for any $t\in(0,T]$.

We first prove that $u$ vanishes on $V\times[0,T]$ by showing that
$u(x,\cdot)$ is analytic with respect to $t$. For the corresponding arguments
on Riemannian manifolds see for example [Z21]. For the reader’s convenience we
sketch the argument here.

As in [LZ19], for any $(x_{0},t_{0})$ with $0<t_{0}<T$, $k>0$ and $j\leq k$,
we consider

(1.15) $\displaystyle H_{j}^{1}$
$\displaystyle=B(x_{0},\frac{j\sqrt{t_{0}}}{\sqrt{2k}})\times[t_{0}-\frac{jt_{0}}{2k},t_{0}+\frac{jt_{0}}{2k}],$
(1.16) $\displaystyle H_{j}^{2}$
$\displaystyle=B(x_{0},\frac{(j+0.5)\sqrt{t_{0}}}{\sqrt{2k}})\times[t_{0}-\frac{(j+0.5)t_{0}}{2k},t_{0}+\frac{(j+0.5)t_{0}}{2k}].$

So that $u$ is defined on these sets, we extend $u$ to be a solution of (0.1)
on $X\times[0,\infty)$.

By [MN19], see also [CN12], we may choose a cutoff function $\phi_{1}(x,t)$
supported in $H^{2}_{j}$ such that $\phi_{1}=1$ on $H^{1}_{j}$, satisfying

(1.17)
$\displaystyle|\nabla\phi_{1}(x,t)|^{2}+|\partial_{t}\phi_{1}(x,t)|\leq\frac{Ck}{t_{0}}.$

Indeed, $\phi_{1}$ may be chosen to be a product of two cutoff functions on
the spatial and time coordinates respectively, so that the spatial and time
derivatives are well-defined without further measure theoretic arguments. It
follows that

$\displaystyle\int_{H^{2}_{j}}|\partial_{t}u|^{2}\phi_{1}^{2}\,d\mathfrak{m}\,dt$
$\displaystyle=-\int_{H_{j}^{2}}(\langle\nabla\partial_{t}u,\nabla
u\rangle\phi_{1}^{2}+\partial_{t}u\langle\nabla
u,\nabla\phi_{1}^{2}\rangle)\,d\mathfrak{m}\,dt$
$\displaystyle=-\int_{H_{j}^{2}}(\frac{1}{2}\partial_{t}|\nabla
u|^{2}\phi_{1}^{2}+\partial_{t}u\langle\nabla
u,\nabla\phi_{1}^{2}\rangle)\,d\mathfrak{m}\,dt$
$\displaystyle\leq\int_{H^{2}_{j}}(\frac{3}{4}|\partial_{t}u|^{2}\phi_{1}^{2}+\frac{Ck}{t_{0}}|\nabla
u|^{2})\,d\mathfrak{m}\,dt,$

where the last inequality follows from integration by parts, Young’s
inequality and (1.17). This gives that

(1.18)
$\displaystyle\int_{H_{j}^{1}}|\partial_{t}u|^{2}\,d\mathfrak{m}\,dt\leq\frac{Ck}{t_{0}}\int_{H_{j}^{2}}|\nabla
u|^{2}\,d\mathfrak{m}\,dt.$

Similarly, we may choose a cutoff function $\phi_{2}(x,t)$ supported in
$H^{1}_{j+1}$ such that $\phi_{2}=1$ on $H^{2}_{j}$, satisfying

(1.19)
$\displaystyle|\nabla\phi_{2}(x,t)|^{2}+|\partial_{t}\phi_{2}(x,t)|\leq\frac{Ck}{t_{0}}.$

Arguing as before, we have

$\displaystyle\int_{H_{j+1}^{1}}|\nabla u|^{2}\phi_{2}^{2}\,d\mathfrak{m}\,dt$
$\displaystyle=-\int_{H_{j+1}^{1}}u\,\partial_{t}u\,\phi_{2}^{2}+u\langle\nabla
u,\nabla\phi_{2}^{2}\rangle\,d\mathfrak{m}\,dt$
$\displaystyle=-\int_{H_{j+1}^{1}}\frac{1}{2}\,\partial_{t}u^{2}\,\phi_{2}^{2}+u\langle\nabla
u,\nabla\phi_{2}^{2}\rangle\,d\mathfrak{m}\,dt$ (1.20)
$\displaystyle\leq\int_{H_{j+1}^{1}}(\frac{1}{4}|\nabla
u|^{2}\phi^{2}_{2}+\frac{Ck}{t_{0}}u^{2})\,d\mathfrak{m}\,dt,$

and so

(1.21) $\displaystyle\int_{H_{j}^{2}}|\nabla
u|^{2}\,dx\,dt\leq\frac{Ck}{t_{0}}\int_{H_{j+1}^{1}}u^{2}\,dx\,dt.$

By using (1.18),(1.21) inductively on $j$, we obtain

(1.22)
$\displaystyle\int_{H_{1}^{1}}(\partial_{t}^{k}u)^{2}\,dx\,dt\leq\frac{(Ck)^{2k}}{t_{0}^{2k}}\int_{H^{1}_{k+1}}u^{2}\,dx\,dt\leq\frac{(Ck)^{2k}}{t_{0}^{2k-1}},$

where $C$ at the end also depends on the $L^{2}$-norm of $u_{0}$.

From [HK00, BB11, R12] we know that the $(p,p)$-Poincaré inequality holds on
$\operatorname{RCD}(K,N)$ space for any $p\geq 1$ (see, for example, [K15,
Section 4] for a careful discussion of Poincaré inequalties in the
$\operatorname{RCD}(K,N)$ setting). Thus Moser iteration ([HK00, Lemma 3.10])
shows that for $R<1$ there exist a mean value inequality

(1.23) $\displaystyle|\partial_{t}^{k}u(x_{0},t_{0})|^{2}\leq
C(\frac{R^{\nu_{2}}}{|B(x_{0},R)|})\frac{1}{R^{\nu_{2}(1+\frac{2}{\nu_{2}})}}\int_{B(x,R)\times(t-R^{2},t)}|\partial_{t}^{k}u|^{2}\,d\mathfrak{m}\,dt.$

Thus by taking $R=\frac{\sqrt{t_{0}}}{\sqrt{2k}}$ and using (1.22), we obtain

(1.24)
$\displaystyle|\partial_{t}^{k}u(x_{0},t_{0})|^{2}\leq\frac{C}{|B(x_{0},\frac{\sqrt{t_{0}}}{\sqrt{2k}})|(\frac{\sqrt{t_{0}}}{\sqrt{2k}})^{2}}\int_{H_{1}^{1}}|\partial_{t}^{k}u|^{2}\,d\mathfrak{m}\,dt\leq\frac{C}{|B(x_{0},\frac{\sqrt{t_{0}}}{\sqrt{2k}})|}\frac{(Ck)^{2k+1}}{t_{0}^{2k}},\,\,t_{0}<k.$

Using volume comparison, this implies that $u(x_{0},\cdot)$ is analytic with
respect to $t$. In particular, since $u$ vanishes on $V\times(t_{1},t_{2})$ by
assumption, $u$ also vanishes on $V\times[0,T]$.

We now prove the second assertion of the theorem by contradiction. Assume $X$
is compact and $u\not\equiv 0$.

Let $\phi_{k}$ be eigenfunctions of $-\Delta$ corresponding to eigenvalues
$\lambda_{k}$ with $\|\phi_{k}\|_{L^{2}}=1$ and
$0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq...\to\infty$, see [H18] for a
discussion on this. It follows from the estimates obtained in the appendix of
[AHTP18] that $u$ admits the representation

(1.25) $\displaystyle
u(x,t)=\sum_{k=0}^{\infty}a_{k}e^{-\lambda_{k}t}\phi_{k}(x).$

Let $j$ be the first index for which $a_{j}\neq 0$. As it does not affect the
argument, we assume for simplicity that the dimension of the eigenspace
corresponding to $\lambda_{j}$ is $1$. This gives that for any $x_{0}\in V$,

(1.26) $\displaystyle
0=\mathop{lim}\limits_{t\to\infty}e^{\lambda_{j}t}u(x_{0},t)=a_{j}\phi_{j}(x_{0}),$

where we have used [AHTP18, Proposition 7.1] to bound the values of
$\phi_{k}(x_{0})$ for $k>j$ to obtain the second equality. As we assumed that
$a_{j}\neq 0$, we conclude that $\phi_{j}(x_{0})=0$. This shows that
$\phi_{j}(x_{0})=0$ for all $k$ and $x_{0}\in V$. From [DZ21], this implies
that $\phi_{j}\equiv 0$, which is a contradiction. It follows that $u\equiv
0$. Note that similar arguments also work for Dirichlet problem on non-compact
space. ∎

######  Remark .

The previous argument actually shows the time analyticity of caloric functions
with respect to time on any $\operatorname{RCD}(K,N)$ space, since that part
of the argument does not require any assumptions on dimension.

######  Remark .

In [L90], Fourier transform was used to reduce the problem to a solution of an
elliptic equation on $X\times\mathbb{R}$. In this case, one does not have weak
unique continuation for elliptic equations on $\operatorname{RCD}(K,3)$
spaces, so a different argument had to be used.

Finally, we recall a well-known counterexample given by Miller [M74] which
indicates that in general we cannot expect weak unique continuation for
parabolic operators with time-dependent coefficients even if the time-slices
of the corresponding metric have a uniform Ricci curvature bound.

######  Proposition .

([M74]) There exists a smooth function
$u:\mathbb{R}^{2}\times[0,\infty)\to\mathbb{R}$ such that:

(1.27) $\displaystyle
u_{t}=((1+A(t)+a)u_{x})_{x}+(bu_{y})_{x}+(bu_{x})_{y}+((1+C(t)+c)u_{y})_{y},$

where $A=C=a=b=c=u=0$ on $t\geq T$, $a,b,c,u$ are smooth on
$\mathbb{R}^{2}\times[0,\infty)$, $A(t),C(t)$ are smooth on $(0,T)$ and Hölder
on $[0,\infty)$. Moreover, $u,a,b,c$ are period in x and y with period $2\pi$.

##  2\. Elliptic Frequency estimate on Metric horn

In this section we will give a frequency estimate on the metric horn, which
allows us to prove a form of unique continuation. Recall from [DZ21] that the
standard formulation of strong unique continuation does not hold at the horn
tip. The form of unique continuation we will prove in this section will
therefore assume a higher order of decay at the horn tip, see Remark 2 at the
end of the section for more details.

On a weighted warped product
$(X,dr^{2}+f^{2}(r)g_{S^{n-1}},e^{-\psi(r)}dvol)$, given function $\varphi$,
we have that

(2.1)
$\displaystyle\Delta\varphi=\partial_{r}^{2}\varphi+(n-1)\partial_{r}\varphi(log\,f)_{r}+f^{-2}\Delta_{S^{n-1}}\varphi-\partial_{r}\varphi\partial_{r}\psi,$

away from $r=0$. In the case of the standard metric horn,

(2.2) $\displaystyle f(r)$ $\displaystyle=\frac{1}{2}r^{1+\epsilon},$ (2.3)
$\displaystyle\psi(r)$ $\displaystyle=-(N-n)(1-\eta)log(r),$

and so,

(2.4)
$\displaystyle\Delta\varphi=\partial_{r}^{2}\varphi+\partial_{r}\varphi(\frac{(n-1)(1+\epsilon)+(N-n)(1-\eta)}{r})+\frac{4}{r^{2+2\epsilon}}\Delta_{S^{n-1}}\varphi.$

From the equation of Laplacian on metric horn, in particular, for
$\varphi=r^{\alpha}$, we have that

(2.5)
$\displaystyle\Delta\varphi=r^{\alpha-2}\alpha(\alpha+N-2+(n-1)\epsilon-(N-n)\eta)$

and

(2.6)
$\displaystyle{\text{Hess}}\,\varphi=\alpha(\alpha-1)r^{\alpha-2}dr\otimes
dr+(\frac{r^{1+\epsilon}}{2})^{2}\alpha(1+\epsilon)r^{\alpha-2}g_{S^{n-1}}.$

If we take $\alpha=2$, then

(2.7)
$\displaystyle{\text{Hess}}\,\varphi=2(dr^{2}+(1+\epsilon)(\frac{r^{1+\epsilon}}{2})^{2}g_{S^{n-1}}).$

Given an eigenfunction u with $\Delta u=\lambda u$, define scale-invariant
quantities $I(r),E(r)$ and the frequency function $U(r)$ with respect to the
level sets of the function $d_{p}$, where $p$ is the horn tip. For $r>0$, we
denote

(2.8) $\displaystyle B_{r}=\\{x\,|\,d_{p}(x)<r\\},$ (2.9)
$\displaystyle\partial B_{r}=\\{x\,|\,d_{p}(x)=r\\},$

and define

(2.10) $\displaystyle I(r)=r^{1-n}\int_{\partial
B_{r}}u^{2}d\mathfrak{m}_{r},$ (2.11) $\displaystyle
E(r)=r^{2-n}\int_{B_{r}}|\nabla u|^{2}+\lambda
u^{2}d\mathfrak{m}=r^{2-n}\int_{\partial B_{r}}\frac{u}{|\nabla
d_{p}|}\,\langle\nabla u,\nabla d_{p}\rangle d\mathfrak{m}_{r},$ (2.12)
$\displaystyle U(r)=\frac{E(r)}{I(r)},$

where $\mathfrak{m}=e^{\psi}(r)dvol$ is the weighted volume measure of the
metric horn and $\mathfrak{m}_{r}=e^{\psi}(r)dvol_{r}$ is the corresponding
weighted area measure on $\partial B_{r}$.

We first compute the derivative of $I$. We remark that all computations are
done away from the cone tip $p$, so no regularity issues will arise from $p$
itself. Let $\phi$ be any smooth function compactly supported on $(0,\infty)$,
we have

$\displaystyle-\int I(t)\phi^{\prime}(t)dt=-\int t^{1-n}\int_{\partial
B_{t}}u^{2}\phi^{\prime}(t)dt=-\int u^{2}d^{1-n}_{p}\phi^{\prime}(d_{p})$
$\displaystyle=-\int u^{2}d^{1-n}_{p}\langle\nabla
d_{p},\nabla\phi(d_{p})\rangle=\int{\text{div}}(u^{2}d^{1-n}_{p}\nabla
d_{p})\phi(d_{p})$ $\displaystyle=\int\frac{1}{2-n}(2u\langle\nabla u,\nabla
d_{p}^{2-n}\rangle+u^{2}\Delta d_{p}^{2-n})\phi(d_{p})$
$\displaystyle=\int_{t}\int_{\partial B_{t}}\frac{1}{(2-n)|\nabla
d_{p}|}(2u\langle\nabla u,\nabla d^{2-n}_{p}\rangle+u^{2}\Delta
d^{2-n}_{p})\phi(d_{p})dt.$

This shows that

(2.13) $\displaystyle I^{\prime}(r)$ $\displaystyle=\int_{\partial
B_{r}}\frac{1}{(2-n)|\nabla d_{p}|}(2u\langle\nabla u,\nabla
d^{2-n}_{p}\rangle+u^{2}\Delta d^{2-n}_{p})$
$\displaystyle=\frac{2E(r)}{r}+\int_{\partial B_{r}}\frac{1}{(2-n)|\nabla
d_{p}|}u^{2}\Delta d^{2-n}_{p},\quad a.e.\,\,t\in(0,1),$

and so by using (2.5),

(2.14) $\displaystyle(\log
I)^{\prime}(r)-\frac{2U(r)}{r}=\frac{N-n+(n-1)\epsilon-(N-n)\eta}{r}.$

For $E$, from the definition (2.11) and the coarea formula, we have that

(2.15) $\displaystyle E^{\prime}(r)=(2-n)\frac{E(r)}{r}+r^{2-n}\int_{\partial
B_{r}}({|\nabla u|^{2}}+\lambda u^{2}).$

Now consider the vector field $X=\frac{\langle\nabla(d_{p}^{2}),\nabla
u\rangle\nabla u-\frac{1}{2}|\nabla u|^{2}\nabla(d_{p}^{2})}{|\nabla
d_{p}|^{3}}$, we have that

(2.16) $\displaystyle{\text{div}}(X)={{\text{Hess}}(d_{p}^{2})(\nabla
u,\,\nabla u)-\frac{1}{2}|\nabla u|^{2}\Delta d_{p}^{2}}+\lambda
u\langle\nabla(d_{p}^{2}),\,\nabla u\rangle$

By Integrating both sides in $B_{r}$ we have that

(2.17) $\displaystyle\int_{\partial B_{r}}\frac{2r}{|\nabla
d_{p}|^{3}}|\langle\nabla d_{p},\,\nabla u\rangle|^{2}-r\frac{|\nabla
u|^{2}}{|\nabla d_{p}|}$ $\displaystyle=\int_{\partial B_{r}}\langle
X,\,\nabla d_{p}\rangle$
$\displaystyle=\int_{B_{r}}{{\text{Hess}}(d_{p}^{2})(\nabla u,\,\nabla
u)-\frac{1}{2}|\nabla u|^{2}\Delta d_{p}^{2}}+\lambda
u\langle\nabla(d_{p}^{2}),\,\nabla u\rangle.$

Combining (2.15) and (2.17) we have that

(2.18) $\displaystyle E^{\prime}(r)=$
$\displaystyle(2-n)\frac{E(r)}{r}+r^{2-n}\int_{\partial B_{r}}\frac{2}{|\nabla
d_{p}|^{3}}|\langle\nabla d_{p},\,\nabla u\rangle|^{2}+\lambda u^{2}$
$\displaystyle-r^{1-n}\int_{B_{r}}({{\text{Hess}}(d_{p}^{2})(\nabla u,\,\nabla
u)-\frac{1}{2}|\nabla u|^{2}\Delta d_{p}^{2}}+\lambda
u\langle\nabla(d_{p}^{2}),\,\nabla u\rangle).$

Now consider the frequency function $U(r)$. We have that

(2.19) $\displaystyle\frac{d}{dr}log\,U(r)=$
$\displaystyle\frac{E^{\prime}(r)}{E(r)}-\frac{I^{\prime}(r)}{I(r)}$
$\displaystyle=$
$\displaystyle\frac{1}{E(r)}(-\frac{2E^{2}(r)}{rI(r)}+r^{2-n}\int_{\partial
B_{r}}|2\langle\nabla d_{p},\,\nabla u\rangle|^{2}+\lambda
u^{2})-\frac{1}{I(r)}\int_{\partial B_{r}}\frac{1}{2-n}u^{2}\Delta
d^{2-n}_{p}+\frac{2-n}{r}$
$\displaystyle-\frac{1}{E(r)}(r^{1-n}\int_{B_{r}}{\text{Hess}}(d_{p}^{2})(\nabla
u,\,\nabla u)-\frac{1}{2}|\nabla u|^{2}\Delta d_{p}^{2}+\lambda
u\langle\nabla(d_{p}^{2}),\,\nabla u\rangle).$

From Cauchy-Schwarz inequality we know that

(2.20) $\displaystyle-\frac{2E^{2}(r)}{rI(r)}+2r^{2-n}\int_{\partial
B_{r}}|\langle\nabla d_{p},\,\nabla u\rangle|^{2}\geq 0.$

Combining (2.19) and (2.20) we have that

(2.21) $\displaystyle\frac{d}{dr}log\,U(r)\geq$ $\displaystyle\frac{\lambda
r}{U(r)}-\frac{1}{I(r)}\int_{\partial B_{r}}\frac{1}{2-n}u^{2}\Delta
d^{2-n}_{p}+\frac{2-n}{r}$
$\displaystyle-\frac{1}{E(r)}(r^{1-n}\int_{B_{r}}{\text{Hess}}(d_{p}^{2})(\nabla
u,\,\nabla u)-\frac{1}{2}|\nabla u|^{2}\Delta
d_{p}^{2}-\frac{\lambda}{2}u^{2}\Delta d_{p}^{2}).$

As we have that

(2.22) $\displaystyle-\frac{1}{2-n}\int_{\partial B_{r}}u^{2}\Delta
d^{2-n}_{p}=-\frac{1}{2}\int_{\partial
B_{r}}u^{2}{\text{div}}(d^{-n}_{p}\nabla
d_{p}^{2})=-\frac{r^{-n}}{2}\int_{\partial B_{r}}u^{2}\Delta
d_{p}^{2}+nr^{-n}\int_{\partial B_{r}}u^{2},$

it follows that

(2.23) $\displaystyle\begin{split}\frac{d}{dr}log\,U(r)&\geq\frac{\lambda
r}{U(r)}+\frac{2}{r}-\frac{1}{E(r)}(r^{1-n}\int_{B_{r}}{\text{Hess}}(d_{p}^{2})(\nabla
u,\,\nabla u))\\\ &\geq\frac{\lambda
r}{U(r)}-\frac{r^{1-n}\int_{B_{r}}2\epsilon r^{2+2\epsilon}g_{S^{n-1}}(\nabla
u,\nabla u)}{r^{2-n}\int_{B_{r}}|\nabla u|^{2}}\\\ &\geq\frac{\lambda
r}{U(r)}-\frac{2\epsilon}{r}.\end{split}$

This shows that

(2.24) $\displaystyle(r^{2\epsilon}U(r))^{\prime}\geq\lambda r^{1+2\epsilon},$

and so

(2.25) $\displaystyle
r^{2\epsilon}U(r)-U(1)\leq\frac{\lambda}{2+2\epsilon}(r^{2+2\epsilon}-1),$

which is equivalent to

(2.26) $\displaystyle U(r)\leq
r^{-2\epsilon}U(1)+\frac{\lambda}{2+2\epsilon}(r^{2}-r^{-2\epsilon})\leq
Cr^{-2\epsilon}.$

Combining with (2.14) we obtain

(2.27) $\displaystyle(\log I)^{\prime}(r)\leq Cr^{-1-2\epsilon}.$

This gives

(2.28) $\displaystyle\log I(r)\geq\log I(1)+C(1-r^{-2\epsilon}),$

and so in all

(2.29) $\displaystyle I(r)\geq Ce^{-Cr^{-2\epsilon}}.$

######  Remark .

From [DZ21], we know that strong unique continuation fails on metric horn. In
particular, any harmonic function with $u(0)=0$ satisfies $u(x)=O(e^{-C(\log
r)^{2}})$. However the discussion above shows that if an eigenfunction, in
particular a harmonic function, vanishes at the tip up to order
$u(x)=O(e^{-Cr^{-2\epsilon}})$, then $u\equiv 0$.

##  3\. Failure of strong unique continuation of eigenfuctions on metric horn

In this section we will prove the following theorem:

######  Theorem .

There exists an $\operatorname{RCD}(K,4)$ space and a non-trivial
eigenfunction on it with eigenvalue $\mu\neq 0$, which vanishes up to infinite
order at one point.

###### Proof.

Consider any modified metric horn $X$ constructed in [DZ21, Section 6], which
uses techniques in [WZ21]. Let us denote the eigenfunctions on $S^{n-1}$ as
$\\{\varphi_{i}\\}_{i=1}^{\infty}$ such that

(3.1)
$\displaystyle\Delta_{S^{n-1}}\varphi_{i}=-\mu_{i}\varphi_{i},\,\,\mu_{i}>0$
(3.2) $\displaystyle\int_{S^{n-1}}\varphi_{i}^{2}dS=1.$

Assume $\varphi$ is an $L^{2}$ function which is smooth away from the tip,
then $\varphi$ may be decomposed as

(3.3)
$\varphi(r,\theta)=f_{0}(r)+\sum_{i=1}^{\infty}f_{i}(r)\varphi_{i}(\theta).$

Therefore, for any eigenfunction on metric horn with eigenvalue $-\mu$ we
have, for the decomposition (3.3) and each $i$,

(3.4) $\displaystyle
f_{i}^{\prime\prime}(r)+f_{i}^{\prime}(r)(\frac{(n-1)(1+\epsilon)+(N-n)(1-\eta)}{r})+\frac{4}{r^{2+2\epsilon}}f_{i}(r)(-\mu_{i})+\mu
f_{i}(r)=0,$

for $r$ sufficiently close to $0$. We note that we do not have this formula
for arbitrary $r$ since the modified metric horn was constructed using a
gluing procedure.

For the radial part, that is for $\mu_{0}=0$,

(3.5) $\displaystyle
f_{0}^{\prime\prime}(r)+f_{0}^{\prime}(r)(\frac{(n-1)(1+\epsilon)+(N-n)(1-\eta)}{r})+\mu
f_{0}(r)=0,$

Denoting $c=(n-1)(1+\epsilon)+(N-n)(1-\eta)$, we have

(3.6) $\displaystyle
f_{0}(r)=k_{1}r^{\frac{1-c}{2}}J_{\frac{c-1}{2}}(r\sqrt{\mu})+k_{2}r^{\frac{1-c}{2}}Y_{\frac{c-1}{2}}(r\sqrt{\mu}),$

where $J_{\nu},\,Y_{\nu}$ are Bessel functions. For a discussion of the
properties of Bessel functions, see [M10, Appendix B]. We will use the fact
that $J_{\frac{c-1}{2}}(r)\approx r^{\frac{c-1}{2}}$ and
$Y_{\frac{c-1}{2}}(r)\approx r^{\frac{1-c}{2}}$ as $r\to 0$. Since $\varphi$
is assumed to be in $L^{2}$, we have that $f_{0}$ is in a weighted $L^{2}$
space where the weight for small $r$ is of the form $r^{c}$. Taking this into
consideration with the asymptotics of $J$ and $Y$, we conclude that

(3.7) $\displaystyle
f_{0}(r)=Cr^{\frac{1-c}{2}}J_{\frac{c-1}{2}}(r\sqrt{\mu})\approx
C\mu^{\frac{c-1}{4}}.$

For $\mu_{i}=i(n+i-2)$ and denoting $c=(n-1)(1+\epsilon)+(N-n)(1-\eta)$, we
obtain

(3.8) $\displaystyle
f_{i}^{\prime\prime}(r)+f_{i}^{\prime}(r)(\frac{c}{r})+\frac{4}{r^{2+2\epsilon}}f_{i}(r)(-\mu_{i})+\mu
f_{i}(r)=0.$

Define

(3.9) $\displaystyle f_{i}(r)=g_{i}(r^{-\epsilon}).$

Then

$\displaystyle f_{i}^{\prime}(r)$ $\displaystyle=-\epsilon
r^{-\epsilon-1}g_{i}^{\prime}(r^{-\epsilon})，$ $\displaystyle
f_{i}^{\prime\prime}(r)$
$\displaystyle=\epsilon(\epsilon+1)r^{-\epsilon-2}g_{i}^{\prime}(r^{-\epsilon})+\epsilon^{2}r^{-2\epsilon-2}g_{i}^{\prime\prime}(r^{-\epsilon}).$

Thus

(3.10) $\displaystyle
g_{i}^{\prime\prime}(r)+\frac{\epsilon+1-c}{\epsilon}r^{-1}g_{i}^{\prime}(r)+\frac{-4\mu_{i}}{\epsilon^{2}}g_{i}(r)+\frac{\mu}{\epsilon^{2}}r^{-\frac{2}{\epsilon}-2}g_{i}(r)=0.$

If we further define

(3.11) $\displaystyle g_{i}(r)=k_{i}(r)r^{\frac{c-1-\epsilon}{2\epsilon}},$

then

$\displaystyle g_{i}^{\prime}(r)$
$\displaystyle=k_{i}^{\prime}(r)r^{\frac{c-1-\epsilon}{2\epsilon}}+k_{i}(r)\frac{c-1-\epsilon}{2\epsilon}r^{\frac{c-1-\epsilon}{2\epsilon}-1},$
$\displaystyle g_{i}^{\prime\prime}(r)$
$\displaystyle=k_{i}^{\prime\prime}(r)r^{\frac{c-1-\epsilon}{2\epsilon}}+2k_{i}^{\prime}(r)\frac{c-1-\epsilon}{2\epsilon}r^{\frac{c-1-\epsilon}{2\epsilon}-1}+k_{i}(r)\frac{c-1-\epsilon}{2\epsilon}(\frac{c-1-\epsilon}{2\epsilon}-1)r^{\frac{c-1-\epsilon}{2\epsilon}-2}$

and so

(3.12) $\displaystyle
k_{i}^{\prime\prime}(r)=(\frac{c-1-\epsilon}{2\epsilon}(\frac{c-1-\epsilon}{2\epsilon}+1)r^{-2}+\frac{4\mu_{i}}{\epsilon^{2}}-\frac{\mu}{\epsilon^{2}}r^{-\frac{2}{\epsilon}-2})k_{i}(r).$

If we consider the solution with data
$k_{i}^{(1)}(r_{\mu})=1=(k_{i}^{(1)})^{\prime}(r_{\mu})$, where $r_{\mu}$ is
chosen such that
$1\geq\frac{c-1-\epsilon}{2\epsilon}(\frac{c-1-\epsilon}{2\epsilon}+1)r^{-2}-\frac{\mu}{\epsilon^{2}}r^{-\frac{2}{\epsilon}-2}\geq
0$ for $r\geq r_{\mu}$, that is

$r_{\mu}=\max\\{\sqrt{\frac{c-1-\epsilon}{2\epsilon}(\frac{c-1-\epsilon}{2\epsilon}+1)},(\frac{c-1-\epsilon}{2\mu}(\frac{c-1-\epsilon}{2}+\epsilon))^{\frac{-\epsilon}{2}}\\}.$

So for $r>r_{\mu}$ from
$(\frac{4\mu_{i}}{\epsilon^{2}}+1)k^{(1)}_{i}(r)\geq(k^{(1)}_{i})^{\prime\prime}(r)\geq\frac{4\mu_{i}}{\epsilon^{2}}k^{(1)}_{i}(r)$,
we have

(3.13) $\displaystyle
e^{(\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}+1)(r-r_{\mu})}\geq
k_{i}^{(1)}(r)\geq\frac{1}{\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}}e^{\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}(r-r_{\mu})}.$

Then we have that

(3.14) $\displaystyle
k_{i}^{(2)}(r):=k_{i}^{(1)}(r)\int_{r}^{\infty}\frac{1}{(k_{i}^{(1)}(s))^{2}}ds.$

is also a solution of (3.12). So we get

(3.15)
$\displaystyle\frac{1}{2({\frac{4\mu_{i}}{\epsilon^{2}}}+\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}})}e^{(-\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}-2)(r-r_{\mu})}\leq
k_{i}^{(2)}(r)\leq\frac{\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}}{2}e^{(-\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}+1)(r-r_{\mu})},\,\,\text{for}\,r>r_{\mu}.$

And also

(3.16) $\displaystyle
0<C(\epsilon,i)^{-1}\leq(k_{2}^{(2)})^{2}(r_{\mu})+((k_{2}^{(2)})^{\prime})^{2}(r_{\mu})\leq
C(\epsilon,i).$

From ODE theory we have that $k_{i}^{(1)},\,k_{i}^{(2)}$ form a basis of the
solutions of (3.12). This tells us that

(3.17) $\displaystyle
f_{i}(r)=(ak_{i}^{(1)}+bk_{i}^{(2)})(r^{-\epsilon})r^{-\frac{c-1-\epsilon}{2}}.$

As before, we have that $f_{i}$ is in a weighted $L^{2}$ space from
assumption, this tells us that

(3.18) $\displaystyle
f_{i}(r)=bk_{i}^{(2)}(r^{-\epsilon})r^{-\frac{c-1-\epsilon}{2}}.$

From (3.15) we have the decay rate $f_{i}(r)=O(e^{-Cr^{-\epsilon}})$. This
tells us that there exists eigenfunctions on $\operatorname{RCD}(K,4)$ spaces
which are not zero but vanish up to infinite order at one point.∎

######  Remark .

In fact this argument tells us that any eigenfunction on this metric horn with
zero integral on each link $\\{r=r_{0}\\}$ vanishes up to infinite order.

######  Remark .

Since

$\displaystyle\int_{r=0}^{r_{\mu}^{-\frac{1}{\epsilon}}}e^{2(-\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}-2)r^{-\epsilon}}r^{-(c-1-\epsilon)}2^{1-n}r^{c}\,dr$
$\displaystyle=2^{1-n}\frac{(2(\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}+2))^{\frac{2(1+\epsilon)}{\epsilon}-1}}{\epsilon}\int_{2(\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}+2)r_{\mu}}^{\infty}e^{-r}r^{-\frac{2(1+\epsilon)}{\epsilon}}dr$
(3.19) $\displaystyle\geq
c(n,\epsilon,\mu_{i})e^{2(-\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}-2)r_{\mu}}r_{\mu}^{-\frac{2(1+\epsilon)}{\epsilon}},$

the normalized coefficient $c_{\mu,i}$ for the eigenfunction
$f_{i}(r)\varphi_{i}(\theta)$ with $L^{2}$-norm 1 corresponding to eigenvalue
$\mu$ satisfies

(3.20) $\displaystyle c_{\mu,i}\leq
c(n,\epsilon,\mu_{i})e^{(\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}+2)r_{\mu}}r_{\mu}^{\frac{1+\epsilon}{\epsilon}}.$

######  Remark .

Let us use a more careful argument on $\varphi$. That is, as we know that

(3.21)
$\displaystyle|((k_{i}^{(2)})^{2}+((k_{i}^{(2)})^{\prime})^{2})|=|2(k_{i}^{(2)})^{\prime}(k_{i}^{(2)}+(k_{i}^{(2)})^{\prime\prime})|\leq
2|k_{i}^{(2)}(k_{i}^{(2)})^{\prime}|C(\mu+1)\leq
C(\mu+1)|(k_{i}^{(2)})^{2}+((k_{i}^{(2)})^{\prime})^{2}|.$

Thus if we denote the pasting radius as $\tilde{r}$, by the discussion we have

(3.22) $\displaystyle
C^{-1}e^{-C(\mu+1)(r_{\mu}-\tilde{r}^{-\frac{1}{\epsilon}})}\leq|((k_{i}^{(2)})^{2}+((k_{i}^{(2)})^{\prime})^{2})^{\prime}|(\tilde{r}^{-\frac{1}{\epsilon}})\leq
Ce^{C(\mu+1)(r_{\mu}-\tilde{r}^{-\frac{1}{\epsilon}})}.$

As we know that when transform back

(3.23) $\displaystyle
C^{-1}e^{-C(\mu+1)(r_{\mu}-\tilde{r}^{-\frac{1}{\epsilon}})}\leq|f_{i}^{2}+(f_{i}^{\prime})^{2}|(\tilde{r})\leq
Ce^{C(\mu+1)(r_{\mu}-\tilde{r}^{-\frac{1}{\epsilon}})}.$

As outside $\tilde{r}$ the space is a cone, the function is of the form
$f(r)=k_{1}r^{(1-b)/2}J_{1/2\sqrt{(b-1)^{2}+4a}}(\sqrt{c}r)+k_{2}r^{(1-b)/2}Y_{1/2\sqrt{(b-1)^{2}+4a}}(\sqrt{c}r)$.
From the asymptotics of Bessel functions we can see that if we consider the
eigenfunction on $B_{R}$, the normalized coefficient satisfies

(3.24) $\displaystyle c_{\mu,i,R}\leq
Ce^{C(\mu+1)(\mu^{\frac{\epsilon}{2}}-\tilde{r}^{-\frac{1}{\epsilon}})}R^{-\frac{n-1}{2}}.$

##  4\. parabolic frequency estimate on metric horn

In this section we give a parabolic frequency estimate and the corresponding
unique continuation type result on the metric horn. For simplicity, we
consider the metric horn which is not modified. For previous discussions on
parabolic frequency see for example [P96, CM22].

Consider the metric horn
$(\mathbb{R}^{n},\,\,g_{\epsilon,n},\,\,e^{(N-n)(1-\eta)log(r)}dvol)$ with
$g_{\epsilon,n}=dr^{2}+(\frac{1}{2}r^{1+\epsilon})^{2}g_{S^{n-1}}$. Note that
we will use $X$ to denote the metric horn in the following. Let $p(x,y,t)$ be
the heat kernel. Following [P96], for $x_{0}\in X$ and $t_{0}\geq 0$, we
define the backward heat kernel
$G_{x_{0},t_{0}}:X\times(-\infty,t_{0})\to\mathbb{R}$ as
$G_{x_{0},t_{0}}(x,t):=p(x,x_{0},t_{0}-t)$. For simplicity of notation, we
denote $G(x,t)=G_{o,0}(x,t)$. Let $u$ be a solution of (0.1) on
$X\times[-R_{0}^{2},0]$ for some $R_{0}>0$.

For $R_{0}\geq R>0$, define

(4.1) $\displaystyle I(R)$ $\displaystyle=R^{2}\int_{t=-R^{2}}|\nabla
u|^{2}G\,d\mathfrak{m},$ (4.2) $\displaystyle D(R)$
$\displaystyle=\int_{t=-R^{2}}u^{2}G\,d\mathfrak{m},$ (4.3) $\displaystyle
N(R)$ $\displaystyle=\frac{I(R)}{D(R)}.$

From the definition of $G$, we have that

(4.4) $\displaystyle\frac{\partial\log G}{\partial t}=-\Delta\log
G-|\nabla\log G|^{2},$

and so

(4.5) $\displaystyle\log
G(x,t)=-\frac{(N+(n-1)\epsilon-(N-n)\eta)}{2}\log(-t)+\frac{|x|^{2}}{4t}.$

Therefore,

(4.6) $\displaystyle I^{\prime}(R)=$ $\displaystyle 2R\int_{t=-R^{2}}|\nabla
u|^{2}Gd\mathfrak{m}-2R^{3}\int_{t=-R^{2}}(2\langle\nabla u,\,\nabla
u_{t}\rangle G+|\nabla u|^{2}\partial_{t}G)d\mathfrak{m}$ $\displaystyle=$
$\displaystyle 2R\int_{t=-R^{2}}|\nabla
u|^{2}Gd\mathfrak{m}+4R^{3}\int_{t=-R^{2}}u_{t}^{2}G+u_{t}\langle\nabla
u,\,\nabla G\rangle d\mathfrak{m}$ $\displaystyle-2R^{3}\int_{t=-R^{2}}|\nabla
u|^{2}\partial_{t}Gd\mathfrak{m}.$

Consider the vector field $X=\langle\nabla G,\,\nabla u\rangle\nabla u$, then

(4.7) $\displaystyle div(X)={\text{Hess}}_{G}(\nabla u,\,\nabla
u)+\frac{1}{2}\langle\nabla G,\,\nabla|\nabla u|^{2}\rangle+\langle\nabla
G,\,\nabla u\rangle\Delta u.$

On the metric horn,

(4.8)
$\displaystyle\frac{{\text{Hess}}_{G}}{G}(x,t)=(\frac{|x|^{2}}{4t^{2}}+\frac{1}{2t})dr\otimes
dr+\frac{(\frac{|x|^{1+\epsilon}}{2})^{2}}{2t}(1+\epsilon)g_{S^{n-1}},$

and so

(4.9) $\displaystyle\int{\text{Hess}}_{G}(\nabla u,\,\nabla u)+\langle\nabla
G,\,\nabla u\rangle u_{t}d\mathfrak{m}=\int\frac{1}{2}\Delta G|\nabla
u|^{2}d\mathfrak{m}.$

So combining with (4.6) we have that

$\displaystyle I^{\prime}(R)$
$\displaystyle=4R^{3}\int_{t=-R^{2}}(\frac{x}{2t}\cdot\nabla
u+u_{t})^{2}G(x,t)d\mathfrak{m}+4R^{3}\int_{t=-R^{2}}\frac{(\frac{|x|^{1+\epsilon}}{2})^{2}}{2t}\epsilon
g_{S^{n-1}}(\nabla u,\,\nabla u)G(x,t)d\mathfrak{m}$ (4.10) $\displaystyle\geq
4R^{3}\int_{t=-R^{2}}(\frac{x}{2t}\cdot\nabla
u+u_{t})^{2}G(x,t)d\mathfrak{m}-\frac{2\epsilon}{R}I(R).$

Moreover,

(4.11) $\displaystyle
D^{\prime}(R)=-2R\int_{t=-R^{2}}(2uu_{t}G+u^{2}\partial_{t}G)d\mathfrak{m}=-4R\int_{t=-R^{2}}u(u_{t}+\frac{x}{2t}\cdot\nabla
u)Gd\mathfrak{m},$

so from integration by parts we have that

(4.12) $\displaystyle I(R)=-R^{2}\int_{t=-R^{2}}u(\partial_{t}u+\nabla
u\cdot\frac{x}{2t})Gd\mathfrak{m}=\frac{R}{4}D^{\prime}(R).$

From Cauchy-Schwarz inequality we obtain

(4.13) $\displaystyle(\int_{t=-R^{2}}(\frac{x}{2t}\cdot\nabla
u+u_{t})^{2}G(x,t)d\mathfrak{m})(\int_{t=-R^{2}}u^{2}G(x,t)d\mathfrak{m})\geq(\int_{t=-R^{2}}u(\frac{x}{2t}\cdot\nabla
u+u_{t})G(x,t)d\mathfrak{m})^{2},$

So we have that

(4.14) $\displaystyle(\log
N)^{\prime}(R)=\frac{I^{\prime}}{I}(R)-\frac{D^{\prime}}{D}(R)\geq-\frac{2\epsilon}{R},$

and so

(4.15) $\displaystyle\log N(R)\leq\log N(1)-2\epsilon\log R,$

which is equivalent to that

(4.16) $\displaystyle N(R)\leq CR^{-2\epsilon}.$

Since

(4.17) $\displaystyle(\log
D)^{\prime}(R)=\frac{4I(R)}{RD(R)}=\frac{4}{R}N(R)\leq CR^{-1-2\epsilon},$

we obtain

(4.18) $\displaystyle\log D(R)\geq\log D(1)+C(1-R^{-2\epsilon}).$

So finally

(4.19) $\displaystyle D(R)\geq Ce^{-CR^{-2\epsilon}}.$

This gives a strong unique continuation type result for solution of heat
equation on the metric horn.

##  5\. Failure of Strong unique continuation property of caloric on metric
horn

In this section we will prove the following theorem:

######  Theorem .

There exists an $\operatorname{RCD}(K,4)$ space and a non-stationary solution
of heat equation on it which vanishes up to infinite order at one point.

Similar as in the elliptic case, the heat equation becomes like

(5.1) $\displaystyle
0=\Delta\varphi-\partial_{t}\varphi=\partial_{r}^{2}\varphi+\partial_{r}\varphi(\frac{(n-1)(1+\epsilon)+(N-n)(1-\eta)}{r})+\frac{4}{r^{2+2\epsilon}}\Delta_{S^{n-1}}\varphi-\partial_{t}\varphi.$

Denote the eigenfunctions on $S^{n-1}$ as $\\{\varphi_{i}\\}_{i=1}^{\infty}$
such that

(5.2)
$\displaystyle\Delta_{S^{n-1}}\varphi_{i}=-\mu_{i}\varphi_{i},\,\,\mu_{i}>0$
(5.3) $\displaystyle\int_{S^{n-1}}\varphi_{i}^{2}dS=1.$

We can decompose it as
$\varphi(r,\theta,t)=f_{0}(r,t)+\sum_{i=1}^{\infty}f_{i}(r,t)\varphi_{i}(\theta)$.
So we have

(5.4)
$\displaystyle\partial_{r}^{2}f_{i}(r,t)+\partial_{r}f_{i}(r,t)(\frac{(n-1)(1+\epsilon)+(N-n)(1-\eta)}{r})+\frac{4}{r^{2+2\epsilon}}f_{i}(r,t)(-\mu_{i})-\partial_{t}f_{i}(r,t)=0$

For $\mu_{i}=i(n+i-2)$ and denote $c=(n-1)(1+\epsilon)+(N-n)(1-\eta)$

(5.5)
$\displaystyle\partial_{r}^{2}f_{i}(r,t)+\partial_{r}f_{i}(r,t)(\frac{c}{r})+\frac{4}{r^{2+2\epsilon}}f_{i}(r,t)(-\mu_{i})-\partial_{t}f_{i}(r,t)=0.$

We first consider the solution of heat equation on the compact modified metric
horn constructed in [DZ21, Section 6], where $p$ is the tip of the metric
horn. We denote the eigenfunctions with eigenvalue $\nu_{j}$ with spherical
components $\varphi_{i}$ as
$q_{\nu_{j}}(r,\theta)=g_{j}(r)\varphi_{i}(\theta)$.

Then we have that

(5.6) $\displaystyle f_{i}(r,t)=\sum_{j}c_{j}e^{-\nu_{j}t}g_{j}(r).$

Denote

$r_{\mu}=\max\\{\sqrt{\frac{c-1-\epsilon}{2\epsilon}(\frac{c-1-\epsilon}{2\epsilon}+1)},(\frac{c-1-\epsilon}{2\mu}(\frac{c-1-\epsilon}{2}+\epsilon))^{\frac{-\epsilon}{2}}\\}.$

By [ZZ21] we have $C_{1}j^{\frac{2}{N}}\leq\nu_{j}\leq C_{2}j^{2}$, thus
$C_{1}j^{\frac{\epsilon}{N}}\leq r_{\nu_{j}}\leq C_{2}j^{\epsilon}.$

From (3.15) we have that given any $100r_{\nu_{k}}\leq
r^{-\epsilon}<100r_{\nu_{k+1}}$, which indicates that
$\frac{C_{1}}{k+1}<r<C_{2}k^{-\frac{1}{N}}$, we have

$\displaystyle|f_{i}(r,t)|$
$\displaystyle\leq\sum_{j=1}^{k}|c_{j}|e^{-\nu_{j}t}c(n,\epsilon,\mu_{i})e^{(\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}+2)r_{\nu_{j}}}r_{\nu_{j}}^{\frac{1+\epsilon}{\epsilon}}e^{(-\sqrt{\frac{4\mu_{i}}{\epsilon^{2}}}+1)r^{-\epsilon}}+\sum_{j=k+1}^{\infty}Ce^{-\nu_{j}t}\nu_{j}$
(5.7) $\displaystyle\leq Ck\cdot
k^{1+\epsilon}e^{-cr^{-\epsilon}}+\sum_{j=k+1}^{\infty}Ce^{-Ctj^{\frac{2}{N}}}j^{2}\leq
Cr^{-(2+\epsilon)N}e^{-cr^{-\epsilon}}+Ce^{-Ctr^{-\frac{2}{N}}}r^{-2}\leq
e^{-cr^{-\epsilon}}.$

As the estimate is independent of k, we can see that $f_{i}$ vanishes up to
infinite order at tip.

######  Remark .

The argument above actually shows that all caloric functions on the modified
metric horn which does not contain the radial part $f_{0}(r,t)$ vanishes up to
infinite order at the tip.

## References

  * [AGMR15] L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, _Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure_, Trans. Amer. Math. Soc., 367 (2015), 4661–4701.
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