\label{thm:backbone}
There exist constants $C_1,C_2>0$ such that for any $\varepsilon\in(0,\frac{1}{10})$,
\begin{equation}\label{eq:bb-proba}
\frac{C_1}{\log|\log\varepsilon|}\le\mathbbm{P}[\Bac_\varepsilon]\le\frac{C_2}{\log|\log\varepsilon|}.
\end{equation}

\begin{theorem}\label{thm:backbone}
There exist constants $C_1,C_2>0$ such that for any $\varepsilon\in(0,\frac{1}{10})$,
\begin{equation}\label{eq:bb-proba}
\frac{C_1}{\log|\log\varepsilon|}\le\P[\Bac_\varepsilon]\le\frac{C_2}{\log|\log\varepsilon|}.
\end{equation}
\end{theorem}

\begin{equation}\label{eq:def-backbone}
\Bac_\varepsilon=\{\exists\ \text{two disjoint subpaths on the trajectory } B[0,\tau_{\D}] \text{ joining } \varepsilon\S^1 \text{ and } \frac{1}{2}\S^1\}.
\end{equation}

\begin{aligned}
F(\varepsilon)&\le \frac{C_0C'}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-n\log2}+q_0^n\\
&\le  \frac{C_0C'}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-\frac{\log2}{|\log q_0|}(\log\log|\log\varepsilon|-\log C')}+\frac{C'}{\log|\log\varepsilon|}
\le\frac{C_2}{\log|\log\varepsilon|}
\end{aligned}
$$
for some positive constant $C_2$, as desired.
\end{proof}

\appendix
\section{Proof of Theorem~\ref{thm-cr-mod-relation}}
\label{appendix}

In this appendix, we provide the proof of Theorem~\ref{thm-cr-mod-relation}, which is implicit in~\cite{ARS-Annulus}. For convenience, we consider the upper half plane $\hH:=\{z\in\C:\text{Im}(z)>0\}$, and let $\sm$ be the restriction of the $\SLE_{8/3}$ loop measure on $\C$ to the loops that are contained in $\hH$ and surround $i$. According to the conformal restriction~\cite{werner-loops}, if $\Phi:\D\to\hH$ is a conformal map with $\Phi(0)=i$, then $\sm=\Phi_*\SLE_{8/3,\D}^{\lp}$. Therefore, to prove Theorem~\ref{thm-cr-mod-relation}, it suffices to show the following
\begin{theorem}
\label{thm-appendix}
There is a constant $C>0$ such that for any measurable function $f:\R_+\to\R_+$ and $\lambda\ge0$, we have
$$
\int f\left(\Mod(\hH\backslash\overline{D(\eta)})\right)\psi_\eta'(i)^\lambda\sm(d\eta)=
C\frac{\sqrt{12\lambda-1}}{\sinh\left(\frac{\pi}{3}\sqrt{12\lambda-1}\right)}\int_0^\infty e^{-(2\lambda-\frac{1}{6})\pi\tau}\eta(2i\tau)f(\tau)d \tau.
$$
Here $\psi_\eta$ is the conformal map from $\hH$ to $D(\eta)$ such that $\psi_\eta(i)=i$ and $\psi_\eta'(i)>0$.
\end{theorem}

The proof of Theorem~\ref{thm-appendix} is based on the SLE-coupled Liouville quantum gravity with parameter $\sqrt{\frac{8}{3}}$. In the following we fix $\gamma=\sqrt{\frac{8}{3}}$ and let $Q=\frac{\gamma}{2}+\frac{2}{\gamma}$. Let $\P_{\hH}$ be the law of the free boundary Gaussian free field (GFF) on $\hH$ with mean zero on the upper semi-circle $\S^1\cap\hH$.

\begin{definition}[Liouville field on $\hH$]
\label{def-LF-H}
Let $(h,c)$ be sampled from $\P_{\hH}\times[e^{-Qc}dc]$, and let $\phi(z)=h(z)-2Q\log|z|_++c$ (here $|z|_+:=\max\{|z|,1\}$). Denote the law of $\phi$ by $\LF_{\hH}$.

For $\alpha\in\R$, let $\LF_{\hH}^{(\alpha,i)}(d\phi):=\lim_{\varepsilon\to0}\varepsilon^{\frac{\alpha^2}{2}}e^{\alpha\phi_\varepsilon(i)}\LF_{\hH}(d\phi)$ (the limit exists in the vague topology, see~\cite[Lemma 2.2]{ARS-FZZ}). Here $\phi_\varepsilon(i)$ is the average of $\phi$ over the circle $\partial B_\varepsilon(i)$.
\end{definition}

For a sample $\phi$ from $\LF_{\hH}^{(\alpha,i)}$, we can define its boundary Gaussian multiplicative chaos (GMC) measure $\nu_{\phi}:=\lim\limits_{\varepsilon\to0}\varepsilon^{\frac{\gamma^2}{4}}e^{\frac{\gamma}{2}\phi_\varepsilon(x)}dx$, where $\phi_\varepsilon(x)$ is the average of $\phi$ over the semi-circle $\partial B_\varepsilon(x)\cap\hH$.
Then denote $\{\LF_{\hH}^{(\alpha,i)}(\ell)\}_{\ell>0}$ to be the disintegration of $\LF_{\hH}^{(\alpha,i)}$ over $\nu_\phi(\R)$; i.e.~$\LF_{\hH}^{(\alpha,i)}=\int_0^\infty\LF_{\hH}^{(\alpha,i)}(\ell)d\ell$. We refer readers to see e.g.~\cite[Section 2.2]{ARS-FZZ} for further details. In particular, by~\cite[Lemma 2.7]{ARS-FZZ}, for $\alpha>\frac{\gamma}{2}$,
there are constants $C_{\alpha,\gamma}>0$ such that $|\LF_{\hH}^{(\alpha,i)}(\ell)|=C_{\alpha,\gamma}\ell^{\frac{2}{\gamma}(\alpha-Q)-1}$ for any $\ell>0$.

We also need the Liouville field on the annulus. For $\tau>0$, let $\mathcal{C}_\tau=[0,\tau]\times[0,1]/\mathord\sim$ be the horizontal cylinder with modulus $\tau$, where under $\sim$ we identify $(x,0)$ and $(x,1)$ for each $x\in[0,\tau]$. Let $\P_{\tau}$ be the free boundary GFF on $\mathcal{C}_\tau$
with zero mean on the circle $\{\frac{\tau}{2}\}\times[0,1]/\mathord\sim$.
\begin{definition}[{\cite[Definition 2.2]{ARS-Annulus}}]
\label{def-LF-annulus}
Let $(h,c)$ be sampled from $\P_\tau\times dc$. Then denote the law of $\phi=h+c$ by $\LF_\tau$ (which is an infinite measure on $H^{-1}(\mathcal{C}_\tau)$).
\end{definition}

For a sample $\phi$ from $\LF_\tau$,
we can similarly define its boundary GMC measures $\nu^1_\phi, \nu_\phi^2$ on the two boundaries $\partial_1\mathcal{C}_\tau=\{0\}\times[0,1]\mathord\sim$ and $\partial_2\mathcal{C}_\tau=\{1\}\times[0,1]\mathord\sim$, respectively. We also let $\{\LF_\tau(\ell_1,\ell_2)\}_{\ell_1,\ell_2>0}$ be the disintegration of $\LF_\tau$ over $\nu_\phi^1(\partial\cC_1)$ and $\nu_\phi^2(\partial\cC_2)$, i.e.~$\LF_\tau=\iint_{\R_+^2}\LF_\tau(\ell_1,\ell_2)d\ell_1d\ell_2$. The following result from~\cite{ARS-Annulus} gives the exact solvability of $|\LF_\tau(\ell_1,\ell_2)|$, which is based on~\cite{wu-annulus}.

\begin{proposition}[{\cite[Equation (3.6)]{ARS-Annulus}}]
 \label{prop-LF-integrability}
For $\tau>0$ and $y\in(-1,\frac{4}{\gamma^2})$, we have
$$
\iint_{\R_+^2}\ell_1e^{-\ell_1}\ell_2^{y}|\LF_\tau(\ell_1,\ell_2)|d\ell_1d\ell_2=\frac{\pi\gamma y\Gamma(1+y)}{2\sin(\frac{\gamma^2}{4}\pi y)}e^{\frac{\pi}{4}\gamma^2\tau y^2}.
$$
\end{proposition}

For two domains $D,\wt{D}\subset\C$ and a conformal map $g:D\to\wt{D}$, when $h$ is a distribution on $D$, we define $g\bullet h:=h\circ g^{-1}+Q\log|(g^{-1})'|$ (which is a distribution on $\wt D$). Let $\Omega$ be the space of simple loops in $\hH$ surrounding $i$, and $\Conf(\hH,i)$ be the group of conformal automorphisms of $\hH$ that fix $i$. For $(\phi,\eta,g,\theta)\in H^{-1}(\hH)\times\Omega\times \Conf(\hH,i)\times[0,1]$, define a measurable map $F$ by
$$
F(\phi,\eta,g,\theta):=\left((g\circ \psi_\eta^{-1})\bullet \phi|_{D(\eta)}, f_{\eta}^\theta\bullet \phi|_{\hH\backslash\overline{D(\eta)}},\Mod(\hH\backslash\overline{D(\eta)})\right).
$$
Here $\psi_\eta:\hH\to D(\eta)$ is the conformal map such that $\psi_\eta(i)=i$ and $\psi_\eta'(i)>0$, and
$f_{\eta}^\theta:\hH\backslash\overline{D(\eta)}\to\mathcal{C}_\tau$ is the conformal map such that $f_{\eta}^\theta(0)=\theta i$ with $\tau=\Mod(\hH\backslash\overline{D(\eta)})$.

Let $\Haar_{(\hH,i)}$ be the Haar measure on $\Conf(\hH,i)$ such that $|\Haar_{(\hH,i)}|=1$, and let $\Unif_{[0,1]}$ be the uniform measure on $[0,1]$.
The following proposition, which is essentially from~\cite{ARS-Annulus}, describes the law of $\LF_\hH^{(\gamma,i)}$ when cut by a simple loop sampled from $\sm$.

\begin{proposition}\label{prop-sle-weld}
There is a constant $C>0$ such that for any $\ell_1>0$,
$$
F_*\left(\LF_{\hH}^{(\gamma,i)}(\ell_1)\times\sm\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)
=C\int_0^\infty \left(\int_0^\infty \ell_2\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi_1)\times\LF_\tau(\ell_2,\ell_1)(d\phi_2)d\ell_2\right)
\eta(2i\tau)d\tau.
$$
Here $F_*$ stands for the pushforward of measures, and we view the right side as a measure on $(\phi_1,\phi_2,\tau)$.
\end{proposition}
\begin{proof}
By~\cite{ARS-Annulus}, the $\SLE_{8/3}$ loop cut the Brownian disk into an independent pair of a (smaller) Brownian disk and a Brownian annulus; see~\cite[Proposition 4.5]{cfsx} for the precise statement and proof. The result then follows from that the uniform embedding of the Brownian disk on $\hH$ gives the Liouville field on $\hH$~\cite[Theorem 3.4]{ARS-FZZ}.
\end{proof}

For $\alpha\in\R$, let $\Delta_\alpha=\frac{\alpha}{2}(Q-\frac{\alpha}{2})$. Define the measure $\sm^\alpha$ via $\frac{d\sm^\alpha}{d\sm}(\eta)=\psi_\eta'(i)^{2\Delta_\alpha-2}$, where $\psi_\eta:\hH\to D(\eta)$ is the conformal map as above. The following proposition is obtained from Proposition~\ref{prop-sle-weld} by a standard reweighting argument. Such an argument was first developed in~\cite{AHS-SLE-integrability}, and appeared in many recent papers on the integrability of SLE/CLE, see e.g.~\cite{ARS-FZZ,ACSW24b,ARS-Annulus,nolin2024backboneexponenttwodimensionalpercolation}.

\begin{proposition}
 \label{prop-sle-reweight}
There is a constant $C>0$ such that for any $\ell_1>0$ and $\alpha\in\R$,
$$
F_*\left(\LF_{\hH}^{(\alpha,i)}(\ell_1)\times\sm^\alpha\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)
=C\int_0^\infty \left(\int_0^\infty \ell_2\LF_{\hH}^{(\alpha,i)}(\ell_2)(d\phi_1)\times\LF_\tau(\ell_2,\ell_1)(d\phi_2)d\ell_2\right)
\eta(2i\tau)d\tau.
$$
As in Proposition~\ref{prop-sle-weld}, we view the right side above as a measure on $(\phi_1,\phi_2,\tau)$.
\end{proposition}

\begin{proof}
For $\phi\in H^{-1}(\hH)$, $\eta\in\Omega$ and $g\in\Conf(\hH,i)$, let $\phi_1=(g\circ \psi_\eta^{-1})\bullet \phi|_{D(\eta)}$. Recall that for $\varepsilon>0$, $(\phi_1)_\varepsilon(i)$ is the average of $\phi_1$ over $\partial B_\varepsilon(i)$.
By~\cite[Lemma 4.8]{ARS-FZZ},
as $\varepsilon\to0$, we have
$$
\begin{aligned}
&F_*\left(\varepsilon^{\frac{1}{2}(\alpha^2-\gamma^2)}e^{(\alpha-\gamma)(\phi_1)_\varepsilon(i)}\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi)\times\sm(d\eta)\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)\\
=~&F_*\left(\varepsilon^{\frac{1}{2}(\alpha^2-\gamma^2)}e^{(\alpha-\gamma)(\phi\circ\psi_\eta\circ g^{-1})_\varepsilon(i)}\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi)\times|\psi_\eta'(i)|^{(\alpha-\gamma)Q}\sm(d\eta)\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)\\
\to~& F_*\left(\LF_{\hH}^{(\alpha,i)}(\ell_2)\times\sm^\alpha\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right).
\end{aligned}

\label{section-intro}
\subsection{Overview and the main result}

There is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\ge 1$, the Brownian intersection exponent $\zeta_n=\frac{4n^2-1}{12}$~\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\cite{lsw-restriction}: both of the \emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\kappa=\frac{8}{3}$.

Note that in the critical percolation, one can also define the \emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.

To be precise, let $(B_t)_{t\ge0}$ be a planar Brownian motion starting from $0$. Denote $\mathbbm{S}^1:=\{z\in\mathbbm{C}:|z|=1\}$ and $\mathbbm{D}:=\{z\in\mathbbm{C}:|z|<1\}$ to be the unit circle and the unit disk, respectively. Let $\tau_\mathbbm{D}$ be the first hitting time of $\mathbbm{S}^1$ for $(B_t)_{t\ge0}$.
For each $\varepsilon\in(0,\frac{1}{2})$, consider the event
\begin{equation}\label{eq:def-backbone}
\Bac_\varepsilon=\{\exists\ \text{two disjoint subpaths on the trajectory } B[0,\tau_{\mathbbm{D}}] \text{ joining } \varepsilon\mathbbm{S}^1 \text{ and } \frac{1}{2}\mathbbm{S}^1\}.
\end{equation}
Namely, $\Bac_\varepsilon$ happens if there are two continuous curves $\gamma^1,\gamma^2:[0,1]\to\mathbbm{C}$ such that $\gamma^1[0,1]\cap\gamma^2[0,1]=\emptyset$, $\gamma^i[0,1]\subset B[0,\tau_{\mathbbm{D}}]$, $\gamma^i(0)\in\varepsilon\mathbbm{S}^1$, and $\gamma^i(1)\in\frac{1}{2}\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\Bac_\varepsilon$ the \emph{backbone event} for the Brownian trajectory $B[0,\tau_\mathbbm{D}]$.

The main result in this paper is the following iterated logarithmic decay for the probability $\mathbbm{P}[\Bac_\varepsilon]$.

The main result in this paper is the following iterated logarithmic decay for the probability $\P[\Bac_\varepsilon]$.

Let $f$ be the conformal map from $D(\ell_1)$ to $\D$ with $f(0)=0$, $f'(0)>0$ and let $\rho=f(\wt\ell_1)$. Note that $\CR(\rho,0)=\frac{\CR(\wt\ell_1,0)}{\CR(\ell_1,0)}$.
The main result of this section is the following exact law of $\CR(\rho,0)$, which is crucial to the final proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.
\begin{proposition}
\label{prop-cr-rho}
For $\lambda\ge0$, we have
\begin{equation}\label{eq:cr-rho}
\E[\CR(\rho,0)^\lambda]=\frac{\sinh(\frac{\pi}{2}\sqrt{12\lambda-1})}{\sqrt{12\lambda-1}}-\frac{2\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{12\lambda-1})}{\sqrt{12\lambda-1}\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}\right)}.
\end{equation}
When $\lambda\in[0,\frac{1}{12})$, the right side of~\eqref{eq:cr-rho} is defined by analytic continuation; see the end of Section~\ref{section-intro}.
\end{proposition}

Let $\nu_\D$ be the law of the loop chosen from the counting measure over $\{\ell_i\}_{i\ge1}$; namely, $\nu_\D$ is such that $\int F(\eta)\nu_\D(d\eta)=\P\left[\sum_{i\ge1}F(\ell_i)\right]$ for any bounded measurable function $F$. The following result, based on~\cite{cfsx} and Theorem~\ref{thm-cr-mod-relation}, gives the explicit law of the conformal radius under $\nu_\D$.
\begin{lemma}\label{lem-CR-BM-counting-ell}
There is a constant $C>0$ such that for $\lambda\ge0$, we have
\begin{equation}
 \label{eq-CR-BM-counting-ell}
 \int\CR(\eta,0)^\lambda\nu_{\D}(d\eta)=C\frac{\sqrt{12\lambda-1}}{\sinh(\frac{\pi}{3}\pi\sqrt{12\lambda-1})}\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}\right).
 \end{equation}
When $\lambda\in(0,\frac{1}{12})$, the right side of~\eqref{eq-CR-BM-counting-ell} is defined via analytic continuation as before.
\end{lemma}
\begin{proof}
By combining~\cite[Lemma 5.3]{cfsx} with~\cite[Theorem 1.3]{cfsx}, we have $\frac{d\nu_{\D}}{d\SLE^{\lp}_{8/3,\D}}(\eta)=\frac{C_1}{\Mod(\eta,\S^1)}$ for some constant $C_1>0$.
Then by Theorem~\ref{thm-cr-mod-relation}, we find
\begin{equation}\label{eq:cr-integral}
\int\CR(\eta,0)^\lambda\nu_{\D}(d\eta)=C
 \frac{\sqrt{12\lambda-1}}{\sinh(\frac{\pi}{3}\sqrt{12\lambda-1})}\int_0^\infty e^{-(2\lambda-\frac{1}{6})\pi\tau}\frac{\eta(2i\tau)}{\tau}d\tau.
\end{equation}
for some constant $C>0$.
Note that for $a>-\frac{\pi}{6}$, we have $\int_0^\infty e^{-a\tau}\eta(2i\tau)d\tau=\sqrt{\frac{\pi}{2a}}\frac{\sinh\left(\sqrt{\frac{2}{3}\pi a}\right)}{\cosh\left(\sqrt{\frac{3}{2}\pi a}\right)}$ (see e.g.~\cite[Equation (A.4)]{ARS-Annulus}) and
 $$
 \frac{d}{da}\left[-\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}{(2-\sqrt{3})^2+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}\right)\right]=\sqrt{\frac{\pi}{2a}}\frac{\sinh\left(\sqrt{\frac{2}{3}\pi a}\right)}{\cosh\left(\sqrt{\frac{3}{2}\pi a}\right)}.
 $$
Therefore, we obtain
 $$
 \int_0^\infty e^{-a\tau}\frac{\eta(2i\tau)}{\tau}d\tau=\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}\right).
 $$
for $a>-\frac{\pi}{6}$. Combined with the right side of~\eqref{eq:cr-integral}, we conclude.
\end{proof}

The upper bound of $\P[\mathsf{Bac}_\varepsilon]$ then relies on the following iterative inequality.
\begin{proposition}
 \label{prop-upper-bound-backbone}
 Let $q_0=\P[\wt\ell_1\cap\A_{1/2}\neq\emptyset]\in(0,1)$. Then
 there is a constant $C_0>0$ such that for any $\varepsilon\in(0,10^{-4})$,
 $$
 \P[\Bac_\varepsilon]\le \frac{C_0}{\log|\log\varepsilon|}+q_0\P[\Bac_{5\sqrt{\varepsilon}}].
 $$
\end{proposition}
\begin{proof}
Note that
 \begin{align}
 \P[\Bac_\varepsilon]&\le\P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\Bac_\varepsilon]\nonumber\\
 &\le \P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\Bac_\varepsilon]\nonumber\\
 &= \P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset, \Bac_\varepsilon]\label{eq:upper-1}
 \end{align}
Here the last line follows from that $(\wt\ell_1\cap(5\sqrt{\varepsilon}\D\cup\A_{1/2})=\emptyset)\cap\Bac_\varepsilon=\emptyset$.
By Lemma~\ref{lem-CR-BM-ell1}, Corollary~\ref{cor-estimate-cr-rho} and using Koebe's 1/4 theorem, there exist some positive constants $C,C'$ and $C_0$ such that
\begin{align}
\P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]&\le\P[\CR(\ell_1,0)<20\sqrt{\varepsilon}]+\P[\CR(\wt\ell_1,0)<20\sqrt{\varepsilon}]\nonumber\\
 & \le C\varepsilon^{\frac{1}{8}}+C'\frac{1}{\log|\log\varepsilon|}\le \frac{C_0}{\log|\log\varepsilon|}.\label{eq:upper-2}
\end{align}

Let $q_0:=\P[\wt\ell_1\cap\A_{1/2}\neq\emptyset]\in(0,1)$. Then
\begin{equation}\label{eq:upper-3}
\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset,\Bac_\varepsilon]\le q_0\P[\Bac_\varepsilon|\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset].
\end{equation}
Let $g:D(\wt\ell_1)\to\D$ be the conformal map such that $g(0)=0,g(B_{s_1})=1$.
By Lemma~\ref{lem-independent-decomposition-BM}, we see that $(g(B_t))_{t\in[0,s_1]}$ is independent of $B[s_1,\tau_\D]$, and has the same law (up to a time change) as $B[0,\tau_{\D}]$ conditioned on $B_{\tau_\D}=1$. Hence, if we denote $\wt\P$ to be the law of $(g(B_t))_{t\in[0,s_1]}$ conditioned on the event $\{\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset\}\cap\{\wt\ell_1\cap\A_{1/2}\neq\emptyset\}$, then $\wt\P$ is still the same as (up to a time change) the law of $B[0,\tau_{\D}]$ conditioned on $B_{\tau_\D}=1$.
By Lemma~\ref{lem-variant-Koebe-1/4},
on the event $\{\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset\}\cap\{\wt\ell_1\cap\A_{1/2}\neq\emptyset\}$,
we have $\overline{g(\varepsilon\D)}\subset5\sqrt{\varepsilon}\D$ and
$g(\frac{1}{2}\D\cap D(\wt\ell_1))\supset \frac{1}{2}\D$. Then we find
\begin{equation}\label{eq:upper-4}
\P[\Bac_\varepsilon|\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset]\le \wt\P[\Bac_{5\sqrt{\varepsilon}}]=\P[\Bac_{5\sqrt{\varepsilon}}],
\end{equation}
where the last equality follows from the rotational invariance of $B[0,\tau_\D]$. The result then follows from combining~\eqref{eq:upper-1},~\eqref{eq:upper-2},~\eqref{eq:upper-3} and~\eqref{eq:upper-4}.
\end{proof}

Now we prove the upper bound for $\P[\Bac_\varepsilon]$, finishing the proof of Theorem~\ref{thm:backbone}.
\begin{proof}[Proof of Theorem~\ref{thm:backbone}, the upper bound]
Let $\varepsilon\in(0,10^{-4})$ and $F(\varepsilon):=\P[\Bac_\varepsilon]$. Note that the function $x\mapsto\frac{1}{\log|\log x|}$ is increasing on $(0,10^{-4})$ and there is a constant $C>0$ such that $\frac{1}{\log|\log(25x)|}\le\frac{C}{\log|\log x|}$ for any $x\in(0,10^{-4})$. Then for each $n\in\mathbbm{Z}_+$ with $25\varepsilon^{\frac{1}{2^n}}<10^{-4}$, by applying Proposition~\ref{prop-upper-bound-backbone} for $n$ times, we find
\begin{align}
 F(\varepsilon)
 &\le C_0\sum\limits_{k=0}^{n-1}\frac{q_0^k}{\log|\log(5^{2-\frac{1}{2^{k-1}}}\varepsilon^{\frac{1}{2^k}})|}+q_0^nF(5^{2-\frac{1}{2^{n-1}}}\varepsilon^{\frac{1}{2^n}})\nonumber\\
 &\le \frac{C_0}{1-q_0}\frac{1}{\log|\log(25\varepsilon^{\frac{1}{2^n}})|}+q_0^n\le\frac{C_0C}{1-q_0}\frac{1}{\log|\log(\varepsilon^{\frac{1}{2^n}})|}+q_0^n\nonumber\\
 &=\frac{C_0C}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-n\log2}+q_0^n.\label{eq-proof-upper-bound-backbone}
\end{align}

\begin{remark}
The same result also holds when $\frac{1}{2}\mathbbm{S}^1$ in~\eqref{eq:def-backbone} is changed to any fixed $r\mathbbm{S}^1$ with $r\in(0,1)$ (and for sufficiently small $\varepsilon$), except that the corresponding constants in~\eqref{eq:bb-proba} will depend on $r$.
\end{remark}

Note that the backbone event $\Bac_\varepsilon$ is closely related to the spatial distribution of the \emph{cut points} of $B[0,\tau_\mathbbm{D}]$. Indeed, our proof of Theorem~\ref{thm:backbone} is based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}. We mention that~\cite{cfsx} heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG). It would be quite interesting to see if there is a derivation of Theorem~\ref{thm:backbone} without relying on LQG.

Theorem~\ref{thm:backbone} is related to a certain kind of special points of Brownian motion. Namely, let
\[
\mathcal{B}:=\{B_s:s\in[0,\tau_\mathbbm{D}) \text{ and there exists } \varepsilon>0 \text{ such that } (B_{s+u})_{0\le u\le\varepsilon} \text{ does not have a cut point}\}.
\]
Then Theorem~\ref{thm:backbone} indicates that $\mathcal{B}$ is non-empty and has Hausdorff dimension 2. Furthermore, it also suggests that the Hausdorff measure of $\mathcal{B}$ with the gauge $r\mapsto r^2\log\frac{1}{r}\log\log\frac{1}{r}\log\log\log\frac{1}{r}$~would exist and be non-trivial (since the Hausdorff gauge of $B[0,\tau_\mathbbm{D}]$ itself is $r\mapsto r^2\log\frac{1}{r}\log\log\log\frac{1}{r}$, see e.g.~\cite{LG85}). In particular, due to that planar Brownian motion a.s.~has no double cut points~\cite{BL}, we know that $\mathcal{B}$ contains the set of double points of $(B_t)_{0\le t\le\tau_\mathbbm{D}}$, which has the Hausdorff gauge $r\mapsto r^2(\log\frac{1}{r}\log\log\log\frac{1}{r})^2$~\cite{LG87}.

\subsection{Outlook and discussions}
\label{section-discussion}

Here we give several remarks and related questions before going into the proof.

\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}

\item One can similarly consider the probability that there exist $2n$ disjoint subpaths on $B[0,\tau_\mathbbm{D}]$ joining $\varepsilon\mathbbm{S}^1$ and $\frac{1}{2}\mathbbm{S}^1$ with $n\ge2$. Note that for $n\ge2$, such $2n$-arm probability is not straightforwardly related to the cut points of $B[0,\tau_\mathbbm{D}]$. Instead, one need to consider the $(2n-1)$-tuples of the local cut points such that removing these $(2n-1)$ cut points from the trajectory $B[0,\tau_\mathbbm{D}]$, $0$ and $B_{\tau_\mathbbm{D}}$ are not in the same connected component of the remaining set. However, analyzing such tuples of local cut points (e.g.~the counterpart of the layer structure in Section~\ref{section-backbone-1}) becomes much more complicated, and it seems difficult to solve them explicitly. We mention that similar difficulty also appears in deriving the monochromatic $k$-arm exponents of critical planar percolation for $k\ge3$, see~\cite[Remark 2.3]{nolin2024backboneexponenttwodimensionalpercolation}.

\item
There is also a natural half-plane variant of our setup. Namely, let $(e_t)_{t\ge0}$ be the Brownian excursion on the upper half plane $\mathbbm{H}$ from $0$ to $\infty$, and define $R\mathbbm{S}^+:=R\mathbbm{S}^1\cap\mathbbm{H}$ for $R>0$. Then for $n\ge1$, consider the asymptotic probability that
there exists $(2n+1)$ disjoint subpaths on the trajectory of $(e_t)$ joining $\mathbbm{S}^+$ and $R\mathbbm{S}^+$ as $R\to\infty$.
Note that the case $n=1$ can be similarly related to the cut points of $(e_t)$, which has been proven in~\cite[Theorem 4]{brownian-beads}.

\item In our forthcoming work~\cite{cx}, we will extend the result of this paper to the Brownian loop soup cases. Let $(B_t)_{0\le t\le\tau_\mathbbm{D}}$ be as before, and let $\mathcal{L}$ be an independent Brownian loop soup on $\mathbbm{D}$ with intensity $\frac{c}{2}$ for $c\in(0,1]$.
For a subset $A\subset\ol\mathbbm{D}$, denote $\mathcal{C}(A)$ to be the union of $A$ and all loop-soup clusters in $\mathcal{L}$ intersecting with $A$,
and let $H:=\mathcal{C}(B[0,\tau_\mathbbm{D}])$.
However, unlike the Brownian motion case,
there is a positive probability that $0$ itself can have two disjoint subpaths (except on $0$) on $H$
connected to $\frac{1}{2}\mathbbm{S}^1$. Moreover, in~\cite{cx} we will show the following
\begin{theorem}
We say $t\in(0,\tau_\mathbbm{D}]$ is a cut time of $H$ if $\mathcal{C}(B[0,t])\cap \mathcal{C}(B[t,\tau_\mathbbm{D}])=B_t$. Let $t_1$ be the largest cut time of $H$ such that the boundary of $\mathcal{C}(B[0,t_1])$ is a simple loop, and denote $E$ to be the event that $t_1$ is the smallest cut time of $H$.
Then for $c\in(0,1)$,
\begin{equation}\label{eq:p_e}
\mathbbm{P}[E]=2^{\frac{c+1}{2}}\sqrt{\frac{6}{1-c}}\left(\int_0^\infty\tau^{-1-\frac{c}{2}}e^{\frac{1-c}{6}\pi\tau}\eta(2i\tau)^{1-c}d\tau\right)^{-1}
\end{equation}
where $\eta(2i\tau):=e^{-\frac{\pi}{6}\tau}\prod_{n=1}^{\infty}\left(1-e^{-4\pi n\tau}\right)$ is the Dedekind eta function.
\end{theorem}
Note that the right side of~\eqref{eq:p_e} tends to $1$ as $c\uparrow1$, and tends to $0$ as $c\downarrow0$.

\end{itemize}

\medskip
\noindent\textbf{Organization of the paper.} In Section~\ref{section-SLE loop}, we review the explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops. Such relation was implicitly established in~\cite{ARS-Annulus}, and we will provide its proof in Appendix~\ref{appendix} for completeness. Then in Section~\ref{section-backbone-1}, we define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and solve it explicitly, based on the results of~\cite{cfsx} and Section~\ref{section-SLE loop}. Finally, we finish the proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.

\medskip
\noindent\textbf{Basic notations.}
For two compact sets $A,B\subset\mathbbm{C}$, let $\mathrm{dist}(A,B):=\inf\{|a-b|:a\in A,b\in B\}$.
For a simple loop $\ell\subset\mathbbm{C}$, let $D(\ell)$ be the bounded connected component of $\mathbbm{C}\backslash\ell$. For a simple loop $\ell\subset\mathbbm{C}$ with $0\in D(\ell)$, we denote the conformal radius of $D(\ell)$ seen from 0 by $\mathrm{CR}(\ell,0)$, i.e. if $f:\mathbbm{D}\to D(\ell)$ is a conformal map that fixes the origin, then $\mathrm{CR}(\ell,0)=|f'(0)|$.

For $0<r<1$, let $\A_r$ be the standard annulus $\{z\in\mathbbm{C}: r<|z|<1\}$. For each annular domain $A\subset\mathbbm{C}$,
there exists a unique $\tau>0$ such that $A$ and $\A_{e^{-2\pi\tau}}$ are conformally equivalent. We call $\tau$ the \textit{modulus} of $A$ and write $\mathrm{Mod}(A):=\tau$. For two simple loops $\eta_1,\eta_2$ such that $\overline{D(\eta_1)}\subset D(\eta_2)$, we also write the modulus of the annular domain $D(\eta_2)\setminus\overline{D(\eta_1)}$ as $\mathrm{Mod}(\eta_1,\eta_2)$ for simplicity.

We will frequently deal with elementary functions such as $\sqrt{x},\sinh(\sqrt{x}),\cosh(\sqrt{x})$, and $\tanh(\sqrt{x})$. In the following, we view them as functions defined on $\mathbbm{R}$ by taking $i\sqrt{|x|},i\sin(\sqrt{|x|}),\cos(\sqrt{|x|})$, and $i\tan(\sqrt{|x|})$ for $x<0$, respectively.

\medskip
\noindent\textbf{Acknowledgment.}
We are grateful to Xin Sun for many helpful discussions and suggestions on the early draft of this paper.
G.C. and Z.X.\ were partially supported by National Key R\&D Program of China (No.\ 2023YFA1010700). G.C. was partially supported by National Key R\&D Program of China (No. 2021YFA1002700).

Our proof of Theorem~\ref{thm:backbone} is described as follows: it is “based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Section~\ref{section-SLE loop} reviews an “explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops”; Section~\ref{section-backbone-1} then defines the layer structure and “solve[s] it explicitly, based on the results of~\cite{cfsx} and Section~\ref{section-SLE loop}”; finally “we finish the proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.”

Our proof of the main theorem is described as follows: it is “based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Next we review an “explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops”; we then define the layer structure and “solve[s] it explicitly, based on the results of~\cite{cfsx} and the preceding review”; finally “we finish the proof of the main theorem later.”

Existential–Universal
Inequality or Bound

Question:
Let $(B_t)_{t\ge 0}$ be planar Brownian motion started from $0$, let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, and let $\tau_{\mathbb D}$ be the first time that $B_t$ hits the unit circle $\mathbb S^1=\{z\in\mathbb C:|z|=1\}$. For $\varepsilon\in(0,\tfrac12)$, define the backbone event
\[
\mathrm{Bac}_\varepsilon=
\Big\{\text{there exist two disjoint subpaths of }B[0,\tau_{\mathbb D}]\text{ joining }\varepsilon\mathbb S^1\text{ and }\tfrac12\mathbb S^1\Big\}.
\]
Equivalently, there exist continuous curves $\gamma^1,\gamma^2:[0,1]\to\mathbb C$ with disjoint images, each contained in the Brownian trajectory $B[0,\tau_{\mathbb D}]$, such that $\gamma^i(0)\in \varepsilon\mathbb S^1$ and $\gamma^i(1)\in \tfrac12\mathbb S^1$ for $i=1,2$. Which quantitative estimate holds for $\mathbb P[\mathrm{Bac}_\varepsilon]$ for small $\varepsilon$?

There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\frac{C_1}{\log|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]

Choices:
B.
There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
C_1\,\frac{1}{|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le C_2\,\frac{1}{|\log \varepsilon|}.
\]
C.
There exists a constant $C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]
D.
There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac12)$,
\[
\frac{C_1}{\log|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]
E.
For every $\delta>0$, there exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\frac{C_1}{(\log|\log \varepsilon|)^{1+\delta}}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{(\log|\log \varepsilon|)^{1-\delta}}.
\]

The stem defines the event carefully but does not reveal the asymptotic form or explicitly hint at the iterated-log behavior. The correct answer is not leaked by wording in the question.

This is close to theorem recall: the stem asks directly which quantitative estimate holds, and the correct option is essentially the theorem statement. However, the alternatives introduce competing nearby formulations, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish the exact two-sided estimate from weaker, overextended, or slightly perturbed variants. Still, the item mainly tests recognition/recall of the precise result rather than generating a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: confusing log with iterated log, accepting a weaker true statement, overextending the epsilon range, or allowing flexible exponents. They are distinct and well-calibrated.

A strong technical MCQ with no answer leakage and high-quality distractors, but it is somewhat theorem-recall-heavy and only moderately tests generative reasoning.


[Main Theorem]
\label{mainThm}
For some $a_0 \ge 1$ and $N \in (0, 2)$, let $V : \R^2 \to \R$ satisfy
\begin{equation}
\label{Vbound}
\left\vertV(w)\right\vert \le a_0^2 \left< w \right>^{-N}.
\end{equation}
Assume that $u : \R^2 \to \R$ is a solution to
\begin{equation}
\label{ellipEq}
- \Delta u + V u = 0 \, \text{ in } \R^2
\end{equation}
with the properties that
\begin{equation}
\label{solNorm}
\left\vertu(0)\right\vert = 1
\end{equation}
and for each $w \in \R^2$,
\begin{equation}
\label{uBound}
\left\vertu(w)\right\vert \le \exp\pr{c_0 \left\vertw\right\vert^{1 - \frac N 2}}.
\end{equation}
For every $\varepsilon \in \left( 0, \frac N 2 \right) $, there exists ${R}_0(N, a_0, c_0, \varepsilon) > 0$ so that whenever $R \ge {R}_0$, it holds that
\begin{equation}
\label{uLower}
\inf_{\left\vertw_0\right\vert = R} \left\| u\right\|_{L^\infty\left( B_1(w_0) \right) } \ge \exp\pr{-R^{1 - \frac N 2 + \varepsilon}}.
\end{equation}

\begin{thm}[Main Theorem]
\label{mainThm}
For some $a_0 \ge 1$ and $N \in (0, 2)$, let $V : \R^2 \to \R$ satisfy
\begin{equation}
\label{Vbound}
\abs{V(w)} \le a_0^2 \innp{w}^{-N}.
\end{equation}
Assume that $u : \R^2 \to \R$ is a solution to
\begin{equation}
\label{ellipEq}
- \LP u + V u = 0 \, \text{ in } \R^2
\end{equation}
with the properties that
\begin{equation}
\label{solNorm}
\abs{u(0)} = 1
\end{equation}
and for each $w \in \R^2$,
\begin{equation}
\label{uBound}
\abs{u(w)} \le \exp\pr{c_0 \abs{w}^{1 - \frac N 2}}.
\end{equation}
For every $\eps \in \pr{0, \frac N 2}$, there exists ${R}_0(N, a_0, c_0, \eps) > 0$ so that whenever $R \ge {R}_0$, it holds that
\begin{equation}
\label{uLower}
\inf_{\abs{w_0} = R} \norm{u}_{L^\iny\pr{B_1(w_0)}} \ge \exp\pr{-R^{1 - \frac N 2 + \eps}}.
\end{equation}
\end{thm}

\begin{align}
\label{propApp}
\sup_{B_{\tilde r} \setminus \bigcup 3 \WT D_j} \abs{\tilde h}
\ge \pr{\frac{16 {\tilde r}} {\WT R}}^{\tilde \kappa} \sup_{B_{\WT T} \setminus \bigcup 3 \WT D_j} \abs{\tilde h},
\end{align}
where
\begin{align*}
\tilde \kappa
:= \kappa\pr{\WT R, \WT S, M}
= \max \set{6C_H C_1 R \sqrt{\log R}, 2^{13} \pr{c_1 R + L + \frac {c_d}{\log R}} RS^{-1}}.
\end{align*}
Since $R \ge 2^{10}$ implies $\frac R 8 > 1$, while $R \ge R_2$ implies $\sqrt{\frac{98 \cdot 16 R}{C_1 \sqrt{\log R}}} > 1$, then $r_0 := \min\set{\frac R 8, \sqrt{\frac{98 \cdot 16 R}{C_1 \sqrt{\log R}}}} > 1$.
Assume that $R \ge \hat R_0$ and $r < r_0$.
Combining \eqref{ufComparison}, \eqref{smallBallBound}, \eqref{propApp}, \eqref{scaleBounds}, and \eqref{midBallBound} shows that
\begin{align*}
\frac 1{1 - c_b \eps^2} \sup_{B_r} \abs{v}
&\ge \sup_{B_r} \abs{f}
\ge \sup_{B_{\tilde r} \setminus \bigcup 3 \WT D_j} \abs{\tilde h}
\ge \pr{\frac{16 {\tilde r}} {\WT R}}^{\tilde \kappa} \sup_{B_{\WT T} \setminus \bigcup 3 \WT D_j} \abs{\tilde h}
\ge \pr{\frac{r} {R}}^{2\tilde \kappa} \frac {e^{- L}} {1 + c_b \eps^2} .
\end{align*}
Since $C_1 =  \frac {a} {32} \sqrt{\frac{2 C_K }{\ln 2}}$, $\disp \frac{1 - c_b \eps^2}{1 + c_b \eps^2} \ge e^{- \frac{c_d}{\log R}}$, and $e < 4 \le \pr{2^{10}}^{\frac 1 5} \le \pr{\frac{R}{r}}^{\frac 1 5}$, then \eqref{propConc} follows, as required.
\end{proof}

\section{Transformation maps}
\label{TransMaps}

In this section, we introduce the transformation maps that serve as real-valued versions of the conformal maps $z \mapsto z^\al$.
Once the maps are defined, we show how they transform solutions to elliptic PDEs and how they transform balls.
These results will be used at the beginning and at the end of the proof of Theorem \ref{mainThm}.

Given $z = (x,y) \in \R^2$ in Cartesian coordinates, the polar coordinates for $z$ are $(r, \te) \in \R_+ \times (-\pi, \pi]$, where $r = \sqrt{x^2 + y^2}$ and $\te = \sgn(y) \arccos\pr{\frac x r}$.
Let $\R^2_+ = \set{z = (r,\te) : r > 0, \te \in (- \frac \pi 2, \frac \pi 2)}$ denote the right half-plane.
For $\al \in (1, \iny)$, we define $T_\al : \R^2 \to \R^2$ to be the map associated to the conformal transformation on $\C$ given by $z \mapsto z^\al$.
In polar coordinates, $T_\al$ is described as
\begin{equation}
\label{TalDefn}
T_\al(r, \te) = (r^\al, \al \te).
\end{equation}
To avoid continuity issues at $\te = \pi$, we restrict the domain and only consider $T_\al : \R^2_+ \to \R^2$.

\begin{lem}[Transformation of PDEs]
\label{transformEqLemma}
If $u : \R^2 \to \R$ is a solution to \eqref{ellipEq}, then with $v : \R^2_+ \to \R$ and $W : \R^2_+ \to \R$ defined by $v(z) = u(T_\al(z))$ and $W(z) = \al^2 \abs{z}^{2\al - 2} V(T_\al(z))$, it holds that
$$-\LP v + W v = 0 \; \text{ in } \R^2_+.$$
\end{lem}

\begin{proof}
Let $w := T_\al(z)$ be given in polar coordinates by $\pr{\rho, \vp}$.
Since $u$ is defined on $\R^2$, then in polar coordinates, $u(\rho, \vp)$ can be defined for all $\rho \in \R^+$ and all $\vp \in \R$ using periodicity.
We then see that $v(z) = v(r, \te) = u(r^\al, \al \te) = u(T_\al(z))$ is well-defined on $\R^2_+$.
Since
\begin{align*}
\del_r v(r, \te) &= \al r^{\al - 1} \del_\rho u(\rho, \vp) \\
\del_{r}^2 v(r, \te) &= \al^2 r^{2\al - 2} \del_{\rho}^2 u(\rho, \vp) + \al \pr{\al - 1} r^{\al - 2} \del_\rho u(\rho, \vp) \\
\del_{\te}^2 v(r, \te) &= \al^2 \del_{\vp}^2 u(\rho, \vp),
\end{align*}
then
\begin{align*}
\LP v(z)
&= \del_{r}^2 v(r, \te) + \frac 1 r \del_{r} v(r, \te) + \frac 1 {r^2} \del_{\te}^2 v(r, \te) \\
&= \al^2 r^{2\al - 2} \del_{\rho}^2 u(\rho, \vp) + \al \pr{\al - 1} r^{\al - 2} \del_\rho u(\rho, \vp)
+ \al r^{\al - 2} \del_\rho u(\rho, \vp)
+ r^{-2} \al^2 \del_{\vp}^2 u(\rho, \vp) \\
&= \al^2 r^{2\al - 2} \brac{\del_\rho^2 u(\rho, \vp) + \rho^{-1} \del_\rho u(\rho, \vp) + \rho^{-2}\del_{\vp}^2 u(\rho, \vp)} \\
&= \al^2 r^{2\al - 2} \LP u(w)
=  \al^2 r^{2\al - 2} V(w) u(w),
\end{align*}
where we have used \eqref{ellipEq}.
With $W(z) = \al^2 \abs{z}^{2\al - 2} V(T_\al(z)) = \al^2 r^{2\al - 2} V(w)$ as given, the conclusion follows.
\end{proof}

\begin{lem}[Ball containment]
\label{ballContainLemma}
Let $z_0 = (r_0, 0)$ and set $\disp \tilde r = \frac{r_0^{1 - \al}}{2 \sqrt 3 \al}$.
There exists $r_\al > 0$ so that whenever  with $r_0 \ge r_\al$, it holds that $T_\al(B_{\tilde r}(z_0)) \su B_1(T_\al(z_0))$ and $B_1(T_\al(z_0)) \su T_\al\pr{B_1(z_0)}$.
\end{lem}

\begin{proof}
With $\tilde r$ as given, it can be shown that
\begin{equation}
\label{ballIn}
\begin{aligned}
B_{\tilde r}(z_0)
&\su \set{r \in [r_0 - \tilde r, r_0 + \tilde r], \abs{\te} \le \frac {2\tilde r} {\sqrt 3 r_0}} \\
&= \set{r \in \brac{r_0\pr{1 -  \frac{r_0^{- \al}}{2 \sqrt 3 \al}}, r_0\pr{1 +  \frac{r_0^{- \al}}{2 \sqrt 3 \al}}}, \abs{\te} \le \frac {1} {3 \al r_0^\al}}.
\end{aligned}
\end{equation}
With $w_0 = T_\al(z_0)= \pr{r_0^\al, 0} = \pr{\rho_0, 0}$, it follows from \eqref{ballIn} that
\begin{align*}
T_\al(B_{\tilde r}(z_0))
&\su \set{r^\al \in \brac{r_0^\al\pr{1 -  \frac{r_0^{- \al}}{2 \sqrt 3 \al}}^\al, r_0^\al \pr{1 +  \frac{r_0^{- \al}}{2 \sqrt 3 \al}}^\al}, \abs{\al \te} \le \frac {\al} {3 \al r_0^\al}} \\
&= \set{\rho \in \brac{\rho_0 \pr{1 - \frac{\rho_0^{- 1}}{2 \sqrt 3 \al}}^\al, \rho_0 \pr{1 + \frac{\rho_0^{- 1}}{2 \sqrt 3 \al} }^\al}, \abs{\vp} \le \frac {1} {3 \rho_0} }.
\end{align*}
Since
\begin{align*}
\rho_0 \pr{1 \pm \frac{\rho_0^{- 1}}{2 \sqrt 3 \al} }^\al
&= \rho_0 \pm \frac{1}{2 \sqrt 3} + \frac{(1 - \frac 1 \al)}{24} r_0^{- \al} + \ldots
\end{align*}
then there exists $r_1(\al) \gg 1$ so that whenever $r_0 \ge r_1$, we have $\rho_0 - \frac 1 2 \le \rho_0 \pr{1 - \frac{\rho_0^{- 1}}{2 \sqrt 3 \al} }^\al$ and $\rho_0 \pr{1 + \frac{\rho_0^{- 1}}{2 \sqrt 3 \al} }^\al \le \rho_0 +\frac 1 2$.
Because,
\begin{align}
&B_1(w_0)  \supset \set{\rho \in \brac{\rho_0 - \frac 1 2, \rho_0 + \frac 1 2}, \abs{\vp} \le \frac 1 {3\rho_0} },
\label{ballOf}
\end{align}

\begin{equation}
\label{uLower}
\inf_{\abs{w_0} = R} \norm{u}_{L^\iny\pr{B_1(w_0)}} \ge \exp\pr{-R^{1 - \frac N 2 + \eps}}.
\end{equation}
\end{thm}

The results of \cite{Dav14} prove an estimate of the form \eqref{est} with $\be = \frac{4 - 2N}{3} = \frac 4 3 \pr{1 - \frac N 2} > 1 - \frac N 2 + \eps$.
In that article, the assumptions are the same as those in Theorem \ref{mainThm}, except that $u$ may be complex-valued.
Thus, as in the case of bounded $V$, Theorem \ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.

As illustrated by the following example, Theorem \ref{mainThm} is sharp (up to $\eps$) for all $N \in (0, 2)$.
Fix $N \in (0, 2)$, then set $u(z) = \exp\pr{- \abs{z}^{1 - \frac N 2}}$.
A computation shows that $u$ satisfies \eqref{ellipEq} where
$$V(z) := \pr{1 - \frac N 2}^2 \pr{\abs{z}^{1 - \frac N 2} - 1}\abs{z}^{-1 - \frac N 2}$$
satisfies $\abs{V(z)} \lesssim \abs{z}^{-N}$.
On the other hand, for any $\be > 0$, with $u(z) = \abs{z}^{-\be}$ on $\abs{z} > 1$, we see that $\disp \LP u = \be^2 \abs{z}^{-\be-2}$ and therefore $u$ satisfies \eqref{ellipEq} on an exterior domain with $V(z) := \be^2\abs{z}^{-2}$.
In particular, we may not have exponential behavior when $V$ decays fast enough, which explains why we restrict ourselves to $N < 2$.

To prove Theorem \ref{mainThm}, we use an iterative argument that is reminiscent of the one in \cite{Dav14}, see also \cite{LW14, DKW19, Dav20a, Dav25}.
To initialize the iteration, we apply a quantitative estimate of the form \eqref{est} with $\be = 1$.
This result, which verifies Landis' conjecture in the real-valued planar setting, was originally proved by Logunov, Malinnikova, Nadirashvili, and Nazarov in \cite{LMNN20}, and we formulate it in Theorem \ref{LandisGrowth} below.
The iteration argument then relies on repeated applications of Proposition \ref{InductiveProp} which is proved using the ideas from \cite{LMNN20}.
Roughly speaking, Proposition \ref{InductiveProp} shows that if an estimate like \eqref{est} holds with $\be = \be_0$, then for some $x_1$ with $\abs{x_1} \gg \abs{x_0}$, another estimate like \eqref{est} holds with $x_0$ replaced by $x_1$ and $\be = \be_1 \in \brac{1, \be_0}$.
When $\be_0 = 1$, Proposition \ref{InductiveProp} isn't useful, but when $\be_0 > 1$, we can decrease the exponent, i.e., make $\be_1 < \be_0$.
Therefore, to benefit from the iteration argument, we need to transform to a situation where $\be_0 > 1$.
We observe that if $u$ is composed with the real-variable version of the conformal transformation $z \mapsto z^\al$, then the new function also satisfies a Schr\"odinger equation.
By choosing $\al > 1$ appropriately, we can ensure that the new potential function is bounded and that the new solution function satisfies a version of \eqref{est} with $\be > 1$.
By repeatedly applying Proposition \ref{InductiveProp} to the transformed equation, we can make $\be$ arbitrarily close to $1$.
Finally, to reach the conclusion, we undo the change of variables.

We use the notation $B_r(z)$ to denote a ball of radius $r > 0$ centered at the point $z$, abbreviated by $B_r$ when the center is clear.
Generic constants are denoted by $c, C$ and may change from line to line without comment.
Specific constants will be indicated by subscripts.

The article is organized as follows.
In Section \ref{harmonic}, we present a unique continuation theorem for harmonic functions in punctured domains.
The content of this section is very similar to \cite[Section 5]{LMNN20} and \cite[Section 2]{Dav24}.
The iterative result described by Proposition \ref{InductiveProp} is the content of Section \ref{localProof}.
Proposition \ref{InductiveProp} is a three-ball inequality for solutions to Schr\"odinger equations and its proof relies on the results from Section \ref{harmonic}. In Section \ref{TransMaps}, we introduce the real-valued versions of $z \mapsto z^\al$ and record some of their properties.
In particular, we show how solutions behave when they are composed with these transformations.
Finally, the proof of Theorem \ref{mainThm} is presented in Section \ref{MainProof}.

\section{Decay properties of harmonic functions in punctured domains}
\label{harmonic}

In this section, we present and prove quantitative unique continuation results (in the form of three-ball inequalities) for harmonic functions in punctured domains.
We recall the following application of the Harnack inequality which appears in \cite[Claim 5.2]{LMNN20} and is repeated in \cite[Lemma 2.1]{Dav24}.

\begin{lem}[Harnack application]
\label{discBounds}
Let $\set{D_j}$ be a finite collection of $100$-separated unit disks in the plane.
Assume that $h$ is real-valued and harmonic in $\R^2 \setminus \bigcup D_j$ and that for each index $j$, $h$ doesn't change sign in $5D_j \setminus D_j$.
There exists an absolute constant $C_H \ge 10$ for which
\begin{enumerate}
\item $\disp \max_{\del \pr{3 D_j}} \abs{h} \le C_H \min_{\del \pr{3 D_j}} \abs{h}$
\item $\disp \max_{\del \pr{3 D_j}} \abs{\gr h} \le C_H \min_{\del \pr{3 D_j}} \abs{h}$.
\end{enumerate}
\end{lem}

\begin{proof}
An application of the Harnack inequality shows that there exists $C_H > 0$ so that for every $j$
\begin{align*}
\max_{\del \pr{3 D_j}} \abs{h}
&\le \sup_{4 D_j \setminus 2 D_j} \abs{h}
\le C_H \inf_{4 D_j \setminus 2 D_j} \abs{h}
\le C_H \min_{\del \pr{3 D_j}} \abs{h}.
\end{align*}
For each $z \in \del \pr{3 D_j}$, since $h$ doesn't change signs in $B_2(z)$, an application of Cauchy's inequality as in \cite[Lemma 1.11]{HL11} shows that $\abs{\gr h(z)} \le \abs{h(z)}$ and the conclusion follows.
\end{proof}

Now we state and prove the main result of this section.
The following is a slight modification of \cite[Lemma 2.1]{Dav24}, which resembles the result \cite[Theorem 5.3]{LMNN20}.

\begin{prop}[Three-ball inequality for harmonic function in punctured domain]
\label{harmonicProp}
Let $\set{D_j}$ be a finite collection of $100$-separated unit disks in the plane for which $0 \notin \bigcup 3 D_j$.
For some $R \ge 2^{10}$, let $h$ be a harmonic function in $B_R \setminus \bigcup D_j$ with the property that for each index $j$, $h$ doesn't change sign in $\pr{5D_j \setminus D_j} \cap B_R$.
Assume that for some $S \in \pr{2^8, \frac R 2}$ and some $M > \log\pr{16R}$, with $T := R - \frac S {32}$,
it holds that
\begin{equation}
\label{normalization}
\sup_{B_T \setminus \bigcup 3 D_j} \abs{h}  \ge e^{-M} \sup_{B_R \setminus \bigcup 3 D_j} \abs{h}.
\end{equation}

\begin{align}
\label{j2j0Comp}
C_Hm_{j_2} \abs{z_{j_2}}^{-k}
&\ge \sup_{\del \pr{3 D_{j_2}}} \abs{h} \abs{z_{j_2}}^{-k}
\ge \abs{h(s z_{j_0})} \abs{s z_{j_0}}^{-k}
\ge \frac{m_{j_0}}{2} s^{-k} \abs{z_{j_0}}^{-k}.
\end{align}
Since $z_{j_0} \in \del \pr{3 D_{j_0}}$ and $s z_{j_0} \in \del \pr{3 D_{j_2}}$ where $j_0 \ne j_2$, and the balls $\set{D_j}$ are of unit radius and $100$-separated, then $\abs{z_{j_0} - s z_{j_0}} \ge 96$.
After rearrangement, we see that $s^{-k} \ge \pr{1 - \frac{96}{R}}^{-k}$.
Since $10 \le C_H$, $2C_HR \le k$, and $\frac{96}{R} < - \log\pr{1 - \frac{96}{R}}$, then
\begin{align*}
\log\pr{2C_H}
&< 96 \cdot 2 C_H
\le \frac{96}{R} k
< - k \log\pr{1 - \frac{96}{R}}
\le - k \log s,
\end{align*}
from which it follows that $s^{-k} > 2C_H$.
We then conclude from \eqref{j2j0Comp} that $m_{j_2} \abs{z_{j_2}}^{-k} > m_{j_0} \abs{z_{j_0}}^{-k}$ which contradicts \eqref{j0Defn} and gives the desired contradiction.
In other words, \eqref{contraBound} fails to hold and we see that
\begin{align*}
\sup_{B_r \setminus \bigcup 3 D_j} \abs{h}
&> \pr{\frac{16r} R}^{3k}
= \pr{\frac {16r} R}^{3k} \sup_{B_T \setminus \bigcup 3 D_j} \abs{h},
\end{align*}
which implies \eqref{lowerBound} by our choice of $k$.
\end{proof}

\section{The iterative proposition}
\label{localProof}

Here we present the proposition that is used repeatedly in the iteration argument.
This unique continuation result takes the form of a three-ball inequality for solutions to Schr\"odinger equations.
The techniques used to prove this theorem are very similar to those that appear in the proofs of \cite[Theorem 2.2]{LMNN20} and \cite[Theorem 1.2]{Dav24}.

\begin{prop}[Iterative Proposition]
\label{InductiveProp}
Given $a > 0$ and $R \ge \hat R_0(a)$, let $W : B_R \to \R$ satisfy $\norm{W}_{L^\iny(B_R)} \le a^2$ and let $v : B_R \to \R$ be a solution to
$$-\LP v + W v = 0 \; \text{ in } B_R$$
with
\begin{equation}
\label{propUB}
\norm{v}_{L^\iny(B_R)} \le e^{c_1 R}
\end{equation}
for some $c_1 \ge 1$.
Assume that for some $S \in \pr{2^8, \frac R 2}$, there exists $z_0 \in \overline{B}_{R-S}$ and $L \ge 0$ such that
\begin{equation}
\label{propLB}
\abs{v(z_0)} \ge e^{- L}.
\end{equation}
Then there exists $r_0(a, R) > 1$ so that whenever $r \in (0, r_0)$, it holds that
\begin{align}
\label{propConc}
\norm{v}_{L^\iny(B_r)}
&\ge \pr{\frac{r} {R}}^{\tau},
\end{align}

\begin{align}
&B_1(w_0)  \supset \set{\rho \in \brac{\rho_0 - \frac 1 2, \rho_0 + \frac 1 2}, \abs{\vp} \le \frac 1 {3\rho_0} },
\label{ballOf}
\end{align}
then it follows that $T_\al(B_{\tilde r}(z_0))  \su B_1(w_0)$.

On the other hand, with $T_\al^{-1}(B_1(w_0)) = \set{z : T_\al(z) \in B_1(w_0)}$, \eqref{ballIn} (with $\tilde r$ replaced by $1$) implies that
\begin{align*}
T_\al^{-1}(B_1(w_0))
&\su \set{r^\al \in [\rho_0 - 1, \rho_0 + 1], \abs{\al \te} \le \frac {2} {\sqrt 3 \rho_0}} \\
&= \set{r \in \brac{\pr{r_0^\al - 1}^{\frac 1 \al}, \pr{r_0^\al + 1}^{\frac 1 \al}}, \abs{\te} \le \frac {2} {\sqrt 3 \al r_0^\al}}.
\end{align*}
Since
\begin{align*}
\pr{r_0^\al \pm 1}^{\frac 1 \al}
&= r_0 \pr{1 \pm \frac 1 {r_0^\al}}^{\frac 1 \al}
= r_0 \pm \frac 1 {\al r_0^{\al-1}} - \frac{1 - \frac 1 \al}{2 \al r_0^{2\al-1}}  + \ldots
\end{align*}
then there exist $r_2(\al) \gg 1$ so that whenever $r_0 \ge r_2$, we have $\pr{r_0^\al + 1}^{\frac 1 \al} \le r_0 + \frac 1 2$ and $\pr{r_0^\al - 1}^{\frac 1 \al} \ge r_0 - \frac 1 2$.
If $r_0 \ge r_3 := \pr{\frac {2 \sqrt 3} { \al }}^{\frac 1 {\al - 1}}$, then
\begin{align*}
T_\al^{-1}(B_1(w_0))
&\su \set{r \in \brac{r_0 - \frac 1 2, r_0 + \frac 1 2}, \abs{\te} \le \frac {1} {3 r_0}}
\su B_1(z_0),
\end{align*}
where we have used \eqref{ballOf}.
It follows that $B_1(w_0) \su T^\al(B_1(z_0))$.
To complete the proof, choose $r_\al = \max\set{r_1, r_2, r_3}$.
\end{proof}

\section{Proof of Theorem \ref{mainThm}}
\label{MainProof}

In this section, we prove the main result, Theorem \ref{mainThm}.
Before proceeding to the rigorous details, we describe the main steps:
\begin{itemize}
\item Initialize the iterative argument by applying Theorem \ref{LandisGrowth} to the solution function $u$.
\item Compose the solution with the transformation map given in \eqref{TalDefn} to get a solution $v$ to a different Schr\"odinger equation, see Lemma \ref{transformEqLemma}.
\item Apply the iterative result, Proposition \ref{InductiveProp}, to $v$ many times to reduce the exponent.
\item Undo the change of variables to get the desired estimate for $u$.
\end{itemize}
We recall the following result originally proved in \cite{LMNN20}, see also \cite[Theorem 1.1]{Dav24} in the case where $N = 0$ for this formulation.
This estimate serves as the initialization step in our iteration argument.

\begin{thm}[Initialization step]
\label{LandisGrowth}
For some $a_0 > 0$, let $V : \R^2 \to \R$ satisfy $\norm{V}_{L^\iny} \le a_0^2$.
Assume that $u : \R^2 \to \R$ is a solution to \eqref{ellipEq} that satisfies \eqref{solNorm} and for each $w \in \R^2$,
$$\abs{u(w)} \le \exp\pr{c_0 \abs{w}}.$$
Then there exists constants $\overline{C}_0 = \overline{C}_0\pr{a_0, c_0} > 0$ and $\overline{R}_0 > 0$ so that whenever $R \ge \overline{R}_0$, it holds that
\begin{equation}
\label{uLower0}
\inf_{\abs{w_0} = R}\norm{u}_{L^\iny\pr{B_1(w_0)}} \ge \exp\pr{- \overline{C}_0 R \log^{\frac 3 2} R}.
\end{equation}
\end{thm}

Observe that if $u$ satisfies the hypothesis of Theorem \ref{mainThm}, then it also satisfies the hypotheses of Theorem \ref{LandisGrowth}.
We now have all of the tools we need to prove Theorem \ref{mainThm}.

\begin{proof}[Proof of Theorem \ref{mainThm}]

With $N \in (0, 2)$ as given, let $\al =\pr{1 - \frac N 2}^{-1} = \frac 2 {2-N} \in (1, \iny)$.
Set $\de = \min\set{\frac{\eps}{2 - N}, 1}$.
With $r_\al$ from Lemma \ref{ballContainLemma}, $\overline{C}_0(a_0, c_0)$ and $\overline{R}_0$ from Theorem \ref{LandisGrowth}, and $\hat R_0(a)$ from Proposition \ref{InductiveProp}, define $\bar{r}_1 \ge \max\set{r_\al, \overline{R}_0^{\frac 1 \al}, \hat R_0\pr{a_0 \al}}$ as small as possible so that each of the following conditions hold:
\begin{align}
& r^{\de} \ge \overline{C}_0 \al^{\frac 3 2} \log^{\frac 3 2} r
\label{rBig1} \\
&r \log r \ge c_d
\label{rBig2} \\
& r^{\frac \de {1 + \de}} \ge \frac{3 a_0 \al C_H}{2^{14} \pr{5c_0 + 4} } \sqrt{\frac{C_K }{32\ln 2}} \sqrt{\log r}
\label{rBig3} \\
& r^{\frac {\de^2} 2 } \ge \brac{2^{14} \pr{5c_0 + 4} + \frac 2 5 }\log r
\label{rBig4} \\
& r^{\frac {5\de^2} 6 } \ge \brac{2^{14} \pr{5c_0 + 4} + \frac 2 5}\log \brac{\frac{4\al}{ \sqrt 3}  \pr{\frac 3 2 r}^{\al} }
\label{rBig5}.
\end{align}

This article is concerned with the quantitative unique continuation of solutions to elliptic Schr\"odinger equations in the plane.
We consider equations with potentials that exhibits pointwise decay at infinity.
The main result is a nearly sharp estimate for the optimal rate of decay at infinity for real-valued solutions.
This theorem may be interpreted as a quantitative Landis-type theorem.

In the late 1960s, E.~M.~Landis \cite{KL88} conjectured that if $u$ and $V$ are bounded functions that satisfy
\begin{equation}
\label{ePDE}
-\Delta u + V u = 0 \; \text{ in } \, \R^n,
\end{equation}
and $u$ decays faster than exponentially, i.e., $\left\vertu(x)\right\vert \lesssim \exp\pr{- c \left\vertx\right\vert^{1+}}$, then it follows that $u \equiv 0$.
This conjecture was later disproved by Meshkov \cite{M92} who constructed non-trivial complex-valued functions $u$ and $V$ that solve \eqref{ePDE} in $\R^2$, where $V$ is bounded and $\left\vertu(x)\right\vert \lesssim \exp\pr{- c \left\vertx\right\vert^{4/3}}$.
Using Carleman estimate techniques, Meshkov also proved a \textit{qualitative unique continuation} result:
If \eqref{ePDE} holds, where $V$ is bounded and $u$ satisfies a decay estimate of the form $\abs{u\left( x \right) } \lesssim \exp\pr{- c \left\vertx\right\vert^{4/3+}}$, then necessarily $u \equiv 0$.

In their work on Anderson localization \cite{BK05}, Bourgain and Kenig established a quantitative version of Meshkov's result.
As a first step in their proof, they used three-ball inequalities derived from Carleman estimates to establish \textit{order of vanishing} estimates for local solutions to Schr\"odinger equations of the form \eqref{ePDE}.
Then, through a scaling argument, they proved a \textit{quantitative unique continuation} result.
More specifically, they showed that if $u$ and $V$ are bounded and satisfy \eqref{ePDE}, and $u$ is normalized so that $\left\vertu(0)\right\vert \ge 1$, then for sufficiently large values of $R$,
\begin{equation}
 \inf_{|x_0| = R}\left\| u\right\|_{L^\infty\left( B_1(x_0) \right) } \ge \exp{(-CR^{\beta}\log R)},
\label{est}
\end{equation}
where $\beta = \frac 4 3$.
Since $ \frac 4 3 > 1$, the constructions of Meshkov, in combination with the qualitative and quantitative unique continuation theorems just described, indicate that Landis' conjecture cannot be true for complex-valued solutions in $\R^2$.
However, at the time, Landis' conjecture still remained open in the real-valued and higher-dimensional settings.

The first significant step towards resolving Landis' conjecture in the real-valued planar setting was made by Kenig, Silvestre and Wang in \cite{KSW15} where they proved a quantitative form of Landis' conjecture under the assumption that the zeroth-order term satisfies $V \ge 0$ a.e.
The techniques in \cite{KSW15} exploit the relationship between real-valued solutions to second-order elliptic PDEs in the plane and solutions to complex-valued Beltrami equations.
Using similar ideas, analogous results were established in the settings with drift terms \cite{KW15}, variable coefficients \cite{DKW17}, singular lower-order terms \cite{DW20}, and when $V_-$ exhibits decay at infinity \cite{DKW19, Dav20a}.
Then, in \cite{LMNN20}, Logunov, Malinnikova, Nadirashvili, and Nazarov proved Landis' conjecture in the real-valued planar setting.
Their proof uses the nodal structure of the domain along with a domain reduction technique to eliminate any sign condition on the zeroth-order term.
Many of the ideas from \cite{LMNN20} are used in this article.

In \cite{Dav14}, we studied the quantitative unique continuation properties of solutions to elliptic equations of the form
$$\Delta u + W \cdot \nabla u + V u = \lambda u \; \text{ in } \; \R^n,$$
where $V$ and $W$ exhibit pointwise decay at infinity, and $\lambda \in \C$.
With $\left< x \right> = \sqrt{1 + \left\vertx\right\vert^2}$, it was shown that if $\abs{V\left( x \right) } \lesssim \left< x \right>^{-N}$ and $\abs{W\left( x \right) } \lesssim \left< x \right>^{-P}$ for $N, P \ge 0$, then the quantitative estimate \eqref{est} holds with $\beta = \max \set{1, \frac{4-2N}{3}, 2 - 2P}$ and $\log R$ replaced by a different slowly-decaying function.
These quantitative estimates were generalized in \cite{LW14}, where Lin and Wang proved analogous estimates for solutions to the corresponding equations with variable-coefficient leading terms.
Qualitative estimates for similar equations are given in \cite{CS97}.
The constructions presented in \cite{Dav14, Dav15} show that the estimates described in this paragraph are sharp.

In \cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\left\vertV(z)\right\vert \lesssim \left\vertz\right\vert^N$ for $N > 0$.
The techniques in that article rely heavily on the ideas in \cite{LMNN20} and careful scaling arguments.
Here, we address the more difficult setting where the potential exhibits decay at infinity.
The precise statement of the main theorem is as follows.

\begin{equation}
\label{ePDE}
-\LP u + V u = 0 \; \text{ in } \, \R^n,
\end{equation}

\begin{equation}
 \inf_{|x_0| = R}\norm{u}_{L^\iny\pr{B_1(x_0)}} \ge \exp{(-CR^{\be}\log R)},
\label{est}
\end{equation}

In \cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\abs{V(z)} \lesssim \abs{z}^N$ for $N > 0$.
The techniques in that article rely heavily on the ideas in \cite{LMNN20} and careful scaling arguments.
Here, we address the more difficult setting where the potential exhibits decay at infinity.
The precise statement of the main theorem is as follows.

The results of \cite{Dav14} prove an estimate of the form \eqref{est} with $\be = \frac{4 - 2N}{3} = \frac 4 3 \pr{1 - \frac N 2} > 1 - \frac N 2 + \eps$.
In that article, the assumptions are the same as those in Theorem \ref{mainThm}, except that $u$ may be complex-valued.
Thus, as in the case of bounded $V$, Theorem \ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.

\begin{prop}[Iterative Proposition]
\label{InductiveProp}
Given $a > 0$ and $R \ge \hat R_0(a)$, let $W : B_R \to \R$ satisfy $\norm{W}_{L^\iny(B_R)} \le a^2$ and let $v : B_R \to \R$ be a solution to
$$-\LP v + W v = 0 \; \text{ in } B_R$$
with
\begin{equation}
\label{propUB}
\norm{v}_{L^\iny(B_R)} \le e^{c_1 R}
\end{equation}
for some $c_1 \ge 1$.
Assume that for some $S \in \pr{2^8, \frac R 2}$, there exists $z_0 \in \overline{B}_{R-S}$ and $L \ge 0$ such that
\begin{equation}
\label{propLB}
\abs{v(z_0)} \ge e^{- L}.
\end{equation}
Then there exists $r_0(a, R) > 1$ so that whenever $r \in (0, r_0)$, it holds that
\begin{align}
\label{propConc}
\norm{v}_{L^\iny(B_r)}
&\ge \pr{\frac{r} {R}}^{\tau},
\end{align}
with
$$\tau(R, S, L, a, c_1) = \max \set{3 a C_H\sqrt{\frac{C_K }{32\ln 2}} R \sqrt{\log R}, 2^{14} \pr{c_1 R + L + \frac {c_d}{\log R}} RS^{-1}} +\frac 1 5 \pr{L + \frac{c_d}{\log R}},$$
where $C_K$ and $c_d$ are universal constants, and $C_H$ is the Harnack constant from Lemma \ref{discBounds}.
\end{prop}

Define $\Om = B_R \setminus \pr{F_0 \bigcup F_1}$ and $\Om_1 = B_R \setminus F_1$.
As shown in \cite[\S 3.1]{LMNN20}, there exists a universal constant $c_P$ (that depends on $c_s$) so that $\Om$ has Poincar\'e constant bounded above by $c_P \rho^2$.
In particular, since $c_P \rho^2 \norm{W}_{L^\iny(B_R)} \le c_P \rho^2 a^2$, then by choosing $\rho \le \rho_0$, a universal constant, we can apply \cite[Lemma 3.2]{LMNN20}.
For $\eps \ll 1$ to be defined later on, let
\begin{equation}
\label{rhoDef}
\rho = \eps a^{-1}.
\end{equation}
An application of the arguments in \cite[\S 3.2]{LMNN20} then shows that there exists $\phi : \Om \to \R$ with the properties that
\begin{align}
&\LP \phi - W \phi = 0 \text{ in } \Om
\nonumber \\
&\phi -1 \in W^{1,2}_0(\Om)
\nonumber \\
&\norm{\phi -1}_\iny \le c_b \pr{\rho a}^2 = c_b \eps^2,
\label{vpuBound}
\end{align}
where $c_b$ is a universal constant that depends on $c_P$, and we have used \eqref{rhoDef}.
By extending $\phi$ to equal $1$ across $F_0 \bigcup F_1$, it is then shown in \cite[Lemma 4.1]{LMNN20} that $\disp f := \frac v \phi \in W^{1,2}_{\loc}(B_R)$ is a weak solution to the divergence-form equation
$$\di\pr{\phi^2 \gr f} = 0 \; \text{ in } \Om_1.$$
Moreover, the bound in \eqref{vpuBound} implies that for any $z \in B_R$,
\begin{equation}
\label{ufComparison}
\pr{1 - c_b \eps^2} \abs{f(z)} \le \abs{v(z)} \le \pr{1 + c_b \eps^2} \abs{f(z)}.
\end{equation}

We introduce the Beltrami coefficient $\mu$, defined as follows:
\begin{equation*}
\mu = \left\{ \begin{array}{ll}
\frac{1 - \phi^2}{1 + \phi^2} \frac{f_x + i f_y}{f_x - i f_y} & \text{ in } \Om_1 \text{ when } \, \gr f \ne 0 \\
0 & \text{ otherwise}
\end{array} \right..
\end{equation*}
Since $\abs{\mu} \lesssim \eps^2$, then as shown in \cite{AIM09}, there exists a $K$-quasiconformal homeomorphism of the complex plane where $K \le 1 + C_K \eps^2$ and $C_K$ depends on $c_b$.
That is, there exists some $w \in W^{1,2}_{\loc}$ that satisfies the Beltrami equation $\disp \frac{\del w}{\del \overline{z}} = \mu \frac{\del w}{\del z}$.
In fact, an application of the Riemann uniformization theorem shows that there exists a $K$-quasiconformal homeomorphism $g: B_R \to B_R$ that is onto with $g(0) = 0$.
Moreover, the function $h : = f \circ g^{-1}$ is harmonic in $g(\Om_1)$.

\begin{thm}[Initialization step]
\label{LandisGrowth}
For some $a_0 > 0$, let $V : \R^2 \to \R$ satisfy $\norm{V}_{L^\iny} \le a_0^2$.
Assume that $u : \R^2 \to \R$ is a solution to \eqref{ellipEq} that satisfies \eqref{solNorm} and for each $w \in \R^2$,
$$\abs{u(w)} \le \exp\pr{c_0 \abs{w}}.$$
Then there exists constants $\overline{C}_0 = \overline{C}_0\pr{a_0, c_0} > 0$ and $\overline{R}_0 > 0$ so that whenever $R \ge \overline{R}_0$, it holds that
\begin{equation}
\label{uLower0}
\inf_{\abs{w_0} = R}\norm{u}_{L^\iny\pr{B_1(w_0)}} \ge \exp\pr{- \overline{C}_0 R \log^{\frac 3 2} R}.
\end{equation}
\end{thm}

Now assume that \eqref{claim} holds for $k-1 \in \N$.
Define $S_k = \frac 1 2 \abs{z_{k-1}}$ and $R_k = \abs{z_{k-1}}^{\al_{k-1} +\de}$.
Comparing with \eqref{zkNorm}, we see that $\abs{z_k} = R_k + S_k - 1$.
Since $B_1(z_{k-1}) \su B_{R_k - S_k}(z_k)$, then the inductive hypothesis shows that there exists $z_0 \in \overline{B}_{R_k - S_k}(z_k)$ so that
\begin{align*}
\abs{v(z_0)}
\ge \exp\pr{- \abs{z_{k-1}}^{\al_{k-1} + \de}}
= \exp\pr{- R_k}.
\end{align*}
The bound in \eqref{uBound} shows that $\disp \abs{v(z)} = \abs{u(T_\al(z))} \le \exp\pr{c_0 \abs{z}^{\al\pr{1 - \frac N 2}}} = \exp\pr{c_0 \abs{z}}$ from which it follows that
\begin{align*}
\norm{v}_{L^\iny\pr{B_{R_k}(z_k)}}
\le \exp\pr{c_0 \pr{\abs{z_k} + R_k}}
\le \exp\pr{\frac {5c_0} 2 R_k}.
\end{align*}
Let $W : \R^2_+ \to \R$ be given by $W(z) = \al^2 \abs{z}^{2\al - 2} V(T_\al(z))= \al^2 \abs{w}^{2 - \frac 2 \al} V(w)$ so that by \eqref{Vbound},
\begin{align*}
\abs{W(z)} = \al^2 \abs{w}^{2 - \frac 2 \al} \abs{V(w)} \le a_0^2 \al^2 \abs{w}^{2 - N - \frac 2 \al} = \pr{a_0 \al}^2.
\end{align*}
An application of Lemma \ref{transformEqLemma} with assumption \eqref{ellipEq} shows that
\begin{equation*}
- \LP v + W v = 0 \; \text{ in } \R^2_+.
\end{equation*}
As $R_k \ge \bar{r}_1$ implies that $R_k \ge \hat R_0(a_0 \al)$ and $S_k \in \pr{2^8, \frac {R_k} 2}$, Proposition \ref{InductiveProp} is applicable with $a = a_0 \al$, $R= R_k$, $S= S_k$, $L = R_k$, and $c_1 = \frac {5c_0} 2$.
With $r = 1$, Proposition \ref{InductiveProp} shows that
\begin{align}
\label{vkBd}
\norm{v}_{L^\iny(B_1(z_k))}
&\ge \exp\pr{-\tau_k(R_k) \log R_k},
\end{align}
where
\begin{equation}
\label{taukDefn}
\begin{aligned}
\tau_k(R_k) &= \max \set{3 a_0 \al C_H\sqrt{\frac{C_K }{32\ln 2}} R_k \sqrt{\log R_k}, 2^{15} \pr{\frac {5c_0} 2 + 1 + \frac {c_d}{R_k \log R_k}} R_k^{2 - \frac 1 {\al_{k-1} + \de}}} \\
&+\frac 1 5 \pr{R_k + \frac{c_d}{\log R_k}}.
\end{aligned}
\end{equation}
Because $R_k \ge \bar{r}_1$ and $\al_{k - 1} > 1$, conditions \eqref{rBig2} and \eqref{rBig3} imply that
\begin{align}
\label{tauKBound}
\tau_k(R_k) &\le \brac{2^{14} \pr{5c_0 + 4} + \frac 2 5} R_k^{2 - \frac 1 {\al_{k-1} + \de}}
\end{align}
while condition \eqref{rBig4} implies that
$$ \brac{2^{14} \pr{5c_0 + 4} + \frac 2 5} \log R_k \le R_k^{\frac {\de^2} 2}.$$
Since $\al > 1$ and $\de \le 1$ implies that $2 - \frac{1}{\al + \de} + \frac{\de^2}2 < 2 - \frac 1 \al + \de$, then combining these bounds shows that
$$\tau_k(R_k) \log R_k \le R_k^{2 - \frac 1 {\al_{k-1}} + \de} = R_k^{\al_{k} + \de}.$$
Returning to \eqref{vkBd}, we conclude that
\begin{align*}
\norm{v}_{L^\iny(B_1(z_k))}
&\ge \exp\pr{-R_k^{\al_{k} + \de}}
\ge \exp\pr{-\abs{z_k}^{\al_{k} + \de}},
\end{align*}
establishing the claim given by \eqref{claim}.

The results of \cite{Dav14} prove an estimate of the form \eqref{est} with $\beta = \frac{4 - 2N}{3} = \frac 4 3 \left( 1 - \frac N 2 \right)  > 1 - \frac N 2 + \varepsilon$.
In that article, the assumptions are the same as those in Theorem \ref{mainThm}, except that $u$ may be complex-valued.
Thus, as in the case of bounded $V$, Theorem \ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.

As illustrated by the following example, Theorem \ref{mainThm} is sharp (up to $\varepsilon$) for all $N \in (0, 2)$.
Fix $N \in (0, 2)$, then set $u(z) = \exp\pr{- \left\vertz\right\vert^{1 - \frac N 2}}$.
A computation shows that $u$ satisfies \eqref{ellipEq} where
$$V(z) := \left( 1 - \frac N 2 \right) ^2 \pr{\left\vertz\right\vert^{1 - \frac N 2} - 1}\left\vertz\right\vert^{-1 - \frac N 2}$$
satisfies $\left\vertV(z)\right\vert \lesssim \left\vertz\right\vert^{-N}$.
On the other hand, for any $\beta > 0$, with $u(z) = \left\vertz\right\vert^{-\beta}$ on $\left\vertz\right\vert > 1$, we see that $\displaystyle \Delta u = \beta^2 \left\vertz\right\vert^{-\beta-2}$ and therefore $u$ satisfies \eqref{ellipEq} on an exterior domain with $V(z) := \beta^2\left\vertz\right\vert^{-2}$.
In particular, we may not have exponential behavior when $V$ decays fast enough, which explains why we restrict ourselves to $N < 2$.

To prove Theorem \ref{mainThm}, we use an iterative argument that is reminiscent of the one in \cite{Dav14}, see also \cite{LW14, DKW19, Dav20a, Dav25}.
To initialize the iteration, we apply a quantitative estimate of the form \eqref{est} with $\beta = 1$.
This result, which verifies Landis' conjecture in the real-valued planar setting, was originally proved by Logunov, Malinnikova, Nadirashvili, and Nazarov in \cite{LMNN20}, and we formulate it in Theorem \ref{LandisGrowth} below.
The iteration argument then relies on repeated applications of Proposition \ref{InductiveProp} which is proved using the ideas from \cite{LMNN20}.
Roughly speaking, Proposition \ref{InductiveProp} shows that if an estimate like \eqref{est} holds with $\beta = \be_0$, then for some $x_1$ with $\left\vertx_1\right\vert \gg \left\vertx_0\right\vert$, another estimate like \eqref{est} holds with $x_0$ replaced by $x_1$ and $\beta = \be_1 \in \left[1, \be_0\right]$.
When $\be_0 = 1$, Proposition \ref{InductiveProp} isn't useful, but when $\be_0 > 1$, we can decrease the exponent, i.e., make $\be_1 < \be_0$.
Therefore, to benefit from the iteration argument, we need to transform to a situation where $\be_0 > 1$.
We observe that if $u$ is composed with the real-variable version of the conformal transformation $z \mapsto z^\alpha$, then the new function also satisfies a Schr\"odinger equation.
By choosing $\alpha > 1$ appropriately, we can ensure that the new potential function is bounded and that the new solution function satisfies a version of \eqref{est} with $\beta > 1$.
By repeatedly applying Proposition \ref{InductiveProp} to the transformed equation, we can make $\beta$ arbitrarily close to $1$.
Finally, to reach the conclusion, we undo the change of variables.

We use the notation $B_r(z)$ to denote a ball of radius $r > 0$ centered at the point $z$, abbreviated by $B_r$ when the center is clear.
Generic constants are denoted by $c, C$ and may change from line to line without comment.
Specific constants will be indicated by subscripts.

The article is organized as follows.
In Section \ref{harmonic}, we present a unique continuation theorem for harmonic functions in punctured domains.
The content of this section is very similar to \cite[Section 5]{LMNN20} and \cite[Section 2]{Dav24}.
The iterative result described by Proposition \ref{InductiveProp} is the content of Section \ref{localProof}.
Proposition \ref{InductiveProp} is a three-ball inequality for solutions to Schr\"odinger equations and its proof relies on the results from Section \ref{harmonic}. In Section \ref{TransMaps}, we introduce the real-valued versions of $z \mapsto z^\alpha$ and record some of their properties.
In particular, we show how solutions behave when they are composed with these transformations.
Finally, the proof of Theorem \ref{mainThm} is presented in Section \ref{MainProof}.

To prove Theorem~\ref{mainThm}, the paper uses “an iterative argument that is reminiscent of the one in \cite{Dav14}.”

Key steps (as described):
\begin{itemize}
\item \textbf{Initialize the iteration.} “To initialize the iteration, we apply a quantitative estimate of the form \eqref{est} with $\beta = 1$,” namely the real-valued planar Landis growth result proved in \cite{LMNN20} and stated as Theorem~\ref{LandisGrowth}.
\item \textbf{Inductive improvement via a three-ball type step.} The iteration “relies on repeated applications of Proposition~\ref{InductiveProp}.” Roughly, this proposition shows that if an estimate like \eqref{est} holds with exponent $\beta=\be_0$, then for some $x_1$ with “$|x_1|\gg |x_0|$,” another estimate like \eqref{est} holds with $x_0$ replaced by $x_1$ and “$\beta=\be_1\in[1,\be_0]$.” Moreover, “when $\be_0>1$, we can decrease the exponent, i.e., make $\be_1<\be_0$.”
\item \textbf{Create a regime with $\beta>1$ via a power map.} Since “when $\be_0=1$, Proposition~\ref{InductiveProp} isn’t useful,” they “transform to a situation where $\be_0>1$” by composing $u$ with “the real-variable version of the conformal transformation $z\mapsto z^\alpha$,” noting the composed function still “satisfies a Schr\"odinger equation.” Choosing “$\alpha>1$ appropriately,” they ensure “the new potential function is bounded” and the new solution satisfies a version of \eqref{est} “with $\beta>1$.”
\item \textbf{Iterate to push $\beta$ down to $1$.} “By repeatedly applying Proposition~\ref{InductiveProp} to the transformed equation, we can make $\beta$ arbitrarily close to $1$.”
\item \textbf{Conclude by reversing the change of variables.} “Finally, to reach the conclusion, we undo the change of variables.”
\end{itemize}

To prove the main theorem, the paper uses “an iterative argument that is reminiscent of the one in \cite{Dav14}.”

Key steps (as described):
\begin{itemize}
\item \textbf{Initialize the iteration.} “To initialize the iteration, we apply a quantitative estimate of the form
\begin{equation}
 \inf_{|x_0| = R}\norm{u}_{L^\iny\pr{B_1(x_0)}} \ge \exp{(-CR^{\be}\log R)},
\label{est}
\end{equation}
with $\beta = 1$,” namely the real-valued planar Landis growth result proved in \cite{LMNN20} and stated as
\begin{thm}[Initialization step]
\label{LandisGrowth}
For some $a_0 > 0$, let $V : \R^2 \to \R$ satisfy $\norm{V}_{L^\iny} \le a_0^2$.
Assume that $u : \R^2 \to \R$ is a solution to \eqref{ellipEq} that satisfies \eqref{solNorm} and for each $w \in \R^2$,
$$\abs{u(w)} \le \exp\pr{c_0 \abs{w}}.$$
Then there exists constants $\overline{C}_0 = \overline{C}_0\pr{a_0, c_0} > 0$ and $\overline{R}_0 > 0$ so that whenever $R \ge \overline{R}_0$, it holds that
\begin{equation}
\label{uLower0}
\inf_{\abs{w_0} = R}\norm{u}_{L^\iny\pr{B_1(w_0)}} \ge \exp\pr{- \overline{C}_0 R \log^{\frac 3 2} R}.
\end{equation}
\end{thm}

\item \textbf{Inductive improvement via a three-ball type step.} The iteration “relies on repeated applications of
\begin{prop}[Iterative Proposition]
\label{InductiveProp}
Given $a > 0$ and $R \ge \hat R_0(a)$, let $W : B_R \to \R$ satisfy $\norm{W}_{L^\iny(B_R)} \le a^2$ and let $v : B_R \to \R$ be a solution to
$$-\LP v + W v = 0 \; \text{ in } B_R$$
with
\begin{equation}
\label{propUB}
\norm{v}_{L^\iny(B_R)} \le e^{c_1 R}
\end{equation}
for some $c_1 \ge 1$.
Assume that for some $S \in \pr{2^8, \frac R 2}$, there exists $z_0 \in \overline{B}_{R-S}$ and $L \ge 0$ such that
\begin{equation}
\label{propLB}
\abs{v(z_0)} \ge e^{- L}.
\end{equation}
Then there exists $r_0(a, R) > 1$ so that whenever $r \in (0, r_0)$, it holds that
\begin{align}
\label{propConc}
\norm{v}_{L^\iny(B_r)}
&\ge \pr{\frac{r} {R}}^{\tau},
\end{align}
with
$$\tau(R, S, L, a, c_1) = \max \set{3 a C_H\sqrt{\frac{C_K }{32\ln 2}} R \sqrt{\log R}, 2^{14} \pr{c_1 R + L + \frac {c_d}{\log R}} RS^{-1}} +\frac 1 5 \pr{L + \frac{c_d}{\log R}},$$
where $C_K$ and $c_d$ are universal constants, and $C_H$ is the Harnack constant from Lemma \ref{discBounds}.
\end{prop}
Roughly, this proposition shows that if an estimate like the equation above holds with exponent $\beta=\be_0$, then for some $x_1$ with “$|x_1|\gg |x_0|$,” another estimate like the equation above holds with $x_0$ replaced by $x_1$ and “$\beta=\be_1\in[1,\be_0]$.” Moreover, “when $\be_0>1$, we can decrease the exponent, i.e., make $\be_1<\be_0$.”

\item \textbf{Create a regime with $\beta>1$ via a power map.} Since “when $\be_0=1$, the iterative proposition above isn’t useful,” they “transform to a situation where $\be_0>1$” by composing $u$ with “the real-variable version of the conformal transformation $z\mapsto z^\alpha$,” noting the composed function still “satisfies a Schr\"odinger equation.” Choosing “$\alpha>1$ appropriately,” they ensure “the new potential function is bounded” and the new solution satisfies a version of the equation above “with $\beta>1$.”

\item \textbf{Iterate to push $\beta$ down to $1$.} “By repeatedly applying the iterative proposition above to the transformed equation, we can make $\beta$ arbitrarily close to $1$.”

\item \textbf{Conclude by reversing the change of variables.} “Finally, to reach the conclusion, we undo the change of variables.” This completes the proof of the main theorem.
\end{itemize}

Question:
Let $a_0\ge 1$ and $N\in(0,2)$. Suppose $V:\mathbb{R}^2\to\mathbb{R}$ satisfies
\[
|V(w)|\le a_0^2\langle w\rangle^{-N}\quad\text{for all }w\in\mathbb{R}^2,
\]
where $\langle w\rangle=\sqrt{1+|w|^2}$. Let $u:\mathbb{R}^2\to\mathbb{R}$ solve
\[
-\Delta u+Vu=0\quad\text{in }\mathbb{R}^2,
\]
with
\[
|u(0)|=1
\]
and the growth bound
\[
|u(w)|\le \exp\big(c_0|w|^{1-N/2}\big)\quad\text{for every }w\in\mathbb{R}^2.
\]
Here $B_1(w_0)$ denotes the unit ball centered at $w_0$. Under these assumptions, which quantitative estimate holds?

For every $\varepsilon\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0,\varepsilon)>0$ such that for every $R\ge R_0$,
\[
\inf_{|w_0|=R}\|u\|_{L^\infty(B_1(w_0))}\ge \exp\big(-R^{1-N/2+\varepsilon}\big).
\]

Choices:
B.
For every $\varepsilon\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0)>0$ such that for every $R\ge R_0$,
\[
\inf_{|w_0|=R}\|u\|_{L^\infty(B_1(w_0))}\ge \exp\big(-R^{1-N/2}\big).
\]
C.
There exists $R_0=R_0(N,a_0,c_0)>0$ such that for every $R\ge R_0$,
\[
\inf_{|w_0|=R}\|u\|_{L^\infty(B_1(w_0))}\ge \exp\big(-R\big).
\]
D.
For every $\varepsilon\in(0,N/2)$ and every $R_0>0$, there exists $R\ge R_0$ such that
\[
\inf_{|w_0|=R}\|u\|_{L^\infty(B_1(w_0))}\ge \exp\big(-R^{1-N/2+\varepsilon}\big).
\]
E.
For every $\varepsilon\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0,\varepsilon)>0$ such that for every $R\ge R_0$,
\[
\inf_{|w_0|\le R}\|u\|_{L^\infty(B_1(w_0))}\ge \exp\big(-R^{1-N/2+\varepsilon}\big).
\]

replace sharp exponent $1-N/2+\varepsilon$ by weaker linear exponent

eventual-for-all-$R$ conclusion weakened to infinitely-many-$R$ statement

sphere condition $|w_0|=R$ replaced by ball condition $|w_0|\le R$

The stem gives the hypotheses and asks for the correct large-radius conclusion, but it does not explicitly reveal the exact bound, quantifier structure, or the infimum/supremum form of the answer.

This is close to a theorem-recall item: the assumptions are essentially those of a specific quantitative unique continuation result, and the correct option is the theorem’s conclusion. However, the presence of nearby alternatives with altered exponents and quantifiers makes it slightly more than a verbatim restatement.

Some reasoning is needed to distinguish the sharp statement from stronger, weaker, or quantifier-tampered alternatives. Still, the item primarily tests recognition of the exact theorem statement rather than generating a conclusion from first principles.

The distractors are strong: one is an overstrong sharpness trap, one is a weaker true statement, one alters quantifier dependence, and one changes infimum to supremum. These reflect realistic mathematical failure modes.

A solid theorem-identification MCQ with high-quality distractors and no answer leakage, but it leans more toward precise theorem recall than genuinely generative reasoning.