[Rigidity of lattices]
    If $\Gamma$ is a finitely generated group quasi-isometric to an irreducible lattice in a semisimple Lie group $G$, then there is a short exact sequence
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda \ar[r] & 1
    \end{tikzcd}
    \end{equation*}
    where $\Lambda$ is a lattice in $G$, and $F$ is a finite group.

\subsection{Background}

One theme of geometric group theory is the rich relationship between the algebraic structure of groups and the large-scale geometry of spaces on which they act. By the ``large-scale geometry" of a space we mean the metric structure that is preserved by quasi-isometries, maps which preserve distances up to a controlled error. Gromov proposed in a 1983 ICM address \cite{gromov-icm} a broad research program of studying finitely generated groups as geometric objects and classifying them up to quasi-isometry. One aspect involves identifying instances of quasi-isometric rigidity, the phenomenon which occurs when algebraic properties of a finitely generated group are determined by its large-scale geometry. For example, a celebrated theorem by Gromov \cite{Gromov-polynomial} states that finitely generated groups of polynomial growth are virtually nilpotent (the converse is also true \cite{wolf1968growth}). Since growth rate is invariant under quasi-isometry, it follows that any finitely generated group which is quasi-isometric to a space of polynomial growth necessarily has a nilpotent subgroup of finite index. Thus we have an instance of when the quasi-isometry type of a group determines some of its algebraic structure.

One of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,

One theme of geometric group theory is the rich relationship between the algebraic structure of groups and the large-scale geometry of spaces on which they act. By the ``large-scale geometry" of a space we mean the metric structure that is preserved by quasi-isometries, maps which preserve distances up to a controlled error. Gromov proposed in a 1983 ICM address \cite{gromov-icm} a broad research program of studying finitely generated groups as geometric objects and classifying them up to quasi-isometry. One aspect involves identifying instances of quasi-isometric rigidity, the phenomenon which occurs when algebraic properties of a finitely generated group are determined by its large-scale geometry. For example, a celebrated theorem by Gromov \cite{Gromov-polynomial} states that finitely generated groups of polynomial growth are virtually nilpotent (the converse is also true \cite{wolf1968growth}). Since growth rate is invariant under quasi-isometry, it follows that any finitely generated group which is quasi-isometric to a space of polynomial growth necessarily has a nilpotent subgroup of finite index. Thus we have an instance of when the quasi-isometry type of a group determines some of its algebraic structure.

One of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,

One of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,

One of the major breakthroughs leading to this general classification was the work of Schwartz \cite{Schwartz} on non-uniform lattices in rank one semisimple Lie groups. These Lie groups agree, up to index 2, with the isometry groups of the negatively curved symmetric spaces: real, complex, quaternionic hyperbolic space, and the Cayley hyperbolic plane.

\begin{theorem}[Schwartz]\label{Schwartz-theorem}
 Let $X$ be a negatively curved symmetric space other than the real hyperbolic plane $\bH^2$. If $\Gamma$ is a finitely generated group quasi-isometric to a non-uniform lattice $\Lambda$ in $\Isom{X}$, then there exists a short exact sequence
 \[
 \begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda' \ar[r] & 1.
 \end{tikzcd}
 \]
 where $\Lambda' \le \Isom{X}$ is a non-uniform lattice commensurable to $\Lambda$, and $F$ is a finite group.
\end{theorem}

\begin{theorem}[Frigerio--Lafont--Sisto]\label{FLS}
 If $\Gamma$ is a finitely generated group quasi-isometric to $\pi_1(M) \times \bZ^d$, then there exist short exact sequences
 \begin{equation*}
 \begin{tikzcd}
 1 \ar[r] & \bZ^d \ar[r,"j"] & \Gamma' \ar[r] & \Delta \ar[r] & 1
 \end{tikzcd}
 \end{equation*}
 \begin{equation*}
 \begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Delta \ar[r] & \pi_1(M') \ar[r] & 1
 \end{tikzcd}
 \end{equation*}
 where $\Gamma' \le \Gamma$ has finite index, $M'$ is a finite-sheeted covering of $M$, $\Delta$ is a group, and $F$ is a finite group. Moreover, $j(\bZ^d)$ is contained in the center of $\Gamma'$. In other words, $\Gamma$ is virtually a central extension by $\bZ^d$ of a finite extension of $\pi_1(M')$.
\end{theorem}

\begin{thmx}\label{Theorem A}
 Let $X \ne \bH^2$ be a negatively curved symmetric space. Let $\Lambda$ be a non-uniform lattice in $\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\Gamma$ is a finitely generated group quasi-isometric to $\Lambda \times L$, then there exist short exact sequences
 \begin{equation}\label{virtually nilcentral extension}
 \begin{tikzcd}
 1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
 \end{tikzcd}
 \end{equation}
 \begin{equation*}
 \begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
 \end{tikzcd}
 \end{equation*}
 where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.
\end{thmx}

\begin{theorem}
Let $X \ne \bH^2$ be a negatively curved symmetric space. Let $\Lambda$ be a non-uniform lattice in $\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\Gamma$ is a finitely generated group quasi-isometric to $\Lambda \times L$, then there exist short exact sequences
 \begin{equation}
 \begin{tikzcd}
 1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
 \tag{\ref{virtually nilcentral extension}}
 \end{tikzcd}
 \end{equation}
 \begin{equation*}
 \begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
 \end{tikzcd}
 \end{equation*}
 where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.
\end{theorem}

\begin{proof}
By Proposition \ref{kernel qi to N}, $\ker\theta$ is quasi-isometric to $N$, where $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. In particular $\ker\theta$ has polynomial growth. Then by Gromov's polynomial growth theorem, $\ker\theta$ is virtually nilpotent. So $\ker\theta$ has a finite-index, hence finitely generated, nilpotent subgroup $K$. Such groups are virtually torsion-free, so take a finite-index torsion-free subgroup $K' \le K$. Since $\ker\theta$ is finitely generated, it has only finitely many subgroups with the same index as $K'$. Let $L'$ denote their intersection. Then $L'$ is a finite-index characteristic subgroup of $\ker\theta$. Moreover, $L'$ is finitely generated, nilpotent, and torsion-free. A theorem of Malcev \cite{malcev} then says that $L'$ embeds as a lattice in a simply connected nilpotent Lie group $N'$, known as the (real) Malcev completion of $L'$ (see also \cite[Theorem 2.18]{raghunathan1972discrete}). Since $L'$ is characteristic in $\ker\theta$, it is normal in $\Gamma$. By construction, $\Gamma/L'$ is an extension of $\Lambda_\Gamma = \Gamma/\ker\theta$ by the finite group $F = \ker\theta/L'$. By Corollary \ref{commensurable}, $\Lambda_\Gamma$ is virtually a finite-index subgroup $\Lambda' \le \Lambda$ which is also a non-uniform lattice in $\text{Isom}(X)$. Let $\Gamma' = \theta^{-1}(\Lambda')$ and $\Delta = \Gamma'/L'$. Then we obtain the desired short exact sequences.

Now suppose we are in the situation of Theorem \ref{Theorem A} and we have the following two short exact sequences.
\begin{equation}\label{vne}
\begin{tikzcd}
 1 \ar[r] & N \ar[r] & \Gamma \ar[r] & \Delta \ar[r] & 1
\end{tikzcd}
\end{equation}
\begin{equation}\label{finite extension}
\begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda \ar[r] & 1
\end{tikzcd}
\end{equation}
where $N$ is a nilpotent lattice (we are now using $N$ to denote the lattice, not the ambient Lie group), $F$ is finite, and $\Lambda$ is a non-uniform lattice in $\text{Isom}(X)$, where $X \ne \bH^2$ is a negatively curved symmetric space. Since $N$ is finitely generated, nilpotent, and torsion-free, $Z(N/Z_i) = Z_{i+1}/Z_i$ is a finitely generated free abelian group. That is, $Z(N/Z_i) = \bZ^{d_i}$ for some $d_i$, and so $\Sigma(N) = \max_i d_i$. We now focus on the situation when $X$ is quaternionic hyperbolic space or the Cayley hyperbolic plane, and $\text{Isom}(X)$ has sufficiently large dimension.

\begin{thmx}\label{Theorem A}
    Let $X \ne \bH^2$ be a negatively curved symmetric space. Let $\Lambda$ be a non-uniform lattice in $\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\Gamma$ is a finitely generated group quasi-isometric to $\Lambda \times L$, then there exist short exact sequences
    \begin{equation}\label{virtually nilcentral extension}
    \begin{tikzcd}
        1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
    \end{tikzcd}
    \end{equation}
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
    \end{tikzcd}
    \end{equation*}
    where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.
\end{thmx}

\begin{equation}\label{virtually nilcentral extension}
    \begin{tikzcd}
        1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
    \end{tikzcd}
    \end{equation}

One of the major breakthroughs leading to this general classification was the work of Schwartz \cite{Schwartz} on non-uniform lattices in rank one semisimple Lie groups. These Lie groups agree, up to index 2, with the isometry groups of the negatively curved symmetric spaces: real, complex, quaternionic hyperbolic space, and the Cayley hyperbolic plane.

\begin{theorem}[Schwartz]\label{Schwartz-theorem}
    Let $X$ be a negatively curved symmetric space other than the real hyperbolic plane $\mathbb{H}^2$. If $\Gamma$ is a finitely generated group quasi-isometric to a non-uniform lattice $\Lambda$ in $\textup{Isom}({X})$, then there exists a short exact sequence
    \[
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda' \ar[r] & 1.
    \end{tikzcd}
    \]
    where $\Lambda' \le \textup{Isom}({X})$ is a non-uniform lattice commensurable to $\Lambda$, and $F$ is a finite group.
\end{theorem}

In the case of $X = \mathbb{H}^2$, non-uniform lattices in $\Isom{\mathbb{H}^2} = \text{PSL}(2,\mathbb{R})$ are virtually the fundamental groups of complete hyperbolic surfaces with finitely many punctures, and hence are virtually free. Any group quasi-isometric to a virtually free group is also virtually free, and every free group can be realized as a non-uniform lattice in $\text{PSL}(2,\mathbb{R})$, so quasi-isometric rigidity does hold in this case. However, the additional conclusion of commensurability fails. One reason for this failure is that commensurability preserves arithmeticity, and $\text{PSL}(2,\mathbb{R})$ contains both arithmetic and non-arithmetic non-uniform lattices. But since these lattices are virtually free, they are quasi-isometric to each other.

Following Schwartz, lattices such as $\Lambda$ and $\Lambda'$ shall from now on be called \textit{non-uniform rank one lattices}. Then an informal summary might be: any group quasi-isometric to a non-uniform rank one lattice is almost a commensurable non-uniform rank one lattice.

In an extensive study on high-dimensional graph manifolds \cite{FLS}, Frigerio--Lafont--Sisto prove, among many other things, a quasi-isometric rigidity result for products of the form $\pi_1(M) \times \mathbb{Z}^d$, where $M$ is a complete non-compact finite-volume hyperbolic $m$-manifold, $m\ge3$.

\begin{theorem}[Frigerio--Lafont--Sisto]\label{FLS}
    If $\Gamma$ is a finitely generated group quasi-isometric to $\pi_1(M) \times \mathbb{Z}^d$, then there exist short exact sequences
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & \mathbb{Z}^d \ar[r,"j"] & \Gamma' \ar[r] & \Delta \ar[r] & 1
    \end{tikzcd}
    \end{equation*}
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Delta \ar[r] & \pi_1(M') \ar[r] & 1
    \end{tikzcd}
    \end{equation*}
    where $\Gamma' \le \Gamma$ has finite index, $M'$ is a finite-sheeted covering of $M$, $\Delta$ is a group, and $F$ is a finite group. Moreover, $j(\mathbb{Z}^d)$ is contained in the center of $\Gamma'$. In other words, $\Gamma$ is virtually a central extension by $\mathbb{Z}^d$ of a finite extension of $\pi_1(M')$.
\end{theorem}

Observe that the case $d = 0$ is covered by Schwartz' theorem, and indeed, the proof of this theorem applies many of the ideas and results from \cite{Schwartz}.

\subsection{Main results}

Our main contribution is a generalization of Theorem \ref{FLS} to products $\Lambda \times L$, where $\Lambda$ is a non-uniform rank one lattice and $L$ is a lattice in a simply connected nilpotent Lie group. From now on, a lattice such as $L$ is called a \textit{nilpotent lattice}. Our first theorem says that up to finite noise, any group quasi-isometric to $\Lambda \times L$ is an extension of a non-uniform rank one lattice commensurable to $\Lambda$ by a nilpotent lattice quasi-isometric to $L$.

\begin{thmx}\label{Theorem A}
    Let $X \ne \mathbb{H}^2$ be a negatively curved symmetric space. Let $\Lambda$ be a non-uniform lattice in $\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\Gamma$ is a finitely generated group quasi-isometric to $\Lambda \times L$, then there exist short exact sequences
    \begin{equation}\label{virtually nilcentral extension}
    \begin{tikzcd}
        1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
    \end{tikzcd}
    \end{equation}
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
    \end{tikzcd}
    \end{equation*}
    where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.
\end{thmx}

The general outline of the proof is similar to that of the proof of Theorem \ref{FLS}. First, the quasi-isometry between $\Gamma$ and $\Lambda \times L$ induces a quasi-action of $\Gamma$ on $B \times N$, where $B \subset X$ is the neutered space associated to $\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \cite{KKL} guarantees that quasi-isometries $B \times N \to B \times N$ project to quasi-isometries $B \to B$, up to bounded error. Thus the quasi-action of $\Gamma$ on $B\times N$ induces a quasi-action of $\Gamma$ on $B$. A key result in \cite{Schwartz} is that quasi-isometries of the neutered space $B$ have finite distance (with respect to the sup norm) from isometries of $X$. Thus we obtain a homomorphism $\theta : \Gamma \to \text{Isom}(X)$, and we show that the image $\text{im} \hspace{.8mm} \theta$ is a non-uniform lattice commensurable to $\Lambda$. The action $\theta : \Gamma \to \text{Isom}(X)$ came from a quasi-action on $B$, which itself was coarsely projected from a quasi-action on $B \times N$. Hence we are able to show that the kernel of $\theta$ is quasi-isometric to $N$. Then $\Gamma = \text{im} \hspace{.8mm} \theta / \ker \theta$, so we pass to finite-index subgroups as necessary to obtain the desired short exact sequences.

Theorem \ref{FLS} asserts that $\mathbb{Z}^d$  can be made central in the group extension. In our more general setting, however, the extension is by a nilpotent group. So we define the notion of a nilcentral extension, analogous to that of a central extension, and find conditions which are sufficient to guarantee that (\ref{virtually nilcentral extension}) may be taken to be a nilcentral extension. Given a nilpotent group $G$ with upper central series $1 = Z_0 \lhd Z_1 \lhd \dots \lhd Z_n = G$, define
\[
    \Sigma(G) \vcentcolon= \max_i \text{rank}(Z_{i+1}/Z_i).
\]

\begin{thmx}\label{Theorem B}
    Assume the hypotheses of Theorem \ref{Theorem A} and let $L'$ be the nilpotent lattice obtained from the conclusion of the theorem. If $X$ is either quaternionic hyperbolic space or the Cayley hyperbolic plane, and $\dim\textup{Isom}(X) > \Sigma(L')$, then the group extension (\ref{virtually nilcentral extension}) in the conclusion of Theorem \ref{Theorem A} is virtually nilcentral.
\end{thmx}

Our proof of this theorem relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T). The precise statement we apply is in \cite{fisher2012strengthening}.

First, the quasi-isometry between $\Gamma$ and $\Lambda \times L$ induces a quasi-action of $\Gamma$ on $B \times N$, where $B \subset X$ is the neutered space associated to $\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \cite{KKL} implies that quasi-isometries $B \times N \to B \times N$ project to quasi-isometries $B \to B$ (up to bounded error), so the quasi-action on $B\times N$ induces a quasi-action on $B$.

A key result in \cite{Schwartz} says quasi-isometries of the neutered space $B$ have finite distance (in the sup norm) from isometries of $X$. Therefore one obtains a homomorphism $\theta: \Gamma \to \Isom(X)$, and one shows $\mathrm{im}\,\theta$ is a non-uniform lattice commensurable to $\Lambda$.

Because $\theta$ comes from the quasi-action on $B$ that was coarsely projected from the quasi-action on $B\times N$, one can show $\ker\theta$ is quasi-isometric to $N$. Since “$\Gamma = \mathrm{im}\,\theta / \ker\theta$”, one then passes to finite-index subgroups as necessary to obtain the desired short exact sequences (as in Theorem~\ref{Theorem A}).

For Theorem~\ref{Theorem B}, after noting that in this setting the extension is “by a nilpotent group” (not necessarily central), the authors “define the notion of a nilcentral extension” and “find conditions” (involving $\Sigma(L')$ and $\dim\Isom(X)$) ensuring that \eqref{virtually nilcentral extension} is “virtually nilcentral”; the proof “relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T).”

First, the quasi-isometry between $\Gamma$ and $\Lambda \times L$ induces a quasi-action of $\Gamma$ on $B \times N$, where $B \subset X$ is the neutered space associated to $\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \cite{KKL} implies that quasi-isometries $B \times N \to B \times N$ project to quasi-isometries $B \to B$ (up to bounded error), so the quasi-action on $B\times N$ induces a quasi-action on $B$.

A key result in \cite{Schwartz} says quasi-isometries of the neutered space $B$ have finite distance (in the sup norm) from isometries of $X$. Therefore one obtains a homomorphism $\theta: \Gamma \to \Isom(X)$, and one shows $\mathrm{im}\,\theta$ is a non-uniform lattice commensurable to $\Lambda$.

Because $\theta$ comes from the quasi-action on $B$ that was coarsely projected from the quasi-action on $B\times N$, one can show $\ker\theta$ is quasi-isometric to $N$. Since “$\Gamma = \mathrm{im}\,\theta / \ker\theta$”, one then passes to finite-index subgroups as necessary to obtain the desired short exact sequences, namely the conclusion that there exist short exact sequences
\begin{equation}\label{virtually nilcentral extension}
    \begin{tikzcd}
        1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
    \end{tikzcd}
    \end{equation}
\begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
    \end{tikzcd}
    \end{equation*}
where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.

For Theorem~\ref{Theorem B}, after noting that in this setting the extension is “by a nilpotent group” (not necessarily central), the authors “define the notion of a nilcentral extension” and “find conditions” (involving $\Sigma(L')$ and $\dim\Isom(X)$) ensuring that the equation above is “virtually nilcentral”; the proof “relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T).”

Question: Let $G$ be a semisimple Lie group, and let an irreducible lattice in $G$ mean a lattice (that is, a discrete subgroup of finite covolume) which is irreducible in the usual sense. Suppose $\Gamma$ is a finitely generated group that is quasi-isometric to an irreducible lattice in $G$. Which existence statement holds?

There exist a finite group $F$ and a lattice $\Lambda$ in $G$ such that $\Gamma$ fits into a short exact sequence
\[
1 \to F \to \Gamma \to \Lambda \to 1.
\]

Choices:
B.
There exist a finite-index subgroup $\Gamma'\le \Gamma$, a finite group $F$, and an irreducible lattice $\Lambda$ in $G$ such that $\Gamma'$ fits into a short exact sequence
\[
1 \to F \to \Gamma' \to \Lambda \to 1.
\]
C. There exists a lattice $\Lambda$ in $G$ and a finite normal subgroup $F \trianglelefteq \Gamma$ such that $\Gamma/F \cong \Lambda$.
D.
There exist a finite group $F$ and an irreducible lattice $\Lambda$ in $G$ such that $\Gamma$ fits into a short exact sequence
\[
1 \to F \to \Gamma \to \Lambda \to 1.
\]
E.
There exist a lattice $\Lambda$ in $G$ and a finite-index subgroup $\Lambda'\le \Lambda$ such that $\Gamma$ fits into a short exact sequence
\[
1 \to \Lambda' \to \Gamma \to F \to 1
\]
for some finite group $F$.

explicit short-exact-sequence phrasing replaced by quotient formulation


[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}

Existential–Universal
Inequality or Bound

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 $|w_0|\le R$