\label{main}
Let $H$ be a split classical group over $F$,
and let $\phi \in \Phi_\mathrm{temp}(H(F))$.
Suppose that one of the following holds:
\begin{description}
\item[(B)]
$H = \SO_{2n+1}$, $p > 6n+3$ and $\phi$ is arbitrary;
\item[(C)]
$H = \Sp_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$;
\item[(D)]
$H = \SO_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$.
\end{description}
Set $\psi = \widehat\phi$.

\begin{enumerate}
\item
There exists $\pi \in \Pi_\psi$ such that
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi) \in \bar\mathcal{N}(\pi)^{\max}.
\]
\item
For $\pi \in \Pi_\psi$ and for $\oo_H^\mathrm{st} \in \bar\mathcal{N}(\pi)$,
if $\oo_H^\mathrm{st} \not= d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$, then
\[
\dim(\oo_H^\mathrm{st}) < \dim(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)).
\]
Here, $\dim(\oo_H^\mathrm{st})$ is the Zariski dimension of the orbit as an algebraic variety.
\item
If we further assume Hypothesis \ref{hypo} below, then Conjecture \ref{upper} is true for $\phi$, i.e.,
\[
\bar\mathcal{N}(\pi)^{\max} \leq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)
\]
for any $\pi \in \Pi_\psi$.
\end{enumerate}

\begin{thm}\label{main}
Let $H$ be a split classical group over $F$,
and let $\phi \in \Phi_\temp(H(F))$.
Suppose that one of the following holds:
\begin{description}
\item[(B)]
$H = \SO_{2n+1}$, $p > 6n+3$ and $\phi$ is arbitrary;
\item[(C)]
$H = \Sp_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$;
\item[(D)]
$H = \SO_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$.
\end{description}
Set $\psi = \widehat\phi$.

\begin{enumerate}
\item
There exists $\pi \in \Pi_\psi$ such that
\[
d_{H^\vee}(\Ad(H^\vee)N_\phi) \in \bar\NN(\pi)^{\max}.
\]
\item
For $\pi \in \Pi_\psi$ and for $\oo_H^\st \in \bar\NN(\pi)$,
if $\oo_H^\st \not= d_{H^\vee}(\Ad(H^\vee)N_\phi)$, then
\[
\dim(\oo_H^\st) < \dim(d_{H^\vee}(\Ad(H^\vee)N_\phi)).
\]
Here, $\dim(\oo_H^\st)$ is the Zariski dimension of the orbit as an algebraic variety.
\item
If we further assume Hypothesis \ref{hypo} below, then Conjecture \ref{upper} is true for $\phi$, i.e.,
\[
\bar\NN(\pi)^{\max} \leq d_{H^\vee}(\Ad(H^\vee)N_\phi)
\]
for any $\pi \in \Pi_\psi$.
\end{enumerate}
\end{thm}

This work is motivated by the elusive relation
between the irreducible admissible Harish-Chandra characters of a reductive $p$-adic group
and the Langlands-Arthur parameters.
Arguably the best-known instance of this relation is the Hiraga--Ichino--Ikeda formal degree formula
for square-integrable characters in terms of adjoint $\gamma$-factors \cite{HII}.
The formal degree concerns the coefficient of the distribution attached to the zero nilpotent orbit
in the Harish-Chandra--Howe local character expansion.
At the opposite end,
the largest orbits that contribute to the character expansion determine the wavefront set of the character distribution.
They are the focus of the present paper.
Under a technical assumption, which is unnecessary for $\SO_{2n+1}$, and for $p$ large (see Theorem \ref{main} for the precise conditions), we prove that
the largest-dimensional orbits in the union of the geometric wavefront sets of the representations
in the Aubert--Zelevinsky dual of a tempered $L$-packet for a split classical $p$-adic group
{\it equal} the Spaltenstein dual of the corresponding tempered nilpotent Langlands parameter.
\par

A relation of this sort, Conjecture \ref{upper},  was proposed in \cite{CK,HLLS},
motivated by the results of \cite{CMBO1,CMBO2} for unipotent representations of split (and `inner-to-split') $p$-adic groups.
The fact that such a link should exist had already been apparent from
the foundational work of Adams--Barbasch--Vogan on the definition and construction of microlocal Arthur packets \cite{ABV}.
In Theorem \ref{main}, we also show that the full Conjecture \ref{upper} follows by further assuming Hypothesis \ref{hypo}.
\vskip 10pt

We fix a non-archimedean local field $F$ of characteristic zero and of residue characteristic $p \gg 0$.
Let $\overline{F}$ be a fixed algebraic closure of $F$.
In this section, we state our main theorem (Theorem \ref{main})
and explain our idea for the proof.

\subsection{Wavefront set}
Let $H$ be a connected reductive group over $F$ with the Lie algebra $\mathfrak{h}$.
We denote by $\mathcal{N}(\mathfrak{h}(F))$ (resp.~ $\mathcal{N}(\mathfrak{h}(\overline{F}))$)
the set of nilpotent $\mathrm{Ad}(H(F))$-orbits (resp.~ $\mathrm{Ad}(H(\overline{F}))$-orbits)
in $\mathfrak{h}(F)$ (resp.~ $\mathfrak{h}(\overline{F})$).
For an irreducible admissible representation $\pi$ of $H(F)$,
we denote its distribution character by $\Theta_\pi$.
The Harish-Chandra--Howe local character expansion (\cite[Theorem 16.2]{HC}) states that
\[
\Theta_\pi(\exp(X)) = \sum_{\oo_H \in \mathcal{N}(\mathfrak{h}(F))} c_{\oo_H}(\pi) \hat\mu_{\oo_H}(\exp(X))
\]
for $X \in \mathfrak{h}(F)$ sufficiently near to $0$,
where $\hat\mu_{\oo_H}$ is the Fourier transform of the orbital integral,
and $c_{\oo_H}(\pi)$ is a constant.
Set $\mathcal{N}(\pi) = \{ \oo_H \in \mathcal{N}(\mathfrak{h}(F)) \;|\; c_{\oo_H}(\pi)\not= 0\}$
and let $\bar\mathcal{N}(\pi)$ be the set of $\oo_H^\mathrm{st} \in \mathcal{N}(\mathfrak{h}(\overline{F}))$
such that $\oo_H^\mathrm{st}$ contains a rational orbit $\oo_H$ belonging to $\mathcal{N}(\pi)$.
We denote by $\bar\mathcal{N}(\pi)^{\max}$ the subset of $\bar\mathcal{N}(\pi)$ consisting of maximal orbits
with respect to the closure ordering
\[
\oo_H^\mathrm{st} \leq \oo_H'^\mathrm{st} \overset{\text{def}}{\iff} \oo_H^\mathrm{st} \subset \overline{\oo_H'^\mathrm{st}}.
\]
We call $\bar\mathcal{N}(\pi)^{\max}$ the \emph{(geometric) wavefront set} of $\pi$.

\subsection{$L$-parameters}
Denote by $H^\vee$ the complex Langlands dual group,
and by $\mathfrak{h}^\vee$ the Lie algebra of $H^\vee$.
Let $W_F$ be the Weil group of $F$.
An \emph{$L$-parameter} for $H(F)$ is a homomorphism
\[
\phi \colon W_F \times \SL_2(\mathbb{C}) \rightarrow H^\vee
\]
such that $\phi(W_F)$ consists of semisimple elements, $\phi|_{W_F}$ is smooth and $\phi|_{\SL_2(\mathbb{C})}$ is algebraic.
It is called \emph{tempered} if $\phi(W_F)$ is bounded.
Let $\Phi(H(F))$ be the set of $H^\vee$-conjugacy classes of $L$-parameters for $H(F)$,
and let $\Phi_\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters.
For $\phi \in \Phi(H(F))$, by the derivative of $\phi|_{\SL_2(\mathbb{C})}$,
we have a linear map
\[
d\phi|_{\SL_2(\mathbb{C})} \colon \sl_2(\mathbb{C}) \rightarrow \mathfrak{h}^\vee.
\]
We denote by $N_{\phi} \in \mathfrak{h}^\vee$ the image of
$\begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix}$ under this map.
The $\mathrm{Ad}(H^\vee)$-orbit of $N_\phi$ is denoted by $\mathrm{Ad}(H^\vee) N_\phi$.
\par

Let $\mathcal{N}(\mathfrak{h}^\vee)$ be the set of nilpotent $\mathrm{Ad}(H^\vee)$-orbits in $\mathfrak{h}^\vee$.
We denote by
\[
d_{H} \colon \mathcal{N}(\mathfrak{h}(\overline{F})) \rightarrow \mathcal{N}(\mathfrak{h}^\vee), \quad
d_{H^\vee} \colon \mathcal{N}(\mathfrak{h}^\vee) \rightarrow \mathcal{N}(\mathfrak{h}(\overline{F}))
\]
the Spaltenstein duality maps (\cite[Section 10]{Sp}).
The images of these maps are the sets of \emph{special nilpotent conjugacy classes}.
Moreover, $\oo_H^\mathrm{st} \leq d_{H^\vee}(d_H(\oo_H^\mathrm{st}))$,
and the equality holds if and only if $\oo_H^\mathrm{st}$ is special.
See \cite[Appendix A]{BV} for more details.
An $L$-parameter $\phi \in \Phi(H(F))$ gives
a nilpotent orbit
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee) N_\phi) \in \mathcal{N}(\mathfrak{h}(\overline{F})).
\]

\subsection{$L$-packets and $A$-packets}
Now suppose that $H$ is a split classical group over $F$.
Namely, $H$ is a symplectic group or a special orthogonal group.
Let $\mathrm{Irr}(H(F))$ be the set of equivalence classes of irreducible representations of $H(F)$,
and let $\Irr_\mathrm{temp}(H(F))$ be its subset consisting of tempered representations.
The local Langlands conjecture proven by Arthur \cite[Theorem 2.2.1]{Ar} gives a canonical surjective map
\[
\mathrm{Irr}(H(F)) \rightarrow \Phi(H(F))
\]
with finite fibers, and preserving the temperedness.
The fiber of $\phi \in \Phi(H(F))$ is denoted by $\Pi_\phi$
and is called the \emph{$L$-packet} associated to $\phi$.
\par

We say that an irreducible representation $\pi$ of $H(F)$ is \emph{co-tempered}
if its Zelevinsky--Aubert dual $\hat\pi$ is tempered.
For $\phi \in \Phi_\mathrm{temp}(H(F))$, its \emph{dual $A$-parameter}
\[
\psi = \widehat\phi \colon W_F \times \SL_2(\mathbb{C}) \times \SL_2(\mathbb{C}) \rightarrow H^\vee
\]
is defined by $\psi(w,g_1,g_2) = \phi(w,g_2)$.
Its \emph{$A$-packet} is given by
\[
\Pi_\psi = \{\pi \in \mathrm{Irr}(H(F)) \;|\; \hat\pi \in \Pi_\phi\}.
\]

\subsection{Conjecture and the main result}
The following conjecture was independently formulated by Kim and the second author \cite[Conjectures 1.1, 1.9]{CK}
and Hazeltine--Liu--Lo--Shahidi \cite[Conjecture 1.1]{HLLS}.

\begin{conj}\label{upper}
Let $\phi \in \Phi(H(F))$.
Then for any $\pi \in \Pi_\phi$, we have
\[
\bar\mathcal{N}(\hat\pi)^{\max} \leq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]
Moreover, if $\phi$ is tempered,
then the equality holds for some $\pi \in \Pi_\phi$.
\end{conj}

This is relevant for any connected reductive group $H$
provided the local Langlands correspondence for $H(F)$ is assumed.
This conjecture is known in the following cases.

\begin{itemize}
\item
$H=\GL_m$, \cite{MW}, see \cite[Theorem 1.4]{CK}.

\item
$\pi$ is any depth-zero simple supercuspidal representation of a classical group (\cite[Theorem 1.8]{CK}).

\item
{$H$ is an inner form of a split group and}
$\pi$ is any unipotent representation of $H(F)$ with real infinitesimal character
(\cite[Theorem 1.4.1]{CMBO2}).

\item
$H$ is an exceptional group $G_2$ (\cite[Theorem 1.10]{CK}).
\end{itemize}

As explained in \cite[Theorems 1.5, 1.8]{HLLS},
Conjecture \ref{upper} implies
a part of Jiang's conjecture and a part of the Generalized Shahidi Conjecture on local ABV packets
(see \cite[Conjectures 1.4, 1.7]{HLLS}).

\begin{rem}
An analogue of Conjecture \ref{upper} for covering groups
has been formulated by Gao--Liu--Lo--Shahidi \cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\GL_m$ in \cite[Theorem 1.3]{GLLS}.
\end{rem}

In this paper, we give a result toward Conjecture \ref{upper} for split classical groups under some conditions.
We focus on the case where $\phi$ is tempered
since Conjecture \ref{upper} is reduced to this case by \cite[Proposition 2.2]{CK} or \cite[Theorem 1.2]{HLLS}.
For $\phi \in \Phi_\mathrm{temp}(H(F))$, we consider the following condition:
\begin{description}
\item[($\star$)]
for any irreducible self-dual subrepresentation $\phi_0 \subset \phi$,
its determinant $\det(\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.
\end{description}
Our main result is stated as follows.

\subsection{Wavefront set}
Let $H$ be a connected reductive group over $F$ with the Lie algebra $\mathfrak{h}$.
We denote by $\mathcal{N}(\mathfrak{h}(F))$ (resp.~ $\mathcal{N}(\mathfrak{h}(\overline{F}))$)
the set of nilpotent $\mathrm{Ad}(H(F))$-orbits (resp.~ $\mathrm{Ad}(H(\overline{F}))$-orbits)
in $\mathfrak{h}(F)$ (resp.~ $\mathfrak{h}(\overline{F})$).
For an irreducible admissible representation $\pi$ of $H(F)$,
we denote its distribution character by $\Theta_\pi$.
The Harish-Chandra--Howe local character expansion (\cite[Theorem 16.2]{HC}) states that
\[
\Theta_\pi(\exp(X)) = \sum_{\oo_H \in \mathcal{N}(\mathfrak{h}(F))} c_{\oo_H}(\pi) \hat\mu_{\oo_H}(\exp(X))
\]
for $X \in \mathfrak{h}(F)$ sufficiently near to $0$,
where $\hat\mu_{\oo_H}$ is the Fourier transform of the orbital integral,
and $c_{\oo_H}(\pi)$ is a constant.
Set $\mathcal{N}(\pi) = \{ \oo_H \in \mathcal{N}(\mathfrak{h}(F)) \;|\; c_{\oo_H}(\pi)\not= 0\}$
and let $\bar\mathcal{N}(\pi)$ be the set of $\oo_H^\mathrm{st} \in \mathcal{N}(\mathfrak{h}(\overline{F}))$
such that $\oo_H^\mathrm{st}$ contains a rational orbit $\oo_H$ belonging to $\mathcal{N}(\pi)$.
We denote by $\bar\mathcal{N}(\pi)^{\max}$ the subset of $\bar\mathcal{N}(\pi)$ consisting of maximal orbits
with respect to the closure ordering
\[
\oo_H^\mathrm{st} \leq \oo_H'^\mathrm{st} \overset{\text{def}}{\iff} \oo_H^\mathrm{st} \subset \overline{\oo_H'^\mathrm{st}}.
\]
We call $\bar\mathcal{N}(\pi)^{\max}$ the \emph{(geometric) wavefront set} of $\pi$.

\subsection{$L$-parameters}
Denote by $H^\vee$ the complex Langlands dual group,
and by $\mathfrak{h}^\vee$ the Lie algebra of $H^\vee$.
Let $W_F$ be the Weil group of $F$.
An \emph{$L$-parameter} for $H(F)$ is a homomorphism
\[
\phi \colon W_F \times \SL_2(\mathbb{C}) \rightarrow H^\vee
\]
such that $\phi(W_F)$ consists of semisimple elements, $\phi|_{W_F}$ is smooth and $\phi|_{\SL_2(\mathbb{C})}$ is algebraic.
It is called \emph{tempered} if $\phi(W_F)$ is bounded.
Let $\Phi(H(F))$ be the set of $H^\vee$-conjugacy classes of $L$-parameters for $H(F)$,
and let $\Phi_\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters.
For $\phi \in \Phi(H(F))$, by the derivative of $\phi|_{\SL_2(\mathbb{C})}$,
we have a linear map
\[
d\phi|_{\SL_2(\mathbb{C})} \colon \sl_2(\mathbb{C}) \rightarrow \mathfrak{h}^\vee.
\]
We denote by $N_{\phi} \in \mathfrak{h}^\vee$ the image of
$\begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix}$ under this map.
The $\mathrm{Ad}(H^\vee)$-orbit of $N_\phi$ is denoted by $\mathrm{Ad}(H^\vee) N_\phi$.
\par

Let $\mathcal{N}(\mathfrak{h}^\vee)$ be the set of nilpotent $\mathrm{Ad}(H^\vee)$-orbits in $\mathfrak{h}^\vee$.
We denote by
\[
d_{H} \colon \mathcal{N}(\mathfrak{h}(\overline{F})) \rightarrow \mathcal{N}(\mathfrak{h}^\vee), \quad
d_{H^\vee} \colon \mathcal{N}(\mathfrak{h}^\vee) \rightarrow \mathcal{N}(\mathfrak{h}(\overline{F}))
\]
the Spaltenstein duality maps (\cite[Section 10]{Sp}).
The images of these maps are the sets of \emph{special nilpotent conjugacy classes}.
Moreover, $\oo_H^\mathrm{st} \leq d_{H^\vee}(d_H(\oo_H^\mathrm{st}))$,
and the equality holds if and only if $\oo_H^\mathrm{st}$ is special.
See \cite[Appendix A]{BV} for more details.
An $L$-parameter $\phi \in \Phi(H(F))$ gives
a nilpotent orbit
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee) N_\phi) \in \mathcal{N}(\mathfrak{h}(\overline{F})).
\]

\begin{conj}\label{upper}
Let $\phi \in \Phi(H(F))$.
Then for any $\pi \in \Pi_\phi$, we have
\[
\bar\mathcal{N}(\hat\pi)^{\max} \leq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]
Moreover, if $\phi$ is tempered,
then the equality holds for some $\pi \in \Pi_\phi$.
\end{conj}

\begin{rem}
An analogue of Conjecture \ref{upper} for covering groups
has been formulated by Gao--Liu--Lo--Shahidi \cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\GL_m$ in \cite[Theorem 1.3]{GLLS}.
\end{rem}

In this paper, we give a result toward Conjecture \ref{upper} for split classical groups under some conditions.
We focus on the case where $\phi$ is tempered
since Conjecture \ref{upper} is reduced to this case by \cite[Proposition 2.2]{CK} or \cite[Theorem 1.2]{HLLS}.
For $\phi \in \Phi_\mathrm{temp}(H(F))$, we consider the following condition:
\begin{description}
\item[($\star$)]
for any irreducible self-dual subrepresentation $\phi_0 \subset \phi$,
its determinant $\det(\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.
\end{description}
Our main result is stated as follows.

\begin{hyp}\label{hypo}
Let $\oo_H^\st \in \NN(\h(\overline{F}))$ be a special nilpotent orbit.
Define $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$ by
\[
\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)).
\]
Then for any constants $\{c_{\oo_H} \;|\; \oo_H \subset \oo_H^\st\}$ such that
$\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}$ is stable,
there exist constants $\{a_{\oo_G} \;|\; \oo_G \subset \oo_G^\st\}$ such that
\[
\sum_{\oo_G \subset \oo_G^\st} a_{\oo_G} \hat\mu_{\oo_G}
\quad\text{is equal to the transfer of}\quad
\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
\end{hyp}

\begin{conj}\label{upper}
Let $\phi \in \Phi(H(F))$.
Then for any $\pi \in \Pi_\phi$, we have
\[
\bar\NN(\hat\pi)^{\max} \leq d_{H^\vee}(\Ad(H^\vee)N_\phi).
\]
Moreover, if $\phi$ is tempered,
then the equality holds for some $\pi \in \Pi_\phi$.
\end{conj}

\begin{thm}\label{twist}
Let $\phi \colon W_F \times \SL_2(\C) \rightarrow G^\vee$
be a self-dual tempered $L$-parameter for $G(F) = \GL_m(F)$,
and let $\pi_\psi$ be the Zelevinsky dual of the irreducible tempered representation
$\pi_\phi$ corresponding to $\phi$.
Then
\[
\bar\NN_\theta(\tl\pi_\psi)^{\max} = \{d_{G^\vee}(\Ad(G^\vee) N_\phi)\}.
\]
Moreover, $c_{\oo_G}(\tl\pi_\psi) = c_{\oo_G'}(\tl\pi_\psi)$
for any rational orbits $\oo_G$ and $\oo_G'$ contained in $d_{G^\vee}(\Ad(G^\vee) N_\phi)$.
\end{thm}
\begin{proof}
Recall that $\bar\NN(\pi_\psi)^{\max} = \{d_{G^\vee}(\Ad(G^\vee) N_\phi)\}$
by M{\oe}glin--Waldspurger \cite{MW} as explained in \cite[Theorem 1.4]{CK}.
Then Varma showed in \cite[Lemma 5.29]{V} that
for any $\oo_G^\st \in \bar\NN_\theta(\tl\pi_\psi)^{\max}$,
the corresponding degenerate Whittaker model is nonzero.
Hence we have
\[
\oo_G^\st \leq d_{G^\vee}(\Ad(G^\vee) N_\phi).
\]
On the other hand, by Konno \cite[Theorem 4.1]{K}
together with the multiplicity one result of degenerate Whittaker models by M{\oe}glin--Waldspurger \cite{MW}
(cf., see \cite[Proposition 4.4]{K}),
we have
\[
d_{G^\vee}(\Ad(G^\vee) N_\phi) \in \bar\NN_\theta(\tl\pi_\psi)^{\max}.
\]
Hence we obtain the first assertion.
The last assertion follows from \cite[Theorem 4.1(2), Remark 4.2]{K}.
\end{proof}

As a consequence of Theorem \ref{transfer}, we obtain the following.
\begin{cor}\label{cor1}
We have
\[
d_{H^\vee}(\Ad(H^\vee) N_\phi) \in \bar\NN(\psi).
\]
Moreover, for $\oo_H^\st \in \bar\NN(\psi)$,
if $\oo_H^\st \not= d_{H^\vee}(\Ad(H^\vee) N_\phi)$, then
\[
\dim(\oo_H^\st) < \dim(d_{H^\vee}(\Ad(H^\vee) N_\phi)).
\]
\end{cor}
\begin{proof}
Set $d = \dim(d_{H^\vee}(\Ad(H^\vee) N_\phi))$.
As in the proof of Theorem \ref{transfer},
consider the unique $L$-parameter $\phi_0 \in \Phi_\temp(H(F))$
such that $\phi_0$ is trivial on $W_F$ and
\[
\Ad(H^\vee) N_{\phi_0} = \Ad(H^\vee) N_\phi.
\]
Set $\psi_0 = \widehat\phi_0$.
For a constant $e \in \C^\times$
the transfer of $S\Theta_\psi - eS\Theta_{\psi_0}$ is
\[
\Theta_{\tl\pi_\psi}-e\Theta_{\tl\pi_{\psi_0}}
= \sum_{\oo_G \in \NN(\g^\theta(F))} \left( c_{\oo_G}(\tl\pi_\psi)-e c_{\oo_G}(\tl\pi_{\psi_0}) \right) \hat\mu_{\oo_G}.
\]
Since $d_{G^\vee}(\Ad(G^\vee) N_\phi) = d_{G^\vee}(\Ad(G^\vee) N_{\phi_0})$,
by Theorem \ref{twist}, we can choose $e \in \C^\times$ such that
for $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$,
if $c_{\oo_G}(\tl\pi_\psi)-e c_{\oo_G}(\tl\pi_{\psi_0}) \not= 0$ for some $\oo_G \subset \oo_G^\st$,
then $\oo_G^\st < d_{G^\vee}(\Ad(G^\vee) N_\phi)$.
This together with Theorem \ref{transfer} implies that
\[
\sum_{\substack{\oo_H^\st \in \NN(\h(\overline{F})) \\ \dim(\oo_H^\st) \geq d}}
\left( \sum_{\oo_H \subset \oo_H^\st} c_{\oo_H}(\psi) \hat\mu_{\oo_H}\right)
=
e\sum_{\substack{\oo_H^\st \in \NN(\h(\overline{F})) \\ \dim(\oo_H^\st) \geq d}}
\left( \sum_{\oo_H \subset \oo_H^\st} c_{\oo_H}(\psi_0) \hat\mu_{\oo_H}\right)
\]
By \cite[Theorem 1.4.1]{CMBO2},
in the right-hand side, only $\oo_H^\st = d_{H^\vee}(\Ad(H^\vee) N_\phi)$ can contribute.
By the linear independence of $\hat\mu_{\oo_H}$ (\cite[Theorem 5.11]{HC}),
the same holds for the left-hand side.
\end{proof}

\begin{prop}\label{refine}
Assume Hypothesis \ref{hypo}.
Then for any $\oo_H^\st \in \bar\NN(\psi)$,
we have
\[
\oo_H^\st \leq d_{H^\vee}(\Ad(H^\vee) N_\phi).
\]
\end{prop}
\begin{proof}
For each special orbit $\oo_H^\st \in \NN(\h(\overline{F}))$,
let $\{a_{\oo_G}(\psi) \;|\; \oo_G \subset d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st))\}$
be the constants in Hypothesis \ref{hypo}
given by $\{c_{\oo_H}(\psi) \;|\; \oo_H \subset \oo_H^\st\}$.
Then by considering the transfer of $S\Theta_\psi$,
we see that the local character expansion of $\Theta_{\tl\pi_\psi}$ is given by
\[
\Theta_{\tl\pi_\psi} = \sum_{\oo_H^\st}
\sum_{\oo_G \subset \oo_G^\st} a_{\oo_G}(\psi) \hat\mu_{\oo_G},
\]
where $\oo_H^\st$ runs over special nilpotent orbits in $\NN(\h(\overline{F}))$
and $\oo_G$ runs over rational orbits contained in
$\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st))$.
In particular, if $\oo_H^\st \in \bar\NN(\psi)$,
then $d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)) \in \bar\NN_\theta(\tl\pi_\psi)$.
By Theorem \ref{twist}, we have
\[
d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)) \leq d_{G^\vee}(\Ad(G^\vee) N_\phi).
\]
Hence $\Ad(G^\vee)d_{H}(\oo_H^\st) \geq \Ad(G^\vee) N_\phi$.
Note that
\[
\Ad(G^\vee)d_{H}(\oo_H^\st) \cap H^\vee = d_{H}(\oo_H^\st),
\quad
\Ad(G^\vee) N_\phi \cap H^\vee = \Ad(H^\vee) N_\phi.
\]
(Here, if $H=\SO_{2n}$, we consider $\O_{2n}(\C)$-orbits actually. See Remark \ref{OvsSO}.)
Hence $d_{H}(\oo_H^\st) \geq \Ad(H^\vee) N_\phi$,
and we conclude that
$\oo_H^\st \leq d_{H^\vee}(d_{H}(\oo_H^\st)) \leq d_{H^\vee}(\Ad(H^\vee) N_\phi)$.
\end{proof}

Now we assume that we have
\begin{itemize}
\item
$\oo_H^\st \in \NN(\h(\overline{F}))$
with $\dim(\oo_H^\st) \geq \dim(d_{H^\vee}(\Ad(H^\vee)N_\phi))$;
\item
$\oo_H \in \NN(\h(F))$ with $\oo_H \subset \oo_H^\st$;
\item
$\pi_0 \in \Pi_\psi$ with $\psi = \widehat\phi$
\end{itemize}
such that $c_{\oo_H}(\pi_0) \not= 0$.
The goal is to show that $\oo_H^\st = d_{H^\vee}(\Ad(H^\vee)N_\phi)$.
\par

By the induction hypothesis, we have
$\dim(\oo_{H_i}^\st) \leq \dim (d_{H_i^\vee}(\Ad(H_i^\vee)N_{\phi_i}))$ for $i \in \{1,2\}$.
Hence by Proposition \ref{WS} and Lemma \ref{dimW},
we have
\begin{align*}
\dim(\oo_H^\st) &= \dim\left(W(\oo_{H_1}^\st, \oo_{H_2}^\st)\right)
\\&\leq \dim\left(
W(d_{H_1^\vee}(\Ad(H_1^\vee)N_{\phi_1}), d_{H_2^\vee}(\Ad(H_2^\vee)N_{\phi_2}))
\right)
\\&\leq
\dim\left(
d_{H^\vee}(\Ad(H^\vee)\xi(
d_{H_1}(d_{H_1^\vee}(\Ad(H_1^\vee)N_{\phi_1})), d_{H_2}(d_{H_2^\vee}(\Ad(H_2^\vee)N_{\phi_2}))
))
\right)
\\&\leq
\dim\left(
d_{H^\vee}(\Ad(H^\vee)\xi(\Ad(H_1^\vee)N_{\phi_1}, \Ad(H_2^\vee)N_{\phi_2}))
\right)
\\&=
\dim\left(d_{H^\vee}(\Ad(H^\vee)N_{\phi})\right).
\end{align*}
Here, we used the facts
that $d_{H_i}(d_{H_i^\vee}(\Ad(H_i^\vee)N_{\phi_i})) \geq \Ad(H_i^\vee)N_{\phi_i}$
and that $d_{H^\vee}$ is an order-reversing map.
Since we assume that $\dim(\oo_H^\st) \geq \dim\left(d_{H^\vee}(\Ad(H^\vee)N_{\phi})\right)$,
this must be an equality.
Moreover, it together with Lemma \ref{dimW} implies that
$\oo_{H_i}^\st = d_{H_i^\vee}(\Ad(H_i^\vee)N_{\phi_i})$ for $i \in \{1,2\}$.
Then by proposition \ref{WS}, we have
\begin{align*}
\oo_{H^\st}
&\leq d_{H^\vee}(\Ad(H^\vee)\xi(d_{H_1}(\oo_{H_1}^\st), d_{H_2}(\oo_{H_2}^\st)))
\\&\leq d_{H^\vee}(\Ad(H^\vee)\xi(\Ad(H_1^\vee)N_{\phi_1}, \Ad(H_2^\vee)N_{\phi_2}))
\\&= d_{H^\vee}(\Ad(H^\vee)N_{\phi}).
\end{align*}
Since $\dim(\oo_H^\st) = \dim\left(d_{H^\vee}(\Ad(H^\vee)N_{\phi})\right)$,
we conclude that $\oo_H^\st = d_{H^\vee}(\Ad(H^\vee)N_\phi)$,
as desired.
\par

The proof of Theorem \ref{main} (3) is similar.
Suppose that $\oo_H^\st \in \NN(\h(\overline{F}))$ contains a rational orbit $\oo_H$
such that $c_{\oo_H}(\pi_0) \not= 0$ for some $\pi_0 \in \Pi_\psi$.
Consider $s \in \Sc_\psi$ satisfying ($\ast$).
If $s = 1$, then Proposition \ref{refine} implies that $\oo_H^\st \leq d_{H^\vee}(\Ad(H^\vee)N_\phi)$.
Otherwise, we can write $\oo_H^\st = W(\oo_{H_1}^\st, \oo_{H_2}^\st)$ as above.
Then $\oo_{H_i}^\st \leq d_{H_i^\vee}(\Ad(H_i^\vee)N_{\phi_i})$ by the induction hypothesis.
By Proposition \ref{WS} and Lemma \ref{Worder} below, we have
\begin{align*}
\oo_H^\st &= W(\oo_{H_1}^\st, \oo_{H_2}^\st)
\\&\leq W(d_{H_1^\vee}(\Ad(H_1^\vee)N_{\phi_1}), d_{H_2^\vee}(\Ad(H_2^\vee)N_{\phi_2}))
\\&\leq
d_{H^\vee}(\Ad(H^\vee)\xi(
d_{H_1}(d_{H_1^\vee}(\Ad(H_1^\vee)N_{\phi_1})), d_{H_2}(d_{H_2^\vee}(\Ad(H_2^\vee)N_{\phi_2}))
))
\\&\leq
d_{H^\vee}(\Ad(H^\vee)\xi(\Ad(H_1^\vee)N_{\phi_1}, \Ad(H_2^\vee)N_{\phi_2}))
\\&=
d_{H^\vee}(\Ad(H^\vee)N_{\phi}).
\end{align*}
This completes the proof of Theorem \ref{main}.
\end{proof}

\subsection{Outline of the proof}\label{sec.outline}
Throughout the paper, we assume that $p \gg 0$ as in Theorem \ref{main}.
The proof of Theorem \ref{main} is similar to the proofs of the Generic Packet Conjecture
by Konno \cite{K} and Varma \cite{V}.
The main idea is to compare local character expansions via
\emph{endoscopic character identities}.
\par

Fix $\phi \in \Phi_\mathrm{temp}(H(F))$ and set $\psi = \widehat\phi$.
Set $G = \GL_m$, where $m$ is the dimension of the standard representation $H^\vee \rightarrow G^\vee = \GL_m(\mathbb{C})$.
By composing with this map, we can regard $\psi$ as an $A$-parameter for $G(F)$,
and obtain an irreducible self-dual representation $\pi_\psi$ of $G(F)$.
Define an involution $\theta$ on $G(F)$ by
\[
\theta(g) = J{}^tg^{-1}J^{-1},
\quad
J = \begin{pmatrix}
&&&1 \\
&&-1&\\
&\iddots&& \\
(-1)^{m-1}&&
\end{pmatrix} \in G(F).
\]
We can extend $\pi_\psi$ to an irreducible representation $\tl\pi_\psi$ of $G \rtimes \langle \theta \rangle$.
The twisted character of $\tl\pi_\psi$ is denoted by $\Theta_{\tl\pi_\psi}$.
\par

Let $\Sc_\psi$ be the component group for $\psi$,
and let $\widehat\Sc_\psi$ be its Pontryagin dual.
Recall that Arthur defined a map
\[
\Pi_\psi \rightarrow \widehat\Sc_\psi, \;
\pi \mapsto \langle \cdot, \pi \rangle_\psi.
\]
Notice that this is bijective since $\psi = \widehat\phi$.
Set $s_\psi = \psi(\1_{W_F}, \mathbf{1}, -\mathbf{1}) = \phi(\1_{W_F}, -\mathbf{1})$.
In this setting, \cite[Theorem 2.2.1]{Ar} asserts that
\begin{description}
\item[(T1)]
the distribution
\[
S\Theta_{\psi} = \sum_{\pi \in \Pi_\psi} \langle s_\psi, \pi \rangle_\psi \Theta_\pi
\]
is stable, and $\Theta_{\tl\pi_\psi}$ is the transfer of $S\Theta_\psi$;
\item[(T2)]
Each $s \in \Sc_\psi$ with $s \not= 1$ gives an elliptic endoscopic group $H_1 \times H_2$ of $H$
and an $A$-parameter $\psi_i$ for $H_i$
(of the form $\psi_i = \widehat\phi_i$ for some $\phi_i \in \Phi_\mathrm{temp}(H_i)$) such that
\[
S\Theta_{\psi, s} = \sum_{\pi \in \Pi_\psi} \langle s \cdot s_\psi, \pi \rangle_\psi \Theta_\pi
\]
is the transfer of $S\Theta_{\psi_1} \otimes S\Theta_{\psi_2}$.
\end{description}
\par

The first step of the proof of Theorem \ref{main} is to establish an analogue for the twisted $\GL_m$ case.
By the twisted local character expansion established by Clozel \cite[Theorem 3]{C},
we can expand
\[
\Theta_{\tl\pi_\psi}
= \sum_{\oo_G \in \mathcal{N}(\mathfrak{g}^\theta(F))} c_{\oo_G}(\tl\pi_\psi) \hat\mu_{\oo_G}.
\]
See Section \ref{sec.TLCE} below for the notations.
We define $\NN_\theta(\tl\pi_\psi) \subset \mathcal{N}(\mathfrak{g}^\theta(F))$
and $\bar\NN_\theta(\tl\pi_\psi) \subset \mathcal{N}(\mathfrak{g}^\theta(\overline{F}))$
similar to $\mathcal{N}(\pi)$ and $\bar\mathcal{N}(\pi)$, respectively.
In Theorem \ref{twist} below, we will show that
\[
\bar\NN_\theta(\tl\pi_\psi)^{\max} = \{d_{G^\vee}(\mathrm{Ad}(G^\vee) N_\phi)\}.
\]
This is an immediate consequence of
results of Konno \cite{K}, Varma \cite{V}, and M{\oe}glin--Waldspurger \cite{MW}.
Moreover, Konno's result \cite[Theorem 4.1(2)]{K} says that
the constant $c_{\oo_G}(\tl\pi_\psi)$ for $\oo_G \subset d_{G^\vee}(\mathrm{Ad}(G^\vee) N_\phi)$
does not depend on the choice of the rational orbit.
\par

Set $\oo_H^\mathrm{st} = d_{H^\vee}(\mathrm{Ad}(H^\vee) N_\phi)$ and $\oo_G^\mathrm{st} = d_{G^\vee}(\mathrm{Ad}(G^\vee) N_\phi)$.
The second step of the proof of Theorem \ref{main} is to show that
there exists a constant $\gamma_{\oo_H}$ for each rational orbit $\oo_H \subset \oo_H^\mathrm{st}$ such that
\[
\sum_{\oo_G \subset \oo_G^\mathrm{st}} \hat\mu_{\oo_G}
\quad
\text{is equal to the transfer of}
\quad
\sum_{\oo_H \subset \oo_H^\mathrm{st}} \gamma_{\oo_H}\hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
See Theorem \ref{transfer} below.
Hypothesis \ref{hypo} is an analogue of this result.
These claims could be proven by similar arguments to the standard case explained in the next paragraph,
but there is no explicit reference in the literature.
Thus we will prove Theorem \ref{transfer} in this paper.
The proof uses Conjecture \ref{upper} in the case established in \cite{CMBO2},
which requires that $H$ is split over $F$.
\par

An analogous result for the standard endoscopy with unramified endoscopic groups
is known by Waldspurger \cite{W_pave}.
Let $H_1 \times H_2$ be an unramified endoscopic group of $H$.
For a special nilpotent orbit $\oo_{H_i}^\mathrm{st} \in \mathcal{N}(\h_i(\overline{F}))$,
Waldspurger explicitly defined a nilpotent orbit
\[
\oo_H^\mathrm{st} = W(\oo_{H_1}^\mathrm{st}, \oo_{H_2}^\mathrm{st}) \in \mathcal{N}(\mathfrak{h}(\overline{F})),
\]
and proved a certain transfer result in \cite[XII.9 Th\'eor\`eme]{W_pave}.
The third step of the proof of Theorem \ref{main} is to relate the Waldspurger map $W(\oo_{H_1}^\mathrm{st}, \oo_{H_2}^\mathrm{st})$
and the Spaltenstein duality (Proposition \ref{WS}).
\par

Combining the two transfer results, we can relate the local character expansion of $S\Theta_\psi$ (resp.~ $S\Theta_{\psi,s}$)
with the one of $\Theta_{\tl\pi_\psi}$ (resp.~ $S\Theta_{\psi_1} \otimes S\Theta_{\psi_2}$).
Then by induction on $m$,
we can deduce Theorem \ref{main} in Section \ref{sec.proof}.
To use \cite[XII.9 Th\'eor\`eme]{W_pave}, we need the condition ($\star$) in Cases (C) and (D),
which is equivalent to saying that all endoscopic groups $H_1 \times H_2$ arising from $s \in \Sc_\psi$ are unramified.

\begin{rem}\label{OvsSO}
The local Langlands correspondence for $H = \SO_{2n}$ as proven by Arthur,
is given up to $\O_{2n}(F)$-conjugation.
Thus, in turn, we should replace $\mathrm{Ad}(H^\vee)N_\phi$ by $\mathrm{Ad}(\O_{2n}(\mathbb{C}))N_\phi$ in our results.
We will ignore this discrepancy since it is not an important difference for the geometric wavefront set results.
By replacing $\SO_{2n}$ with $\O_{2n}$, we can use the following properties:
\begin{enumerate}
\item
the distributions on $\SO_{2n}(F)$ considered in the paper are all $\O_{2n}(F)$-invariant;
\item
the transfer of stable distributions on $H(F)$ to $G(F) \rtimes \theta$ is injective;
\item
$\mathrm{Ad}(G^\vee)\oo_{H^\vee} \cap H^\vee = \oo_{H^\vee}$ for $\oo_{H^\vee} \in \mathcal{N}(\mathfrak{h}^\vee)$.
\end{enumerate}
Note that (2) and (3) hold when $H = \SO_{2n+1}$ or $H = \Sp_{2n}$.
\end{rem}

The proof of Theorem~\ref{main} follows Konno~\cite{K} and Varma~\cite{V}: the "main idea is to compare local character expansions via \emph{endoscopic character identities}." Fix $\phi\in\Phi_\mathrm{temp}(H(F))$, set $\psi=\widehat\phi$, and embed into $G=\GL_m$ via the standard map $H^\vee\to G^\vee$. This yields a self-dual $\pi_\psi$ of $G(F)$, extended to $\tilde\pi_\psi$ on $G\rtimes\langle\theta\rangle$, whose twisted character $\Theta_{\tilde\pi_\psi}$ matches Arthur’s stable distribution
\[
S\Theta_{\psi}=\sum_{\pi\in\Pi_\psi}\langle s_\psi,\pi\rangle_\psi\,\Theta_\pi
\]
by endoscopic character identities (Arthur’s (T1)), and similarly for $S\Theta_{\psi,s}$ (Arthur’s (T2)).

(1) "The first step" is an analogue in the twisted $\GL_m$ case: using Clozel’s twisted local character expansion
\[
\Theta_{\tilde\pi_\psi}=\sum_{\oo_G\in\mathcal N(\mathfrak g^\theta(F))} c_{\oo_G}(\tilde\pi_\psi)\,\hat\mu_{\oo_G},
\]
one defines $\bar\mathcal N_\theta(\tilde\pi_\psi)^{\max}$ and proves (Theorem~\ref{twist})
\[
\bar\mathcal N_\theta(\tilde\pi_\psi)^{\max}=\{d_{G^\vee}(\mathrm{Ad}(G^\vee)N_\phi)\},
\]
using results of Konno, Varma, and M\oe glin--Waldspurger; additionally, Konno’s result implies the constants $c_{\oo_G}(\tilde\pi_\psi)$ for $\oo_G\subset d_{G^\vee}(\mathrm{Ad}(G^\vee)N_\phi)$ do not depend on the rational orbit.

(2) Setting $\oo_H^{\mathrm{st}}=d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$ and $\oo_G^{\mathrm{st}}=d_{G^\vee}(\mathrm{Ad}(G^\vee)N_\phi)$, "the second step" is a transfer statement (Theorem~\ref{transfer}): for each rational orbit $\oo_H\subset\oo_H^{\mathrm{st}}$ there is a constant $\gamma_{\oo_H}$ such that on a small neighborhood of $\theta$,
\[
\sum_{\oo_G\subset\oo_G^{\mathrm{st}}}\hat\mu_{\oo_G}\quad\text{equals the transfer of}\quad\sum_{\oo_H\subset\oo_H^{\mathrm{st}}}\gamma_{\oo_H}\hat\mu_{\oo_H}.
\]
Hypothesis~\ref{hypo} is described as an analogue of this transfer result.

(3) "The third step" relates Waldspurger’s explicit map for standard endoscopy,
\[
\oo_H^{\mathrm{st}}=W(\oo_{H_1}^{\mathrm{st}},\oo_{H_2}^{\mathrm{st}}),
\]
to Spaltenstein duality (Proposition~\ref{WS}).

Finally, "combining the two transfer results," one relates the local character expansions of $S\Theta_\psi$ and $S\Theta_{\psi,s}$ to those of $\Theta_{\tilde\pi_\psi}$ and $S\Theta_{\psi_1}\otimes S\Theta_{\psi_2}$, and then "by induction on $m$" deduces Theorem~\ref{main} (in Section~\ref{sec.proof}). The condition $(\star)$ in Cases (C) and (D) is needed to apply Waldspurger’s theorem, equivalently ensuring the relevant endoscopic groups $H_1\times H_2$ are unramified.

The proof of Theorem~\ref{main} follows Konno~\cite{K} and Varma~\cite{V}: the "main idea is to compare local character expansions via \emph{endoscopic character identities}." Fix $\phi\in\Phi_\mathrm{temp}(H(F))$, set $\psi=\widehat\phi$, and embed into $G=\GL_m$ via the standard map $H^\vee\to G^\vee$. This yields a self-dual $\pi_\psi$ of $G(F)$, extended to $\tilde\pi_\psi$ on $G\rtimes\langle\theta\rangle$, whose twisted character $\Theta_{\tilde\pi_\psi}$ matches Arthur’s stable distribution
\[
S\Theta_{\psi}=\sum_{\pi\in\Pi_\psi}\langle s_\psi,\pi\rangle_\psi\,\Theta_\pi
\]
by endoscopic character identities (Arthur’s (T1)), and similarly for $S\Theta_{\psi,s}$ (Arthur’s (T2)).

(1) "The first step" is an analogue in the twisted $\GL_m$ case: using Clozel’s twisted local character expansion
\[
\Theta_{\tilde\pi_\psi}=\sum_{\oo_G\in\mathcal N(\mathfrak g^\theta(F))} c_{\oo_G}(\tilde\pi_\psi)\,\hat\mu_{\oo_G},
\]
one defines $\bar\mathcal N_\theta(\tilde\pi_\psi)^{\max}$ and proves the following theorem.
\begin{thm}\label{twist}
Let $\phi \colon W_F \times \SL_2(\C) \rightarrow G^\vee$
be a self-dual tempered $L$-parameter for $G(F) = \GL_m(F)$,
and let $\pi_\psi$ be the Zelevinsky dual of the irreducible tempered representation
$\pi_\phi$ corresponding to $\phi$.
Then
\[
\bar\NN_\theta(\tl\pi_\psi)^{\max} = \{d_{G^\vee}(\Ad(G^\vee) N_\phi)\}.
\]
Moreover, $c_{\oo_G}(\tl\pi_\psi) = c_{\oo_G'}(\tl\pi_\psi)$
for any rational orbits $\oo_G$ and $\oo_G'$ contained in $d_{G^\vee}(\Ad(G^\vee) N_\phi)$.
\end{thm}
This uses results of Konno, Varma, and M\oe glin--Waldspurger.

(2) Setting $\oo_H^{\mathrm{st}}=d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$ and $\oo_G^{\mathrm{st}}=d_{G^\vee}(\mathrm{Ad}(G^\vee)N_\phi)$, "the second step" is the following transfer statement.
\begin{thm}\label{transfer}
Let $\oo_H^\st \in \NN(\h(\overline{F}))$ be a special nilpotent orbit.
Define $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$ by
\[
\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)).
\]
Then for each rational orbit $\oo_H \subset \oo_H^\st$,
there exists a constant $\gamma_{\oo_H} \in \C$ such that
\[
\sum_{\oo_G \subset \oo_G^\st} \hat\mu_{\oo_G}
\quad\text{is equal to the transfer of }\quad
\sum_{\oo_H \subset \oo_H^\st} \gamma_{\oo_H} \hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
\end{thm}
The following hypothesis is described as an analogue of this transfer result.
\begin{hyp}\label{hypo}
Let $\oo_H^\st \in \NN(\h(\overline{F}))$ be a special nilpotent orbit.
Define $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$ by
\[
\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)).
\]
Then for any constants $\{c_{\oo_H} \;|\; \oo_H \subset \oo_H^\st\}$ such that
$\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}$ is stable,
there exist constants $\{a_{\oo_G} \;|\; \oo_G \subset \oo_G^\st\}$ such that
\[
\sum_{\oo_G \subset \oo_G^\st} a_{\oo_G} \hat\mu_{\oo_G}
\quad\text{is equal to the transfer of}\quad
\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
\end{hyp}

(3) "The third step" relates Waldspurger’s explicit map for standard endoscopy,
\[
\oo_H^{\mathrm{st}}=W(\oo_{H_1}^{\mathrm{st}},\oo_{H_2}^{\mathrm{st}}),
\]
to Spaltenstein duality using the proposition below.
\begin{prop}\label{WS}
For $i \in \{1,2\}$, let $\oo_{H_i}^\st \in \NN(\h_i(\overline{F}))$ be a special nilpotent orbit.
Then
\[
W(\oo_{H_1}^\st, \oo_{H_2}^\st)
\leq d_{H^\vee}(\Ad(H^\vee)\xi(d_{H_1}(\oo_{H_1}^\st), d_{H_2}(\oo_{H_2}^\st))).
\]
\end{prop}

Finally, "combining the two transfer results," one relates the local character expansions of $S\Theta_\psi$ and $S\Theta_{\psi,s}$ to those of $\Theta_{\tilde\pi_\psi}$ and $S\Theta_{\psi_1}\otimes S\Theta_{\psi_2}$, and then "by induction on $m$" establishes the main theorem (later in the paper). The condition $(\star)$ in Cases (C) and (D) is needed to apply Waldspurger’s theorem, equivalently ensuring the relevant endoscopic groups $H_1\times H_2$ are unramified.

\label{main}
Let $H$ be a split classical group over $F$,
and let $\phi \in \Phi_\mathrm{temp}(H(F))$.
Suppose that one of the following holds:
\begin{description}
\item[(B)]
$H = \SO_{2n+1}$, $p > 6n+3$ and $\phi$ is arbitrary;
\item[(C)]
$H = \Sp_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$;
\item[(D)]
$H = \SO_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$.
\end{description}
Set $\psi = \widehat\phi$.

\begin{enumerate}
\item
There exists $\pi \in \Pi_\psi$ such that
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi) \in \bar\mathcal{N}(\pi)^{\max}.
\]
\item
For $\pi \in \Pi_\psi$ and for $\oo_H^\mathrm{st} \in \bar\mathcal{N}(\pi)$,
if $\oo_H^\mathrm{st} \not= d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$, then
\[
\dim(\oo_H^\mathrm{st}) < \dim(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)).
\]
Here, $\dim(\oo_H^\mathrm{st})$ is the Zariski dimension of the orbit as an algebraic variety.
\item
If we further assume the following hypothesis, then the corresponding upper-bound conjecture holds for $\phi$, i.e.,
\[
\bar\mathcal{N}(\pi)^{\max} \leq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)
\]
for any $\pi \in \Pi_\psi$.

\begin{hyp}\label{hypo}
Let $\oo_H^\st \in \NN(\h(\overline{F}))$ be a special nilpotent orbit.
Define $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$ by
\[
\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)).
\]
Then for any constants $\{c_{\oo_H} \;|\; \oo_H \subset \oo_H^\st\}$ such that
$\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}$ is stable,
there exist constants $\{a_{\oo_G} \;|\; \oo_G \subset \oo_G^\st\}$ such that
\[
\sum_{\oo_G \subset \oo_G^\st} a_{\oo_G} \hat\mu_{\oo_G}
\quad\text{is equal to the transfer of}\quad
\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
\end{hyp}

\begin{conj}\label{upper}
Let $\phi \in \Phi(H(F))$.
Then for any $\pi \in \Pi_\phi$, we have
\[
\bar\NN(\hat\pi)^{\max} \leq d_{H^\vee}(\Ad(H^\vee)N_\phi).
\]
Moreover, if $\phi$ is tempered,
then the equality holds for some $\pi \in \Pi_\phi$.
\end{conj}
\end{enumerate}

Question:
Let $H$ be a split classical group over a nonarchimedean local field $F$, and let $\phi\in \Phi_{\mathrm{temp}}(H(F))$ be a tempered $L$-parameter, where an $L$-parameter is a homomorphism $\phi:W_F\times \mathrm{SL}_2(\mathbb C)\to H^\vee$ with $\phi(W_F)$ semisimple, $\phi|_{W_F}$ smooth, and $\phi|_{\mathrm{SL}_2(\mathbb C)}$ algebraic. Let
\[
N_\phi=d\phi\bigg|_{\mathrm{SL}_2(\mathbb C)}\!egin{pmatrix}0&1\\0&0\end{pmatrix}\in \mathfrak h^\vee,
\]
and let $d_{H^\vee}$ be the Spaltenstein duality map from nilpotent $\mathrm{Ad}(H^\vee)$-orbits in $\mathfrak h^\vee$ to nilpotent stable orbits in $\mathfrak h(\overline F)$. For an irreducible admissible representation $\pi$ of $H(F)$, let $\bar{\mathcal N}(\pi)^{\max}$ denote its geometric wavefront set, i.e. the set of maximal nilpotent stable orbits in $\mathfrak h(\overline F)$ whose stable orbit contains a rational orbit occurring with nonzero coefficient in the Harish-Chandra--Howe local character expansion of $\pi$. Set $\psi=\widehat\phi$, and let $\Pi_\psi$ be the packet attached to $\psi$.

Assume one of the following:
- (B) $H=\mathrm{SO}_{2n+1}$, $p>6n+3$, and $\phi$ is arbitrary;
- (C) $H=\mathrm{Sp}_{2n}$, $p>6n$, and $\phi$ satisfies $(\star)$;
- (D) $H=\mathrm{SO}_{2n}$, $p>6n$, and $\phi$ satisfies $(\star)$,

where $(\star)$ means: for every irreducible self-dual subrepresentation $\phi_0\subset \phi$, the determinant $\det(\phi_0)$ is trivial on the inertia subgroup $I_F\subset W_F$.

Under these assumptions, which existence statement holds?

There exists a representation $\pi\in \Pi_\psi$ such that the nilpotent stable orbit $d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$ belongs to the geometric wavefront set of $\pi$, i.e.
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\in \bar{\mathcal N}(\pi)^{\max}.
\]

Choices:
B.
For every representation $\pi\in \Pi_\psi$, the nilpotent stable orbit $d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$ belongs to the geometric wavefront set of $\pi$, i.e.
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\in \bar{\mathcal N}(\pi)^{\max}.
\]
C.
There exists a representation $\pi\in \Pi_\psi$ such that the nilpotent stable orbit $d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$ lies in $\bar{\mathcal N}(\pi)$, i.e.
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\in \bar{\mathcal N}(\pi).
\]
D.
There exists a representation $\pi\in \Pi_\psi$ such that the geometric wavefront set of $\pi$ is exactly the singleton
\[
\bar{\mathcal N}(\pi)^{\max}=\left\{d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\right\}.
\]
E.
For every nilpotent stable orbit $\oo_H^{\mathrm{st}}\in \bar{\mathcal N}(\pi)$ attached to some representation $\pi\in \Pi_\psi$, if
\[
\dim(\oo_H^{\mathrm{st}})=\dim\bigl(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\bigr),
\]
then there exists a representation $\pi\in \Pi_\psi$ such that
\[
\bar{\mathcal N}(\pi)^{\max}=\{\oo_H^{\mathrm{st}}\}=\left\{d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\right\}.
\]

existential packet statement strengthened to uniform over all $\pi\in\Pi_\psi$

maximality in the geometric wavefront set

membership in $\bar{\mathcal N}(\pi)^{\max}$ strengthened to exact singleton equality

dimension-equality conclusion from part (2) converted into an existence-and-uniqueness wavefront statement


\begin{theorem}\label{thm}
Assume $1\leq p <2$, $d\geq 1$ and the pressure $P$ satisfies $P'(\bar{\rho})>0$ for $\bar{\rho}>0$. There is a small constant $\delta_1>0$ such that if
$
\|(n_0,\bv_0)\|_{\dB_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon}
+\varepsilon\|(n_0,\bv_0)\|_{\dB_{2,1}^{\frac{d}{2}+1}}^{h,J_\varepsilon}
\leq \delta_1,
$
then the Cauchy problem \eqref{1.2}-\eqref{1.2initial} admits a uniform global unique solution in $E_\infty^{J_\varepsilon}$ for all $\varepsilon>0$.
\end{theorem}

\begin{equation}\label{DefHLB}
\|f\|_{\dB_{q_1,1}^{s_1}}^{h,J_{\varepsilon}}\triangleq \sum_{j\geq J_{\varepsilon}}
2^{s_1j}\|\ddj f\|_{q_1}\quad \mbox{and}\quad
\|f\|_{\dB_{q_2,1}^{s_2}}^{\ell,J_{\varepsilon}}\triangleq \sum_{j\leq J_{\varepsilon}-1}
2^{s_2j}\|\ddj f\|_{q_2}.
\end{equation}

This work studies the compressible Euler equations with linear damping, a model describing gas flow through a porous medium where the solid matrix exerts a frictional force proportional to and opposite to the fluid momentum. The governing system is given by
\begin{eqnarray}\label{1.1}
\left\{\aligned
&\d_t \rho+\mbox{\rm div}\;\! (\rho \bu)=0, \\
&\d_t (\rho \bu)+\nabla \cdot (\rho \bu\otimes \bu)+\nabla P(\rho)+\alpha\rho \bu=0
\endaligned\right.
\end{eqnarray}
for $t\geq0,\ \bm{x}\in \R^d(d\geq 1)$ and a damping coefficient $\alpha>0$
. Here $\rho\in\mathbb{R}^{+}$, $\bu=(u^1, u^2,\cdots, u^d)^\top$ ($\top$ represents transpose) denote the density and velocity of fluid flow, respectively. The pressure function $P=P(\rho)$ is assumed to be smooth around the constant reference density $\bar{\rho}>0$.

There are lots of mathematical results about the existence and asymptotic behavior of system \eqref{1.1} in sobolev space (see \cite{HL,HuangFM2,Wangdehua,Sui,Tan3,XY} ). In critical Besov space, Fang and Xu \cite{FXu} (with improvements in \cite{Jiu}) studied the existence and asymptotic behavior of classical solutions. Later, Xu and Wang \cite{XuWang} justified the relaxation convergence from \eqref{1.1} to the porous medium equation. And also, there are some results on hyperbolic system for balance laws including \eqref{1.1} (see \cite{XuKawaARM14,XuKawaARM15}). Recently, Crin-Barat and Danchin \cite{BaratDanchin,BaratDanchin2,BaratDanchin3} improved those results \cite{XuKawaARM14,XuKawaARM15,XuWang} in the $L^2$-$L^p$ hybrid Besov spaces with $2\leq p\leq\max\{4,\frac{2d}{d+2}\}$, where the low frequencies are bounded in $L^p$-type spaces and the high frequencies in $L^2$-type spaces with a specific linearity assumption. And then, the first author and Xu \cite{XuZhang} remove the assumption and to show that the results of \cite{BaratDanchin,BaratDanchin2} hold true for the general pressure function. Very recently, Crin-Barat and Song \cite{BSong} extend the work of \cite{BaratDanchin,BaratDanchin2,XuZhang} to the case $p\in[2,\infty)$. And a natural question is how about $p<2$?

Introducing a new unknown called ``enthalpy"
$n(\rho)\triangleq \int_1^{\rho} \frac{P'(s)}{s} ds.$ System \eqref{1.1} can be rewritten as
\begin{eqnarray}\label{1.2}
\left\{\aligned
&\d_t n + \bv \cdot \nabla n + P'(\bar{\rho})\mbox{\rm div}\;\! \bv + G(n)\mbox{\rm div}\;\! \bv = 0 \\
&\d_t \bv+ \bv\cdot \nabla \bv+\nabla n= -\frac{1}{\varepsilon} \bv
\endaligned\right.
\end{eqnarray}
with the initial data
\begin{eqnarray}\label{1.2initial}
 (n,\bv)|_{t=0}=(n_0, \bv_0).
\end{eqnarray}
Here $P'(\bar{\rho})>0$ for $\bar{\rho}>0$, $\bar{\rho}=P'(\bar{\rho})=1$ and the composite function $G(n)$ is smooth.

Define $J_\varepsilon=-[\log_2\varepsilon]+k$ being the threshold between high and low frequencies with a suitable integer $k$ is to be determined later, and the space $E_T^{J_\varepsilon}$ as follows
\begin{equation*}
\begin{split}
&E_T^{J_\varepsilon}=\bigg\{(n,\bv)|(n^{\ell,J_\varepsilon},\bv^{\ell,J_\varepsilon})\in \cC_b([0,T];\dB_{p,1}^{\frac{d}{p}}),\
\varepsilon n^{\ell,J_\varepsilon}\in \tL_T^1(\dB_{p,1}^{\frac{d}{p}+2}),\
\bv^{\ell,J_\varepsilon}\in \tL_T^1(\dB_{p,1}^{\frac{d}{p}+1}),
\varepsilon^{-\frac{1}{2}} \bv\in \tL_T^2(\dB_{p,1}^{\frac{d}{p}}),\\
&\qquad\qquad
(\varepsilon n^{h,J_\varepsilon}, \varepsilon\bv^{h,J_\varepsilon})\in \cC_b([0,T];\dB_{2,1}^{\frac{d}{2}+1}),\
(n^{h,J_\varepsilon},\bv^{h,J_\varepsilon}) \in \tL_T^1(\dB_{2,1}^{\frac{d}{2}+1}),
\varepsilon^{-1}\bv+\nabla n\in \tL_T^1(\dB_{p,1}^{\frac{d}{p}})
\bigg\}.
\end{split}
\end{equation*}
When $T=\infty$ we also use $E_\infty^{J_\varepsilon}$ for convenience.
For the reader's convenience, We introduce the notations $\|\cdot\|_{\dB_{q_1,1}^s}^{h,J_{\varepsilon}}$ and
$\|\cdot\|_{\dB_{q_2,1}^s}^{\ell,J_{\varepsilon}}$ to denote Besov semi-norms with respect to the threshold $J_{\varepsilon}$, that is,
\begin{equation}\label{DefHLB}
\|f\|_{\dB_{q_1,1}^{s_1}}^{h,J_{\varepsilon}}\triangleq \sum_{j\geq J_{\varepsilon}}
2^{s_1j}\|\dot \Delta_j f\|_{q_1}\quad \mbox{and}\quad
\|f\|_{\dB_{q_2,1}^{s_2}}^{\ell,J_{\varepsilon}}\triangleq \sum_{j\leq J_{\varepsilon}-1}
2^{s_2j}\|\dot \Delta_j f\|_{q_2}.
\end{equation}
It is not difficult to deduce that for all $\sigma_0>0$,
\begin{equation}\label{HLEst}
\|f\|_{\dB_{q_1,1}^{s_1}}^{h,J_{\varepsilon}}
\leq
2^{-\sigma_0J_{\varepsilon}}\|f\|_{\dB_{q_1,1}^{s_1+\sigma_0}}^{h,J_{\varepsilon}}\quad \mbox{and}\quad
\|f\|_{\dB_{q_1,1}^{s_1}}^{\ell,J_{\varepsilon}}
\leq
2^{\sigma_0J_{\varepsilon}}\|f\|_{\dB_{q_1,1}^{s_1-\sigma_0}}^{\ell,J_{\varepsilon}}.
\end{equation}

The main goal of this paper is to broaden the assumption on $p$ and to show that the results of \cite{BaratDanchin,BaratDanchin2,XuZhang} hold true for the case $p<2$. Our main result is stated as follows.

\begin{eqnarray}\label{1.1}
\left\{\aligned
&\d_t \rho+\div (\rho \bu)=0, \\
&\d_t (\rho \bu)+\nabla \cdot (\rho \bu\otimes \bu)+\nabla P(\rho)+\alpha\rho \bu=0
\endaligned\right.
\end{eqnarray}

\begin{eqnarray}\label{1.2}
\left\{\aligned
&\d_t n + \bv \cdot  \nabla n + P'(\bar{\rho})\div \bv + G(n)\div \bv = 0 \\
&\d_t \bv+ \bv\cdot \nabla \bv+\nabla n= -\frac{1}{\varepsilon} \bv
\endaligned\right.
\end{eqnarray}

Introducing a new unknown called ``enthalpy"
$n(\rho)\triangleq \int_1^{\rho} \frac{P'(s)}{s} ds.$ System \eqref{1.1} can be rewritten as
\begin{eqnarray}\label{1.2}
\left\{\aligned
&\d_t n + \bv \cdot \nabla n + P'(\bar{\rho})\div \bv + G(n)\div \bv = 0 \\
&\d_t \bv+ \bv\cdot \nabla \bv+\nabla n= -\frac{1}{\varepsilon} \bv
\endaligned\right.
\end{eqnarray}
with the initial data
\begin{eqnarray}\label{1.2initial}
 (n,\bv)|_{t=0}=(n_0, \bv_0).
\end{eqnarray}
Here $P'(\bar{\rho})>0$ for $\bar{\rho}>0$, $\bar{\rho}=P'(\bar{\rho})=1$ and the composite function $G(n)$ is smooth.

\begin{remark}
Based on Theorem \ref{thm}, the relaxation limit for the case $p<2$ can be performed by the same procedure as \cite{BaratDanchin2}.
\end{remark}

\section{Global a priori estimates}
In this section, we only give the key a priori estimate for the case $p<2$, which lead to the global existence and uniqueness of solutions in $E_\infty^{J_\varepsilon}$. See \cite{BShouZhang} for more details. For convenience, we use $ f\lesssim g $ to denote that there exists a generic constant $C>0$ independent on $\varepsilon$ such that $f\leq C g$ in this section.

Finally, by classical remainder estimates (see \cite[Theorem 2.85]{chemin}) we can directly obtain
\begin{equation}\nonumber
\begin{aligned}
\|R[a,b]\|_{\dB_{p,1}^{s_1}}^{\ell,J_\varepsilon}
\lesssim
\|b\|_{\dB_{\frac{2p}{2-p},1}^{s_1-\frac{d}{2}}}\|a\|_{\dB_{2,1}^{\frac{d}{2}}}
\lesssim
\left(\|b\|_{\dB_{p,1}^{s_1}}^{\ell,J_\varepsilon}+ 2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{\dB_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \right)\|a\|_{\dB_{2,1}^{\frac{d}{2}}}.
\end{aligned}
\end{equation}
Adding above three inequality together, we can finally obtain \eqref{newproduct}.
\end{proof}
For simplicity, we define $\cX(T):=\|(n,\bv)\|_{E_T^{J\varepsilon}}$. The proof of Theorem \ref{thm} reduces to establishing a global-in-time a priori estimate. Specifically, we claim that if
\begin{equation*}\begin{split}
\|n\|_{L^\infty}+\|\bv\|_{L^\infty} \ll 1 \quad \text{on} \quad [0,T],
\end{split}
\end{equation*}
then there exists a constant $C$, independent of $T$ and $\varepsilon$, such that
\begin{equation}\label{APrioriEstimate}
\cX(T)\leq C\big( \cX(0)+\cX^2(T)\big) \qquad\text{for all } \varepsilon>0.
\end{equation}

Next, we will deal with the nonlinear part. For the first nonlinear term, it follows the Lemma \ref{NonClassicalProLaw1} that
\begin{equation*}
\begin{split}
\|\bv\cdot \nabla n\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
&\lesssim
\left(
\|\nabla n\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+ \|\nabla n\|_{\tL_T^2(\dB_{2,1}^{\frac{d}{2}})}^{h,J_\varepsilon}
\right)
\|\bv\|_{\tL_T^2(\dB_{2,1}^{\frac{d}{2}})}.
\end{split}
\end{equation*}
Then the interpolation inequalities (see Lemma 3.2 in \cite{XuZhang}) implies
\begin{equation}\label{NL1}
\begin{split}
\|\bv\cdot \nabla n\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
&\lesssim
\left(\left(\| n\|_{\tL_T^\infty(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
\|n\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon}\right)^{\frac{1}{2}}
+\left(\|n\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}
\|n\|_{\tL_T^1(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right)^{\frac{1}{2}}
\right)\\
&\quad
\times\left(\|\bv\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\varepsilon\left(\|\bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}
\|\bv\|_{\tL_T^1(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right)^{\frac{1}{2}}\right)\\
&\lesssim
\cX^2(T)+\cX(T)\varepsilon^{-\frac{1}{2}}\|\bv\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}.
\end{split}
\end{equation}
And similarly, for the convection term we can deduce
\begin{equation}\label{NL2}
\begin{split}
\|\bv\cdot \nabla \bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
\lesssim
\cX^2(T).
\end{split}
\end{equation}
For the last nonlinear term, the inequality \eqref{newproduct} implies
\begin{equation*}
\begin{split}
\|G(n)\div \bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
&\lesssim
\|G(n)\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}
\left(\|\div \bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+ \|\div \bv\|_{\tL_T^1(\dB_{2,1}^{\frac{d}{2}})}^{h,J_\varepsilon}\right).
\end{split}
\end{equation*}
By the classical composition estimate (see \cite{chemin}), we can obtain
\begin{equation}\label{NL3}
\begin{split}
\|G(n)\div \bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
\lesssim
\|n\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}
\left(\|\bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}+1})}^{\ell,J_\varepsilon} +\|\bv\|_{\tL_T^1(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right)
\lesssim
\cX^2(T).
\end{split}
\end{equation}

Inserting \eqref{NL1}-\eqref{NL3} into \eqref{LowEst3}, we can obtain
\begin{equation*}
\begin{split}
&\|(n, \varepsilon \bz)\|_{\tL_T^\infty(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\bigg(\varepsilon\|n\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon}
+\|\bz\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg)\\
&\quad\leq C\bigg(
\|(n_0, \varepsilon \bz_0)\|_{\dot B_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon}
+\cX^2(T)+\cX(T)\varepsilon^{-\frac{1}{2}}
\|\bv\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg) ,
\end{split}
\end{equation*}
which eventually leads to
\begin{equation}\label{LowEst4}
\begin{split}
&\|n\|_{\tL_T^\infty(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\|\bv\|_{\tL_T^\infty(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\bigg(\varepsilon\|n\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon}+\|\varepsilon^{-1}\bv+\nabla n\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\|\bv\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}+1})}^{\ell,J_\varepsilon}
+\varepsilon^{-\frac{1}{2}}
\|\bv\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg)\\
&\quad\lesssim
\cX(0)+\cX^2(T).
\end{split}
\end{equation}

Since $2^{J_\varepsilon}\approx 2^{k}\varepsilon^{-1}$, by the spatial embedding
$\dB_{p,1}^{\frac{d}{p}}(\R^d)\hookrightarrow \dB_{2,1}^{\frac{d}{2}}(\R^d)\hookrightarrow L^\infty(\R^d)$, the classical estimation on smooth functions (see Corollary 2.65 in \cite{chemin}) and \eqref{HLEst} we can obtain that
\begin{equation}\label{HNonLinEst1}
\begin{split}
\varepsilon\|(\nabla G(n),\nabla \bv)\|_{L_T^\infty(L^\infty)}
\lesssim
\varepsilon\|(n, \bv)\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}+1})}
\lesssim\cX(T).
\end{split}
\end{equation}
By the continuity equation, it is not difficult to get
\begin{equation}\label{HNonLinEst2}
\begin{split}
\varepsilon\|\d_t G(n)\|_{L_T^\infty(L^\infty)}
&\lesssim
\varepsilon \|\div \bv+\bv \cdot \nabla n
+ G(n)\div\bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\\
&\lesssim
\varepsilon\|\bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}+1})}
+\|\bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\varepsilon \|\nabla n\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\
+\|n\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\varepsilon \|\div \bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\\
&\lesssim
\cX(T)+\cX^2(T).
\end{split}
\end{equation}

\begin{remark}
Note that $\dot{B}_{p,1}^{\frac{d}{p}}\hookrightarrow \dot{B}_{2,1}^{\frac{d}{2}}$ when $p<2$, that means the existence and uniqueness of solutions hold in a smaller space rather than escaping to the complement $\dot{B}_{2,1}^{\frac{d}{2}}\setminus \dot{B}_{p,1}^{\frac{d}{p}}$. On the other hand, \eqref{DefHLB} implies that a smaller space for $p$ corresponds to more singular and concentrated data, which naturally arises in many physical problems (such as point sources and vortex filaments).
\end{remark}

\begin{remark}
Based on Theorem \ref{thm}, the relaxation limit for the case $p<2$ can be performed by the same procedure as \cite{BaratDanchin2}.
\end{remark}

The major difficulty of proof of Theorem \ref{thm} lies in dealing with the nonlinear terms. To achieve it, a new product estimation will be developed, see Lemma \ref{NonClassicalProLaw1} below. As a matter of fact, the new tool could be applied to investigate other systems, such as the hyperbolic-parabolic chemotaxis system (see \cite{BaratShou2}) in $L^2-L^p$ framework and Navier-Stokes equations (see \cite{Danchin} ) in $L^p-L^2$ framework for $p<2$.

The post-theorem introduction only indicates that "[t]he major difficulty of proof of Theorem \ref{thm} lies in dealing with the nonlinear terms" and that, "[t]o achieve it, a new product estimation will be developed, see Lemma \ref{NonClassicalProLaw1} below." No further proof steps are described.

The post-theorem introduction only indicates that "[t]he major difficulty of proof of the main theorem lies in dealing with the nonlinear terms" and that, "[t]o achieve it, a new product estimation will be developed," namely the following lemma.

\begin{lemma}\label{NonClassicalProLaw1}
Let $0<s_1<\frac{d}{2}$, $1\leq p<2$. Then, we have the following inequality
\begin{align}\label{newproduct}
\|ab\|_{\dB_{p,1}^{s_1}}^{\ell,J_\varepsilon}
\leq C
\left(\|b\|_{\dB_{p,1}^{s_1}}^{\ell,J_\varepsilon}+ 2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{\dB_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \right)\|a\|_{\dB_{2,1}^{\frac{d}{2}}}
\end{align}
with $C$ independent on $\varepsilon$.
\end{lemma}

No further proof steps are described.

\label{thm}
Assume $1\leq p <2$, $d\geq 1$ and the pressure $P$ satisfies $P'(\bar{\rho})>0$ for $\bar{\rho}>0$. There is a small constant $\delta_1>0$ such that if
$
\|(n_0,\bv_0)\|_{\dB_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon}
+\varepsilon\|(n_0,\bv_0)\|_{\dB_{2,1}^{\frac{d}{2}+1}}^{h,J_\varepsilon}
\leq \delta_1,
$
then the Cauchy problem
\begin{eqnarray}\label{1.2}
\left\{\aligned
&\d_t n + \bv \cdot  \nabla n + P'(\bar{\rho})\div \bv + G(n)\div \bv = 0 \\
&\d_t \bv+ \bv\cdot \nabla \bv+\nabla n= -\frac{1}{\varepsilon} \bv
\endaligned\right.
\end{eqnarray}
with initial data
\begin{eqnarray}\label{1.2initial}
 (n,\bv)|_{t=0}=(n_0, \bv_0).
\end{eqnarray}
admits a uniform global unique solution in $E_\infty^{J_\varepsilon}$ for all $\varepsilon>0$.

Question:
Let $1\le p<2$, $d\ge 1$, and let the pressure law $P$ satisfy $P'(\bar\rho)>0$ for $\bar\rho>0$. For each $\varepsilon>0$, set $J_\varepsilon=-[\log_2\varepsilon]+k$ for a fixed suitable integer $k$. If $f=\sum_j \dot\Delta_j f$ is the homogeneous Littlewood--Paley decomposition, write $f^{\ell,J_\varepsilon}$ and $f^{h,J_\varepsilon}$ for the low- and high-frequency parts relative to $J_\varepsilon$, with seminorms
$$\|f\|_{\dot B_{q,1}^s}^{\ell,J_\varepsilon}=\sum_{j\le J_\varepsilon-1}2^{sj}\|\dot\Delta_j f\|_{L^q},\qquad \|f\|_{\dot B_{q,1}^s}^{h,J_\varepsilon}=\sum_{j\ge J_\varepsilon}2^{sj}\|\dot\Delta_j f\|_{L^q}.$$
Define $E_T^{J_\varepsilon}$ to be the set of pairs $(n,\mathbf v)$ such that
$$(n^{\ell,J_\varepsilon},\mathbf v^{\ell,J_\varepsilon})\in \mathcal C_b([0,T];\dot B_{p,1}^{d/p}),\quad \varepsilon n^{\ell,J_\varepsilon}\in \widetilde L_T^1(\dot B_{p,1}^{d/p+2}),\quad \mathbf v^{\ell,J_\varepsilon}\in \widetilde L_T^1(\dot B_{p,1}^{d/p+1}),$$
$$\varepsilon^{-1/2}\mathbf v\in \widetilde L_T^2(\dot B_{p,1}^{d/p}),\quad (\varepsilon n^{h,J_\varepsilon},\varepsilon\mathbf v^{h,J_\varepsilon})\in \mathcal C_b([0,T];\dot B_{2,1}^{d/2+1}),\quad (n^{h,J_\varepsilon},\mathbf v^{h,J_\varepsilon})\in \widetilde L_T^1(\dot B_{2,1}^{d/2+1}),$$
$$\varepsilon^{-1}\mathbf v+\nabla n\in \widetilde L_T^1(\dot B_{p,1}^{d/p}),$$
and let $E_\infty^{J_\varepsilon}$ denote this space with $T=\infty$. Consider on $[0,\infty)\times\mathbb R^d$ the Cauchy problem
$$\begin{cases}
\partial_t n+\mathbf v\cdot\nabla n+P'(\bar\rho)\,\operatorname{div}\mathbf v+G(n)\,\operatorname{div}\mathbf v=0,\\
\partial_t \mathbf v+\mathbf v\cdot\nabla \mathbf v+\nabla n=-\dfrac1\varepsilon\mathbf v,
\end{cases}$$
with initial data $(n,\mathbf v)|_{t=0}=(n_0,\mathbf v_0)$, where $G(n)$ is the smooth function arising in the enthalpy formulation. There exists a small constant $\delta_1>0$ such that whenever
$$\|(n_0,\mathbf v_0)\|_{\dot B_{p,1}^{d/p}}^{\ell,J_\varepsilon}+\varepsilon\|(n_0,\mathbf v_0)\|_{\dot B_{2,1}^{d/2+1}}^{h,J_\varepsilon}\le \delta_1,$$
which existence statement holds?

For every $\varepsilon>0$, the Cauchy problem admits a uniform global unique solution $(n,\mathbf v)$ in $E_\infty^{J_\varepsilon}$.

Choices:
B. For every $\varepsilon>0$, the Cauchy problem admits a uniform global unique solution $(n,\mathbf v)$ in $E_\infty^{J_\varepsilon}$, provided one assumes in addition that $0<\frac{d}{p}<\frac d2$.
C. For every $\varepsilon>0$, the Cauchy problem admits a uniform global solution $(n,\mathbf v)$ in $E_\infty^{J_\varepsilon}$.
D.
There exists a small constant $\delta_1>0$, independent of $\varepsilon$, such that whenever
$$\|(n_0,\mathbf v_0)\|_{\dot B_{p,1}^{d/p}}^{\ell,J_\varepsilon}+\|(n_0,\mathbf v_0)\|_{\dot B_{2,1}^{d/2+1}}^{h,J_\varepsilon}\le \delta_1,$$
then for every $\varepsilon>0$ the Cauchy problem admits a uniform global unique solution $(n,\mathbf v)$ in $E_\infty^{J_\varepsilon}$.
E.
For every $\varepsilon>0$, the Cauchy problem admits a uniform global unique solution $(n,\mathbf v)$ in $E_\infty^{J_\varepsilon}$ whenever
$$\|(n_0,\mathbf v_0)\|_{\dot B_{p,1}^{d/p}}^{\ell,J_\varepsilon}+\varepsilon\|(n_0,\mathbf v_0)\|_{\dot B_{p,1}^{d/p+1}}^{h,J_\varepsilon}\le \delta_1.$$

range needed for low-frequency product lemma

epsilon-weight on high-frequency smallness

high-frequency norm/exponent switched from $\dot B_{2,1}^{d/2+1}$ to $\dot B_{p,1}^{d/p+1}$


\begin{theorem}\label{thm:main}
Consider any $T \ge 1$ and $L,\alpha>0$. Then there exist an $\alpha$-strongly convex set $S$ and an $L$-smooth, $L$-strongly convex function $f$ in dimension $d=2(T+1)$ such that every \texttt{FO-LMO} method applied to $(f, S)$ with $x_0 = 0$ has $x_T$ infeasible or
\[
f(x_T) - \min_{x\in S}f(x) \geq \frac{1}{528}\frac{L\, \mathrm{diam}(S)^2}{(T+1)^2}.
\]
Consequently, for any $\varepsilon>0$, there exist problem instances where $T\geq \sqrt{\frac{L\, \mathrm{diam}(S)^2}{528\varepsilon}} - 1$ iterations are required for any \texttt{FO-LMO} method to reach a suboptimality of $f(x_T) - \min_{x\in S}f(x) \leq \varepsilon$.
\end{theorem}

In this work, we consider convex constrained optimization problems of the form
$$ \begin{cases}
\min & f(x)\\
\mathrm{s.t.\ } & x\in S \subseteq \mathbb{R}^d.
\end{cases}$$
In particular, we consider high-dimensional problems where $d$ may be arbitrarily large.
Frank-Wolfe methods and the broader family of ``projection-free'' algorithms using a linear minimization subroutine have found renewed interest due to their scalability. See the survey~\cite{braun2025conditionalgradientmethods} and references therein. In a basic form, these methods iterate from an initialization $x_0$, producing for $k=0,1,\dots$
\begin{align*}
    p_k &= \nabla f(x_k),\\
    z_{k+1} & \in\argmin_{x\in S}\langle p_k, x\rangle,\\
    x_{k+1} & \in \operatorname{conv}\{x_k, z_{k+1}\}.
\end{align*}
As examples, an exact line search implementation of Frank-Wolfe would set $x_{k+1}$ as the minimizer of $f$ on the segment $[x_k,z_{k+1}]$. Similarly, a fixed ``open-loop'' stepsize implementation would fix $x_{k+1} = \theta_k x_k + (1-\theta_k)z_{k+1}$ for predetermined $\theta_k\in [0,1]$.
The two key computational oracles assumed here are access to gradients of the objective $f$ to compute $p_k$ and access to a Linear Minimization Oracle (LMO) to produce $z_{k+1}$. As a shorthand, we denote $\mathtt{LMO}_S(p) \in \argmin_{x\in S}\langle p, x\rangle$ as an oracle providing a selection of this minimizer (potentially selected adversarially).

In this work, we provide complexity lower bounds for the family of all deterministic first-order methods using a linear minimization oracle, containing the above Frank-Wolfe methods. Our focus is on lower bounds for problem classes where the constraint set $S$ possesses structural properties like strong convexity or smoothness. Below, we formalize these classes of problems and algorithms.

\paragraph{The Families of Smooth and Strongly Convex Constraint Sets and Functions.} A problem instance is defined by a function $f\colon \mathbb{R}^d\rightarrow \mathbb{R}$ and a set $S\subseteq\mathbb{R}^d$ as well as an initialization $x_0$, which we will set as $x_0=0$ without loss of generality. A problem class is defined by a set of allowable $f$ and $S$ values. We consider the standard family of convex objective functions parameterized by $0\leq\mu\leq L$, defined as
\begin{align*}
    \text{$f$ is $\mu$-strongly convex if\quad }& f(\lambda x + (1-\lambda)y)\leq \lambda f(x) + (1-\lambda)f(y) - \frac{\lambda(1-\lambda)\mu}{2}\|y-x\|_2^2\\
    &\hskip7.0cm \forall x,y\in \mathbb{R}^d, \lambda\in[0,1], \\
    \text{$f$ is $L$-smooth if\quad } & \|\nabla f(y)-\nabla f(x)\|_2\leq L\|y-x\|_2 \quad \forall x,y\in \mathbb{R}^d.
\end{align*}
If $L=\mu$, then $f$ must be a quadratic function of the form $\frac{L}{2}\|x-x_\star\|_2^2$ up to an additive constant. Such simple quadratics will suffice for our theoretical development.

We consider compact convex sets $S$ with diameter $\mathrm{diam}(S)=\max_{x,y\in S}\|x-y\|_2$ with at least one of the natural parallel notions of strong convexity and smoothness for constraint sets. We say
$$ \text{$S$ is $\alpha$-strongly convex if\quad } \lambda x + (1-\lambda)y + B\left(0,\ \frac{\lambda(1-\lambda)\alpha}{2}\|y-x\|_2^2\right) \subseteq S $$
for all $x,y\in S, \lambda\in [0,1]$ where $B(0,r)=\{x\in\mathbb{R}^d \mid \|x\|_2\leq r\}$ denotes the closed ball of radius $r$. Normal vectors $n \in N_S(x) := \{n \mid \langle n, y-x\rangle \leq 0\ \forall y\in S\}$ at each $x\in\operatorname{bdry}S$ serve as the analogues of gradients. Considering any unit length normal vectors, we define
\begin{align*}
    \text{$S$ is $\beta$-smooth if\quad } \|n_y - n_x\|_2 \leq \beta \|y-x\|_2 \quad \forall &x,y\in\operatorname{bdry}S,\\
    &n_y\in \{n\in N_S(y) \mid \|n\|_2=1\},\\
    &n_x\in \{n\in N_S(x) \mid \|n\|_2=1\}.
\end{align*}
Note that the above definition requires that smooth sets have a unique unit normal vector at each boundary point.
For a classical reference providing equivalent characterizations of the strong convexity of a set, see~\cite{vial1982strong}. A modernized treatment of smoothness and strong convexity of sets was given by~\cite{liu2023gauges}. Therein, parallels to the smoothness and strong convexity of functions are explored.

\paragraph{The Family of First-Order Linear Minimization Oracle Methods (\texttt{FO-LMO}).}
For a given problem instance $(f,S)$ and initialization $x_0 \in S$, a \texttt{FO-LMO} method generates sequences of search directions $p_k\in\mathbb{R}^d$, linear minimization solutions $z_{k+1} = \mathtt{LMO}_S(p_k)\in\argmin_{x\in S}\langle p_k, x\rangle$, and iterates $x_{k+1}$. We require that $p_k$ is a deterministic function of the oracle responses $\{(f(x_i),\nabla f(x_i))\}^k_{i=0}$ and $\{z_{i}\}^k_{i=1}$ so far and that $x_{k+1}$ is a deterministic function of the responses $\{(f(x_i),\nabla f(x_i))\}^k_{i=0}$ and $\{z_{i}\}^{k+1}_{i=1}$. Note that we do not require that $x_{k+1}$ lie in the convex hull of $x_0$ and $\{z_{i}\}_{i=1}^{k+1}$. However, when outside this convex hull, the iterates may fail to be feasible.
This \texttt{FO-LMO} family of methods includes, for example, the Away-step Frank–Wolfe methods~\cite{lacoste2015global,beck2017linearly}, Pairwise Frank–Wolfe methods~\cite{lacoste2015global,tsuji2022pairwise}, Fully Corrective Frank–Wolfe methods~\cite{holloway1974extension,lacoste2015global}, and Blended Pairwise Frank–Wolfe methods~\cite{braun2019blended}.

In this model, each iteration of the algorithm can make one call to a first-order oracle, returning $(f(x_k), \nabla f(x_k))$, and one call to the LMO. Our theory then bounds iteration complexity to measure the minimum number of such pairs of oracle calls needed to reach a target suboptimality.

For the minimization of an $L$-smooth convex function $f$ over a compact convex set $S$, Frank-Wolfe methods~\cite{frank1956algorithm,jaggi2013revisiting} are known to provide convergence rates of the following form
$$ f(x_T) - \min_{x\in S}f(x) \leq \mathcal{O}\left(\frac{L\, \mathrm{diam}(S)^2}{T}\right). $$
This provides an upper bound on the iteration complexity of computing a point with $\varepsilon$-suboptimality (i.e., $f(x_T)-\min_{x\in S}f(x)\leq\varepsilon$) of $\mathcal{O}(L\, \mathrm{diam}(S)^2/\varepsilon)$. A matching complexity lower bound was provided by Lan~\cite{lan2013complexity}, establishing this as the order of the optimal LMO complexity.
Garber and Hazan~\cite{garber2015faster} showed that given the additional structure that $f$ and $S$ are strongly convex, this rate can be accelerated to
$$ f(x_T) - \min_{x\in S}f(x) \leq \mathcal{O}\left(\frac{L\, \mathrm{diam}(S)^2}{T^2}\right) $$
giving an iteration complexity of $\mathcal{O}(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$. This accelerated result poses the natural question of whether further acceleration is possible or if this complexity is the optimal strongly convex order.

Providing recent progress on this question, Halbey et al.~\cite{halbey2026lower} showed that two particular variants of Frank-Wolfe (exact line search and the short stepsize procedure) cannot improve upon this $\mathcal{O}(1/T^2)$ rate. However, these results do not preclude the possibility of other \texttt{FO-LMO} methods exceeding this rate.
Similar limitations in unconstrained minimization were overcome by the foundational work~\cite{nemirovski1983problem}, establishing information/oracle complexity lower bounds against the whole family of gradient methods. This was accomplished by the design of hard functions whose gradient reveals only one new coordinate of information per step. This property, known as a ``zero-chain'' property, can prevent {\it any gradient-span method} from having made substantial progress until dimension-many steps have been taken. By combining this with an adversarial ``resisting oracle'', lower bounds against {\it all deterministic gradient methods} were achieved.

\paragraph{Our Contributions.} We extend the classical zero-chain approach to LMOs to derive complexity lower bounds on any \texttt{FO-LMO} method over the problem classes corresponding to minimizing a quadratic $f(x)=\frac{L}{2}\|x-x_\star\|_2^2$ over structured sets, possessing either strong convexity or smoothness. Our primary result shows that over $\alpha$-strongly convex constraints, no \texttt{FO-LMO} method can guarantee $\varepsilon$-suboptimality in fewer than $\Omega(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ iterations. By matching the upper bounding guarantee of~\cite{garber2015faster}, this establishes $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ as the optimal order of complexity for this class. Formally, we prove the following in Section~\ref{sec:general}.

\paragraph{The Families of Smooth and Strongly Convex Constraint Sets and Functions.} A problem instance is defined by a function $f\colon \mathbb{R}^d\rightarrow \mathbb{R}$ and a set $S\subseteq\mathbb{R}^d$ as well as an initialization $x_0$, which we will set as $x_0=0$ without loss of generality. A problem class is defined by a set of allowable $f$ and $S$ values. We consider the standard family of convex objective functions parameterized by $0\leq\mu\leq L$, defined as
\begin{align*}
 \text{$f$ is $\mu$-strongly convex if\quad }& f(\lambda x + (1-\lambda)y)\leq \lambda f(x) + (1-\lambda)f(y) - \frac{\lambda(1-\lambda)\mu}{2}\|y-x\|_2^2\\
 &\hskip7.0cm \forall x,y\in \mathbb{R}^d, \lambda\in[0,1], \\
 \text{$f$ is $L$-smooth if\quad } & \|\nabla f(y)-\nabla f(x)\|_2\leq L\|y-x\|_2 \quad \forall x,y\in \mathbb{R}^d.
\end{align*}
If $L=\mu$, then $f$ must be a quadratic function of the form $\frac{L}{2}\|x-x_\star\|_2^2$ up to an additive constant. Such simple quadratics will suffice for our theoretical development.

We consider compact convex sets $S$ with diameter $\mathrm{diam}(S)=\max_{x,y\in S}\|x-y\|_2$ with at least one of the natural parallel notions of strong convexity and smoothness for constraint sets. We say
$$ \text{$S$ is $\alpha$-strongly convex if\quad } \lambda x + (1-\lambda)y + B\left(0,\ \frac{\lambda(1-\lambda)\alpha}{2}\|y-x\|_2^2\right) \subseteq S $$
for all $x,y\in S, \lambda\in [0,1]$ where $B(0,r)=\{x\in\mathbb{R}^d \mid \|x\|_2\leq r\}$ denotes the closed ball of radius $r$. Normal vectors $n \in N_S(x) := \{n \mid \langle n, y-x\rangle \leq 0\ \forall y\in S\}$ at each $x\in\operatorname{bdry}S$ serve as the analogues of gradients. Considering any unit length normal vectors, we define
\begin{align*}
 \text{$S$ is $\beta$-smooth if\quad } \|n_y - n_x\|_2 \leq \beta \|y-x\|_2 \quad \forall &x,y\in\operatorname{bdry}S,\\
 &n_y\in \{n\in N_S(y) \mid \|n\|_2=1\},\\
 &n_x\in \{n\in N_S(x) \mid \|n\|_2=1\}.
\end{align*}
Note that the above definition requires that smooth sets have a unique unit normal vector at each boundary point.
For a classical reference providing equivalent characterizations of the strong convexity of a set, see~\cite{vial1982strong}. A modernized treatment of smoothness and strong convexity of sets was given by~\cite{liu2023gauges}. Therein, parallels to the smoothness and strong convexity of functions are explored.

For the minimization of an $L$-smooth convex function $f$ over a compact convex set $S$, Frank-Wolfe methods~\cite{frank1956algorithm,jaggi2013revisiting} are known to provide convergence rates of the following form
$$ f(x_T) - \min_{x\in S}f(x) \leq \mathcal{O}\left(\frac{L\, \mathrm{diam}(S)^2}{T}\right). $$
This provides an upper bound on the iteration complexity of computing a point with $\varepsilon$-suboptimality (i.e., $f(x_T)-\min_{x\in S}f(x)\leq\varepsilon$) of $\mathcal{O}(L\, \mathrm{diam}(S)^2/\varepsilon)$. A matching complexity lower bound was provided by Lan~\cite{lan2013complexity}, establishing this as the order of the optimal LMO complexity.
Garber and Hazan~\cite{garber2015faster} showed that given the additional structure that $f$ and $S$ are strongly convex, this rate can be accelerated to
$$ f(x_T) - \min_{x\in S}f(x) \leq \mathcal{O}\left(\frac{L\, \mathrm{diam}(S)^2}{T^2}\right) $$
giving an iteration complexity of $\mathcal{O}(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$. This accelerated result poses the natural question of whether further acceleration is possible or if this complexity is the optimal strongly convex order.

Providing recent progress on this question, Halbey et al.~\cite{halbey2026lower} showed that two particular variants of Frank-Wolfe (exact line search and the short stepsize procedure) cannot improve upon this $\mathcal{O}(1/T^2)$ rate. However, these results do not preclude the possibility of other \texttt{FO-LMO} methods exceeding this rate.
Similar limitations in unconstrained minimization were overcome by the foundational work~\cite{nemirovski1983problem}, establishing information/oracle complexity lower bounds against the whole family of gradient methods. This was accomplished by the design of hard functions whose gradient reveals only one new coordinate of information per step. This property, known as a ``zero-chain'' property, can prevent {\it any gradient-span method} from having made substantial progress until dimension-many steps have been taken. By combining this with an adversarial ``resisting oracle'', lower bounds against {\it all deterministic gradient methods} were achieved.

\paragraph{Our Contributions.} We extend the classical zero-chain approach to LMOs to derive complexity lower bounds on any \texttt{FO-LMO} method over the problem classes corresponding to minimizing a quadratic $f(x)=\frac{L}{2}\|x-x_\star\|_2^2$ over structured sets, possessing either strong convexity or smoothness. Our primary result shows that over $\alpha$-strongly convex constraints, no \texttt{FO-LMO} method can guarantee $\varepsilon$-suboptimality in fewer than $\Omega(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ iterations. By matching the upper bounding guarantee of~\cite{garber2015faster}, this establishes $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ as the optimal order of complexity for this class. Formally, we prove the following in Section~\ref{sec:general}.

As a secondary result, we provide a partial generalization to $\beta$-smooth sets. These bounds are meaningful in the regime of only modestly smooth sets, having $\beta=\Omega(1/\sqrt{\varepsilon})$. In this regime, no \texttt{LMO-span} method (see Section~\ref{sec:SC} for a formal definition) can improve past the optimal compact convex set complexity of $\Theta(L\, \mathrm{diam}(S)^2/\varepsilon)$ or over the optimal $\alpha$-strongly convex set complexity of $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$. Theorems~\ref{thm:smooth} and~\ref{thm:mixed} formalize these limits on acceleration due to smoothness.

First, Section~\ref{subsec:Construction-SC} constructs our candidate hard problem instance for each dimension and verifies its validity (computing its strong convexity constant and diameter). Then Section~\ref{subsec:zero-chain-SC} establishes a key ``zero-chain'' property of these hard instances, showing that any \texttt{LMO-span} method applied will only discover one new coordinate per iteration. Finally, by leveraging this property, Section~\ref{subsec:proof-SC} proves this section's main result, Theorem~\ref{thm:SC}, showing that no method in $T$ steps can guarantee a suboptimality less than $\Omega(L\, \mathrm{diam}(S)^2/T^2)$. Combined with the upper bound of~\cite{garber2015faster}, this establishes the optimal iteration complexity for strongly convex constrained \texttt{LMO-span} optimization as $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$.

Any considered \texttt{FO-LMO} method must set $x_T$ deterministically as a function of the history of oracle responses. Crucially, while the algorithm must fix $(x_T)_i$ for $i \in U$, the resisting oracle remains free to select any completion of the permutation $\pi$ over $U$. Let $\mathcal{P}_U\subseteq \mathcal{P}_d$ denote all such permutations. If some completion $\pi\in\mathcal{P}_U$ has $x_T\not\in S_\pi$, then the adversarial oracle can select that $\pi$, making $x_T$ infeasible and the theorem hold trivially. Hence, we can assume for all $\pi\in \mathcal{P}_U$ that $x_T\in S_\pi$. From the strong convexity of $f$ and optimality of $x^{(\pi)}_\star$ guaranteeing $\langle \nabla f(x^{(\pi)}_\star), x_T - x^{(\pi)}_\star\rangle\geq 0$ since $x_T\in S_\pi$, the suboptimality for any $\pi\in\mathcal{P}_U$ is bounded by
\begin{equation}\label{eq:sc-lower-bound}
 f(x_T) - f(x_\star^{(\pi)}) \ge \frac{1}{2}\|x_T - x_\star^{(\pi)}\|_2^2 \ge \frac{1}{2} \sum_{j\in U} \left( (x_T)_{j} - (x^{(\pi)}_\star)_{j} \right)^2.
\end{equation}
Let $\pi_\star \in \mathcal{P}_U$ denote the permutation choice maximizing this lower bound. Then, we have
\begin{align*}
 f(x_T) - f(x_\star^{(\pi_\star)}) &\geq \frac{1}{2} \sum_{j\in U} \left( (x_T)_{j} - (x^{(\pi_\star)}_\star)_{j} \right)^2\\
 &\geq \frac{1}{|\mathcal{P}_U|}\sum_{\pi\in\mathcal{P}_U} \left(\frac{1}{2}\sum_{j\in U} \left( (x_T)_{j} - (x^{(\pi)}_\star)_{j} \right)^2\right)\\
 & = \frac{1}{2}\sum_{j\in U} \frac{1}{|\mathcal{P}_U|}\sum_{\pi\in\mathcal{P}_U} \left( (x_T)_{j} - (x^{(\pi)}_\star)_{j} \right)^2\\
 &\geq \frac{m}{2}\mathrm{Var}\left\{ \rho(w_d - w_k) \right\}_{k = d-m+1}^d \\
 &=\frac{\rho^2 m}{2} \mathrm{Var}\{w_{k}\}_{k = d-m+1}^d\\
 &= \rho^2 m C^2 \mathrm{Var}\{\sqrt{k}\}_{k = d-m+1}^d
\end{align*}
where the first inequality is by~\eqref{eq:sc-lower-bound}, the second inequality lower bounds this maximal $\pi_\star$ by the average lower bound over $\mathcal{P}_U$, and the third inequality notes that each inner sum is minimized when $(x_T)_{j}$ is the average of $\{\rho(w_d - w_{d-m+1}),\dots, \rho(w_d - w_{d})\}$ yielding the stated variance lower bound.

\begin{theorem}\label{thm:smooth}
For any $d \ge 1$ and $\beta > 0$, there exist a convex $\beta$-smooth set $S_\beta$ and a $1$-smooth, $1$-strongly convex function $f_\beta$ such that every \texttt{LMO-span} method applied to $(f_\beta, S_\beta)$ starting from $x_0=0$ has for all $t\leq d-1$
 \begin{equation}
 f_\beta(x_t) - \min_{x\in S_\beta}f_\beta(x) \geq \max \left(0, \sqrt{\frac{d-t}{2}}\left(\frac{\mathrm{diam}(S_\beta) - 2/\beta}{\sqrt{2}d}+\frac{1}{\beta\sqrt{d}}\right)-\frac{1}{\sqrt{2}\beta}\right)^2 .
 \end{equation}
 In particular, for any fixed budget $T \ge 1$, there exist $S_\beta$ and $f_\beta$ in dimension $d=2T$ such that
 \[
 f_\beta(x_T) - \min_{x\in S_\beta}f_\beta(x) \geq \max\left(0,\frac{\mathrm{diam}(S_\beta) - 2/\beta}{4\sqrt{T}} - \frac{\sqrt{2}-1}{2\beta }\right)^2 .
 \]
\end{theorem}
\begin{theorem}\label{thm:mixed}
 For any $d \ge 1$ and $\beta > 0$, there exist a $\beta$-smooth, $1/(1+1/\beta)$-strongly convex set $S_\beta$ and a $1$-smooth, $1$-strongly convex function $f_\beta$ such that every \texttt{LMO-span} method applied to $(f_\beta, S_\beta)$ starting from $x_0 = 0$ has for all $t\leq d-1$
 \begin{equation}
 f_\beta(x_t) - \min_{x\in S_\beta}f_\beta(x) \geq \max\left(0,\sqrt{\frac{2}{5}\frac{(d-t)(\mathrm{diam}(S_\beta) - 2/\beta)^2}{(d+2)^3}} - \frac{1}{\sqrt{2}\beta}\right)^2 .
 \end{equation}
 In particular, for any fixed budget $T \ge 1$, there exist $S_\beta$ and $f_\beta$ in dimension $d=2(T+1)$ such that
 \[
 f_\beta(x_T) - \min_{x\in S_\beta}f_\beta(x) \geq \max\left(0,\frac{\mathrm{diam}(S_\beta) - 2/\beta}{\sqrt{20}(T+2)} - \frac{1}{\sqrt{2}\beta }\right)^2 .
 \]
\end{theorem}
\noindent In both of these theorems, the lower bounds are only meaningful in the regime of $\beta=\Omega(1/\sqrt{\varepsilon})$ as otherwise, the two-term maximums above will take value zero, making the lower bound vacuous.

Inductively applying Lemma~\ref{lem:approx-chain}, the iterate $x_t \in \operatorname{conv}\{x_0, z_1, \dots, z_t\}$ must have final $d-t$ coordinates $i\geq t+1$ equal with value bounded by $|(x_t)_i| \leq \max_{1 \le k \le t} |\delta_k| \le \frac{1}{\beta \sqrt{d-t+1}}$. Then the suboptimality $f_\beta(x_t) - f_\beta(x_\star) = \frac{1}{2}\|x_t-x_\star\|^2_2$ is lower bounded by
\begin{align*}
 f_\beta(x_t) - f_\beta(x_\star) &\ge \frac{1}{2} \sum_{i=t+1}^d \left((x_t)_i - \left(\frac{1}{d}+\frac{1}{\beta \sqrt{d}}\right)\right)^2 \\
 &\ge \frac{d-t}{2}\max \left(0, \frac{1}{d}+\frac{1}{\beta\sqrt{d}}-\frac{1}{\beta\sqrt{d-t+1}}\right)^2 \\
 &\ge \max \left(0, \sqrt{\frac{d-t}{2}}\left(\frac{\mathrm{diam}(S_\beta) - 2/\beta}{\sqrt{2}d}+\frac{1}{\beta\sqrt{d}}\right)-\frac{1}{\sqrt{2}\beta}\right)^2.
\end{align*}
Specializing to $t=T$ and $d=2T$, this provides a lower bound of
$$ f_\beta(x_T) - f_\beta(x_\star) \geq \max\left(0,\frac{\mathrm{diam}(S_\beta) - 2/\beta}{4\sqrt{T}} - \frac{\sqrt{2}-1}{2\beta }\right)^2 . $$
\subsection{Proof of Theorem~\ref{thm:mixed}}
Given any $d>0$, consider the set $S$ previously defined in~\eqref{eq:S-simple} and the parameter $\nu$ defined by~\eqref{eq:nu-def}. Fix any $\beta > 0$ and define
$$ S_\beta := S + B(0,1/\beta) . $$
From Lemma~\ref{lem:diam-SC}, it is immediate that the diameter of $S_\beta$ is $\mathrm{diam}(S_\beta) = \mathrm{diam}(S)+2/\beta = 2C(2-\sqrt{2}) + 2/\beta$. Further, $S_\beta$ is $\beta$-smooth and $\alpha = 1/(1+1/\beta)$-strongly convex by the calculus rules for Minkowski sums~\cite[Lemma 9]{liu2023gauges}.

While Theorem~\ref{thm:main} applies for any $L,\alpha>0$, it does not allow a free selection of $\mathrm{diam}(S)$. Rather, our constructed hard ``zero-chain'' set $S$ has $\mathrm{diam}(S) = \Theta(1/\alpha d)$. One cannot arbitrarily select $\alpha$ and $\mathrm{diam}(S)$ as they must satisfy, for example, that $\mathrm{diam}(S)\leq 2/\alpha$. In the limit where $\mathrm{diam}(S)= 2/\alpha$, the set $S$ is forced to be (up to translation) the ball $B(0,1/\alpha)$. Such forced structure enables faster algorithms\footnote{For example, when $\mathrm{diam}(S)= 2/\alpha$ and $S = B(0,1/\alpha)$, an LMO can be used to explicitly compute orthogonal projections onto the feasible region $S$. Then the class of \texttt{FO-LMO} methods includes projected gradient methods, which are known to converge linearly for smooth, strongly convex objectives.}. The development of hard instances for any selection of $\mathrm{diam}(S) \leq 2/\alpha$ and resulting more nuanced complexity bounds is left as an important future direction.

As a secondary result, we provide a partial generalization to $\beta$-smooth sets. These bounds are meaningful in the regime of only modestly smooth sets, having $\beta=\Omega(1/\sqrt{\varepsilon})$. In this regime, no \texttt{LMO-span} method (see Section~\ref{sec:SC} for a formal definition) can improve past the optimal compact convex set complexity of $\Theta(L\, \mathrm{diam}(S)^2/\varepsilon)$ or over the optimal $\alpha$-strongly convex set complexity of $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$. Theorems~\ref{thm:smooth} and~\ref{thm:mixed} formalize these limits on acceleration due to smoothness.

Since our constructions throughout use an $L$-smooth, $L$-strongly convex objective, all of our lower bounds apply against the wider class of problems with smooth, strongly convex objectives having any $0 \leq \mu \leq L$. Hence, our theory highlights a fundamental difficulty of constrained optimization via LMOs: {\it Even on perfectly conditioned objective functions, hard adversarial constraint sets constitute a barrier to linear convergence.} This stands in contrast to gradient methods with an orthogonal projection oracle where convergence is dominated by objective function conditioning.

\paragraph{Outline.} Section~\ref{sec:SC} first derives a lower bound via a novel construction of a strongly convex feasible region that is hard for all \texttt{FO-LMO} methods with an additional span restriction. Section~\ref{sec:general} removes this restriction, proving Theorem~\ref{thm:main}. This construction is the main technical innovation of our work. Section~\ref{sec:smooth} then provides results extending lower bounds to the specialized setting of sets with modest levels of smoothness.

\subsection{Related Work}
Our Theorem~\ref{thm:main} differs from the previously mentioned lower bounding result of~\cite{halbey2026lower} in two aspects, making the result complementary. In terms of algorithmic scope, our bound provides a wider guarantee, establishing a universal lower bound against all \texttt{FO-LMO} methods. In contrast,~\cite{halbey2026lower} provides a hard instance for two standard implementations of Frank-Wolfe, namely those fixing $p_k=\nabla f(x_k)$ and using an exact line search or the short stepsize procedure to select $x_{k+1}$. In terms of problem class scope, our construction requires a nonsmooth feasible region and a large problem dimension $d$ (linear in the number of iterations to be run). Such a high dimensionality assumption (i.e., $d>T$) is classical and widespread in the optimization complexity literature~\cite{nemirovski1983problem}. In contrast,~\cite{halbey2026lower} is able to provide a hard instance using a smooth ball in $d=2$ dimensions. As a result, they provide a stronger illustration of the limitations of the two methods that their theory covers.

As mentioned above, most classical lower bounding results in unconstrained first-order optimization rely on setting the problem dimension larger than the number of iterations to be conducted. This enables ``zero-chain'' arguments where the objective function is designed to reveal only one new coordinate to the given gradient-span algorithm at each iteration. The main technical innovation of our work is the design of a hard strongly convex set where the LMO possesses a similar zero-chain property to these classical hard objective constructions. Such constructions open the possibility to extend proof techniques and insights from existing unconstrained optimization lower bounds to constrained LMO/projection-free settings.

For example, although our strongly convex convergence rate lower bound matches the upper bound of~\cite{garber2015faster} in order, the two differ by constant factors: our lower bound has a universal constant of $1/528$ and their upper bound has $9/2$. This leaves open the question of determining exactly minimax optimal algorithms and hard problem instances. In settings of unconstrained first-order minimization, the Performance Estimation Problem (PEP) techniques pioneered by~\cite{drori2014performance,Interpolation,Interpolation2} have provided such theory. For example, in smooth convex optimization, these facilitated the identification of the Optimized Gradient Method~\cite{kim2016optimized} and an exactly matching lower bound~\cite{drori2017exact}. PEP was extended to cover structured smooth and strongly convex sets by Luner and Grimmer~\cite{luner2024performance}. As a result, future work may leverage PEP to similarly close the constant-factor gap left for \texttt{FO-LMO} methods.

Another important property of most Frank-Wolfe methods is that their trajectory is independent of the choice of inner product used. This differs from projected-gradient methods, where the choice of inner product and notion of orthogonality for projections affects the algorithm trajectory, making good preconditioning important for practical success. The line of work~\cite{Pena2023,Wirth2025,Wirth2026} provided Frank-Wolfe with ``affine-covariant'' convergence theory, matching the method's affine covariant nature by avoiding notions like smoothness and strong convexity defined in terms of a fixed inner product. Developing lower bounding theory in affine-covariant terms is an interesting future direction.

One can view the oracle model of an LMO as assuming first-order access to the support function $p \mapsto \sup\{\langle p, x\rangle \mid x\in S\}$. For sets with $0\in S$, a polar transformation establishes this as dual to assuming first-order access to the Minkowski gauge $y \mapsto \inf\{ \gamma>0 \mid y/\gamma \in S\}$. This gauge model has been studied by the works~\cite{Renegar2016,Grimmer2021-part1,Grimmer2021-part2,Zakaria2022,Lu2023} as an alternative projection-free framework. The works~\cite{liu2023gauges} and~\cite{samakhoana2024scalable} developed accelerated $\mathcal{O}(1/T^2)$ convergence guarantees for gauge methods over smooth sets, complementing the accelerated LMO rates for strongly convex sets. Identifying any structural relationships between the complexity of these dual oracle models and problem settings is another interesting direction.

Question:
Let \(T\ge 1\) and \(L,\alpha>0\). For a set \(S\subseteq \mathbb{R}^d\), write \(\mathrm{diam}(S)=\max_{x,y\in S}\|x-y\|_2\). Say that \(S\) is \(\alpha\)-strongly convex if for all \(x,y\in S\) and \(\lambda\in[0,1]\),
\[
\lambda x+(1-\lambda)y+B\!\left(0,\frac{\lambda(1-\lambda)\alpha}{2}\|y-x\|_2^2\right)\subseteq S,
\]
where \(B(0,r)=\{u\in\mathbb{R}^d:\|u\|_2\le r\}\). A function \(f:\mathbb{R}^d\to\mathbb{R}\) is \(L\)-smooth if \(\|\nabla f(y)-\nabla f(x)\|_2\le L\|y-x\|_2\) for all \(x,y\), and \(L\)-strongly convex if for all \(x,y\in\mathbb{R}^d\) and \(\lambda\in[0,1]\),
\[
f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)-\frac{\lambda(1-\lambda)L}{2}\|y-x\|_2^2.
\]
An \(\texttt{FO-LMO}\) method is an algorithm that uses first-order information about \(f\) together with a linear minimization oracle for \(S\). Starting from \(x_0=0\), which existence statement holds?

There exist an \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^d\) and an \(L\)-smooth, \(L\)-strongly convex function \(f:\mathbb{R}^d\to\mathbb{R}\) in dimension \(d=2(T+1)\) such that every \(\texttt{FO-LMO}\) method applied to \((f,S)\) with \(x_0=0\) either has \(x_T\notin S\) or satisfies
\[
f(x_T)-\min_{x\in S}f(x)\ge \frac{1}{528}\frac{L\,\mathrm{diam}(S)^2}{(T+1)^2}.
\]
Moreover, for every \(\varepsilon>0\), there exist problem instances for which any \(\texttt{FO-LMO}\) method requires at least
\[
\sqrt{\frac{L\,\mathrm{diam}(S)^2}{528\varepsilon}}-1
\]
iterations to achieve \(f(x_T)-\min_{x\in S}f(x)\le \varepsilon\).

Choices:
B.
There exist an \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^d\) and an \(L\)-smooth, \(L\)-strongly convex function \(f:\mathbb{R}^d\to\mathbb{R}\) in dimension \(d=2(T+1)\) such that every \(\texttt{FO-LMO}\) method applied to \((f,S)\) with \(x_0=0\) either has \(x_T\notin S\) or satisfies
\[
f(x_T)-\min_{x\in S}f(x)\ge \frac{1}{528}\frac{L\,\mathrm{diam}(S)^2}{T+1}.
\]
Consequently, for every \(\varepsilon>0\), there exist problem instances for which any \(\texttt{FO-LMO}\) method requires at least
\[
\frac{L\,\mathrm{diam}(S)^2}{528\varepsilon}-1
\]
iterations to achieve \(f(x_T)-\min_{x\in S}f(x)\le \varepsilon\).
C.
There exist an \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^d\) and an \(L\)-smooth, \(L\)-strongly convex function \(f:\mathbb{R}^d\to\mathbb{R}\) in some dimension \(d\) such that every \(\texttt{FO-LMO}\) method applied to \((f,S)\) with \(x_0=0\) either has \(x_T\notin S\) or satisfies
\[
f(x_T)-\min_{x\in S}f(x)\ge \frac{1}{528}\frac{L\,\mathrm{diam}(S)^2}{(T+1)^2}.
\]
D.
There exist an \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^d\) and an \(L\)-smooth, \(L\)-strongly convex function \(f:\mathbb{R}^d\to\mathbb{R}\) in dimension \(d=2(T+1)\) such that for every \(\texttt{FO-LMO}\) method applied to \((f,S)\) with \(x_0=0\), one has \(x_T\in S\) and
\[
f(x_T)-\min_{x\in S}f(x)\ge \frac{1}{528}\frac{L\,\mathrm{diam}(S)^2}{(T+1)^2}.
\]
Consequently, for every \(\varepsilon>0\), there exist problem instances for which any \(\texttt{FO-LMO}\) method requires at least
\[
\sqrt{\frac{L\,\mathrm{diam}(S)^2}{528\varepsilon}}-1
\]
iterations to achieve \(f(x_T)-\min_{x\in S}f(x)\le \varepsilon\).
E.
For every \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^d\) and every \(L\)-smooth, \(L\)-strongly convex function \(f:\mathbb{R}^d\to\mathbb{R}\) in dimension \(d=2(T+1)\), every \(\texttt{FO-LMO}\) method applied to \((f,S)\) with \(x_0=0\) either has \(x_T\notin S\) or satisfies
\[
f(x_T)-\min_{x\in S}f(x)\ge \frac{1}{528}\frac{L\,\mathrm{diam}(S)^2}{(T+1)^2}.
\]
Moreover, for every \(\varepsilon>0\), any \(\texttt{FO-LMO}\) method requires at least
\[
\sqrt{\frac{L\,\mathrm{diam}(S)^2}{528\varepsilon}}-1
\]
iterations to achieve \(f(x_T)-\min_{x\in S}f(x)\le \varepsilon\).