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==== TABLE OF CONTENTS ====
Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//main.tex
Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//preamble.tex
Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/calculus.tex
Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/curvature-change.tex
Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/drift.tex
Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/every-geodesic.tex
Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/introduction.tex
Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//references.bib
===========================
==== BEGIN Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Weak Hessian bounds along every geodesic in RCD spaces}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
On a full-support \(\RCD(K,N)\) space with \(1<N<\infty\), we prove
that a bounded globally Lipschitz function whose distributional
Hessian is bounded above by a bounded continuous function satisfies
the corresponding second-derivative inequality along every
minimizing geodesic.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/calculus}
\input{sections/drift}
\input{sections/curvature-change}
\input{sections/every-geodesic}
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
==== END Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//main.tex ====
==== BEGIN Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//preamble.tex ====
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pdftitle={Weak Hessian bounds along every geodesic in RCD spaces},
pdfauthor={OpenAI},
pdfsubject={Weak Hessians, bounded changes of measure, and geodesic inequalities}}
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==== END Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//preamble.tex ====
==== BEGIN Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/calculus.tex ====
\section{Test functions, approximation and Bochner measures}
\label{sec:preliminaries}
This section records the original-space calculus needed to change
measure. The key points are a compact core for the Laplacian and the
admissibility of Lipschitz multipliers in the Bochner inequality.
All statements concern a full-support \(\RCD(K,N)\) space with
\(1<N<\infty\). They do not involve a weighted curvature bound.
\subsection{A compact core and local approximation}
Besides the class \(\Test(M)\) in \eqref{eq:test-class}, write
\[
\Test^\infty(M)=\{g\in\Test(M):\Delta g\in L^\infty(M)\}.
\]
The superscript denotes a bounded Laplacian. Both test classes are
algebras. Test functions have \(L^2\) Hessians, and the product rule
gives \(\Gamma(g)\in W^{1,2}\) for \(g\in\Test\).
The connection is metric-compatible and torsion-free, with
\(\nabla_{\nabla g}\nabla g=\nabla(\Gamma(g)/2)\)
\cite{Gigli2018,Savare2014}. We denote the \(L^2\) Hessian by
\(H_g\). The graph norm of \(\Delta\) is
\(\|g\|_2+\|\Delta g\|_2\).
\begin{lemma}[Cutoffs and approximation]
\label{pre:approximation}
The following approximations are available.
\begin{enumerate}
\item For a compact set \(E\) in an open set \(U\), there is
\(\chi\in\Test^\infty\) with \(0\le\chi\le1\), equal to one on a
neighborhood of \(E\), and compactly supported in \(U\).
For a fixed basepoint \(o\in M\), there are also exhausting
cutoffs \(\chi_R\in\Test^\infty\),
equal to one on \(B_R(o)\), supported in \(B_{R+1}(o)\), whose
gradient and Laplacian bounds are independent of \(R\).
\item For \(g\in D(\Delta)\),
\begin{equation}
\label{eq:pre-hessian-estimate}
\|H_g\|_2^2\le
\|\Delta g\|_2^2+\max\{-K,0\}\|\nabla g\|_2^2.
\end{equation}
Compactly supported functions in \(\Test^\infty\) form a core
for \(\Delta\).
\item On a fixed compact region, a function \(g\in\Test\) can be
approximated by compactly supported \(g_j\in\Test^\infty\) with
uniformly bounded gradients, with
\[
g_j\to g,\qquad \nabla g_j\to\nabla g,\qquad
\Delta g_j\to\Delta g,\qquad H_{g_j}\to H_g
\quad\hbox{strongly in }L^2
\]
on that region.
\item A compactly supported Lipschitz function \(\xi\) can be
approximated by compactly supported test functions \(\xi_j\)
with a common compact support and a common Lipschitz bound,
uniformly and strongly in \(W^{1,2}\).
If \(\xi\ge0\), the approximants may be chosen nonnegative.
\end{enumerate}
\end{lemma}
\begin{proof}
The good-cutoff construction and the integrated Bochner estimate give
\textup{(i)} and \eqref{eq:pre-hessian-estimate}; see
\cite[Appendix~A, Lemma~A.2 and Equation~(A.13)]{GigliTamanini2021}.
For the local version of the cutoff construction one uses finitely
many small good cutoffs inside \(U\), and a smooth truncation of their
sum. This keeps the Laplacian bounded and Sobolev by the test
algebra and chain rules.
Let \(P_t\) denote the heat semigroup. For \(h\in L^2\cap L^\infty\),
positive-time regularization gives
\(P_th\in\Test^\infty\)
\cite[Appendix~A, Equation~(A.7)]{GigliTamanini2021}.
Spectral calculus and density of bounded \(L^2\) data show that these
regularizations form a core. For each such regularization \(h\),
\[
\Delta(\chi_Rh)
=\chi_R\Delta h+h\Delta\chi_R
+2\langle\nabla\chi_R,\nabla h\rangle.
\]
The uniform cutoff bounds and the \(L^2\) tails imply convergence
to \(\Delta h\) in \(L^2\), as well as convergence of the functions
and gradients. This proves the compact-core assertion.
For \textup{(iii)}, choose nested cutoffs \(\chi,\zeta\) such that
\(\chi=1\) near the region in question and \(\zeta=1\) near
\(\supp\chi\). Put
\[
g_t=\chi P_t(\zeta g).
\]
The function \(\zeta g\) lies in \(\Test\). The product rule,
spectral continuity and the fixed cutoff bounds give convergence
of \(g_t\) to \(\chi\zeta g\) in the Laplacian graph norm.
The Bakry gradient estimate gives
\[
\|\nabla g_t\|_\infty
\le \|\chi\|_\infty e^{-Kt}\|\nabla(\zeta g)\|_\infty
+\|\nabla\chi\|_\infty\|\zeta g\|_\infty.
\]
Apply \eqref{eq:pre-hessian-estimate} to
\(g_t-\chi\zeta g\) to obtain strong Hessian convergence.
Since \(\chi\zeta g=g\) near the region, \textup{(iii)} follows.
For \textup{(iv)}, use \(\xi_t=\chi P_t\xi\), with an outer
cutoff \(\chi=1\) near \(\supp\xi\).
The heat semigroup converges strongly in \(W^{1,2}\);
the uniform Lipschitz estimate and the small-time heat-kernel
moment bound give uniform convergence on the common compact
support. The cutoff product gives the same conclusions for
\(\xi_t\), and positivity is preserved.
\end{proof}
The compact core in Lemma~\ref{pre:approximation} is constructed
using only the original heat flow. It will therefore remain
available when we compare the two generator domains in
Section~\ref{sec:drift}.
\subsection{Weak Hessians and admissible Bochner multipliers}
The weak Hessian in \eqref{eq:weak-hessian} is continuous under weak
local \(W^{1,2}\) convergence of \(F\), with the test gradient and
weight fixed. For \(F\in D(\Delta)\), the integration-by-parts
formula for its \(L^2\) Hessian identifies it with that tensor.
An upper bound for \(H_F(Y,Y)\) means an inequality on every
nonnegative compactly supported Lipschitz weight; it does not
presuppose an \(L^2\) Hessian for \(F\).
The original test class in \eqref{eq:test-class} is contained in
Savar\'e's class \(\mathbb D_\infty\), which requires
\(\Delta g\in W^{1,2}\) but not a bounded Laplacian
\cite[Equation~(3.4)]{Savare2014}.
His measure-valued Bochner calculus
\cite[Lemma~3.2 and Equations~(3.7)--(3.9)]{Savare2014},
together with the Hessian-square estimate
\cite[Theorem~3.3.8]{Gigli2018}, gives
\begin{equation}
\label{eq:bochner-hessian}
\Gamma_2(g):=
\Delta\bigl(\Gamma(g)/2\bigr)
-\Gamma(g,\Delta g)\,m
\ge \bigl(K\Gamma(g)+|H_g|^2\bigr)m.
\end{equation}
The first Laplacian in this expression is measure-valued.
The associated nonnegative Bochner remainder is locally finite
and does not charge sets of zero \(2\)-capacity. Here
\[
\Cap_2(E)=
\inf_{\substack{U\supset E\\U\ \mathrm{open}}}
\ \inf_{\substack{v\in W^{1,2}(M)\\v\ge1\ m\text{-a.e.\ on }U}}
\int_M\bigl(v^2+|\nabla v|^2\bigr)\dd m.
\]
Sobolev multipliers are evaluated using their quasi-continuous
representatives, which are defined outside a set of zero capacity.
In particular, a nonnegative compactly supported Lipschitz
multiplier is admissible in \eqref{eq:bochner-hessian}.
Indeed, approximate it by the nonnegative test functions in
Lemma~\ref{pre:approximation}\textup{(iv)}. Strong
\(W^{1,2}\) convergence gives, along a subsequence, convergence
outside a set of zero capacity. The approximants have a common
bound and compact support, so dominated convergence applies
against the fixed local Bochner measure. In the integration by
parts, strong convergence of the multiplier gradients pairs
with \(\nabla\Gamma(g)\in L^2\). This gives the same identity
and inequality for the Lipschitz multiplier.
\subsection{Metric and heat-flow facts}
A finite-dimensional full-support \(\RCD\) space is proper and
geodesic, has positive finite measure on every nonempty bounded
ball, and has the Sobolev-to-Lipschitz property. In particular, a
Sobolev function with essentially bounded weak gradient has a
Lipschitz representative, with Lipschitz constant bounded by that
gradient bound. Bishop--Gromov volume control implies Gaussian
volume growth: for a fixed basepoint \(o\), there is a constant
\(C>0\) such that
\(m(B_r(o))\le C\exp(Cr^2)\) for \(r\ge1\).
These properties are unchanged under bounded equivalence of the
measure, except that a weighted curvature bound still requires proof.
The original heat semigroup \(P_t\) is Markovian and strongly
continuous on \(W^{1,2}\). Its Bakry gradient and
reverse-Poincar\'e estimates give, for bounded data and positive
times, the bounds used in the proof of
Lemma~\ref{weight:resolvent}. We use the standard estimates in
\cite{Savare2014,AGS2015}; the finite-dimensional heat regularity
and cutoff statements used above are collected in
\cite[Appendix~A]{GigliTamanini2021}.
==== END Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/calculus.tex ====
==== BEGIN Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/curvature-change.tex ====
\section{From a weak Hessian bound to variable curvature}
\label{sec:curvature-change}
We now apply the generator facts to \(m_\lambda=e^{\lambda F}m\).
The weak Hessian upper bound lowers the Ricci bound by
\(\lambda G\). We first prove this identity on compact original
test functions, then close it in the common generator domain. This
order is needed because a Lipschitz drift can destroy the Sobolev
regularity of an original test function's weighted Laplacian.
For a probability measure \(\mu\) with bounded support, define
\[
\Ent_m(\mu)=
\begin{cases}
\displaystyle\int_M\rho\log\rho\dd m,&\mu=\rho m,\\
+\infty,&\mu\not\ll m,
\end{cases}
\qquad 0\log0=0.
\]
The negative part is integrable: a bounded ball containing the
support has finite measure, and \(s\log s\ge-1/e\).
This is the entropy domain used below, also for the bounded
reweightings \(m_\lambda\).
For a bounded continuous function \(k\), we use the variable
curvature-dimension condition \(\CD(k,\infty)\) of
\cite[Definition~1.2]{BHS2021}. In the bounded-support case needed
below, it supplies, for any two finite-entropy probability measures
\(\mu_0,\mu_1\), a dynamical optimal plan \(\pi\) with
\(\mu_t=(e_t)_\#\pi\) such that
\begin{align}
\label{weight:variable-entropy}
\Ent_m(\mu_t)
&\le (1-t)\Ent_m(\mu_0)+t\Ent_m(\mu_1)\notag\\
&\quad-\int_{\Geo(M)}\int_0^1
\bigl(\min\{s,t\}-st\bigr)k(\gamma_s)|\dot\gamma|^2
\dd s\dd\pi(\gamma).
\end{align}
Here \(\Geo(M)\) is the space of constant-speed minimizing paths
on \([0,1]\), and \(e_t(\gamma)=\gamma_t\). A dynamical optimal
plan is a probability measure on this path space whose endpoint
coupling is optimal for quadratic transport. This formula fixes
our Green-kernel and Laplacian normalizations.
\begin{proposition}[A distributional Hessian bound changes curvature]
\label{weight:variable-cd}
Let \((M,d,m)\) be a full-support \(\RCD(K,N)\) space,
where \(K\in\mathbb R\) and \(1<N<\infty\).
Let \(F\colon M\to\mathbb R\) be bounded and globally Lipschitz, and let
\(G\colon M\to\mathbb R\) be bounded and continuous. Suppose
that for every compactly supported \(g\in\Test\) and every
nonnegative \(h\in\Lip_c(M)\),
\[
H_F(\nabla g,\nabla g)(h)
\le\int hG\Gamma(g)\dd m.
\]
For every \(\lambda>0\), the measure
\(m_\lambda=e^{\lambda F}m\) satisfies
\(\CD(K-\lambda G,\infty)\). The metric is unchanged.
\end{proposition}
\begin{proof}
\textbf{Core identity.}
Fix \(\lambda>0\), and write \(V=\lambda F\), \(w=e^V\),
\(k=K-\lambda G\). Take a nonnegative \(\xi\) in
\eqref{weight:multiplier-class}. Lemma~\ref{weight:resolvent}
makes \(\xi\) Lipschitz. Initially let \(g\in\Test\) be
compactly supported, and set
\(q=\Gamma(g)/2\), \(b=\Gamma(V,g)\). Define
\begin{equation}
\label{weight:integrated-bochner}
\mathcal B_V(g,\xi)
=-\int w\Gamma(\xi,q)\dd m
+\int w\bigl[\xi\Delta_Vg+\Gamma(\xi,g)\bigr]
\Delta_Vg\dd m.
\end{equation}
The notation \(\mathcal B_0(g,\eta)\) denotes the same formula
with \(V=0\); it is also meaningful for a bounded Lipschitz
multiplier \(\eta\) after localization near \(\supp g\).
All terms are integrable: \(q\in W^{1,2}\),
\(\Delta_Vg\in L^2\), and \(\xi,|\nabla\xi|\) are bounded.
The original measure-valued Bochner inequality, integrated against
\(w\xi\), gives
\begin{equation}
\label{weight:original-bochner}
\mathcal B_0(g,w\xi)
\ge K\int w\xi\Gamma(g)\dd m.
\end{equation}
Here and below a compact test cutoff equal to one near
\(\supp g\) can be inserted in the multiplier. Its derivatives
do not meet the support of any term. Thus the bounded Lipschitz
multiplier is admissible by the local Bochner-measure calculus
in Section~\ref{sec:preliminaries}.
The product rule and \eqref{weight:generator} give the exact identity
\begin{align}
\label{weight:bochner-difference}
\mathcal B_V(g,\xi)-\mathcal B_0(g,w\xi)
&=\int w\xi\Gamma(V,q)\dd m
+\int wb\bigl[\Gamma(\xi,g)+\xi\Delta_Vg\bigr]\dd m\nonumber\\
&=-H_V(\nabla g,\nabla g)(w\xi).
\end{align}
The second equality is precisely \eqref{eq:weak-hessian}, since
\(\divg(w\xi\nabla g)=w[\Gamma(\xi,g)+\xi\Delta_Vg]\).
Use \eqref{weight:hessian-assumption}, with the same harmless
cutoff, in \eqref{weight:bochner-difference}. Together with
\eqref{weight:original-bochner} this yields
\begin{equation}
\label{weight:core-be}
\mathcal B_V(g,\xi)\ge\int k\xi\Gamma(g)\dd m_V.
\end{equation}
The example in Remark~\ref{weight:domain-warning} explains why
the weighted generator domain needs a separate closure argument.
\smallskip\noindent\textbf{Closure in the generator domain.}
We next close this inequality in the correct generator domain.
Integration by parts in its first term rewrites the left side as
\begin{equation}
\label{weight:graph-form}
\frac12\int (\Delta_V\xi)\Gamma(g)\dd m_V
+\int (\Delta_Vg)\Gamma(\xi,g)\dd m_V
+\int\xi(\Delta_Vg)^2\dd m_V.
\end{equation}
Every term in \eqref{weight:graph-form}, as well as the right
side of \eqref{weight:core-be}, is continuous under convergence
in the \(\Delta_V\) graph norm. Indeed, graph convergence implies
strong \(L^2\) convergence of gradients; use the boundedness of
\(\xi,\Delta_V\xi,|\nabla\xi|\), and \(k\).
By Lemma~\ref{pre:approximation} and graph-norm equivalence,
compact original test functions form a core for \(\Delta_V\).
Consequently \eqref{weight:core-be} in the form
\eqref{weight:graph-form} holds for every \(g\in D(\Delta_V)\).
If in addition \(\Delta_Vg\in W^{1,2}\), the last two terms
in \eqref{weight:graph-form} combine by integration by parts,
and give
\begin{equation}
\label{weight:full-be}
\frac12\int (\Delta_V\xi)\Gamma(g)\dd m_V
-\int\xi\Gamma(g,\Delta_Vg)\dd m_V
\ge\int k\xi\Gamma(g)\dd m_V.
\end{equation}
This is the weighted Bakry--\'Emery inequality on its actual
test domain. In particular it holds on the smaller class also
requiring \(\Gamma(g)\in L^\infty\).
\smallskip\noindent\textbf{Constant curvature.}
We first obtain a weighted \(\RCD\) structure with the same
intrinsic distance \(d\). Replace \(k\) by the constant
\(K_V=\inf_M k> -\infty\), and set
\(\mathcal E_V=2\mathrm{Ch}_V\). The unchanged weak gradients
make \(\mathcal E_V\) a strongly local symmetric Dirichlet form
with carr\'e du champ \(\Gamma\).
In the metric topology \(\tau\), using the completed Borel
\(\sigma\)-algebra, completeness, separability, full support
and finite bounded-ball measures give condition \textup{(MD)}
of \cite{AGS2015}.
The original Sobolev-to-Lipschitz property gives \textup{(ED.a)};
boundedly supported Lipschitz functions with slope at most one
have weak gradient at most one, giving \textup{(ED.b)}.
The converse of \cite[Theorem~3.9]{AGS2015} therefore makes
\((M,\tau,m_V,\mathcal E_V)\) an Energy measure space and
identifies its intrinsic distance with \(d\).
Since \(\mathcal E_V=2\mathrm{Ch}_V\) for this metric,
the necessity direction of \cite[Theorem~3.14]{AGS2015}
gives upper regularity. Bounded equivalence preserves Gaussian
volume growth, hence \textup{(MD.exp)}, so the final assertion
of that theorem gives conservativity.
Sobolev-to-Lipschitz also supplies continuous representatives
for energy functions with \(\Gamma\le1\), as required by
\cite[Definition~3.16]{AGS2015}. Thus this is a Riemannian
Energy measure space. Applying \cite[Theorem~4.17]{AGS2015}
to \eqref{weight:full-be} with the constant \(K_V\) gives
\(\RCD(K_V,\infty)\).
\smallskip\noindent\textbf{Variable curvature.}
On this weighted \(\RCD\) space, \eqref{weight:full-be} is its
\(\mathrm{BE}_2(k,\infty)\) condition, and the bounded continuous
function \(k\) meets its lower semicontinuity and lower-bound
assumptions. Hence \(\CD(k,\infty)\) follows
\cite[Definition~1.4 and Theorem~1.1]{BHS2021}.
\end{proof}
\begin{remark}[Why the domain argument is needed]
\label{weight:domain-warning}
An original test function need not satisfy
\(\Delta_Vg\in W^{1,2}\). Even in Euclidean space, a bounded
Lipschitz \(V\) agreeing with \(-|x_1|\) near the origin and a
smooth compactly supported \(g\) agreeing with \(x_1\) there
give \(\Delta_Vg=-\operatorname{sign}(x_1)\) locally.
The integrated identity \eqref{weight:bochner-difference} and
closure through \eqref{weight:graph-form} avoid differentiating
this drift term. They also explain why no general converse from
an unspecified weak Hessian to geodesic convexity is being used;
compare \cite[Remark~2.4]{BrenaGigli2024}.
\end{remark}
==== END Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/curvature-change.tex ====
==== BEGIN Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/drift.tex ====
\section{Bounded changes of measure and the generator domain}
\label{sec:drift}
Our first task is to control the weighted generator before assigning
any curvature bound to the weighted space. A bounded Lipschitz weight
preserves weak gradients and the generator domain. The less immediate
point is that bounded functions with bounded weighted Laplacian are
Lipschitz; the resolvent argument below supplies this regularity from
the original heat flow.
\begin{lemma}[Bounded drift]
\label{weight:resolvent}
Let \((M,d,m)\) be a full-support \(\RCD(K,N)\) space, with \(1<N<\infty\), and
let \(V\) be bounded and globally Lipschitz. Set
\(m_V=e^V m\), and let \(\Delta_V\) be the generator of the
Cheeger energy for \(m_V\), with the nonpositive sign convention.
The Sobolev spaces and their weak gradients agree, and
\begin{equation}
\label{weight:generator}
D(\Delta_V)=D(\Delta),\qquad
\Delta_Vu=\Delta u+\Gamma(V,u).
\end{equation}
The two generator graph norms are equivalent. Moreover,
\begin{equation}
\label{weight:multiplier-class}
\xi\in D(\Delta_V),\quad
\xi,\Delta_V\xi\in L^\infty
\quad\Longrightarrow\quad
\xi\text{ has a globally Lipschitz representative}.
\end{equation}
\end{lemma}
\begin{proof}
Since \(e^V\) is bounded above and below by positive constants,
the classes of test plans defining weak upper gradients agree:
a compression bound for one measure gives a compression bound
for the other. Thus the weak gradients agree, and the two
\(W^{1,2}\) norms are equivalent. In particular, the weighted
Cheeger energy is quadratic. Multiplication by \(e^V\) or
\(e^{-V}\) preserves \(W^{1,2}\). In the weak definition of the
generator, replacing a test function \(\eta\) by \(e^V\eta\)
gives \eqref{weight:generator} in one direction. Replacing it by
\(e^{-V}\eta\) gives the converse. All drift terms are in \(L^2\)
because \(|\nabla V|\) is bounded.
The energy identity
\[
\|\nabla u\|_{L^2(m)}^2=-\int u\Delta u\dd m
\]
and its weighted counterpart bound the gradient by the respective
graph norm. Together with \eqref{weight:generator}, this proves
graph-norm equivalence.
Here is a resolvent proof of the last assertion, before imposing
any curvature condition on the weighted space. Let \(P_t\) be
the original heat semigroup, and put
\[
R_a h=\int_0^\infty e^{-at}P_t h\dd t=(a-\Delta)^{-1}h.
\]
For \(h\in L^2\cap L^\infty\), spectral calculus, the Markov
property, and the heat gradient estimate give, for all sufficiently
large \(a\),
\begin{align}
\label{weight:resolvent-bounds}
\|R_a h\|_p&\le a^{-1}\|h\|_p\quad(p=2,\infty),\\
\|\nabla R_a h\|_2&\le C a^{-1/2}\|h\|_2,
&\|\nabla R_a h\|_\infty&\le C_Ka^{-1/2}\|h\|_\infty.
\nonumber
\end{align}
For the last estimate one integrates the reverse-Poincar\'e
bound \(\|\nabla P_t h\|_\infty\le
C_K(t^{-1/2}+1)e^{C_Kt}\|h\|_\infty\).
These are estimates for the original \(\RCD\) heat semigroup.
Consider the Banach space
\[
\mathcal E=\{z\in W^{1,2}:z,|\nabla z|\in L^\infty\},\qquad
\|z\|_{\mathcal E}=\|z\|_2+\|\nabla z\|_2
+\|z\|_\infty+\|\nabla z\|_\infty.
\]
Completeness follows from closedness of the weak differential
and completeness of \(L^\infty\). Given \(h\in L^2\cap L^\infty\),
the map
\[
z\longmapsto R_a\bigl(h+\Gamma(V,z)\bigr)
\]
is a contraction on \(\mathcal E\) when \(a\) is sufficiently
large, by \eqref{weight:resolvent-bounds} and
\(\|\Gamma(V,z)\|_p\le\|\nabla V\|_\infty\|\nabla z\|_p\).
Its fixed point lies in \(D(\Delta)\) and solves
\((a-\Delta_V)z=h\). The weighted energy identity gives
uniqueness of this solution in \(D(\Delta_V)\).
For \(\xi\) as in \eqref{weight:multiplier-class}, take
\(h=(a-\Delta_V)\xi\). It belongs to \(L^2\cap L^\infty\),
so uniqueness identifies \(\xi\) with the fixed point.
The original Sobolev-to-Lipschitz property now proves the claim.
\end{proof}
==== END Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/drift.tex ====
==== BEGIN Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/every-geodesic.tex ====
\section{Localizing the entropy inequality to a prescribed geodesic}
\label{sec:every-geodesic}
Proposition~\ref{weight:variable-cd} provides an entropy inequality
for optimal transports between measures. To recover a statement on
one prescribed geodesic, we concentrate the endpoint measures in
shrinking balls. Nonbranching makes every strict interior subsegment
unique, so all the resulting transport paths converge to that
subsegment. At the same time, we increase \(\lambda\) fast enough
that endpoint entropy contributes no error in the limit.
\begin{proof}[Proof of Theorem~\ref{weight:every-geodesic}]
The case \(\ell=0\) is immediate. A finite-dimensional
\(\RCD\) space is nonbranching \cite[Theorem~1.3]{Deng2020}.
Every strict subsegment of \(\sigma\) is therefore the unique
minimizing geodesic between its endpoints. Indeed, replacing
such a subsegment by a different minimizer gives a minimizing
curve between \(\sigma(0)\) and \(\sigma(1)\), with the same
length \(\ell\), agreeing with \(\sigma\) on a nontrivial
initial interval. This contradicts nonbranching.
Fix one strict subsegment, and denote its constant-speed
reparametrization by \(\tau\colon[0,1]\to M\).
For \(\varepsilon_j\downarrow0\), let
\[
\mu_i^j=\frac{m|_{B_{\varepsilon_j}(\tau(i))}}
{m(B_{\varepsilon_j}(\tau(i)))},\qquad
E_i^j=\Ent_m(\mu_i^j)
=-\log m(B_{\varepsilon_j}(\tau(i))),\quad i=0,1.
\]
These quantities are finite by full support and local finiteness.
Choose
\[
\lambda_j\ge j\bigl(1+|E_0^j|+|E_1^j|\bigr).
\]
The two endpoint measures have finite entropy relative to
\(m_{\lambda_j}\), so Proposition~\ref{weight:variable-cd}
provides a dynamical optimal plan \(\pi_j\) satisfying the
variable-curvature entropy inequality. Write
\(\mu_t^j=(e_t)_\#\pi_j\).
All geodesics in these plans lie in one compact set \(Q\) and
have lengths bounded by one constant \(L\). Properness and
Arzel\`a--Ascoli give compactness of this family of paths.
Every limit of a path with these shrinking endpoint constraints
must be \(\tau\), by the uniqueness just proved. Hence
\(\pi_j\) converges weakly to \(\delta_\tau\), even though
the measures \(m_{\lambda_j}\) vary with \(j\).
Set \(g(s,t)=\min\{s,t\}-st\), the nonnegative Dirichlet
Green kernel on \([0,1]\), and define
\begin{align*}
A_t^j&=\int F\dd\mu_t^j,& E_t^j&=\Ent_m(\mu_t^j),\\
C_t^j&=\int_{\Geo(M)}\int_0^1g(s,t)|\dot\gamma|^2
\dd s\dd\pi_j(\gamma),\\
D_t^j&=\int_{\Geo(M)}\int_0^1g(s,t)G(\gamma_s)|\dot\gamma|^2
\dd s\dd\pi_j(\gamma).
\end{align*}
We use the large-parameter entropy comparison appearing in
\cite[Theorem~3.13, Equation~(3.6)]{Han2018Monotonicity}, now with the variable
bound and the concentrating endpoint measures chosen above.
Using \(\Ent_{m_{\lambda_j}}(\mu)=\Ent_m(\mu)
-\lambda_j\int F\dd\mu\), the defining
\(\CD(K-\lambda_jG,\infty)\) inequality becomes
\begin{equation}
\label{weight:entropy-rearranged}
A_t^j\ge(1-t)A_0^j+tA_1^j-D_t^j
+\frac{E_t^j-(1-t)E_0^j-tE_1^j+KC_t^j}{\lambda_j}.
\end{equation}
This uses the Green-kernel normalization of
\cite[Definition~1.2]{BHS2021}. The intermediate entropies
are finite, and Jensen's inequality on \(Q\) gives
\(E_t^j\ge-\log m(Q)\). Also
\(0\le C_t^j\le L^2/8\).
Thus the last term in \eqref{weight:entropy-rearranged} has
lower limit at least zero, uniformly in \(t\).
The continuity of \(F,G\) on \(Q\), compactness of the paths,
and \(\pi_j\to\delta_\tau\) imply
\begin{equation}
\label{weight:green-concavity}
F(\tau_t)\ge(1-t)F(\tau_0)+tF(\tau_1)
-|\dot\tau|^2\int_0^1g(s,t)G(\tau_s)\dd s.
\end{equation}
The argument applies to every strict subsegment of \(\sigma\).
On any compact interval inside \((0,1)\), choose a twice
continuously differentiable function \(a\) with
\(a''(t)=\ell^2G(\sigma_t)\).
The Green formula and \eqref{weight:green-concavity}, applied
on all subintervals, say exactly that \(F\circ\sigma-a\)
lies above each of its chords. It is concave, so its second
distributional derivative is nonpositive. This proves
\eqref{weight:curve-hessian}. Continuity supplies the values
at the ends of any closed subinterval.
\end{proof}
==== END Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/every-geodesic.tex ====
==== BEGIN Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/introduction.tex ====
\section{Introduction}
\label{sec:introduction}
On a smooth Riemannian manifold, an upper bound for the Hessian of a
function bounds its second derivative along every geodesic. The same
question is subtler when the function has only a distributional
Hessian. A weak inequality is tested against the reference measure,
whereas a prescribed geodesic can lie in a set of measure zero. The
goal of this paper is to pass from such a weak inequality to a bound
on every minimizing geodesic in a finite-dimensional \(\RCD\) space.
We work on a complete separable metric measure space \((M,d,m)\)
with full support and measure finite on bounded sets. We use
\(\RCD(K,N)\) to mean the unreduced curvature-dimension condition
\(\CD(K,N)\), together with quadratic Cheeger energy. The parameters
satisfy \(K\in\mathbb R\) and \(1<N<\infty\).
The Laplacian is the nonpositive \(L^2\) generator:
\[
\int_M\langle\nabla u,\nabla v\rangle\dd m
=-\int_M(\Delta u)v\dd m.
\]
Its domain \(D(\Delta)\) is contained in \(W^{1,2}(M)\).
Write \(\Gamma(u,v)=\langle\nabla u,\nabla v\rangle\) and
\(\Gamma(u)=\Gamma(u,u)\). Our global test class is
\begin{equation}
\label{eq:test-class}
\Test(M)=\left\{g\in D(\Delta):
\begin{array}{l}
g\text{ has a bounded globally Lipschitz representative},\\
\Delta g\in W^{1,2}(M)
\end{array}\right\}.
\end{equation}
The second-order \(\RCD\) calculus assigns an \(L^2\) Hessian to a
test function and gives
\(\nabla_{\nabla g}\nabla g=\nabla(\Gamma(g)/2)\)
\cite{Gigli2018}. For \(F\in W^{1,2}_{\rm loc}(M)\), a compactly
supported \(g\in\Test(M)\), \(Y=\nabla g\), and
\(h\in\Lip_c(M)\), define its weak Hessian evaluation by
\begin{equation}
\label{eq:weak-hessian}
H_F(Y,Y)(h)
=-\int_M\divg(hY)\,\langle\nabla F,Y\rangle\dd m
-\int_M h\,\langle\nabla F,\nabla_YY\rangle\dd m.
\end{equation}
Here \(\divg(hY)=\Gamma(h,g)+h\Delta g\), and both integrals
are finite: the relevant derivatives are locally \(L^2\), and
\(h,Y\) are bounded with compact support. This agrees with the
usual \(L^2\) Hessian when \(F\in D(\Delta)\). The definition
itself requires only first derivatives of \(F\).
\begin{theorem}[Weak Hessian bounds along every geodesic]
\label{weight:every-geodesic}
Let \((M,d,m)\) be a full-support \(\RCD(K,N)\) space,
where \(K\in\mathbb R\) and \(1<N<\infty\). Let
\(F\colon M\to\mathbb R\) be bounded and globally Lipschitz,
and let \(G\colon M\to\mathbb R\) be bounded and continuous.
Suppose that for every compactly supported \(g\in\Test(M)\)
and every nonnegative \(h\in\Lip_c(M)\),
\begin{equation}
\label{weight:hessian-assumption}
H_F(\nabla g,\nabla g)(h)
\le\int_M hG\Gamma(g)\dd m.
\end{equation}
Then every constant-speed minimizing geodesic
\(\sigma\colon[0,1]\to M\), of length \(\ell\), satisfies
\begin{equation}
\label{weight:curve-hessian}
(F\circ\sigma)''\le\ell^2G\circ\sigma
\qquad\text{in distributions on }(0,1).
\end{equation}
\end{theorem}
Thus, for every nonnegative \(\varphi\in C_c^\infty(0,1)\), the
conclusion means
\[
\int_0^1 F(\sigma_t)\varphi''(t)\dd t
\le \ell^2\int_0^1 G(\sigma_t)\varphi(t)\dd t.
\]
For constant \(G=c\), it says that
\(t\mapsto F(\sigma_t)-c\ell^2t^2/2\) is concave.
The ambient reference measure can have infinite total mass:
bounded Lipschitz \(F\) is locally Sobolev, and no global
\(L^2(m)\) condition on \(F\) is needed.
\subsection{Background and the analytic obstacle}
The differential calculus of \(\RCD\) spaces makes weak gradients,
Hessians and integration by parts available without a smooth atlas
\cite{Gigli2018}. Savar\'e's self-improvement of the
Bakry--\'Emery condition supplies a measure-valued Bochner calculus
and the multiplier properties used below \cite{Savare2014}.
Ambrosio--Gigli--Savar\'e identify the corresponding constant
Bakry--\'Emery bounds with Riemannian curvature-dimension bounds
\cite{AGS2015}; Braun--Habermann--Sturm develop the analogous
variable-curvature equivalences \cite{BHS2021}.
Changes of measure and large-parameter entropy limits already play
an important role in Hessian-to-convexity arguments. Ketterer uses
an exponential weight and a scaling limit under a Hessian assumption
that includes Sobolev regularity of gradient pairings
\cite[Theorem~7.1 and its proof]{Ketterer2015Obata}. Sturm develops a
vanishing-entropy argument for semiconvex functions
\cite[Lemmas~1 and 2]{Sturm2018}. Particularly close to our transport
step is Han's proof of infinitesimal-to-weak convexity: he applies
entropy convexity for \(e^{-a u}m\), divides by \(a\), and lets
\(a\) tend to infinity \cite[Theorem~3.13, Equation~(3.6)]{Han2018Monotonicity}.
His hypotheses include local second-order Sobolev regularity of
\(u\). The work below establishes the measure-change step under
the distributional hypothesis \eqref{weight:hessian-assumption}
and retains the spatially varying bound \(G\).
Brena--Gigli study fine measure representations of weak Hessians
and discuss the passage from a weak Hessian bound to geodesic
convexity \cite[Remarks~2.3 and 2.4]{BrenaGigli2024}.
In particular, their Remark~2.4 proposes using a change of measure
and a weak Bochner inequality, and identifies the regularity needed
to justify the approximation in that argument. This is a close
antecedent of the method here. Under our bounded Lipschitz
hypotheses, we prove the multiplier regularity needed for the
measure-change argument by a resolvent contraction for the original
heat semigroup.
For bounded globally Lipschitz \(V\), the change of measure
\(m_V=e^V m\) preserves the \(L^2\) generator domain,
but it need not preserve the test class. Its generator has the drift term
\(\Delta_Vu=\Delta u+\Gamma(V,u)\); a Lipschitz \(V\) does not
ensure that \(\Gamma(V,u)\) is Sobolev. The proof therefore closes
the weighted Bochner inequality in the common generator graph
domain before applying any curvature-dimension equivalence.
The example in Remark~\ref{weight:domain-warning} makes this
regularity issue explicit.
With Laplacian-domain regularity, Kapovitch--Ketterer also pass
from an \(L^2\)-Hessian bound to a geodesic convexity inequality
by approximating point endpoints with measures
\cite[Theorem~4.7]{KapovitchKetterer2020}.
The final passage to a prescribed geodesic uses Deng's
nonbranching theorem for finite-dimensional \(\RCD(K,N)\) spaces
\cite[Theorem~1.3]{Deng2020}. It makes every strict interior
subsegment of that geodesic the unique minimizer between its
endpoints. This metric fact allows entropy inequalities for
concentrating measures to recover the chosen subsegment, including
subsegments invisible to an almost-everywhere assertion about paths.
\subsection{Proof structure}
The proof has two stages. First fix \(\lambda>0\) and change the
measure to \(m_\lambda=e^{\lambda F}m\). The weak Hessian bound
implies a weighted Bochner inequality with curvature
\(K-\lambda G\). A resolvent estimate gives Lipschitz regularity
of the weighted multipliers, and the common graph core makes the
Bochner inequality valid on the actual weighted generator domain.
The constant and variable curvature-dimension equivalences then
yield \(\CD(K-\lambda G,\infty)\).
Second, fix a strict subsegment of the geodesic and approximate its
endpoints by normalized measures on small balls. All transport
geodesics between these shrinking supports converge to that unique
subsegment. The identity for Boltzmann entropy
\[
\Ent_{m_\lambda}(\mu)
=\Ent_m(\mu)-\lambda\int F\dd\mu
\]
then turns the variable-curvature inequality into a lower bound for
\(F\) above its endpoint chord, with the Green-kernel correction
determined by \(G\). Choosing \(\lambda\) to dominate the endpoint
entropies removes the remaining entropy term. Applying this bound
on every strict subinterval proves the distributional inequality.
Section~\ref{sec:preliminaries} gives the common calculus and
approximation facts. Section~\ref{sec:drift} establishes the weighted
generator and resolvent estimates. Section~\ref{sec:curvature-change}
proves the variable curvature-dimension bound, and
Section~\ref{sec:every-geodesic} completes the localization argument.
==== END Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces//sections/introduction.tex ====
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