\begin{theorem}\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\Omega}(t)$ has an expansion as $t\to 0$ given by
\begin{multline}\label{eq:heattrace}
H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t} - \frac{|\partial\Omega|}{8\sqrt{\pi t}} + \frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\, ds + \sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)\\
+ \sqrt t\Big(\frac{1}{256\sqrt{\pi}}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n\mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big) +O(t\log t),
\end{multline}
where $\mathcal C_{1/2}$ is a function only of the angle $\alpha$ and the limit curvatures $\kappa_{\pm}$.
\end{theorem}

\begin{theorem}\label{thm:moreid} For $\alpha\ne\pi$, the function $\mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$ has the form
\begin{equation}\label{eq:form} \mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-}) = c_{1/2}(\alpha)r_0(\alpha,\kappa_+,\kappa_-)
\end{equation}
where $c_{1/2}(\alpha)$ is a function depending only on $\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:
\begin{equation}\label{eq:general} \begin{aligned}
        c_{1/2}(\alpha)
        &= -\,\fp
        \iint_{(0,\infty)\times(0,\alpha)}
        \int_{0}^{1} (1-\sigma)^{-1}\sigma^{-1/2}
        \iint_{(0,\infty)\times(0,\alpha)} \\
        &\quad a\left(\frac{R}{\sqrt{1-\sigma}},\theta,
        \frac{\sqrt{\sigma}\,\rho}{\sqrt{1-\sigma}},\phi\right)\,
        b\left(\rho,\phi,\frac{R}{\sqrt{\sigma}},\theta\right)\,
        \dmup\,\dmu.
        \end{aligned}
\end{equation}
Here $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\alpha$, explicitly given by \eqref{eq:a-def} and \eqref{eq:bexpression-clean}, and $\fp$ denotes the Hadamard finite part at $R=\infty$ of the improper integral in $R$.

Moreover, $c_{1/2}(\alpha)$ has the special value
\begin{equation}\label{eq:specialpi2} c_{1/2}(\frac{\pi}{2}) = \frac{1}{16\sqrt\pi}.
\end{equation}
\end{theorem}

\begin{equation}\label{eq:a-def}
    a(R,\theta,R',\theta')
    =\frac12\,\exp \Big[-\frac{R^2+(R')^2}{4}\Big]
    \sum_{j=1}^\infty I_{\mu_j} \Big(\frac{RR'}{2}\Big)\,
    \varphi_j(\theta)\,\varphi_j(\theta').
\end{equation}

\begin{equation}\label{eq:bexpression-clean}
    \begin{aligned}
    b(R,\theta,R',\theta')&=
    \Big(\tfrac12 D^2S\Big)\,a(R,\theta,R',\theta')\\
    &\quad+\,4 R\cos(\theta+\theta_0)\,\Delta_{R,\theta}a(R,\theta,R',\theta')\\
    &\quad+\,\tfrac{1}{2}\,\Delta_{R,\theta}\big(D^2S\cdot a(R,\theta,R',\theta')\big)
    -\tfrac{1}{2}\,(D^2S)\,\Delta_{R,\theta}a(R,\theta,R',\theta').
    \end{aligned}
    \end{equation}

A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\mathbb R^2$ \cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \cite{Cheeger1983,vanDenBergSrisat88,NRS}.

We now explain the setting of this paper. Let $\Omega\subseteq\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\alpha_j>0$, $j=1,\ldots,n$. Let $\kappa$ be the inward-pointing curvature of the boundary, and let $\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\kappa_{j,\pm}$ be the limits of $\kappa(\gamma(s))$ as $s$ approaches $\gamma^{-1}(P_j)$ from above and below respectively.

It is well known that the trace of the Dirichlet heat kernel on $\Omega$, $H^D_{\Omega}(t)$, has an asymptotic expansion as $t\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \cite{Cheeger1983} and \cite{vdBS}, and then extended to the general curvilinear setting \cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \cite{EGS_2017}, based on results in the literature such as \cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.

It is well known that the trace of the Dirichlet heat kernel on $\Omega$, $H^D_{\Omega}(t)$, has an asymptotic expansion as $t\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \cite{Cheeger1983} and \cite{vdBS}, and then extended to the general curvilinear setting \cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \cite{EGS_2017}, based on results in the literature such as \cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.

Moreover, $c_{1/2}(\alpha)$ has the special value
\begin{equation}\label{eq:specialpi2} c_{1/2}(\frac{\pi}{2}) = \frac{1}{16\sqrt\pi}.
\end{equation}
\end{theorem}
\begin{remark} A few comments:
\begin{itemize}
\item The $O(t\log t)$ error in \eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.
\item Each of $a$ and $b$ is a function only of $\alpha$ and the variables of integration, see \eqref{eq:a-def}, \eqref{eq:bexpression-clean}. Thus \eqref{eq:general} is indeed independent of $\kappa_{\pm}$.
\item For inverse spectral results, it is crucial that \eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\alpha=\pi/2$. It would be of significant interest to find out for which $\alpha$ we have $c_{1/2}(\alpha)\ne 0$.
\item Theorem \ref{thm:moreid} does not work for $\alpha=\pi$ because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\alpha=\pi$. So the case $\alpha=\pi$, where the boundary has an abrupt change in second derivative, remains open.
\end{itemize}
\end{remark}

\begin{proposition}
 Suppose that an admissible curvilinear polygon $\Omega$ is Dirichlet isospectral to a polygon. Then $\Omega$ is a polygon.
\end{proposition}
\begin{proof}
For a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\Omega$. But under our assumptions, by Theorems \ref{thm:thm1} and \ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\Omega}(t)$ is given by
 \begin{equation}
 \frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n c_{1/2}(\alpha_j)\sqrt{\kappa_{+,j}^2+\kappa_{-,j}^2}.
 \end{equation}
 Since this coefficient is non-negative and only zero for an exact polygon, $\Omega$ is an exact polygon.
\end{proof}
\begin{remark}
Observe from \eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.
\end{remark}

\subsection{Plan of the paper}
In section \ref{sec:gm} we prove Theorem \ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \cite{NRS}. We show that there is a well-defined corner contribution $\mathcal C_{1/2}\sqrt t$, and then prove that any two corners with the same angle $\alpha$ and limit curvatures $\kappa_{\pm}$ yield the same contribution, regardless of the rest of $\Omega$. To take advantage of this uniformity, in section \ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\mathcal C_{1/2}\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \cite{NRS} and allows us to obtain both \eqref{eq:form} and \eqref{eq:general}.
Finally, in section \ref{sec:specialpi2}, we prove \eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \eqref{eq:heattrace}.

\begin{lemma}\label{lem:local-corner}
Let $\Omega$ and $\Omega'$ be domains with a single corner having the same opening angle
$\alpha$ and the same one‑sided boundary curvatures $\kappa_\pm$ at that corner.
Then $H^D_\Omega-H^D_{\Omega'}$ has order $\le T^{0}$ (at worst $T^0 = t^0$) at ff.
Equivalently, their ff coefficients of orders $T^{-2} = t^{-1}$ and $T^{-1} = t^{-1/2}$ agree.
\end{lemma}

We can now prove Theorem \ref{thm:thm1}.
\begin{proof}[Proof of Theorem \ref{thm:thm1}] As in \cite{NRS}, the heat trace is obtained by restricting $H^{D}_{\Omega}$ to the spatial diagonal and then integrating over $z\in\Omega$. The technical tool used is Melrose's pushforward theorem. We will not repeat all the details here, but the upshot is that each face td, sf, and ff gives a separate contribution to the expansion \eqref{eq:heattrace}. A priori, the expansions can interact to give logarithmic terms, but for the same reasons as in \cite[p. 49]{NRS}, keeping in mind that we now have two orders at ff rather than one, there are no logarithmic terms until at least $O(T^2\log T)=O(t\log t)$ (and potentially much later, if at all). The contribution from td is
\[\frac{|\Omega|}{4\pi t} + O(t^{\infty}).\]
The contributions from sf are
\[-\frac{|\partial\Omega|}{8\sqrt{\pi t}} + \frac{1}{12\pi}\int_{\partial\Omega}\kappa\, ds +\frac{\sqrt t}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\, ds + O(t).\]
And the contributions from ff, since the first is already known and the second depends only on $\alpha$ and $\kappa_{\pm}$, are
\[\sum_{j=1}^{n}\frac{\pi^2-\alpha_j^2}{24\pi\alpha_j} + \sqrt t\sum_{j=1}^{n}\mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-}) + O(t),\]
for some unknown function $\mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$. Summing the three faces gives Theorem \ref{thm:thm1}.
\end{proof}

For $H(t)$, we use the asymptotics of Bessel function zeroes due to McMahon \cite[(10.21.19)]{NIST:DLMF}:
\begin{equation}
j_{0,k}= (k-\frac 14)\pi + \frac{1}{8(k-\frac 14)\pi} + O(k^{-3}),
\end{equation}
which imply
\begin{equation}
j_{0,k}^{2} = (k-\frac 14)^2\pi^2 + \frac 14 + O(k^{-2}).
\end{equation}
Based on this we define the function
\begin{equation}
\tilde H(t) = \sum_{k=1}^{\infty} \exp[-t((k-\frac 14)^2\pi^2 + \frac 14)]
\end{equation}
and estimate its difference with $H(t)$.
\begin{lemma}\label{lem:difference} With all notation as above, as $t\to 0$,
\begin{equation}
H(t) - \tilde H(t) = O(t).
\end{equation}
\end{lemma}
This is helpful because we also have the following lemma.
\begin{lemma}\label{lem:compute} As $t\to 0$,
\begin{equation}
\tilde H(t) = \frac{1}{2\sqrt\pi}t^{-1/2} + c - \frac{1}{8\sqrt\pi}t^{1/2}+ O(t).
\end{equation}
Here $c$ is a constant which is irrelevant to our purposes.
\end{lemma}
From these two Lemmas and \eqref{eq:relationship} we immediately deduce this proposition.
\begin{proposition} As $t\to 0$,
\begin{equation}
\Tr e^{-t\Delta_{\Omega}} = \frac 18 t^{-1} - (\frac{\sqrt\pi}{8}+\frac{1}{4\sqrt\pi})t^{-1/2}-\frac 12c + (\frac{\sqrt\pi}{256}+\frac{1}{16\sqrt\pi})t^{1/2}+ O(t).
\end{equation}
\end{proposition}
\begin{remark} We could reverse engineer $c$ from the known expansion from \cite{NRS} but there is no need.
\end{remark}

We can say substantially more about $\mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$ whenever $\alpha\ne\pi$. Define
\begin{equation}\label{eq:defofr0}
    r_0=r_0(\alpha,\kappa_{+},\kappa_-)=\frac 12\csc\alpha\sqrt{\kappa_+^2+\kappa_-^2-2\kappa_+\kappa_-\cos\alpha}.
\end{equation}
\begin{theorem}\label{thm:moreid} For $\alpha\ne\pi$, the function $\mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$ has the form
\begin{equation}\label{eq:form} \mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-}) = c_{1/2}(\alpha)r_0(\alpha,\kappa_+,\kappa_-)
\end{equation}
where $c_{1/2}(\alpha)$ is a function depending only on $\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:
\begin{equation}\label{eq:general} \begin{aligned}
        c_{1/2}(\alpha)
        &= -\,\operatorname*{fp}
        \iint_{(0,\infty)\times(0,\alpha)}
        \int_{0}^{1} (1-\sigma)^{-1}\sigma^{-1/2}
        \iint_{(0,\infty)\times(0,\alpha)} \\
        &\quad a\left(\frac{R}{\sqrt{1-\sigma}},\theta,
        \frac{\sqrt{\sigma}\,\rho}{\sqrt{1-\sigma}},\phi\right)\,
        b\left(\rho,\phi,\frac{R}{\sqrt{\sigma}},\theta\right)\,
        \,\rho\,d\rho\,d\phi\,\,R\,dR\,d\theta.
        \end{aligned}
\end{equation}
Here $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\alpha$, explicitly given by \eqref{eq:a-def} and \eqref{eq:bexpression-clean}, and $\operatorname*{fp}$ denotes the Hadamard finite part at $R=\infty$ of the improper integral in $R$.

Moreover, $c_{1/2}(\alpha)$ has the special value
\begin{equation}\label{eq:specialpi2} c_{1/2}(\frac{\pi}{2}) = \frac{1}{16\sqrt\pi}.
\end{equation}
\end{theorem}
\begin{remark} A few comments:
\begin{itemize}
\item The $O(t\log t)$ error in \eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.
\item Each of $a$ and $b$ is a function only of $\alpha$ and the variables of integration, see \eqref{eq:a-def}, \eqref{eq:bexpression-clean}. Thus \eqref{eq:general} is indeed independent of $\kappa_{\pm}$.
\item For inverse spectral results, it is crucial that \eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\alpha=\pi/2$. It would be of significant interest to find out for which $\alpha$ we have $c_{1/2}(\alpha)\ne 0$.
\item Theorem \ref{thm:moreid} does not work for $\alpha=\pi$  because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\alpha=\pi$. So the case $\alpha=\pi$, where the boundary has an abrupt change in second derivative, remains open.
\end{itemize}
\end{remark}

\subsection{Inverse spectral applications}

In \cite{EGS_2017}, the authors prove that any curvilinear polygon with straight corners is in fact isospectral to a polygon. They require straight corners because of a lack of a formula in the curved corner case. We can strengthen their result.
\begin{definition}
    A curvilinear polygon $\Omega$ is \emph{admissible} if, for each $j$, either $c_{1/2}(\alpha_j)>0$  or $\Omega$ is straight near $P_j$.
\end{definition}

\begin{proposition}
    Suppose that an admissible curvilinear polygon $\Omega$ is Dirichlet isospectral to a polygon. Then $\Omega$ is a polygon.
\end{proposition}
\begin{proof}
For a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\Omega$. But under our assumptions, by Theorems \ref{thm:thm1} and \ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\Omega}(t)$ is given by
    \begin{equation}
    \frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n c_{1/2}(\alpha_j)\sqrt{\kappa_{+,j}^2+\kappa_{-,j}^2}.
    \end{equation}
 Since this coefficient is non-negative and only zero for an exact polygon, $\Omega$ is an exact polygon.
\end{proof}
\begin{remark}
Observe from \eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.
\end{remark}

\subsection{Plan of the paper}
In section \ref{sec:gm} we prove Theorem \ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \cite{NRS}. We show that there is a well-defined corner contribution $\mathcal C_{1/2}\sqrt t$, and then prove that any two corners with the same angle $\alpha$ and limit curvatures $\kappa_{\pm}$ yield the same contribution, regardless of the rest of $\Omega$. To take advantage of this uniformity, in section \ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\mathcal C_{1/2}\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \cite{NRS} and allows us to obtain both \eqref{eq:form} and \eqref{eq:general}.
Finally, in section \ref{sec:specialpi2}, we prove \eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \eqref{eq:heattrace}.

In the “Plan of the paper” the authors outline the proof of Theorem~\ref{thm:thm1}: in Section~\ref{sec:gm} they “examin[e] the geometric microlocal description of the heat kernel given in \cite{NRS},” “show that there is a well-defined corner contribution $\mathcal C_{1/2}\sqrt t$,” and “prove that any two corners with the same angle $\alpha$ and limit curvatures $\kappa_{\pm}$ yield the same contribution, regardless of the rest of $\Omega$.” Then, “to take advantage of this uniformity,” in Section~\ref{sec:conformal} they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\mathcal C_{1/2}\sqrt t$ in that model,” via “a parametrix construction as in \cite{NRS},” which “allows [them] to obtain both \eqref{eq:form} and \eqref{eq:general}.” Finally, in Section~\ref{sec:specialpi2} they prove \eqref{eq:specialpi2} “by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \eqref{eq:heattrace}.”

In the “Plan of the paper” the authors outline the proof of the main theorem: next they “examin[e] the geometric microlocal description of the heat kernel given in NRS,” “show that there is a well-defined corner contribution $\mathcal C_{1/2}\sqrt t$,” and “prove that any two corners with the same angle $\alpha$ and limit curvatures $\kappa_{\pm}$ yield the same contribution, regardless of the rest of $\Omega$.” Then, “to take advantage of this uniformity,” they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\mathcal C_{1/2}\sqrt t$ in that model,” via “a parametrix construction as in NRS,” which “allows [them] to obtain both
\begin{equation}\label{eq:form} \mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-}) = c_{1/2}(\alpha)r_0(\alpha,\kappa_+,\kappa_-)
\end{equation}
and
\begin{equation}\label{eq:general} \begin{aligned}
        c_{1/2}(\alpha)
        &= -\,\fp
        \iint_{(0,\infty)\times(0,\alpha)}
        \int_{0}^{1} (1-\sigma)^{-1}\sigma^{-1/2}
        \iint_{(0,\infty)\times(0,\alpha)} \\
        &\quad a\left(\frac{R}{\sqrt{1-\sigma}},\theta,
        \frac{\sqrt{\sigma}\,\rho}{\sqrt{1-\sigma}},\phi\right)\,\
        b\left(\rho,\phi,\frac{R}{\sqrt{\sigma}},\theta\right)\,\
        \dmup\,\dmu.
        \end{aligned}
\end{equation}.
Finally, they prove
\begin{equation}\label{eq:specialpi2} c_{1/2}(\frac{\pi}{2}) = \frac{1}{16\sqrt\pi}.
\end{equation}
“by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with
\begin{multline}\label{eq:heattrace}
H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t} - \frac{|\partial\Omega|}{8\sqrt{\pi t}} + \frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\, ds + \sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)\\
+ \sqrt t\Big(\frac{1}{256\sqrt{\pi}}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n\mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big) +O(t\log t),
\end{multline}.”,

\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\Omega}(t)$ has an expansion as $t\to 0$ given by
\begin{multline}\label{eq:heattrace}
H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t} - \frac{|\partial\Omega|}{8\sqrt{\pi t}} + \frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\, ds + \sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)\\
+ \sqrt t\Big(\frac{1}{256\sqrt{\pi}}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n\mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big) +O(t\log t),
\end{multline}
where $\mathcal C_{1/2}$ is a function only of the angle $\alpha$ and the limit curvatures $\kappa_{\pm}..

Question: Let \(\Omega\subseteq\mathbb R^2\) be a curvilinear polygon: its boundary is piecewise smooth, smooth except at finitely many corners \(P_1,\dots,P_n\), where the interior angle at \(P_j\) is \(\alpha_j>0\). Let \(\kappa\) denote the inward-pointing curvature along the smooth boundary arcs, and for each corner let \(\kappa_{j,+}\) and \(\kappa_{j,-}\) be the one-sided limits of \(\kappa\) as the corner is approached along the two incident sides. Let \(H^D_{\Omega}(t)\) be the Dirichlet heat trace of \(\Omega\). As \(t\to 0\), which asymptotic expansion with error bound is valid?

\[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt t\Big(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big)+O(t\log t),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_+,\kappa_-)\) depends only on the corner angle \(\alpha\) and the one-sided limiting curvatures \(\kappa_\pm\).

Choices:
B. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt t\Big(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j)\sqrt{\kappa_{j,+}^2+\kappa_{j,-}^2}\Big)+O(t\log t),\]\nwhere \(\mathcal C_{1/2}(\alpha)\) depends only on the corner angle.
C. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+O(\sqrt t),\]\nas \(t\to 0\).
D. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt t\Big(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big)+O(t),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_+,\kappa_-)\) depends only on the corner angle \(\alpha\) and the one-sided limiting curvatures \(\kappa_\pm\).
E. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt t\Big(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big)+O(t^{3/2}),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_+,\kappa_-)\) may depend on the global geometry of \(\Omega\) in addition to \(\alpha\) and \(\kappa_\pm\).

corner-term dependence on full pair \((\kappa_+,\kappa_-)\) rather than a forced factorization through Euclidean norm

explicit \(\sqrt t\)-coefficient and sharper remainder \(O(t\log t)\)

error term improved from first guaranteed \(O(t\log t)\) to \(O(t)\)

uniform local determination of corner contribution by \(\alpha\) and \(\kappa_\pm\)


Question: Let \(\Omega\subseteq\mathbb R^2\) be a curvilinear polygon with piecewise smooth boundary, smooth except at finitely many corners \(P_j\) with interior angles \(\alpha_j>0\), \(j=1,\dots,n\). Let \(\kappa\) denote the inward-pointing curvature of \(\partial\Omega\), and for each corner \(P_j\), let \(\kappa_{j,+}\) and \(\kappa_{j,-}\) be the one-sided limits of \(\kappa\) along the two boundary arcs meeting at \(P_j\). If \(H^D_{\Omega}(t)\) denotes the trace of the Dirichlet heat kernel on \(\Omega\), then as \(t\to 0\), which asymptotic expansion holds?

\[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\left(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\right)+\sqrt t\left(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\right)+O(t\log t),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_+,\kappa_-)\) depends only on the corner angle \(\alpha\) and the one-sided curvature limits \(\kappa_\pm\).

Choices:
B. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\left(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\right)+\sqrt t\left(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j)\right)+O(t\log t),\]\nwhere \(\mathcal C_{1/2}(\alpha)\) depends only on the corner angle.
D. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\left(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\right)+\sqrt t\left(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\right)+O(t),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_+,\kappa_-)\) depends only on the corner angle \(\alpha\) and the one-sided curvature limits \(\kappa_\pm\).
E. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\left(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\right)+\sqrt t\left(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+}+\kappa_{j,-})\right)+O(t\log t),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_++\kappa_-)\) depends only on the corner angle \(\alpha\) and the sum of the one-sided curvature limits.

corner dependence on both angle and one-sided curvatures

explicit identification of the \sqrt t coefficient and sharper remainder

first possible logarithmic remainder order

separate dependence on \kappa_+ and \kappa_- rather than only their sum


\begin{teo} \label{thm:main}
	Let $\Omega\subset\mathbb{G}$ be an open and bounded set
	with smooth enough boundary $\partial \Omega$, and let $p\in (0,1)$. Then, there exists $\Lambda > 0$
	such that
	\begin{itemize}
		\item[A)]problem \eqref{EqProblem}$_{\lambda}$ does not admit weak solutions
		for every $\lambda>\Lambda$;
		\item[B)] problem \eqref{EqProblem}$_{\lambda}$ admits at least one weak solution for every $\lambda \in (0,\Lambda]$;
		\item[C)] problem \eqref{EqProblem}$_{\lambda}$ admits at least two weak solutions for every $0<\lambda<\Lambda$.
	\end{itemize}
\end{teo}

Let $\mathbb{G}$ be a Carnot group and let $\Omega \subset \mathbb{G}$ be an bounded and connected
open set with smooth enough boundary $\partial \Omega$. Let $q \in (0,1)$, let $2^{\star}_{Q}:=\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\mathbb{G}$, and let $\lambda >0$. We consider the following Dirichlet boundary value problem
\begin{equation}\tag{{$\mathrm{P}$}}\label{EqProblem}
 \left\{\begin{array}{rll}
 -\Delta_{\mathbb{G}}u& = \lambda\, u^{q} + u^{2^{\star}_{Q}-1} & \textrm{ in } \Omega,\\
 u&>0 & \textrm{ in } \Omega,\\
 u&=0 & \textrm{ on } \partial \Omega.
 \end{array}\right.
\end{equation}

We stress that $-\Delta_{\mathbb{G}}$ denotes here the sub-laplacian on $\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\"{o}rmander condition, see e.g. \cite{Hormander}. We refer to Section~\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\

Along the paper it will sometimes be useful to denote the above problem as \eqref{EqProblem}$_\lambda$
to make it clear the choice of the parameter.
We immediately state the main result of this paper. In what follows,
we refer to Definition \ref{def:weak_sub_super_sol} for the precise definition of \emph{weak solution}
of \eqref{EqProblem}$_{\lambda}$.

\begin{equation}\tag{{$\mathrm{P}$}}\label{EqProblem}
	\left\{\begin{array}{rll}
		-\Delta_{\mathbb{G}}u& = \lambda\, u^{q} + u^{2^{\star}_{Q}-1} & \textrm{ in } \Omega,\\
		u&>0 & \textrm{ in } \Omega,\\
		u&=0 & \textrm{ on } \partial \Omega.
	\end{array}\right.
\end{equation}

\begin{defin}\label{def:weak_sub_super_sol}
	Let $\Omega\subseteq\mathbb{G}$ be an open, bounded and connected set.
We say that a function $u \in S^{1}_{0}(\Omega)$ is a weak subsolution (resp. supersolution) of \eqref{EqProblem}$_\lambda$ if it satisfies the following properties:
	\begin{itemize}
		\item[(i)] $u>0$ in $\Omega$.
		\item[(ii)] For every $0\leq \varphi \in C^{\infty}_{0}(\Omega)$, it holds that
		\begin{equation}\label{eq:weak_sub_super_sol}
			\int_{\Omega}\langle \nabla_{\mathbb{G}}u, \nabla_{\mathbb{G}}\varphi \rangle_{\mathfrak{g}_{1}} \leq (\textrm{resp. } \geq)  \int_{\Omega}\left(\lambda u^{q}  + \ u^{2^{\star}_{Q}}\right)\varphi.
		\end{equation}
	\end{itemize}
	Finally, we say that $u \in S^{1}_{0}(\Omega)$ is a weak solution of \eqref{EqProblem}$_\lambda$ if it is both a weak subsolution and a weak supersolution of \eqref{EqProblem}$_\lambda$ without the non-negativity condition on $\varphi$.
\end{defin}

The above theorem is the natural generalization to Carnot groups of classical results of \cite{ABC}. We refer e.g. to \cite{BESS,CCP,CoPe,BV} for further generalizations.

We are now ready to properly set the definition of weak sub/supersolution of \eqref{EqProblem}$_\lambda$.
\begin{defin}\label{def:weak_sub_super_sol}
 Let $\Omega\subseteq\mathbb{G}$ be an open, bounded and connected set.
We say that a function $u \in S^{1}_{0}(\Omega)$ is a weak subsolution (resp. supersolution) of \eqref{EqProblem}$_\lambda$ if it satisfies the following properties:
 \begin{itemize}
 \item[(i)] $u>0$ in $\Omega$.
 \item[(ii)] For every $0\leq \varphi \in C^{\infty}_{0}(\Omega)$, it holds that
 \begin{equation}\label{eq:weak_sub_super_sol}
 \int_{\Omega}\langle \nabla_{\mathbb{G}}u, \nabla_{\mathbb{G}}\varphi \rangle_{\mathfrak{g}_{1}} \leq (\textrm{resp. } \geq) \int_{\Omega}\left(\lambda u^{q} + \ u^{2^{\star}_{Q}}\right)\varphi.
 \end{equation}
 \end{itemize}
 Finally, we say that $u \in S^{1}_{0}(\Omega)$ is a weak solution of \eqref{EqProblem}$_\lambda$ if it is both a weak subsolution and a weak supersolution of \eqref{EqProblem}$_\lambda$ without the non-negativity condition on $\varphi$.
\end{defin}

\begin{lemma}\label{lem:Perron}
 Let $\underline{u}, \overline{u}\in S^{1}_{0}(\Omega)$ be a weak subsolution and a weak supersolution, respectively, of problem \eqref{EqProblem}$_\lambda$.
 We assume that
 \begin{itemize}
 \item[a)] $\underline{u}(g) \leq \overline{u}(g)$ for a.e.\,$g\in \Omega$;
 \item[b)] for every open set $\Oo\Subset\Omega$ there exists
 $C = C(\Oo,\underline{u}) > 0$ such that
 $$\text{$\underline{u}\geq C$ a.e.\,in $\Oo$}.$$
 \end{itemize}
 Then, there exists a weak solution $u \in S^{1}_{0}(\Omega)$ of \eqref{EqProblem}$_{\lambda}$ such that
 $$\text{$\underline{u}(g) \leq u(g) \leq \overline{u}(g)$ for a.e. $g \in \Omega$}.$$
\end{lemma}

\noindent We split the proof of I) in two lemmas.
\begin{lemma}\label{lem:lambda0}
 Let $\Lambda$ as defined in \eqref{eq:DefinitionLambda}. Then $\Lambda >0$.
 \begin{proof}
 We will show that there exists a sufficiently small $\lambda >0$ such that
 \eqref{EqProblem}$_{\lambda}$ has a solution. To this aim, we will use Lemma \ref{lem:Perron} exhibiting both a super and a subsolution.
 Looking for a supersolution, we consider the following auxiliary torsion problem
 \begin{equation}
 \left\{\begin{array}{rl}
 -\Delta_{\mathbb{G}} V = 1 & \textrm{in } \Omega,\\
 V = 0 & \textrm{on } \partial \Omega,
 \end{array}\right.
 \end{equation}
 \noindent whose unique solution is provided by Lax-Milgram Theorem. Moreover, by a classical Stampacchia iteration method, it holds that $V\in L^{\infty}(\Omega)$. Observe further that for any positive constant $C$,
 the function $C\cdot x^p-x$, with $p>1$, has negative value for $x>0$ sufficiently small.
 Therefore, for every $C,C'\in \R^+$
 there exists $\lambda^*>0$ such that
 for every $\lambda<\lambda^*$,
 \[
 \exists \, m_\lambda\in \R^+:\quad \lambda\cdot C'\cdot m_\lambda^q+C\cdot m_\lambda^p-m_\lambda\leq0.
 \]
 We fix $\lambda<\lambda^*$ and set $C=\|V\|_{L^{\infty}(\Omega)}^p$, $C'=\|V\|_{L^{\infty}(\Omega)}^q$.
 We define $\overline u_1:=m_\lambda V$, which weakly verifies
 \[
 \left\{\begin{array}{rll}
 -\Delta_{\G}\overline u_1&=m_\lambda \geq \lambda\overline u_1^q+\overline u_1^p &\ \m{in }\Omega\\
 \overline u_1 &>0 &\ \m{in }\Omega\\
 \overline u_1 &=0 &\ \m{on }\de\Omega,
 \end{array}\right.
 \]
 therefore it is a weak supersolution of \eqref{EqProblem}$_\lambda$.\\
 Regarding the weak subsolution to \eqref{EqProblem}, we choose the unique solution $\underline u_\lambda$ to \eqref{eq:Sublinear_Problem}.
 We can now conclude the proof by appealing Lemma \ref{lem:Perron}. Indeed, by Lemma \ref{lem:Weak_Comparison_Model} with $w=\overline{u}_1$ and $v=\underline{u}_{\lambda}$, we get that
 $\overline{u}_1 \geq \underline{u}_{\lambda}$, which is condition a) of Lemma \ref{lem:Perron}. Regarding b) of Lemma \ref{lem:Perron} it is enough to recall \cite[Corollary 2.3]{BiGaVe}. This closes the proof.
 \end{proof}
\end{lemma}

We now set $\overline{u}= u_{\lambda'}$ and $\underline{u} = w_{\lambda}$,
 and we apply Lemma \ref{lem:Perron}: this immediately yields that problem \eqref{EqProblem}$_{\lambda}$ admits a weak solution $u_{\lambda}$ for every $\lambda \in (0,\Lambda)$. Moreover, recalling the definition of $I_{\lambda}$ in \eqref{eq:Def_Ilambda}, such a solution satisfies that
 $$I_{\lambda}(u_\lambda) = \min\{u\in S_0^1(\Omega):\,w_{\lambda}\leq u\leq u_{\lambda'}\}
 \leq I_{\lambda}(w_{\lambda}).$$
 In particular,
 by Theorem \ref{thm:Sublinear_Problem} we have
 \begin{equation} \label{eq:Ilambdaulambdaneg}
 I_{\lambda}(u_{\lambda}) \leq I_{\lambda}(w_{\lambda}) \leq J_{\lambda}(w_{\lambda}) <0.
 \end{equation}
 It remains to consider the case $\lambda = \Lambda$. The proof
 is rather standard and pretty similar to that of \cite[Lemma 3.5]{BiGaVe}. We report it here for the sake of completeness.
 To begin with, we choose a monotone increasing sequence $\{\lambda_k\}_k\subseteq(0,\Lambda)$ such that
 $\lambda_{k} \to \Lambda$ as $k\to+\infty$. Now, for each $k \in \mathbb{N}$, we set
 $$u_k := u_{\lambda_k}\in S^{1}_{0}(\Omega),$$
 \noindent where $u_{\lambda_k}$ is the weak solution of problem \eqref{EqProblem}$_{\lambda_k}$ constructed
 as above by means of Lemma \ref{lem:Perron}. Thanks to
 \eqref{eq:Ilambdaulambdaneg}, for every $k\geq 1$ we have
 \begin{equation} \label{eq:Ilambdakneg}
 I_{\lambda_k,}(u_{k})
 = \dfrac{1}{2} \int_{\Omega}|\nabla_{\mathbb{G}}u_k|^2 - \dfrac{\lambda_k}{q+1}\int_{\Omega}|u_k|^{q+1} - \dfrac{1}{2^{\star}_{Q}}\int_{\Omega}|u_k|^{2^{\star}_{Q}} <0.
 \end{equation}
 Moreover, by using $\varphi = u_k$ in \eqref{eq:weak_sub_super_sol}, and recalling that $u_k$
 solves \eqref{EqProblem}$_{\lambda_k}$, we get
 \begin{equation} \label{eq:testwithukzero}
 \int_{\Omega}|\nabla_{\mathbb{G}}u_k|^2 -\lambda_k\int_\Omega u_k^{q+1}
 -\int_\Omega u_k^{2^{\star}_{Q}} = 0.
 \end{equation}
 Combining \eqref{eq:Ilambdakneg} with \eqref{eq:testwithukzero},
 we notice that the sequence $\{u_k\}_k$ is bounded in~$S^{1}_{0}(\Omega)$.
 Therefore, we can find a function
 $$u_{\Lambda}\in S^{1}_{0}(\Omega)$$
 such that
 (up to a subsequence and as $k\to+\infty$)
 \begin{itemize}
 \item[a)] $u_k\to u_{\Lambda}$ weakly in $S^{1}_{0}(\Omega)$ and strongly
 in $L^p(\Omega)$ for $1\leq p <2^{\star}_{Q}$;
 \item[b)] $u_k\to u_{\Lambda}$ a.e.\,in $\Omega$.
 \end{itemize}
 We now observe that, being $\{\lambda_k\}_k$ increasing, it follows that $\lambda_k\geq \lambda_1$ for every $k\geq 1$.
 Moreover, arguing as above yields that
 $u_{\lambda_k}\geq w_{\lambda_1}$, and thus
 $$u_{\Lambda} > 0\quad\text{a.e.\,in $\Omega$}.$$
 Moreover, since $u_k$ solves problem \eqref{EqProblem}$_{\lambda_k}$,
 we have
 $$\int_{\Omega}\langle \nabla_{\mathbb{G}}u_k,\nabla_{\mathbb{G}}\varphi\rangle_{\mathfrak{g}_{1}} -\lambda_k\int_\Omega u_k^{q}\varphi
 -\int_\Omega u_k^{2^{\star}_{Q}-1}\varphi = 0\quad\text{for every $\varphi\in S^{1}_{0}(\Omega)$}.$$
Therefore, passing to the limit as $k\to+\infty$ in the above identity, and by dominated convergence, we get that $u_{\Lambda}$ satisfies
 $$\int_{\Omega}\langle \nabla_{\mathbb{G}}u_{\Lambda},\nabla_{\mathbb{G}}\varphi\rangle_{\mathfrak{g}_{1}} -\Lambda\int_\Omega u_{\Lambda}^{q}\varphi
 -\int_\Omega u_{\Lambda}^{2^{\star}_{Q}-1}\varphi = 0\quad\text{for every $\varphi\in S^{1}_{0}(\Omega)$},$$
 \noindent which shows that $u_{\Lambda}$ is actually a weak solution of
 problem \eqref{EqProblem}$_{\Lambda}$.
 This closes the proof.
\end{proof}

The above theorem is the natural generalization to Carnot groups of classical results of \cite{ABC}. We refer e.g. to \cite{BESS,CCP,CoPe,BV} for further generalizations.

The interest in studying existence of positive solutions to critical problems in the Carnot group setting,
 is in the geometric significance of the purely critical problem in the model case of the Heisenberg group.
 Indeed, when $\lambda =0$ and $\Omega = \mathbb{H}^n$, the problem \eqref{EqProblem} becomes the famous CR-Yamabe problem studied by Jerison and Lee \cite{JerisonLee,JerisonLee2,JerisonLee3}. The problem  we are interested in is settled on bounded domains, where tipycally one can prove non-existence of positive solutions, at least in star-shaped domains, by appealing suitable versions of the Pohozaev identity. Because of this, the seminal paper by Brezis and Nirenberg \cite{BN} showed that adding a perturbative term, linear in \cite{BN}, but subsequently extended to much more general perturbations, may allow to prove the existence of one or more positive solutions. A crucial tool in the argument performed in \cite{BN} is provided by the use of the Aubin-Talenti functions, whose analogue in $\mathbb{H}^n$ made its appearance in \cite{JerisonLee2}. This was a key ingredient which gave rise to a prolific study of critical problems in $\mathbb{H}^n$, see e.g. \cite{Citti, GaLa, Ugu1, CitUg, MaUg, FelliUgu, MaMaPi, PaPiTe}.

As long as one needs explicit knowledge of proper replacements of the Aubin-Talenti functions, the only other sub-Riemannian structure where they are known is that of groups of Iwasawa type, see \cite{GaroVa2, GaroVa}. As far as we know, there are no other structures, nor Sobolev inequalities with $p\neq 2$, for which the minimizers are explicitly known.
On the other hand, since the best constant in the Sobolev inequality is achieved in all Carnot groups (see \cite{GaroVa2}), it has been proved to be enough to know  the asymptotic behaviour at infinity of the minimizers. This is now known for $p\neq 2$ as well, see \cite{Loiudice3}, and it paved the way for a series of
existence, multiplicity or non-existence of positive solutions for critical problems  à la Br\'{e}zis-Nirenberg in $\mathbb{G}$: we refer e.g. to \cite{BoUg, Loiudice1, Loiudice2, Loiudice4, BiGaVe}.

\medskip

Let us now briefly describe the proof of Theorem \ref{thm:main}:
\begin{itemize}
	\item in Theorem \ref{thm:Existence_First} we prove the existence of a first solution by means of a variational Perron method which transfers the approach of Struwe \cite{Struwe} to the Carnot group setting. In particular, setting
	\begin{equation*}
	\Lambda := \sup \{ \lambda >0: \eqref{EqProblem}_\lambda \textrm{ admits a weak solution}\},
\end{equation*}
	\noindent we show first that $0<\Lambda<+\infty$, and this immediately provides a threshold for the non-existence of weak solutions. Once this is done, we use the unique solution of the purely sublinear problem \eqref{eq:Sublinear_Problem} as a weak subsolution and we construct a weak supersolution for fixed $\lambda$ by using the weak solution for a bigger $\lambda'$;
	\item we show that for $\lambda \in (0,\Lambda)$ the first solution obtained as described before is a local minimizer in the natural topology associated with problem \eqref{EqProblem}, see Lemma \ref{lem:locmin}. We stress here that in \cite{ABC} the authors made use of a famous result by Brezis and Nirenberg \cite{BNH1C1} which does not have an analog in the Carnot group setting. This is due to the fact that $C^{1,\alpha}$ regularity up to the boundary is still a delicate issue at the so called characteristic points: the first obstructions have been observed by Jerison \cite{Jerison,Jerison2}, but this is still an active field of research, see e.g. \cite{BaCiCu,BaGaMu,AbTr}. For this reason we follow here a more variational approach based on a paper by Alama \cite{Alama}, already used in a different setting in \cite{AbDiVa};
	\item we prove the existence of a second solution following an argument originally due to Tarantello \cite{Tarantello}: this combines the Ekeland variational principle \cite{Ekeland} with the fine asymptotic expansions proved in \cite{Loiudice1}.
\end{itemize}

\medskip

We stress that the multiplicity result obtained in Theorem \ref{thm:main} can be easily extended to cover the convex-case of a Sobolev sub-critical nonlinearity.

\medskip

The paper is organized as follows: in Section \ref{sec:Prel} we recall the basic facts on Carnot groups and we set the variational functional setting necessary for the study of \eqref{EqProblem} We also recall the basic result regarding the purely sublinear problems, like existence and uniqueness of a positive solution and a comparison principle resembling the classical one. In Section \ref{sec:First_Solution} we prove the existence of a first solution as described before, while the existence of a second solution (for $\lambda \in (0,\Lambda)$) is postponed to the final Section \ref{sec:Second_Solution}.

Let us now briefly describe the proof of Theorem \ref{thm:main}:
\begin{itemize}
\item In Theorem \ref{thm:Existence_First} the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \cite{Struwe} to the Carnot group setting.” Defining
\[
\Lambda := \sup\{\lambda>0:\ \eqref{EqProblem}_\lambda\ \textrm{admits a weak solution}\},
\]
one “show[s] first that $0<\Lambda<+\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem \eqref{eq:Sublinear_Problem} as a weak subsolution” and constructs “a weak supersolution for fixed $\lambda$ by using the weak solution for a bigger $\lambda'$.”
\item For $\lambda\in(0,\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with problem \eqref{EqProblem}, see Lemma \ref{lem:locmin}.” Since the Brezis–Nirenberg boundary regularity tool used in \cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \cite{Alama}.”
\item The second solution is obtained by “an argument originally due to Tarantello \cite{Tarantello},” combining “the Ekeland variational principle \cite{Ekeland} with the fine asymptotic expansions proved in \cite{Loiudice1}.”
\end{itemize}

Let us now briefly describe the proof of Theorem \ref{thm:main}:
\begin{itemize}
\item We first prove the following theorem.
\begin{teo}\label{thm:Existence_First}
	Problem \eqref{EqProblem}$_\lambda$ admits at least
	one weak solution $u_\lambda\in S^{1}_{0}(\Omega)$ for every $\lambda \in (0,\Lambda]$.
\end{teo}
In this step, the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \cite{Struwe} to the Carnot group setting.” Defining
\[
\Lambda := \sup\{\lambda>0:\ \begin{equation}\tag{{$\mathrm{P}$}}\label{EqProblem}
	\left\{\begin{array}{rll}
		-\Delta_{\mathbb{G}}u& = \lambda\, u^{q} + u^{2^{\star}_{Q}-1} & \textrm{ in } \Omega,\\
		u&>0 & \textrm{ in } \Omega,\\
		u&=0 & \textrm{ on } \partial \Omega.
	\end{array}\right.
\end{equation}_\lambda\ \textrm{admits a weak solution}\},
\]
one “show[s] first that $0<\Lambda<+\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem
\begin{equation}\label{eq:Sublinear_Problem}
	\left\{\begin{array}{rll}
				-\Delta_{\mathbb{G}}u &= \lambda\, u^{q} & \textrm{ in } \Omega,\\
				u&>0 & \textrm{ in } \Omega,\\
				u&=0 & \textrm{ on } \partial \Omega,
	\end{array}\right.
\end{equation}
as a weak subsolution” and constructs “a weak supersolution for fixed $\lambda$ by using the weak solution for a bigger $\lambda'$.”
\item For $\lambda\in(0,\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with the equation above.” We now record the corresponding statement.
\begin{lemma}\label{lem:locmin}
	For $\lambda\in (0,\Lambda)$, if $u_\lambda$ is the solution
	presented in Theorem \ref{thm:Existence_First}, then
	$u_\lambda$ is a local minimum for $I_\lambda$ in the $S^1_0(\Omega)$-topology,
	meaning that there exists $r_0>0$ such that for any $u\in S^1_0(\Omega)$,
	\[
	I_\lambda(u_\lambda)\leq I_\lambda(u) \quad \textrm{ for all } u\in S^{1}_{0}(\Omega) \textrm{ with } \|u-u_\lambda\|_{S_{0}^{1}(\Omega)}< r_0 .
	\]
\end{lemma}
Since the Brezis–Nirenberg boundary regularity tool used in \cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \cite{Alama}.”
\item The second solution is obtained by “an argument originally due to Tarantello \cite{Tarantello},” combining “the Ekeland variational principle \cite{Ekeland} with the fine asymptotic expansions proved in \cite{Loiudice1}.”
\end{itemize}

\label{thm:main}
	Let $\Omega\subset\mathbb{G}$ be an open and bounded set
	with smooth enough boundary $\partial \Omega$, and let $p\in (0,1)$. Then, there exists $\Lambda > 0$
	such that
	\begin{itemize}
		\item[A)]problem \begin{equation}\tag{{$\mathrm{P}$}}\label{EqProblem}
	\left\{\begin{array}{rll}
		-\Delta_{\mathbb{G}}u& = \lambda\, u^{q} + u^{2^{\star}_{Q}-1} & \textrm{ in } \Omega,\\
		u&>0 & \textrm{ in } \Omega,\\
		u&=0 & \textrm{ on } \partial \Omega.
	\end{array}\right.
\end{equation}$_{\lambda}$ does not admit weak solutions
		for every $\lambda>\Lambda$;
		\item[B)] problem the equation above$_{\lambda}$ admits at least one weak solution for every $\lambda \in (0,\Lambda]$;
		\item[C)] problem the equation above$_{\lambda}$ admits at least two weak solutions for every $0<\lambda<\Lambda$.
	\end{itemize}

Question:
Let $\mathbb{G}$ be a Carnot group, let $\Omega\subset \mathbb{G}$ be an open bounded set with sufficiently smooth boundary $\partial\Omega$, and let $q\in(0,1)$. Let $2_Q^{\star}=\frac{2Q}{Q-2}$, where $Q$ is the homogeneous dimension of $\mathbb{G}$. For $\lambda>0$, consider the Dirichlet problem
\[
\begin{cases}
-\Delta_{\mathbb{G}}u=\lambda u^q+u^{2_Q^{\star}-1} & \text{in }\Omega,\\
u>0 & \text{in }\Omega,\\
u=0 & \text{on }\partial\Omega,
\end{cases}
\]
where $\Delta_{\mathbb{G}}$ is the sub-Laplacian on $\mathbb{G}$. A weak solution means a function $u\in S_0^1(\Omega)$ with $u>0$ in $\Omega$ such that
\[
\int_\Omega \langle \nabla_{\mathbb{G}}u,\nabla_{\mathbb{G}}\varphi\rangle_{\mathfrak g_1}
=\int_\Omega \bigl(\lambda u^q+u^{2_Q^{\star}-1}\bigr)\varphi
\quad\text{for every }\varphi\in S_0^1(\Omega).
\]
Which statement holds?

There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda>\Lambda$, has at least one weak solution for every $\lambda\in(0,\Lambda]$, and has at least two weak solutions for every $\lambda\in(0,\Lambda)$.

Choices:
B. There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda\geq\Lambda$, has at least one weak solution for every $\lambda\in(0,\Lambda)$, and has at least two weak solutions for every $\lambda\in(0,\Lambda)$.
C. There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda>\Lambda$ and has at least one weak solution for every $\lambda\in(0,\Lambda]$.
D. There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda>\Lambda$, has at least one weak solution for every $\lambda\in(0,\Lambda]$, and has at least two weak solutions for every $\lambda\in(0,\Lambda]$.
E. There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda>\Lambda$, and for every $\lambda\in(0,\Lambda]$ the problem admits a unique weak solution.


Question:
Let \(\mathbb G\) be a Carnot group, let \(\Omega\subset \mathbb G\) be an open bounded set with sufficiently smooth boundary \(\partial\Omega\), let \(q\in(0,1)\), and let \(2_Q^*:=\frac{2Q}{Q-2}\) be the critical Sobolev exponent associated with \(\mathbb G\). For \(\lambda>0\), consider the Dirichlet problem
\[
\begin{cases}
-\Delta_{\mathbb G}u=\lambda u^q+u^{2_Q^*-1} & \text{in }\Omega,\\
u>0 & \text{in }\Omega,\\
u=0 & \text{on }\partial\Omega.
\end{cases}
\]
Here \(-\Delta_{\mathbb G}\) is the sub-Laplacian on \(\mathbb G\), and a weak solution means a function \(u\in S_0^1(\Omega)\) with \(u>0\) in \(\Omega\) such that
\[
\int_\Omega \langle \nabla_{\mathbb G}u,\nabla_{\mathbb G}\varphi\rangle\,=\int_\Omega \bigl(\lambda u^q+u^{2_Q^*-1}\bigr)\varphi
\quad\text{for every }\varphi\in S_0^1(\Omega).
\]
Which existence statement holds for this problem?

There exists \(\Lambda>0\) such that: (i) for every \(\lambda>\Lambda\), the problem admits no weak solution; (ii) for every \(\lambda\in(0,\Lambda]\), the problem admits at least one weak solution; and (iii) for every \(0<\lambda<\Lambda\), the problem admits at least two weak solutions.

Choices:
B. There exists \(\Lambda>0\) such that: (i) for every \(\lambda\ge \Lambda\), the problem admits no weak solution; (ii) for every \(\lambda\in(0,\Lambda)\), the problem admits at least one weak solution; and (iii) for every \(0<\lambda<\Lambda\), the problem admits at least two weak solutions.
C. There exists \(\Lambda>0\) such that: (i) for every \(\lambda>\Lambda\), the problem admits no weak solution; and (ii) for every \(\lambda\in(0,\Lambda]\), the problem admits at least one weak solution.
D. There exists \(\Lambda>0\) such that: (i) for every \(\lambda>\Lambda\), the problem admits no weak solution; (ii) for every \(\lambda\in(0,\Lambda]\), the problem admits a unique weak solution; and (iii) for every \(0<\lambda<\Lambda\), the problem admits at least two weak solutions.
E. There exists \(\Lambda>0\) such that: (i) for every \(\lambda>\Lambda\), the problem admits no weak solution; (ii) for every \(\lambda\in(0,\Lambda]\), the problem admits at least one weak solution; and (iii) for every \(0<\lambda\le \Lambda\), the problem admits at least two weak solutions.

endpoint solvability at \(\lambda=\Lambda\)

multiplicity conclusion for \(0<\lambda<\Lambda\)

construction of a second solution below \(\Lambda\) does not imply uniqueness of the first/existing solutions

strict interior range for the second-solution result excludes the endpoint \(\lambda=\Lambda\)


\begin{theorem} \label{thm:main}
Let $\beta\in \{1,2,4\}$. If $p\ge \frac{3}{2}$, then, as $n\to\infty$,
\begin{align}\label{eq:main-1}
\ln \vol(\bpb)
=&-\frac{\beta}{2}\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n\notag\\
&+\frac{\beta}{2}\Big(\frac{1}{2}\ln\frac{4\pi}{\beta}+\frac{3}{4}+\ln A(p)\Big)n^2\notag\\
&-\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}+\frac{1}{p}\Big)n\ln n\notag\\
&+\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}\ln\frac{4}{\beta\pi}+\frac{1}{2}+\frac{1}{2p}+\ln A(p)+\Ent(\mu_p)\Big)n +o(n).
\end{align}
Moreover, if $\beta=2$ and $p\ge 1$, then, for some constant $M_p\in \IR$,
\begin{equation} \label{eq:main-2}
\ln \vol(\IB_{p,2}^n)
=-\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n
+\Big(\frac{1}{2}\ln 2\pi +\frac{3}{4}+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\end{equation}
\end{theorem}

The Schatten or Schatten-von Neumann class consists of all compact operators between two Hilbert spaces whose sequence of singular values belongs to the Lebesgue sequence space $\ell_p$. It can be seen a non-commutative version of $\ell_p$ and is named after Schatten and von Neumann \cite{Sch46,Sch50,SVN46}. We refer to \cite{Pie07,PX03} for an overview of their role in functional analysis and Banach space geometry.

In this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \cite{ASW11,Wil13}. Let $\mathbb{F}_1=\mathbb{R}$, $\mathbb{F}_2=\mathbb{C}$ and $\mathbb{F}_4=\mathbb{H}$ and denote the singular values of a matrix $A\in \mathbb{F}_{\beta}^{n\times n}$, where $\beta\in\{1,2,4\}$, by
\[
s_1(A)\ge \cdots \ge s_n(A)\ge 0.
\]
For $1\le p\le \infty$, the $p$-Schatten norm of $A$ is
\[
\|A\|_p
=
\begin{cases}
	\Big(\sum_{j=1}^{n}s_j(A)^p\Big)^{1/p}&\colon p<\infty\\
	s_1(A) &\colon p=\infty.
\end{cases}
\]

The associated unit ball and its intersection with the subspace of self-adjoint matrices $\cH_n(\IF_{\beta})=\{A\in \IF_{\beta}^{n\times n}\colon A^{*}=A\}$  are denoted by
\[
B_{p,\beta}^n
=\{A\in \IF_{\beta}^{n\times n}\colon \|A\|_p\le 1\}\quad \text{and}\quad
\mathbb{B}_{p,\beta}^n
= B_{p,\beta}^n \cap \cH_n(\IF_{\beta}).
\]
For self-adjoint matrices the Schatten $p$-norm $\|A\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \cite{JKP24}.

We shall identify $\IF_{\beta}^{n\times n}$ with $\mathbb{R}^{\beta n^2}$ and $\cH_n(\IF_{\beta})$ with $\mathbb{R}^{d_n}$, where
\begin{align}\label{eq:dn}
d_n:=\beta\frac{n(n-1)}{2}+n.
\end{align}
Then $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\mathbb{R}^{\beta n^2}$ and $\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\mathrm{vol}(B_{p,\beta}^n)$ and $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\mathbb{B}_{p,\beta}^n$ as $n\to \infty$.

The exact volume of $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ is only known in the cases $p=2$ and $p=\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus
\begin{equation*}
\mathrm{vol}(B_{2,\beta}^n)
=\frac{\pi^{\beta n^2/2}}{\Gamma(1+\frac{\beta n^2}{2})}
\qquad\text{and}\qquad
\mathrm{vol}(\IB_{2,\beta}^n)
=\frac{\pi^{d_n/2}}{\Gamma(1+\frac{d_n}{2})},
\end{equation*}
where $\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}\mathrm{d} t$ denotes the Gamma function. For $p=\infty$ it follows from Saint Raymond's work \cite{SR84} and Selberg's integral formula \cite{And91,Sel44} that
\begin{equation} \label{eq:inf-exact}
\mathrm{vol}(B_{\infty,\beta}^n)
=\frac{\prod_{j=0}^{n-1}\Gamma( 1+j\frac{\beta}{2} )}{\prod_{j=n}^{2n-1}\Gamma( 1+j\frac{\beta}{2})}\pi^{\beta n^2/2},
\end{equation}
and in the self-adjoint case it holds that
\begin{equation} \label{eq:inf-sa-exact}
\mathrm{vol}(\IB_{\infty,\beta}^n)
=2^{d_n}(2\pi)^{\beta n(n-1)/4}\prod_{j=0}^{n-1}\frac{\Gamma( 1+j\frac{\beta}{2})^2\Gamma( (j+1)\frac{\beta}{2} )}{\Gamma( 2+(n+j-1)\frac{\beta}{2} )\Gamma( j\frac{\beta}{2} )}.
\end{equation}
For general $1\le p\le \infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \cite{SR84} derived $A(p)>0$ such that, for $\beta\in\{1,2\}$,
\begin{equation} \label{eq:sim-asym}
\lim_{n\to \infty}n^{1/2+1/p}\mathrm{vol}(B_{p,\beta}^n)^{1/\beta n^2}
=\sqrt{\frac{2\pi}{\beta}}e^{3/4}A(p)
\end{equation}
and determined $A(2)=e^{-1/4}$ and $A(\infty)=\frac{1}{2}$. Gu\'{e}don and Paouris \cite{GP07} showed that $\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)^{1/d_n}, \beta\in\{1,2,4\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \cite{KPT20a} determined
\begin{equation} \label{eq:delta}
A(p)
=\frac{1}{2}\bigg(\frac{p\sqrt{\pi}\Gamma(\frac{p}{2})}{\sqrt{e}\Gamma(\frac{p+1}{2})}\bigg)^{1/p}
\end{equation}
and moreover showed in \cite{KPT20b} that, for $\beta\in \{1,2,4\}$,
\begin{equation} \label{eq:sim-sym}
	\lim_{n\to \infty}n^{1/2+1/p}\mathrm{vol}(\mathbb{B}_{p,\beta}^n)^{1/d_n}
	=\sqrt{\frac{4\pi}{\beta}}e^{3/4}A(p).
\end{equation}
We remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\mathrm{vol}(n^{1/p}\mathbb{B}_{p,\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\'{e}don and Zitt \cite[Remark 4.8]{DFG+23} showed \eqref{eq:sim-asym} for $\beta=4$. Thus, the asymptotics in \eqref{eq:sim-asym} and \eqref{eq:sim-sym} hold in all cases $\beta\in \{1,2,4\}$ and $1\le p\le \infty$.

In case of $p=\infty$, the quantity $A(\infty)=\frac{1}{2}$ can be expressed in terms of the minimal logarithmic energy of probability measures supported in $[-1,1]$ attained by the arcsine distribution, see \cite{ST97}. We refer to \cite{BDF+24} and \cite{Bra24} for related asymptotics for discrete minimal logarithmic energy.

In case of $p<\infty$, the quantity $A(p)$ in \eqref{eq:delta} can be expressed as
\[
A(p)
=\frac{1}{2}\Big(\frac{p\sqrt{\pi}\Gamma(\frac{p}{2})}{\sqrt{e}\Gamma(\frac{p+1}{2})}\Big)^{1/p}
=\frac{1}{2}\Big(\frac{1}{\sqrt{e}\alpha_p}\Big)^{1/p},
\]
where $\alpha_p=\int_{\mathbb{R}} |x|^p\mathrm{d}\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\mu_p$ which has density
\begin{equation} \label{eq:ullman-1}
f_p(x)
= \frac{p}{\pi}\int_{|x|}^1 \frac{t^{p-1}}{\sqrt{t^2-x^2}}\mathrm{d} t, \qquad x\in [-1,1].
\end{equation}

Our main contribution is the following asymptotic expansion for the logarithmic volume $\ln \mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. Recall that the differential entropy of a probability measure $\mu$ on $\mathbb{R}$ is
\[
\mathrm{Ent}(\mu)
= -\int_{\mathbb{R}}\ln(f(x))f(x)\mathrm{d} x,
\]
whenever $\mu$ admits a Lebesgue density $f$ and $\mathrm{Ent}(\mu)=-\infty$ otherwise.

In this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \cite{ASW11,Wil13}. Let $\mathbb{F}_1=\mathbb{R}$, $\mathbb{F}_2=\mathbb{C}$ and $\mathbb{F}_4=\mathbb{H}$ and denote the singular values of a matrix $A\in \mathbb{F}_{\beta}^{n\times n}$, where $\beta\in\{1,2,4\}$, by
\[
s_1(A)\ge \cdots \ge s_n(A)\ge 0.
\]
For $1\le p\le \infty$, the $p$-Schatten norm of $A$ is
\[
\|A\|_p
=
\begin{cases}
 \Big(\sum_{j=1}^{n}s_j(A)^p\Big)^{1/p}&\colon p<\infty\\
 s_1(A) &\colon p=\infty.
\end{cases}
\]

The associated unit ball and its intersection with the subspace of self-adjoint matrices $\cH_n(\IF_{\beta})=\{A\in \IF_{\beta}^{n\times n}\colon A^{*}=A\}$ are denoted by
\[
B_{p,\beta}^n
=\{A\in \IF_{\beta}^{n\times n}\colon \|A\|_p\le 1\}\quad \text{and}\quad
\mathbb{B}_{p,\beta}^n
= B_{p,\beta}^n \cap \cH_n(\IF_{\beta}).
\]
For self-adjoint matrices the Schatten $p$-norm $\|A\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \cite{JKP24}.

We shall identify $\IF_{\beta}^{n\times n}$ with $\mathbb{R}^{\beta n^2}$ and $\cH_n(\IF_{\beta})$ with $\mathbb{R}^{d_n}$, where
\begin{align}\label{eq:dn}
d_n:=\beta\frac{n(n-1)}{2}+n.
\end{align}
Then $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\mathbb{R}^{\beta n^2}$ and $\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\mathrm{vol}(B_{p,\beta}^n)$ and $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\mathbb{B}_{p,\beta}^n$ as $n\to \infty$.

The exact volume of $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ is only known in the cases $p=2$ and $p=\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus
\begin{equation*}
\mathrm{vol}(B_{2,\beta}^n)
=\frac{\pi^{\beta n^2/2}}{\Gamma(1+\frac{\beta n^2}{2})}
\qquad\text{and}\qquad
\mathrm{vol}(\IB_{2,\beta}^n)
=\frac{\pi^{d_n/2}}{\Gamma(1+\frac{d_n}{2})},
\end{equation*}
where $\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}\mathrm{d} t$ denotes the Gamma function. For $p=\infty$ it follows from Saint Raymond's work \cite{SR84} and Selberg's integral formula \cite{And91,Sel44} that
\begin{equation} \label{eq:inf-exact}
\mathrm{vol}(B_{\infty,\beta}^n)
=\frac{\prod_{j=0}^{n-1}\Gamma( 1+j\frac{\beta}{2} )}{\prod_{j=n}^{2n-1}\Gamma( 1+j\frac{\beta}{2})}\pi^{\beta n^2/2},
\end{equation}
and in the self-adjoint case it holds that
\begin{equation} \label{eq:inf-sa-exact}
\mathrm{vol}(\IB_{\infty,\beta}^n)
=2^{d_n}(2\pi)^{\beta n(n-1)/4}\prod_{j=0}^{n-1}\frac{\Gamma( 1+j\frac{\beta}{2})^2\Gamma( (j+1)\frac{\beta}{2} )}{\Gamma( 2+(n+j-1)\frac{\beta}{2} )\Gamma( j\frac{\beta}{2} )}.
\end{equation}
For general $1\le p\le \infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \cite{SR84} derived $A(p)>0$ such that, for $\beta\in\{1,2\}$,
\begin{equation} \label{eq:sim-asym}
\lim_{n\to \infty}n^{1/2+1/p}\mathrm{vol}(B_{p,\beta}^n)^{1/\beta n^2}
=\sqrt{\frac{2\pi}{\beta}}e^{3/4}A(p)
\end{equation}
and determined $A(2)=e^{-1/4}$ and $A(\infty)=\frac{1}{2}$. Gu\'{e}don and Paouris \cite{GP07} showed that $\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)^{1/d_n}, \beta\in\{1,2,4\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \cite{KPT20a} determined
\begin{equation} \label{eq:delta}
A(p)
=\frac{1}{2}\bigg(\frac{p\sqrt{\pi}\Gamma(\frac{p}{2})}{\sqrt{e}\Gamma(\frac{p+1}{2})}\bigg)^{1/p}
\end{equation}
and moreover showed in \cite{KPT20b} that, for $\beta\in \{1,2,4\}$,
\begin{equation} \label{eq:sim-sym}
 \lim_{n\to \infty}n^{1/2+1/p}\mathrm{vol}(\mathbb{B}_{p,\beta}^n)^{1/d_n}
 =\sqrt{\frac{4\pi}{\beta}}e^{3/4}A(p).
\end{equation}
We remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\mathrm{vol}(n^{1/p}\mathbb{B}_{p,\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\'{e}don and Zitt \cite[Remark 4.8]{DFG+23} showed \eqref{eq:sim-asym} for $\beta=4$. Thus, the asymptotics in \eqref{eq:sim-asym} and \eqref{eq:sim-sym} hold in all cases $\beta\in \{1,2,4\}$ and $1\le p\le \infty$.

In case of $p<\infty$, the quantity $A(p)$ in \eqref{eq:delta} can be expressed as
\[
A(p)
=\frac{1}{2}\Big(\frac{p\sqrt{\pi}\Gamma(\frac{p}{2})}{\sqrt{e}\Gamma(\frac{p+1}{2})}\Big)^{1/p}
=\frac{1}{2}\Big(\frac{1}{\sqrt{e}\alpha_p}\Big)^{1/p},
\]
where $\alpha_p=\int_{\mathbb{R}} |x|^p\mathrm{d}\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\mu_p$ which has density
\begin{equation} \label{eq:ullman-1}
f_p(x)
= \frac{p}{\pi}\int_{|x|}^1 \frac{t^{p-1}}{\sqrt{t^2-x^2}}\mathrm{d} t, \qquad x\in [-1,1].
\end{equation}

Our main contribution is the following asymptotic expansion for the logarithmic volume $\ln \mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. Recall that the differential entropy of a probability measure $\mu$ on $\mathbb{R}$ is
\[
\mathrm{Ent}(\mu)
= -\int_{\mathbb{R}}\ln(f(x))f(x)\mathrm{d} x,
\]
whenever $\mu$ admits a Lebesgue density $f$ and $\mathrm{Ent}(\mu)=-\infty$ otherwise.

\begin{equation} \label{eq:sim-asym}
\lim_{n\to \infty}n^{1/2+1/p}\vol(B_{p,\beta}^n)^{1/\beta n^2}
=\sqrt{\frac{2\pi}{\beta}}e^{3/4}A(p)
\end{equation}

\begin{equation} \label{eq:sim-sym}
	\lim_{n\to \infty}n^{1/2+1/p}\vol(\bpb)^{1/d_n}
	=\sqrt{\frac{4\pi}{\beta}}e^{3/4}A(p).
\end{equation}

Our main contribution is the following asymptotic expansion for the logarithmic volume $\ln \vol(\bpb)$. Recall that the differential entropy of a probability measure $\mu$ on $\IR$ is
\[
\Ent(\mu)
= -\int_{\IR}\ln(f(x))f(x)\dd x,
\]
whenever $\mu$ admits a Lebesgue density $f$ and $\Ent(\mu)=-\infty$ otherwise.

In case of $p=2$ it is known that $\Ent(\mu_2)=\ln \pi -\frac{1}{2}$ where $\mu_2$ is the semi-circle law. Otherwise the entropy $\Ent(\mu_p)$ of the Ullman distribution appears to be unknown. Independently, the expansion in \eqref{eq:main-1} was recently obtained by Dworaczek Guera, Memin and Pain \cite{GMP25} in case of $p\ge 2$.

In this context the Ullman distribution arises as the corresponding equilibrium measure minimizing the functional
\[
\mu\mapsto I_p(\mu)
=-\int_{\IR}\int_{\IR} \ln|x-y|\dd\mu(x)\dd\mu(y)+v_p\int_{\IR} |x|^p\dd\mu(x)
\]
over $\mu\in \cP(\IR)$, where $\cP(\IR)$ denotes the space of probability measures on $\IR$. More precisely, if $X=(X_1,\dots,X_n)$ is distributed according to the density $f_{n,p,\beta}$, then the random empirical measure $ \mu_{n,p} =\frac{1}{n}\sum_{i=1}^{n}\delta_{X_i} $ tends to $\mu_p$ as $n\to\infty$. Moreover, large deviation principles due to \cite{BG97} and \cite{HP00} yield
\begin{equation} \label{eq:Z-first}
\lim_{n\to\infty}\frac{2}{\beta n^2}\ln Z_{n,p,\beta}
= I_p(\mu_p)
=\ln 2+\frac{3}{2p}.
\end{equation}
Earlier work \cite{DPS95,Joh98} also identified the limiting constant but with different techniques. In \cite{DFG+23}, the limit \eqref{eq:Z-first} is used to prove \eqref{eq:sim-sym}. The large deviation principle for tagged empirical fields due to Lebl\'e and Serfaty \cite{LS17} allows to refine \eqref{eq:Z-first}. The following result forms the core of the proof of \eqref{eq:main-1} in Theorem~\ref{thm:main}.

\begin{theorem} \label{thm:LS}
Let $\beta>0$ and $p\ge \frac{3}{2}$. As $n\to\infty$,
\[
\ln Z_{n,p,\beta}
=-\frac{\beta}{2}n^2 I_p(\mu_p)+\frac{\beta}{2}n\ln n- C(\beta)n+\Big(1-\frac{\beta}{2}\Big)\Ent(\mu_p)n +o(n),
\]
where
\begin{equation} \label{eq:c-beta}
C(\beta)
=\Big(1-\frac{\beta}{2}\Big)\ln \frac{\beta}{2}-\frac{\beta}{2}\ln 2\pi+\frac{\beta}{2}+\ln \Gamma\Big(\frac{\beta}{2}\Big).
\end{equation}
\end{theorem}

It remains to check Hölder continuity at zero. Let $h\in (0,1)$ and write
\begin{align*}
\frac{\pi}{p}\big(f_p(h)-f_p(0)\big)
&=\int_h^1 \frac{t^{p-1}}{\sqrt{t^2-h^2}}\dd t - \int_0^1 t^{p-2}\dd t\\
&=\int_h^1 t^{p-1}\Big(\frac{1}{\sqrt{t^2-h^2}}-\frac{1}{t}\Big)\dd t -\frac{h^{p-1}}{p-1}.
\end{align*}
The latter integral can be estimated by
\begin{align*}
\int_h^1 t^{p-1}\Big(\frac{1}{\sqrt{t^2-h^2}}-\frac{1}{t}\Big)\dd t
&\le h^2\int_h^1 \frac{t^{p-3}}{\sqrt{t^2-h^2}}\dd t\\
&= h^{p-1}\int_1^{1/h} \frac{u^{p-3}}{\sqrt{u^2-1}}\dd u.
\end{align*}
In particular, if $1< p<2$, then
\begin{equation} \label{eq:hoelder}
0\le \frac{\pi}{p}\frac{f_p(h)-f_p(0)}{h^{p-1}}
\le \int_1^{\infty} \frac{u^{p-3}}{\sqrt{u^2-1}}\dd u
<\infty.
\end{equation}
Thus, in this case we obtain that $f_p$ is Hölder continuous of order $p-1$ at $0$. For $p\ge 2$, a similar argument shows that $f_p$ is continuously differentiable at $0$ with $f_p'(0)=0$. In conclusion, if $p\ge\frac{3}{2}$, the function $f_p$ is Hölder continuous of order $\frac{1}{2}$ on $[-1,1]$. This completes the proof.
\end{proof}

\begin{lemma} \label{lem:cn}
Let $\beta\in \{1,2,4\}$. Then, as $n\to\infty$,
\begin{align*}
\ln c_{n}
=&-\frac{\beta}{4}n^2\ln n + \frac{\beta}{2}\Big(\frac{1}{2}\ln \frac{4\pi}{\beta}+\frac{3}{4}\Big)n^2-\frac{1}{2}\Big(1+\frac{\beta}{2}\Big)n\ln n\\
&+\Big(\frac{1}{2}\Big(1-\frac{\beta}{2}\Big)\ln \beta -\frac{1}{2}\Big(1+\frac{\beta}{2}\Big)(\ln \pi-1)-\ln 2+\ln \Gamma\Big(\frac{\beta}{2}\Big)\Big)n\\
&-\frac{3+\frac{\beta}{2}+\frac{2}{\beta}}{12}\ln n
+a_{\beta}+o(1),
\end{align*}
where $a_{\beta}\in \IR$ is some constant.
\end{lemma}
\begin{proof}
It follows from the definition of $c_{n}$ that
\[
c_{n}
=\frac{1}{n!}\Gamma_2\Big(n+1;\frac{2}{\beta},1\Big)\bigg(\frac{\Gamma(\frac{\beta}{2})}{(\frac{\beta}{2})^{\frac{\beta}{2}}}\bigg)^n\Big(\frac{4\pi}{\beta}\Big)^{\frac{\beta}{4}n^2-(\frac{1}{2}+\frac{\beta}{4})n},
\]
where $\Gamma_2(x;\frac{2}{\beta},1)$ is a particular instance of the Barnes double Gamma function, see \cite{BG24,Spr09}. Adapting \cite[eq. (7.17)]{BG24}, the following asymptotic expansion holds
\begin{align*}
\ln \Gamma_2\Big(n+1;\frac{2}{\beta},1\Big)
=&-\frac{\beta}{4}n^2\ln n+\frac{3\beta}{8}n^2+\frac{1}{2}\Big(1-\frac{\beta}{2}\Big)n\ln n-\frac{1}{2}\Big(1-\frac{\beta}{2}\Big)n\\
&+\bigg(\frac{1}{2} -\frac{3+\frac{\beta}{2}+\frac{2}{\beta}}{12}\bigg)\ln n +a_{\beta}'+o(1),
\end{align*}
where $a_{\beta}'\in \IR$ is some constant. Combined with Stirling's approximation $\ln n!=n\ln n-n+\frac{1}{2}\ln(2\pi n)+o(1)$ we deduce the claimed expansion.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $\beta\in \{1,2,4\}$ and $p\ge \frac{3}{2}$. Recall that $d_n=\frac{\beta}{2}n^2+(1-\frac{\beta}{2})n$. By Lemma~\ref{lem:vol-partition} it holds that
\begin{equation} \label{eq:vol-asymp-parts}
\ln \vol(\bpb)
=\ln c_{n}+\frac{d_n}{p}\ln \Big(\frac{\beta nv_p}{2}\Big)-\ln \Gamma\Big(1+\frac{d_n}{p}\Big)+\ln Z_{n,p,\beta}.
\end{equation}
By Stirling's approximation, as $n\to\infty$,
\begin{align}\label{eq:p-term-1}
\ln \Gamma\Big(1+\frac{d_n}{p}\Big)
&=\frac{d_n}{p}\Big(\ln \frac{d_n}{p}-1\Big)+o(n)\notag\\
&=\frac{d_n}{p}\Big(2\ln n+\ln \frac{\beta}{2p}-1+\Big(1-\frac{\beta}{2}\Big)\frac{2}{\beta n}\Big)+o(n).
\end{align}
Therefore,
\begin{align}\label{eq:p-term-2}
\frac{d_n}{p}\ln\Big(\frac{\beta n v_p}{2}\Big)
-\ln \Gamma\Big(1+\frac{d_n}{p}\Big)
=&-\frac{d_n}{p}\Big(\ln n -\ln(p v_p)-1+\Big(1-\frac{\beta}{2}\Big)\frac{2}{\beta n}\Big)\notag\\
=&-\frac{\beta}{2p}n^2\ln n+\frac{\beta}{2p}\big(\ln(pv_p)+1\big)n^2\notag\\
&-\frac{1}{p}\Big(1-\frac{\beta}{2}\Big)n\ln n+\frac{\ln(pv_p)}{p}\Big(1-\frac{\beta}{2}\Big)n +o(n).
\end{align}

In case of $p=2$ it is known that $\mathrm{Ent}(\mu_2)=\ln \pi -\frac{1}{2}$ where $\mu_2$ is the semi-circle law. Otherwise the entropy $\mathrm{Ent}(\mu_p)$ of the Ullman distribution appears to be unknown. Independently, the expansion in \eqref{eq:main-1} was recently obtained by Dworaczek Guera, Memin and Pain \cite{GMP25} in case of $p\ge 2$.

The proof of Theorem~\ref{thm:main} relies on the fact that the volume of $\mathbb{B}_{p,\beta}^n$ can be expressed in terms of the integral
\begin{equation} \label{eq:Z}
Z_{n,p,\beta}
=\int_{\mathbb{R}^n}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta} e^{-\frac{\beta}{2}n\, v_p\sum_{i=1}^{n}|x_i|^p}\mathrm{d} x,
\end{equation}
where
\[
v_p =\frac{\sqrt{\pi}\Gamma(\frac{p}{2})}{\Gamma(\frac{p+1}{2})}=\frac{1}{p\alpha_p}.
\]
The quantity $Z_{n,p,\beta}$ is the partition function of a one-dimensional $\beta$-ensemble with potential $V(x)=v_p |x|^p$, where eigenvalues/particles are distributed with joint density
\begin{equation} \label{eq:density}
f_{n,p,\beta}(x)
=\frac{1}{Z_{n,p,\beta}}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta} e^{-\frac{\beta}{2}n\, v_p\sum_{i=1}^{n}|x_i|^p}\mathrm{d} x, \qquad x\in \mathbb{R}^n.
\end{equation}
The parameter $\beta$ stands for the inverse temperature and may take any positive value, while for the matrix representation we restrict to $\beta\in \{1,2,4\}$. We refer to \cite{AGZ10} and \cite{For10} for more information on $\beta$-ensembles.

In this context the Ullman distribution arises as the corresponding equilibrium measure minimizing the functional
\[
\mu\mapsto I_p(\mu)
=-\int_{\mathbb{R}}\int_{\mathbb{R}} \ln|x-y|\mathrm{d}\mu(x)\mathrm{d}\mu(y)+v_p\int_{\mathbb{R}} |x|^p\mathrm{d}\mu(x)
\]
over $\mu\in \mathcal{P}(\mathbb{R})$, where $\mathcal{P}(\mathbb{R})$ denotes the space of probability measures on $\mathbb{R}$.  More precisely, if $X=(X_1,\dots,X_n)$ is distributed according to the density $f_{n,p,\beta}$, then the random empirical measure $ \mu_{n,p} =\frac{1}{n}\sum_{i=1}^{n}\delta_{X_i} $  tends to $\mu_p$ as $n\to\infty$. Moreover, large deviation principles due to \cite{BG97} and \cite{HP00} yield
\begin{equation} \label{eq:Z-first}
\lim_{n\to\infty}\frac{2}{\beta n^2}\ln Z_{n,p,\beta}
= I_p(\mu_p)
=\ln 2+\frac{3}{2p}.
\end{equation}
Earlier work \cite{DPS95,Joh98} also identified the limiting constant but with different techniques. In \cite{DFG+23}, the limit \eqref{eq:Z-first} is used to prove \eqref{eq:sim-sym}. The large deviation principle for tagged empirical fields due to Lebl\'e and Serfaty \cite{LS17} allows to refine \eqref{eq:Z-first}. The following result forms the core of the proof of \eqref{eq:main-1} in Theorem~\ref{thm:main}.

\begin{theorem} \label{thm:LS}
Let $\beta>0$ and $p\ge \frac{3}{2}$. As $n\to\infty$,
\[
\ln Z_{n,p,\beta}
=-\frac{\beta}{2}n^2 I_p(\mu_p)+\frac{\beta}{2}n\ln n- C(\beta)n+\Big(1-\frac{\beta}{2}\Big)\mathrm{Ent}(\mu_p)n +o(n),
\]
where
\begin{equation} \label{eq:c-beta}
C(\beta)
=\Big(1-\frac{\beta}{2}\Big)\ln \frac{\beta}{2}-\frac{\beta}{2}\ln 2\pi+\frac{\beta}{2}+\ln \Gamma\Big(\frac{\beta}{2}\Big).
\end{equation}
\end{theorem}

Independently, in the recent work \cite{GMP25} the authors deduced the statement of Theorem~\ref{thm:LS} in case of $p\ge 2$ from a central limit theorem for linear eigenvalue statistics. The restriction to $p\ge 2$ results from regularity requirements of the underlying approach, see also \cite{DFG+23}. The slightly milder restriction on $p$ in Theorem~\ref{thm:LS} is due to the fact that the Ullman density $f_p$ is Hölder continuous of order $\frac{1}{2}$ if and only if $p\ge\frac{3}{2}$. This is required to apply the results in \cite{LS17}.

The term $\mathrm{Ent}(\mu_p)$ in Theorem~\ref{thm:LS} arises via a rescaling property of the specific entropy which is part of the rate function in the large deviation principle in \cite{LS17}. In case of $\beta=2$, the contribution of $\mathrm{Ent}(\mu_p)$ vanishes and more precise asymptotics hold. The following is a consequence of \cite[Prop.~1.1]{CKM23} which is based on \cite{KM99}.

\begin{theorem} \label{thm:CKM}
Let $p\ge 1$. There exists $M_p'\in \mathbb{R}$ such that, as $n\to\infty$,
\[
\ln Z_{n,p,2}
=-n^2 I_p(\mu_p)+n\ln n+(\ln 2\pi -1) n +\frac{5}{12}\ln n +M_p'+o(1).
\]
\end{theorem}

Theorem~\ref{thm:CKM} combined with Lemma~\ref{lem:vol-partition} and Lemma~\ref{lem:cn} below yields \eqref{eq:main-2} in Theorem~\ref{thm:main}. In case of $p$ an even integer, the potential $V(x)=v_p |x|^p$ is an analytic function and results of \cite{BG13} yield a full asymptotic expansion of $\ln Z_{n,p,\beta}$, and consequently of $\ln \mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. In case of $p=2$ the integral $Z_{n,2,\beta}$ is known via a Mehta-Selberg integral. For $p\neq 2$ the coefficients in the full asymptotic expansion are defined in terms of integrals of the Stieltjes transform of $\mu_p$, see also \cite[Section 7]{BG24} and \cite{Shc13,Shc14}.

In the case of non self-adjoint matrices, the volume of the unit ball $B_{p,\beta}^n$ may be written in terms of
\[
\tilde{Z}_{n,p,\beta}
=\int_{(\IR_+)^n}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta}\prod_{i=1}^n x_i^{\beta/2-1} e^{-\beta n\, v_p\sum_{i=1}^{n}x_i^{p/2}}\mathrm{d} x,
\]
see e.g.~\cite[Sec.~4.2]{DFG+23}. This corresponds to the partition function of a Laguerre $\beta$-ensemble with potential $V(x)=v_p|x|^{p/2}$. The leading order asymptotics for $\tilde{Z}_{n,p,\beta}$ can again be obtained via large deviations, leading to \eqref{eq:sim-asym}. However, the results in \cite{LS17} do not apply and it goes beyond the scope of the present work to derive next-order asymptotics for $\tilde{Z}_{n,p,\beta}$ and $B_{p,\beta}^n$.

\begin{remark}
	The unit balls $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ can also be studied for $0<p<1$. Then \eqref{eq:sim-asym} and \eqref{eq:sim-sym} as well as Theorem~\ref{thm:CKM} continue to hold. However, even neglecting regularity assumptions, the proof of Theorem~\ref{thm:main} and the underlying large deviation principle in \cite{LS17} are limited to $p\ge 1$. This is because, while the microscopic limit point process in the bulk is the universal sine$_2$-process for $p\ge 1$, it becomes dependent on $p$ for $p<1$, see \cite{BEY14,CKM23}. Since the minimizer of the rate function is the free energy which is uniquely given by the sine$_{\beta}$-process, see \cite{EHL21,HM25}, the large deviation principle in \cite{LS17} cannot hold in its present form if $p<1$.
\end{remark}

\begin{remark}
	Heuristically, the unit balls $B_{p,\beta}^n$ and $\IB_{p,\beta}^n$ are non-commutative versions of $\IB_p^n=\{x\in \mathbb{R}^n\colon \sum_{i=1}^{n}|x_i|^p\le 1\}$. To compare, note that
\[
\mathrm{vol}(n^{1/p}\IB_{p}^n)
=n^{n/p}\frac{2^n\Gamma(1+\frac{1}{p})^n}{\Gamma(1+\frac{n}{p})}
=\frac{1+o(1)}{\sqrt{2\pi n/p}} e^{\mathrm{Ent}(\nu_p)n},
\]
where $\nu_p$ has density proportional to $e^{-|x|^p/p}$ and maximizes entropy over $\mu\in \mathcal{P}(\mathbb{R})$ with $\int |x|^p\mathrm{d}\mu(x) \le 1$, see also \cite{KP21}. Similarly, the Ullman distribution $\mu_p$ maximizes free entropy over $\mu\in \mathcal{P}(\mathbb{R})$ with $\int |x|^p\mathrm{d}\mu(x)\le \alpha_p$, see \cite[Prop.~5.34]{HP00}. In case of $p=\infty$, one can interpret $\nu_{\infty}$ as the uniform distribution on $[-1,1]$, which maximizes entropy, and $\mu_{\infty}$ as the arcsine distribution, which maximizes free entropy. Formally, see Lemma~\ref{lem:vol-partition} below, the case $\beta=0$ corresponds to the commutative case where a Poisson point process appears in the microscopic limit, see also \cite{AD14,BP15,Leb16}.
\end{remark}

\begin{remark}
Theorem~\ref{thm:main} can be combined with the stochastic representation in [Cor.~4.3]{KPT20b} of a random matrix drawn uniformly from a Schatten ball $\mathbb{B}_{p,\beta}^n$ to obtain information on critical intersections of Schatten $p$-balls in the spirit of \cite{KPT20b} and \cite{ST25}.
\end{remark}

The proof of Theorem~\ref{thm:main} is based on rewriting $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)$ in terms of the partition function
\[
Z_{n,p,\beta}=\int_{\mathbb{R}^n}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta}\, e^{-\frac{\beta}{2}n\, v_p\sum_{i=1}^{n}|x_i|^p}\,\mathrm{d} x,
\]
which is the partition function of a one-dimensional $\beta$-ensemble with potential $V(x)=v_p|x|^p$ and joint density $f_{n,p,\beta}$.

In this setting, the Ullman distribution $\mu_p$ appears as the equilibrium measure minimizing
\[
I_p(\mu)=-\iint\ln|x-y|\,\mathrm{d}\mu(x)\mathrm{d}\mu(y)+v_p\int |x|^p\,\mathrm{d}\mu(x),
\]
and the empirical measure $\mu_{n,p}=\frac1n\sum_{i=1}^n\delta_{X_i}$ (for $X\sim f_{n,p,\beta}$) converges to $\mu_p$. Large deviations yield the leading-order limit
\[
\lim_{n\to\infty}\frac{2}{\beta n^2}\ln Z_{n,p,\beta}=I_p(\mu_p)=\ln 2+\frac{3}{2p}.
\]

To obtain the next-order terms needed for \eqref{eq:main-1}, the argument uses the large deviation principle for tagged empirical fields of Lebl\'e--Serfaty \cite{LS17} to refine this to the expansion stated in Theorem~\ref{thm:LS}:
\[
\ln Z_{n,p,\beta}=-\frac{\beta}{2}n^2 I_p(\mu_p)+\frac{\beta}{2}n\ln n- C(\beta)n+\Big(1-\frac{\beta}{2}\Big)\mathrm{Ent}(\mu_p)n+o(n).
\]
The restriction $p\ge\tfrac32$ is tied to the requirement that the Ullman density is H\"older $1/2$, which is needed to apply \cite{LS17}. The appearance of $\mathrm{Ent}(\mu_p)$ is attributed to a rescaling property of the specific entropy in the rate function of \cite{LS17}.

For the $\beta=2$ refinement \eqref{eq:main-2}, the text explains that the $\mathrm{Ent}(\mu_p)$ contribution vanishes and one invokes Theorem~\ref{thm:CKM} (a consequence of \cite[Prop.~1.1]{CKM23}) giving sharper asymptotics for $\ln Z_{n,p,2}$; then “Theorem~\ref{thm:CKM} combined with Lemma~\ref{lem:vol-partition} and Lemma~\ref{lem:cn} below yields \eqref{eq:main-2} in Theorem~\ref{thm:main}.”

The proof of the main theorem is based on rewriting $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)$ in terms of the partition function
\[
Z_{n,p,\beta}=\int_{\mathbb{R}^n}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta}\, e^{-\frac{\beta}{2}n\, v_p\sum_{i=1}^{n}|x_i|^p}\,\mathrm{d} x,
\]
which is the partition function of a one-dimensional $\beta$-ensemble with potential $V(x)=v_p|x|^p$ and joint density $f_{n,p,\beta}$.

In this setting, the Ullman distribution $\mu_p$ appears as the equilibrium measure minimizing
\[
I_p(\mu)=-\iint\ln|x-y|\,\mathrm{d}\mu(x)\mathrm{d}\mu(y)+v_p\int |x|^p\,\mathrm{d}\mu(x),
\]
and the empirical measure $\mu_{n,p}=\frac1n\sum_{i=1}^n\delta_{X_i}$ (for $X\sim f_{n,p,\beta}$) converges to $\mu_p$. Large deviations yield the leading-order limit
\[
\lim_{n\to\infty}\frac{2}{\beta n^2}\ln Z_{n,p,\beta}=I_p(\mu_p)=\ln 2+\frac{3}{2p}.
\]

To obtain the next-order terms needed for
\begin{align}\label{eq:main-1}
\ln \vol(\bpb)
=&-\frac{\beta}{2}\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n\notag\\
&+\frac{\beta}{2}\Big(\frac{1}{2}\ln\frac{4\pi}{\beta}+\frac{3}{4}+\ln A(p)\Big)n^2\notag\\
&-\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}+\frac{1}{p}\Big)n\ln n\notag\\
&+\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}\ln\frac{4}{\beta\pi}+\frac{1}{2}+\frac{1}{2p}+\ln A(p)+\Ent(\mu_p)\Big)n +o(n).
\end{align}
the argument uses the large deviation principle for tagged empirical fields of Lebl\'e--Serfaty (LS17) to refine this to the following result.

\begin{theorem} \label{thm:LS}
Let $\beta>0$ and $p\ge \frac{3}{2}$. As $n\to\infty$,
\[
\ln Z_{n,p,\beta}
=-\frac{\beta}{2}n^2 I_p(\mu_p)+\frac{\beta}{2}n\ln n- C(\beta)n+\Big(1-\frac{\beta}{2}\Big)\Ent(\mu_p)n +o(n),
\]
where
\begin{equation} \label{eq:c-beta}
C(\beta)
=\Big(1-\frac{\beta}{2}\Big)\ln \frac{\beta}{2}-\frac{\beta}{2}\ln 2\pi+\frac{\beta}{2}+\ln \Gamma\Big(\frac{\beta}{2}\Big).
\end{equation}
\end{theorem}

The restriction $p\ge\tfrac32$ is tied to the requirement that the Ullman density is H\"older $1/2$, which is needed to apply LS17. The appearance of $\mathrm{Ent}(\mu_p)$ is attributed to a rescaling property of the specific entropy in the rate function of LS17.

For the $\beta=2$ refinement
\begin{equation} \label{eq:main-2}
\ln \vol(\IB_{p,2}^n)
=-\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n
+\Big(\frac{1}{2}\ln 2\pi +\frac{3}{4}+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\end{equation}
one notes that the $\mathrm{Ent}(\mu_p)$ contribution vanishes and one invokes the following sharper asymptotics.

\begin{theorem} \label{thm:CKM}
Let $p\ge 1$. There exists $M_p'\in \IR$ such that, as $n\to\infty$,
\[
\ln Z_{n,p,2}
=-n^2 I_p(\mu_p)+n\ln n+(\ln 2\pi -1) n +\frac{5}{12}\ln n +M_p'+o(1).
\]
\end{theorem}

Then the theorem above, combined with the following two lemmas,

\begin{lemma} \label{lem:vol-partition}
Let $\beta\in\{1,2,4\}$ and $1\le p<\infty$. Then
\begin{align*}
\vol(\bpb)
&=c_n\int_{\IB_p^n} \prod_{1\le i< j\le n}|x_i-x_j|^{\beta}\dd x_1 \cdots \dd x_n\\
&=\frac{c_{n}}{\Gamma(1+\frac{d_n}{p})}\Big(\frac{n\beta v_p}{2}\Big)^{d_n/p}Z_{n,p,\beta}
\end{align*}
where $Z_{n,p,\beta}$ and $v_p$ are as in \eqref{eq:Z} and
\[
c_{n}
=\frac{1}{n!}\bigg(\frac{\Gamma(\frac{\beta}{2})}{(2\pi)^{\beta/2}}\bigg)^{n}\prod_{k=1}^n\frac{(2\pi)^{\beta k/2}}{\Gamma(\frac{\beta k}{2})}.
\]
\end{lemma}

\begin{lemma} \label{lem:cn}
Let $\beta\in \{1,2,4\}$. Then, as $n\to\infty$,
\begin{align*}
\ln c_{n}
=&-\frac{\beta}{4}n^2\ln n + \frac{\beta}{2}\Big(\frac{1}{2}\ln \frac{4\pi}{\beta}+\frac{3}{4}\Big)n^2-\frac{1}{2}\Big(1+\frac{\beta}{2}\Big)n\ln n\\
&+\Big(\frac{1}{2}\Big(1-\frac{\beta}{2}\Big)\ln \beta -\frac{1}{2}\Big(1+\frac{\beta}{2}\Big)(\ln \pi-1)-\ln 2+\ln \Gamma\Big(\frac{\beta}{2}\Big)\Big)n\\
&-\frac{3+\frac{\beta}{2}+\frac{2}{\beta}}{12}\ln n
+a_{\beta}+o(1),
\end{align*}
where $a_{\beta}\in \IR$ is some constant.
\end{lemma}

yields the stated $\beta=2$ volume asymptotics, completing that part of the main theorem.

\label{thm:main}
Let $\beta\in \{1,2,4\}$. If $p\ge \frac{3}{2}$, then, as $n\to\infty$,
\begin{align}\label{eq:main-1}
\ln \mathrm{vol}(\mathbb{B}_{p,\beta}^n)
=&-\frac{\beta}{2}\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n\notag\\
&+\frac{\beta}{2}\Big(\frac{1}{2}\ln\frac{4\pi}{\beta}+\frac{3}{4}+\ln A(p)\Big)n^2\notag\\
&-\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}+\frac{1}{p}\Big)n\ln n\notag\\
&+\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}\ln\frac{4}{\beta\pi}+\frac{1}{2}+\frac{1}{2p}+\ln A(p)+\mathrm{Ent}(\mu_p)\Big)n +o(n).
\end{align}
Moreover, if $\beta=2$ and $p\ge 1$, then, for some constant $M_p\in \mathbb{R}$,
\begin{equation} \label{eq:main-2}
\ln \mathrm{vol}(\IB_{p,2}^n)
=-\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n
+\Big(\frac{1}{2}\ln 2\pi +\frac{3}{4}+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\end{equation},

Question: Let \(\mathbb F_1=\mathbb R\), \(\mathbb F_2=\mathbb C\), and \(\mathbb F_4=\mathbb H\). For \(\beta\in\{1,2,4\}\) and \(A\in \mathbb F_\beta^{n\times n}\), let \(s_1(A)\ge\cdots\ge s_n(A)\ge0\) be the singular values and define the Schatten \(p\)-norm by \(\|A\|_p=(\sum_{j=1}^n s_j(A)^p)^{1/p}\) for \(1\le p<\infty\) and \(\|A\|_\infty=s_1(A)\). Let \(\mathcal H_n(\mathbb F_\beta)=\{A\in\mathbb F_\beta^{n\times n}:A^*=A\}\), and let \(\mathbb B_{p,\beta}^n=\{A\in\mathcal H_n(\mathbb F_\beta):\|A\|_p\le1\}\), whose volume is taken in the corresponding real vector space of self-adjoint matrices. Also let \(A(p)=\frac12\Big(\frac{p\sqrt\pi\,\Gamma(p/2)}{\sqrt e\,\Gamma((p+1)/2)}\Big)^{1/p}\) for finite \(p\) (and \(A(\infty)=\tfrac12\)), and let \(\mathrm{Ent}(\mu_p)\) denote the differential entropy of the Ullman distribution \(\mu_p\). Which asymptotic statement for \(\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)\) as \(n\to\infty\) holds for \(p\ge \tfrac32\), and what sharper asymptotic statement holds in the special case \(\beta=2\) and \(p\ge1\)?

As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge \tfrac32\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
-\Big(1-\frac{\beta}{2}\Big)\Big(\frac12+\frac1p\Big)n\ln n
+\Big(1-\frac{\beta}{2}\Big)\Big(\frac12\ln\frac{4}{\beta\pi}+\frac12+\frac{1}{2p}+\ln A(p)+\mathrm{Ent}(\mu_p)\Big)n
+o(n).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\]

Choices:
B.
As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge 1\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
-\Big(1-\frac{\beta}{2}\Big)\Big(\frac12+\frac1p\Big)n\ln n
+\Big(1-\frac{\beta}{2}\Big)\Big(\frac12\ln\frac{4}{\beta\pi}+\frac12+\frac{1}{2p}+\ln A(p)+\mathrm{Ent}(\mu_p)\Big)n
+o(n).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\]
C.
As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge \tfrac32\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
+o(n^2).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
+o(n^2).
\]
D.
As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge \tfrac32\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
-\Big(1-\frac{\beta}{2}\Big)\Big(\frac12+\frac1p\Big)n\ln n
+\Big(1-\frac{\beta}{2}\Big)\Big(\frac12\ln\frac{4}{\beta\pi}+\frac12+\frac{1}{2p}+\ln A(p)\Big)n
+o(n).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\]
E.
As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge \tfrac32\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
-\Big(1-\frac{\beta}{2}\Big)\Big(\frac12+\frac1p\Big)n\ln n
+\Big(1-\frac{\beta}{2}\Big)\Big(\frac12\ln\frac{4}{\beta\pi}+\frac12+\frac{1}{p}+\ln A(p)+\mathrm{Ent}(\mu_p)\Big)n
+o(n).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\]