\int Q_1f\,\d(\e_0)_*\ppi&\geq \limi_{n\to\infty}\int Q_1f_n\,\d(\e_0)_*\ppi_n\geq \int f\,\d(\e_1)_*\ppi+\frac12\iint_0^1|\dot\gamma_t|^2\,\d t\,\d\ppi(\gamma).
%&\geq \limi_{n\to\infty} \int f_n\,\d(\e_1)_*\ppi_n+\limi_{n\to\infty}\frac12\iint_0^1|\dot\gamma_t|^2\,\d t\,\d\ppi_n(\gamma)\\
%&\geq \int f(\gamma_1)+\frac12\int_0^1|\dot\gamma_t|^2\,\d t\,\d\ppi(\gamma)\\
%&\stackrel{*}\geq \int f(\gamma_1)+\frac{\sfd^2(\gamma_0,\gamma_1)}2\,\d\ppi(\gamma)\stackrel{*}\geq \int Q_1f(\gamma_0)\,\d \ppi(\gamma),
\end{split}
\end{equation}
Since  \eqref{eq:unipin}  implies   $(\e_1)_*\ppi\ll\mm$,  we can apply Proposition \ref{prop:unique} and deduce that
\[
|\d Q_tf|=\lip(Q_tf)\qquad (\e_{1-t})_*\ppi-a.e..
\]
Then the arbitrariness of $\rho$ and the very same arguments used at the beginning of the previous step (based on the fact that the estimate \eqref{eq:unipin} passes to the limit in place of \eqref{eq:regestpi}) show that \eqref{eq:slopedqtf}, and in particular {\eqref{eq:coresl}, holds.}

Also, since the inequality $f(\gamma_1)+\frac12\int_0^1|\dot\gamma_t|^2\,\d t\geq f(\gamma_1)+\frac{\sfd^2(\gamma_0,\gamma_1)}2\geq Q_1f(\gamma_0)$ is true for any $\gamma\in C([0,1],\X)$, the bound \eqref{eq:tantiug2}  gives  $\limi_n\int Q_1f_n\,\d(\rho\mm)\geq \int Q_1f\,\d(\rho\mm)$. Now   put $g_n:=\sup_{k\geq n}Q_1f_k$, $g:=\inf_n g_n=\lims_nQ_1f_n$ and observe that,  by dominated convergence,  $g_n\to g$ in $L^1(\X,\rho\mm)$. On the other hand
\[
\lims_n\|g_n-Q_1f_n\|_{L^1(\rho\mm)}=\lims_n\int g_n-Q_1f_n\,\d(\rho\mm)\leq \int \underbrace{ g-Q_1f}_{\leq 0\text{ by \eqref{eq:daglims}}}\,\d(\rho\mm)\leq 0.
\]
This forces at once $Q_1f_n\to g$ in $L^1(\rho\mm)$ and $g=Q_1f$ $\rho\mm$-a.e., hence  the arbitrariness of $\rho$ tells that $(Q_1f_n)$ converges to $Q_1f$ in $L^1_{loc}(\X)$. Now from \eqref{eq:boundbrutti} and    the validity  of \eqref{eq:stimalapqtf} for $f_n$ we get  $\bd Q_1f_n\leq (C''+2K^-C'')\mm$ and according to the stability property \eqref{eq:stablub} we can conclude that $\bd Q_1f\leq (C''+2K^-C'')\mm$. Since by a scaling argument the same inequality is true for $Q_tf$, \eqref{eq:corelap} holds. We thus established \eqref{eq:core}, as desired.

\st{Step 3: wrapping everything up}

\noindent\underline{The case of test functions}  Let $f\in\testi$. Then   clearly $f\in\acm(\X)$. Also, the properties  \eqref{eq:core} are granted by  Lemma \ref{prop:GTT}, hence in this case the conclusion follows from the first step.

\noindent\underline{Intermediate regularity} Assume that $f$ is in $\Lip_\b(\X)\cap D(\Delta_\loc)$ with $\Delta f\in \Lip_b(\X)$. Pick $\bar x\in\X$, let $\phi_n:=(1-\sfd(\cdot,B_n(\bar x)))^+$, define $\eta_n:=\tilde\h_1\phi_n\in\tvar(\X)$ and then $f_n:=\eta_nf$. It is clear from \eqref{eq:calclap} that $f_n\in\testi$ for every $n\in\N$, so by the previous step the conclusions of the theorem are true for it. We claim that the bounds \eqref{eq:boundbrutti} and the $\Gamma-$convergence \eqref{eq:glim} are in place.

The bound \eqref{eq:oscbound} is trivial and since \eqref{eq:apprf}  gives
\[
\sup_n\osc(\eta_n)+\Lip(\eta_n)+\|\Delta\eta_n\|_\infty<\infty,
\]
the bound  \eqref{eq:lapbrutto}  follows from the Leibniz rule for the Laplacian \eqref{eq:calclap}. We pass to \eqref{eq:glim} and   notice that the monotonicity of the heat flow grants that $(\eta_n(x))$ is a bounded non-decreasing sequence for every $x\in\X$, hence it converges to a limit $\eta(x)$. In particular  for $\varphi\in\tvar(\X)$ by dominated convergence we have $\int\eta_n\varphi\,\d\mm\to\int\eta\varphi\,\d\mm$. On the other hand we have
\[
\int \varphi\eta_n\,\d\mm=\int \varphi\tilde\h_1\phi_n\,\d\mm=\int \tilde\h_1\varphi\,\phi_n\,\d\mm\qquad\to\qquad\int \tilde\h_1\varphi\,\d\mm=\int \varphi\,\d\mm
\]
and by the density property \eqref{eq:denstvar} we conclude that $\eta=1$ $\mm$-a.e., and thus everywhere because it is a Lipschitz function (being limit of equiLipschitz functions). It follows that for $x_n\to x$ we have
\[
|\eta_n(x_n)-\eta(x)|\leq \sfd(x_n,x)\sup_n\Lip(\eta_n)+|\eta_n(x)-\eta(x)|\quad\to\quad 0\qquad\text{ as $n\to\infty$}
\]
and thus, since $f\in\Lip_\b(\X)$, that $ f_n(x_n)\to f(x)$. This property is clearly stronger than \eqref{eq:glim}, which therefore is proved. Hence the desired conclusion for $f$ follows from Step 2.

\noindent\underline{The general case}  Let $f:\X\to\R$ be as in the assumptions, put $f_n:=\tilde\h_n f$ and notice that by \eqref{eq:extappr} and \eqref{eq:lintylip} we have $f_n\in \Lip_\b(\X)\cap D(\Delta_\loc)$ with $\Delta f_n\in\Lip_b(\X)$. Then by the previous set the conclusions of the theorem are true for $f_n$, whereas by  \eqref{eq:stabub2}  and the weak maximum principle we see  that the uniform bounds \eqref{eq:boundbrutti} are in place.

According to Step 2 to conclude it is therefore sufficient to show that the $\Gamma$-convergence    \eqref{eq:glim} is in place. To prove \eqref{eq:glimi} we  let $g\in\Lip_{\b}(\X)$, $g_n:=\tilde\h_n g$ and notice that \eqref{eq:BElip} implies $L:=\sup_n\Lip(g_n)<\infty$. Thus if  $g\in \Lip_\b(\X)$, for any   $x_n\to x$  the monotonicity of the heat flow gives
\begin{equation}
\label{eq:perlimi}
\limi_{n\to\infty}f_n(x_n)\geq \limi_{n\to\infty} g_n(x_n)\geq \limi_{n\to\infty} (g_n(x)-L\sfd(x,x_n))= \limi_{n\to\infty}g_n(x)=g(x),
\end{equation}
where in the last equality we used the continuity of $g$, the representation $\h_tg(x)=\int g\,\d\h_t\delta_x$ and the fact that $\h_t\delta_x\weakto\delta_x$ as $t\downarrow0$. From the lower semicontinuity of $f$ it is easy to establish that $f=\sup\{g\, :\, g\in \Lip_b(\X),  g\leq f\}$ (see e.g.\ \cite[Section 5.1]{AmbrosioGigliSavare08}),  thus \eqref{eq:glimi} follows from \eqref{eq:perlimi}.

To prove \eqref{eq:glims} we recall that for some Borel set $E\subset\X$ with $\mm$-negligible complement we have that $f\restr E$ is continuous. Also, using \eqref{eq:acmc} and a diagunalization argument it is clear that it suffices to prove \eqref{eq:glims} for $x\in E$. Fix such $x$, let $\eps>0$, find $r>0$ such that $f\leq f(x)+\eps$ on $B_r(x)\cap E$ and notice that since $\h_t\delta_x\ll\mm$ for every $t>0$ and $\h_t\delta_x(B_r^c)\to 0$ as $t\downarrow0$ we have
\[
\lims_{t\downarrow0}\h_tf(x)=\lims_{t\downarrow0}\int f\,\d\h_t\delta_x=\lims_{t\downarrow0}\int_{B_r(x)\cap E} f\,\d\h_t\delta_x\leq f(x)+\eps.
\]
Since $\eps>0$ was arbitrary, the claim \eqref{eq:glims}  follows by picking $x_n\equiv x$.
\end{proof}
\begin{remark}\label{re:infdim}{\rm
Let  $\X,f$ be as in the previous statement, $T>0$ and $U,U'\subset \X$ be open sets such that $F_T(x)\in U'$ for $\mm$-a.e.\ $x\in U$. Assume also that for some $c\in\R$, in addition to the assumption $\bd f\leq C\mm$ we also have $\bd f\restr U\leq c\mm$. Then from Theorem \ref{prop:varregf} and  Lemmas \ref{le:lappunt}, \ref{le:lapcomb} we get
\begin{equation}
\label{eq:vargen}
\bd Q_Tf\leq ( c\nchi_U+C\nchi_{\X\setminus U}-K{\sf D}(U,U'))\mm.
\end{equation}
Comparing this bound with \cite[Theorem 1.5]{MS22} we see that a similar result, stated for $p=2$ in place of $p=1$, holds without assuming  non-collapsing or finite dimensionality.

We notice also that variants of \eqref{eq:vargen} and Theorem \ref{prop:varregf} for other values of $p$ are quite easy to obtain following the above strategy, at least on $\RCD(K,N)$ spaces where existence of optimal maps for the $L^p$-cost is known ([24], [35], see also [1]). The key ingredients are the `$p$-versions' of Lemma \ref{le:lappunt} - whose validity has already been noticed in [86] - and of Lemma \ref{prop:GTT}  - that can be obtained following the arguments in [61] using the $L^p$-version of Bochner inequality in [94].

\section{Harmonic maps from $\RCD(K,N)$ to  $\CAT(0)$ spaces}\label{ch:main}

\subsection{Reminders: basic definitions and results} \label{se:sobmet}
From now on we shall work under the following assumptions:
\begin{itemize}
\item[-] $(\X,\sfd,\mm)$ is  an $\RCD(K,N)$ space of essential dimension $d\in\N\cap[1,N]$ ([9], [48], [22]).
\item[-] $U\subset\X$ is open, bounded and such that $\mm(\X\setminus U)>0$.
\item[-] $(\Y,\sfd_\Y)$  is a  $\CAT(0)$ space (see e.g.\ [18] for an introduction to the topic).
\end{itemize}
Recall that the space $L^2(U;)$ is the collection of Borel maps $u$ (defined up to $$-a.e.\ equality)  from $U$ to $$ that are essentially separably valued (i.e.\ $(U∖u^-1('))=0$ for some separable subset $'⊂$) and with $∫_U_^2(u(x),ø) (x)<∞$ for some $ø∈$, and thus any by $(U)<∞$.

We start recalling two notions of metric-valued Sobolev functions, referring to [72] for more details on the first and to [79], [63]  for the second.
\begin{definition}[Sobolev space via post-composition] $W^{1,2}(U;\Y)\subset L^2(U;\Y)$ is the collection of those maps $u:U\to\Y$ such that $\varphi\circ u\in W^{1,2}(U)$ for every $\varphi:\Y\to\R$ Lipschitz and so that the function $|\d u|:U\to[0,\infty]$ defined up to $\mm$-a.e.\ equality as
\begin{equation}
\label{eq:defdun}
|\d u|:=\esssup_{\varphi:\Y\to\R\atop\Lip(\varphi)\leq 1}|\d(\varphi\circ u)|
\end{equation}
is in $L^2(U)$.
\end{definition}
\begin{definition}[Sobolev space \`a la Korevaar-Schoen]
For every  $r>0$ define the local energy $\kse[u,U]:U\to\R^+$ of $u$ at scale $r$   as
\begin{equation}
\label{eq:kse}
\kse_{2,r}[u,U] (x) := \begin{cases} \ \Big\vert \fint_{B_r(x)} \frac{\sfd_{\Y}^2(u(x),u(y))}{r^2} \, \d \mm(y) \Big\vert^{1/2} &\text{ if } B_r(x) \subset U, \\ \ 0 &\text{ otherwise.} \end{cases}
\end{equation}
Then $\KS(U;\Y)\subset L^2(U,\Y)$ is the collection of those $u$'s such that
\begin{equation}
\label{eq:defKS}
\E(u):=\limi_{r\downarrow0}\int_U\kse_{2,r}[u,U] ^2\,\d\mm<\infty.
\end{equation}
\end{definition}
We now collect those properties of metric-valued Sobolev functions that are relevant for our discussion, referring to  \cite[Theorem 3.13, Proposition 3.9, Theorem 6.4]{GT20} for the proof:
\begin{proposition}\label{prop:baseKS}With the same assumptions and notations as above we have:
\begin{itemize}
\item[i)] For every $u\in \KS(U;\Y)$  there is a function $\e_2[u]\in L^2(U)$, called \emph{energy density}, such that
\begin{equation}
\label{eq:conven}
\kse_{2,r}[u,U]\quad\to\quad \e_2[u]\qquad\text{ in $L^2(U)$ and $\mm$-a.e.\ as $r\downarrow0$.}
\end{equation}
In particular, the $\limi$ in \eqref{eq:defKS} is a limit.
\item[ii)] We have $W^{1,2}(U;\Y)= \KS(U;\Y)$ with
\begin{equation}
\label{eq:dfu}
\e_2[u]\leq |\d u |\leq c(d)\e_2[u]\qquad\mm-a.e.\ on\ U,
\end{equation}
for some  constant $c(d)$ depending only on the essential dimension $d$, and thus only on $N$.
\item[iii)] For any $u\in \KS(U;\Y)$ there is a unique minimizer $\bar u\in \KS(U;\Y)$ of the Korevaar-Schoen energy $\E$ in the class of functions $v\in \KS(U;\Y)$  such that $x\mapsto \sfd_\Y(u(x),v(x))$ is in $W^{1,2}_0(U)$. Such minimizer is called \emph{harmonic map}.
\end{itemize}
\end{proposition}
In what follows we will need to know that the energy density $_2[u]$ is not only the limit of $_2,r[u,U]$, but also of an appropriate average of $_(u(x),u(·))$ via heat kernel:
\begin{proposition} With the same assumptions and notations as in the beginning of the section, the following holds. Let $u\in \KS(U;\Y)$. Then there is $E\subset U$ Borel with $\mm(U\setminus E)=0$ such that
\begin{equation}
\label{eq:convenvar}
\lim_{t\downarrow0}\frac1t\int \sfd_\Y(u(x),u(y))\,\d\h_t\delta_x(y)= 2(d+2) \e_2[u]^2(x)\qquad\forall x\in E.
\end{equation}
\end{proposition}
The \emph{proof} of this proposition follows very closely that of  \cite[Theorem 3.13]{GT20} from which the result \eqref{eq:conven} recalled above is extracted. Since the arguments for proving  \cite[Theorem 3.13]{GT20} are rather lengthy (and based on few additional concepts such as that of mGH-convergence of spaces, approximate metric differentiability, Hajlasz-Sobolev functions...) and since the modifications needed to obtain \eqref{eq:convenvar} are straightforward, we limit ourselves to pointing out what follows. The basic idea behind the proof of \eqref{eq:conven} is that for $$-a.e.\ $x∈U$ there is a seminorm $md_u(x)$ on $^d$ that acts as a sort of `metric differential' by describing the infinitesimal variation of $u$ around $x$ (see  \cite[Definition 3.3]{GT20}). The energy density $_2[u](x)$ is then defined by the formula
\begin{equation}
\label{eq:defe2}
\begin{split}
\e_2^2[u](x):=&\fint_{B_1^{\R^d}(0)} {\sf md}_u(x)(v)^2\,\d\mathcal L^d(v)\\
=&\frac1{|B^{\R^d}_1|}\int_0^1r^{d+1}\underbrace{\int_{S^{d-1}}{\sf md}_u(x)(v)^2\,\d\mathcal H^{d-1}(v)}_{=:|||{\sf md}_u(x)|||}\,\d r=\frac{|||{\sf md}_u(x)|||}{(d+2)|B^{\R^d}_1|}
\end{split}
\end{equation}
and  \eqref{eq:conven} follows by the `first order expansion of $u$ given by $md_u$' by noticing that the measures $(B_r(x))^-1B_r(x)$  appearing in the definition of $_2,r[u,U]$ converge, after scaling, to  $1/|B^^d_1|ℒ^dB^^d_1(0)$\footnote{Such convergence of measures is in place, by definition, at all points with Euclidean tangent by the very definition of mGH-convergence of pointed spaces. To get the desired \eqref{eq:conven} one uses also that $\RCD$ spaces are doubling and support a Poincar\'e inequality to call into play maximal estimates that are useful to control the term $\tfrac{\sfd_\Y(u(x),u(y))}{r}$ appearing in the definition of $\kse_{2,r}[u,U]$.  Such control is necessary due to the nature of the concept of approximate metric differentiability, that only speaks about $y$ belonging to suitabl charts for which the point $x$ is assumed to be a Lebesgue point, while in the definition of $\kse_{2,r}[u,U]$ all the points $y$ in $B_r(x)$ apear in the integral. See the proof of  \cite[Theorem 3.13]{GT20}  for more details}. Similarly, \eqref{eq:convenvar} is proved recalling that the heat kernels $_tδ_x$ converge, after scaling, to the Gaussian measure $^^d_1δ_0=(4π)^-d/2e^-|·|^2/4ℒ^d$ (this follows from the stability of the heat flow under lower Ricci curvature bounds first noticed in [44] - here we use \cite[Theorem 7.7]{GMS15}). Using such convergence and arguing as for \eqref{eq:conven} we see that the limit in \eqref{eq:convenvar} is equal to
\[
\int  {\sf md}_u(x)(v)^2\,\d \h^{\R^d}_1\delta_0(v)=\frac{|||{\sf md}_u(x)|||}{(4\pi)^{\frac d2}}\int_0^{+\infty} r^{d+1}e^{-\frac{r^2}4}\,\d r,
\]
thus from \eqref{eq:defe2} we conclude that \eqref{eq:convenvar} is ultimately reduced to the verification of the identity
\[
\frac{1}{(4\pi)^{\frac d2}}\int_0^{+\infty} r^{d+1}e^{-\frac{r^2}4}\,\d r=\frac2{|B^{\R^d}_1|}
\]
that in turn is well known (and easy to establish by direct computation, e.g.\ by induction). In all this, the fact that the set $E$ can be chosen to be Borel follows from the construction or, alternatively, noticing that the right hand side of \eqref{eq:convenvar} is a Borel function, that the set of points for which the limit in the left hand side exists is Borel and that so is the associated limit function.

\bigskip

We shall also need the following result, see  \cite[Theorem 4.18]{gn20}:
\begin{theorem}\label{thm:lapcomp}  With the same assumptions and notations as in the beginning of the section, the following holds.
Let $u\in \KS(U;\Y)$ be harmonic,  $f:\Y\to\R$ be  $\lambda$-convex and $\Y'\subset\Y$ be a $\CAT(0)$ subspace with $\mm(U\setminus u^{-1}(\Y'))=0$ and so that $f\restr {\Y'}$ is Lipschitz. Then
\begin{equation}
\label{eq:fconv}
\bd(f\circ u)\restr U\geq \lambda (d+2)\e_2[u]^2\mm.
\end{equation}
\end{theorem}
\begin{remark}{\rm
There are few differences between the statement as given above and that of \cite[Theorem 4.18]{gn20}, let us comment on them.

One concerns the fact that in  \cite[Theorem 4.18]{gn20} we assumed $f$ to be globally Lipschitz, rather than only on an essential image of $u$ as above. The modification we did is inessential in  the proof (just replace $\Y$ with $\Y'$ and notice that $u$ remains harmonic) and useful later on, when, after proving that $u$ is locally bounded, we will choose as $f$ the squared distance from a point.

Another is about the use of $\e_2[u]$ in place of $|\d u|_\HS$ that appeared in \cite[Theorem 4.18]{gn20}. There is a strict relation between these two: in \cite[Proposition 6.7]{GT20} we proved that $(d+2)\e_2[u]^2=|\d u|_{\HS}^2$ so  that \eqref{eq:fconv} is equivalent to
\begin{equation}
\label{eq:fconv2}
\bd(f\circ u)\restr U\geq \lambda |\d u|_{\HS}^2\mm.
\end{equation}
We preferred to state the bound as in \eqref{eq:fconv} because in order to give a meaning to the pointwise Hilbert-Schmidt norm   $|\d u|_\HS$ one needs to rely on the  \emph{universally infinitesimally Hilbertian} of $\CAT(\kappa)$ spaces proved in  [36].  Since, perhaps surprisingly, such result  plays no role in this paper (nor in [54]), we decided to stick to the notation $\e_2[u]$ to emphasize this fact.

The last  difference is made more evident by the writing in \eqref{eq:fconv2}, as in  \cite[Theorem 4.18]{gn20} the right hand side is multiplied by an additional factor $\frac1{d+2}$. This is due to a computational mistake done in [54] that made it all the way up to publication: namely, in deriving the bound \cite[(4.12)]{gn20} from \cite[(4.11)]{gn20} we forgot to multiply one of the terms in the right hand side by $\frac1{d+2}$. This results in an incorrect formula, the correct version being
\[
\lims_{t\downarrow0}\frac{\E(u_t)-\E(u)}t\leq -\frac1{d+2}\int_\Omega \lambda g|\d u|_{\HS}^2+\la\d(f\circ u),\d g \ra\,\d\mm.
\]
Using this bound in place of the original \cite[(4.12)]{gn20} in the proof of  \cite[Theorem 4.18]{gn20} we obtain the correct result reported above.
We conclude recalling the following key regularity statement about subharmonic functions.
\begin{theorem}\label{thm:ellrt}
Let $(\X,\sfd,\mm)$ be an $\RCD(K,N)$ space, $K\in\R$, $N<\infty$, $U\subset\X$ open, $B_R(\bar x)\subset U$, $\alpha,\beta \in\R$ with $\beta\geq 0$ and $f\in W^{1,2}_{loc}(U)$ be with
\begin{equation}
\label{eq:subsol}
\bd f\geq -R^{-2}(\alpha  f+\beta)\mm\qquad\ on\ U.
\end{equation}
Then for every   $\lambda \in(0,1)$ and $r\in(0, R]$  we have
\begin{equation}
\label{eq:ellst}
\|f^+\|_{L^\infty(B_{ \lambda r}(\bar x))}\leq C_1(\alpha^+,\tfrac1{1-\lambda},K^-R^2,N)\fint_{B_{ r}(\bar x)}f^+\,\d\mm+\beta\frac{r^2}{R^2}\,C_2(\tfrac1{1-\lambda},K^-R^2,N) .
\end{equation}
Moreover, $f$ admits an upper semicontinuous representative. More precisely, it  coincides $\mm$-a.e.\ with its   essential upper semicontinuous envelope $f^*$ defined as $f^*(x):=\inf_{r>0}\esssup_{B_r(x)\subset U} f$.
\end{theorem}
This result is folklore in the field and its \emph{proof} can be achieved via Moser's iteration. In this direction let us point out what follows.
\begin{itemize}[leftmargin=*]
\item[-] The constants appearing in Moser's technique depend only on the local doubling constant and on the constant in the Poincar\'e inequality. This has been realized in the smooth category in [93], where it has been proved that doubling \& Poincar\'e imply a Sobolev inequality (needed in Moser's iteration) with an argument that carries over to the present setting without modifications, much like it does so in the setting of Dirichlet forms, as noticed in [101]. Notice that the settings of Dirichlet forms and $\RCD$ spaces are fully compatible by \cite[Theorem 6.10]{AmbrosioGigliSavare11-2} (whose proof requires the chain and Leibniz rules established in [9] and   [48]).
\item[-] Chain rule and integration by parts formula in general metric measure spaces have been provided in [48] and can be used to follow the proof, e.g., of \cite[Theorem 8.17]{GT01} without any relevant modification, even in non-Hilbertian spaces. These tools are not-necessary if $\alpha=\beta=0$ as in this case the inequality $\bd f\geq 0$ can be read in purely variational terms and, notably, this is sufficient to develop a satisfactory theory, see e.g.\ [19] and references therein. In the `dual' setting of Dirichlet forms, in place of the calculus as in [48] one can use that based on the `Carr\'e du champ' operator, see e.g.\ [84], [42] and references therein.
\item[-] The dependance of $C_1,C_2$ on the given parameters can be seen e.g.\ inspecting the proof of \cite[Theorem 8.17]{GT01}. The dependance on $K^-R^2$ can also be proved via a scaling argument: replacing $\sfd$ with $\frac1r\sfd$ the original $\RCD(K,N)$ space becomes $\RCD(Kr^2,N)$, hence $\RCD(-K^-R^2,N)$, inequality \eqref{eq:subsol} becomes $\bd f\geq -\frac{r^2}{R^{2}}(\alpha  f+\beta)\mm$ and everything happens in a ball of radius 1.
\item[-] From  \eqref{eq:ellst} it is clear that $f(x)=f^*(x)$ for  $x$ Lebesgue point of $f$ such that $f(x)>0$. For the general case we  notice that  $f+k$, $k\gg1$, satisfies a similar differential inequality.
\item[-] In \cite[Theorem 8.17]{GT01}  there is an $L^p$ norm, $p>1$, at the right hand side. Still, once such bound is proved, as customary one can use the $L^\infty$ control it provides and the trivial inequality $\|f\|_{L^p}\leq \|f\|^{\frac{p-1}p}_{L^\infty}\|f\|_1^{\frac 1p}$ to get \eqref{eq:ellst}.
\end{itemize}

\subsection{First regularity results}

In this section we shall work under the following assumptions:
\begin{itemize}
\item[-] $(\X,\sfd,\mm)$ is an $\RCD(K,N)$ space of essential dimension $d\in\N$.
\item[-] $U\subset\X$ is open, bounded and such that $\mm(\X\setminus U)>0$.
\item[-] $(\Y,\sfd_\Y)$  is a $\CAT(0)$ space.
\item[-] $u\in\ks(U;\Y)\subset L^2(U,\Y)$ is harmonic. For convenience, we shall also fix  Borel representatives of $u$ and $\e_2[u]$.
\end{itemize}
We recall that the $(0)$ condition implies (see e.g.\ [18]):
\begin{equation}
\label{eq:convdacat3}
\text{for every $p\in\Y$ the map $\sfd_\Y(\cdot,p)$ is convex and the map $\sfd_\Y^2(\cdot,p)$ is 2-convex.}
\end{equation}
This and Theorem \ref{thm:ellrt} give the following important result about the `continuity set' of $u$:
\begin{proposition}\label{prop:boundlinfty}
With the above notation and assumptions, the following holds.

There exists $\cont(u)\subset U$ Borel with
\begin{equation}
\label{eq:mcont}
\mm(U\setminus \cont(u))=0
\end{equation}
such that the restriction of $u$ to $\cont(u)$ is continuous and locally bounded. More precisely, for any $\o\in\Y$, ball $B_R(x)\subset U$,  $\lambda\in(0,1)$ and $r\in(0, R]$ we have
%putting   $B:=B_{r}(x)$ and  $\lambda B:=B_{\lambda r}(x)\subset U$  we have
\begin{equation}
\label{eq:boundest}
\sup_{x\in\cont(u)\cap B_{\lambda r}(x)}\sfd_\Y(u(x),\o)\leq  C(\tfrac1{1-\lambda},K^-R^2,N)\sqrt{\fint_{ B_{ r}(x)}\sfd^2_\Y(u(x),\o)\,\d\mm(x)}.
\end{equation}
\end{proposition}
\begin{proof}
By \eqref{eq:convdacat3} and Theorem \ref{thm:lapcomp} we know that for any $p\in\Y$ we have $\bd\sfd_{\Y}( u(\cdot),p)\geq 0$ on  $U$, thus  by  the second part of Theorem \ref{thm:ellrt} we know that there is a Borel $\mm$-negligible set $N_p\subset U$ such that $\sfd_{\Y}( u(\cdot),p)$ is upper semicontinuous. Now recall that since $u$ is essentially separable valued there is $\Y'\subset\Y$  separable such that $\mm(U\setminus u^{-1}(\Y'))=0$, then let $(p_n)\subset\Y'$ be countable and dense and put
\[
\cont(u):=u^{-1}(\Y')\setminus(\cup_n N_{p_n}).
\]
Then $\cont(u)$ is Borel with $\mm(U\setminus \cont(u))=0$ and
\[
\lims_{y\to x\atop y\in \cont(u)}\sfd_\Y(u(x),u(y))\leq \sfd_\Y(u(x),p_n)+\lims_{y\to x\atop y\in \cont(u)}\sfd_\Y(p_n,u(y))\leq 2\sfd_\Y(u(x),p_n)%\qquad\forall x\in \cont(u),\quad  n\in\N.
\]
for every $x\in \cont(u)$ and $n\in\N$. Taking the $\inf$ in $n$ we obtain the desired continuity, while    estimate  \eqref{eq:boundest}  follows directly from  \eqref{eq:ellst} with $C=C_1(0,\tfrac1{1-\lambda},K^-R^2,N)$.
\end{proof}
We now define the upper semicontinuous function $_u:U×U→$ as
\begin{equation}
\label{eq:defdu}
\sfd_u(x,y):=\lims_{(x',y')\to(x,y)\atop (x',y')\in \cont(u)\times \cont(u)}\sfd_\Y(u(x'),u(y'))
\end{equation}
and notice that trivially from the definition it satisfies
\begin{subequations}
\begin{align}
\label{eq:duu}
\sfd_u(x,y)=\sfd_\Y(u(x),u(y))\qquad\forall x,y\in \cont(u),\\
\label{eq:dut}
\sfd_u(x,y)\leq\sfd_u(x,z)+\sfd_u(z,y)\qquad\forall x,y,z\in U.
\end{align}
\end{subequations}
Another direct consequence of the definition and \eqref{eq:duu}  is that
\begin{equation}
\label{eq:liptilt}
\lip(u)(x)=\lims_{y\to x}\frac{\sfd_\Y(u(y),u(x))}{\sfd(y,x)}=\tilt(-\sfd_u)(x)\qquad\forall x\in\cont(u),
\end{equation}
where the definition of $$ was given in \eqref{eq:tilt}. We shall also need the following properties:
\begin{proposition}\label{prop:lapdu} With the above notation and assumptions, the following holds:
\begin{itemize}
\item[i)] For every $x\in U$ we have  $\sfd_u(x,\cdot)\in W^{1,2}(U)$ and $\bd\sfd_u(x,\cdot)\restr U\geq 0$.
\item[ii)] We have
\begin{equation}
\label{eq:tilte2}
\lip (u)\leq C(N)\,\e_2[u]\qquad\mm-a.e.\ on\ U.
\end{equation}
\end{itemize}
\end{proposition}
\begin{proof}\ \\
\st{(i)} For $x\in\cont(u)$ we have $\sfd_u(x,y)=\sfd_\Y(u(x),u(y))$ for $\mm$-a.e.\ $y\in U$ by \eqref{eq:duu} and \eqref{eq:mcont}. Thus in this case the claims follow from item $(ii)$ in Proposition \ref{prop:baseKS} and Theorem  \ref{thm:lapcomp} in conjunction with  property \eqref{eq:convdacat3}. For the general case we fix $x\in U$ and for $r>0$ put $g_r(y):=\sup_{x'\in B_r(x)\cap \cont(u)}\sfd_u(x',y)$. Then  \eqref{eq:defdu} and \eqref{eq:duu} ensure that
\begin{equation}
\label{eq:perlims}
\sfd_u(x,\cdot)=\inf_{r>0}g_r(\cdot)\qquad \text{on\ $\cont(u)$ and thus $\mm$-a.e.\ on\ $U$}.
\end{equation}
Now notice that $\sfd_\Y(u(x'),\cdot)$ is 1-Lipschitz for every $x'\in U$, thus the uniform estimate \eqref{eq:dfu} - recall also \eqref{eq:defdun} -, the lattice property of Sobolev functions (that in turn follows from the chain rule and the locality of minimal weak upper gradients) and the lower semicontinuity of minimal weak upper gradients easily give that $g_r\in W^{1,2}(U)$ with $|\d g_r|\leq c(d)\e_2[u]$. Then \eqref{eq:perlims} and again the lower semicontinuity of minimal weak upper gradients tell that $\sfd_u(x,\cdot)\in W^{1,2}(U)$.

For the Laplacian estimate we apply  Lemma \ref{le:inflap2}  (with inverted signs) to deduce that $\bd g_r\restr U\geq0$, so that by \eqref{eq:perlims} and  the stability property \eqref{eq:stablubloc} we conclude.

\st{(ii)} Let $x\in\cont(u)$ and $r>0$ be so that $B:=B_r(x)$ and $2B=B_{2r}(x)$ are contained in $U$. Then%  and notice that% \eqref{eq:defdu}, \eqref{eq:duu},\eqref{eq:mcont} give
\[
\sup_{y\in B}\sfd_u(x,y)\stackrel{\eqref{eq:defdu},\eqref{eq:mcont} }=\|\sfd(u(x),u(\cdot))\|_{L^\infty(B)}\stackrel{\eqref{eq:boundest}}\leq C(K^-r^2,N)\sqrt{\fint_{2B}\sfd^2_\Y(u(y),u(x))\,\d\mm(y)},
\]
hence dividing by $r$ and letting $r\downarrow0$ we conclude recalling \eqref{eq:conven}.
%\sup_{y\in B_r(x)}\sfd_u(x,y)=\|\sfd(u(x),u(\cdot))\|_{L^\infty(B_r(x))}\stackrel{\eqref{eq:boundest}}\leq C(Kr^2,N)\sqrt{\fint_{2B}\sfd^2_\Y(u(x),\o)\,\d\mm(x)}.
\end{proof}

Another direct consequence of \eqref{eq:convdacat3} and Theorem \ref{thm:lapcomp} is:
\begin{proposition}[Reverse Poincar\'e inequality]\label{prop:invP}
With the above notation and assumptions, the following holds. Let $B:=B_r(x)\subset U$ and  $\lambda\in(0,1)$. Then
\begin{equation}
\label{eq:invP}
\int_{B_{\lambda r}(x)}\e_2[u]^2+|\d u|^2\,\d\mm\leq \frac{c(d)}{r^2(1-\lambda)^2}\inf_{\o\in\Y}\int_{ B_r(x)}\sfd_\Y^2(u(x),\o)\,\d\mm,
\end{equation}
for some constant $c( d)$ depending only on the essential dimension $d$ of $\X$ (and thus only on $N$).
\end{proposition}
\begin{proof} With an approximation argument we can assume that ${ B}\subset\subset U$. Hence  $u\restr{ B}$ is (essentially) bounded by Proposition \ref{prop:boundlinfty} and clearly harmonic. In particular $u( B)$ is essentially contained in a closed ball $\Y'$ of $\Y$. As it is well-known, and trivial consequence of \eqref{eq:convdacat3}, closed balls in $\CAT(0)$ spaces are convex, and thus  $\CAT(0)$ as well. Then for  $\o\in\Y$ let  $\sfd_\o:\Y'\to\R$ be given by $\sfd_\o(q):=\sfd_\Y(q,\o)$ and notice that by \eqref{eq:convdacat3} $\sfd_\o^2$ is 2-convex on $\Y'$, thus  Theorem \ref{thm:lapcomp} (with $ B$ in place of $U$)  gives $\e_2^2[u]\mm\leq c(d) \bd(\sfd^2_\o\circ u)$ on $ B$.  Therefore \eqref{eq:dfu} gives  $|\d u|^2\mm\leq c(d) \bd(\sfd^2_\o\circ u)$ on $ B$ and for    $\varphi:=(1-\tfrac2{r(1-\lambda)}\sfd(\cdot,B_{\lambda r}(x)))^+$ and $\alpha>0$ we have
\[
\begin{split}
\int_\X\varphi^2|\d u|^2\,\d\mm&\stackrel{\eqref{eq:considerazioni}}\leq  -c(d)\int_\X\d(\varphi^2)\cdot\d(\sfd_\o^2\circ u)\,\d\mm\\
&\stackrel{\phantom{\eqref{eq:considerazioni}}}=-c(d)\int_\X\varphi\,\sfd_\o\circ u\,\d\varphi\cdot\d(\sfd_\o\circ u)\,\d\mm\\
&\stackrel{\,\eqref{eq:defdun}\,}\leq c(d)\int_\X\alpha \varphi^2|\d u|^2+\alpha^{-1}\sfd^2_\o\circ u\,|\d\varphi|^2\,\d\mm.
\end{split}
\]
Picking $\alpha<\frac1{2c(d)}$, using  the trivial bound $|\d\varphi|\leq \frac2{r(\lambda-1)}\nchi_{ B}$ and  \eqref{eq:dfu} we get the conclusion.
\end{proof}

We conclude the section with the following general result:
\begin{proposition}[A Rademacher-type result]\label{prop:rad}
With the above notation and assumptions, the following holds. Let $v\in W^{1,2}(U,\Y)=\KS(U,\Y)$ be locally Lipschitz. Then
\begin{equation}
\label{eq:radelip}
\lip(v)=|\d v|\qquad\mm-a.e.\ on\ U.
\end{equation}
\end{proposition}
\begin{proof}
By \cite[Prop. 2.5, Lemma 3.4]{GT20} and with the terminology therein we have  $\lip(v)(x)=|||{\sf md}_x(v)|||$ for $\mm$-a.e.\ $x\in U$, thus taking into account  \cite[Thm 4.12]{GT20} we need only to verify that
\begin{equation}
\label{eq:ovvio}
|||{\sf md}_\cdot(v)|||=\esssup{\sf md}_\cdot(v)(\mathscr I({\sf v}))\qquad\text{and}\qquad |\d v|=\esssup|\d v({\sf v})|,
\end{equation}
where in both cases the $\mm$-essential supremum is taken among ${\sf v}\in L^0(T\X)$ with $|{\sf v}|\leq 1$ $\mm$-a.e.. The first in \eqref{eq:ovvio} is obvious (see also \cite[Thm 4.9]{GT20} and  [56] for the definition of the isomorphism $\mathscr I$ between the `abstract' $L^0(T\X)$ and `concrete' $L^0(T_{\rm GH}\X)$ tangent modules), while the second comes from the properties  of the abstract differential $\d v$ (see \cite[Prop. 3.5]{GPS18}).
\end{proof}
\begin{remark}\label{re:rad}{\rm This last statement has little to do with the specific geometry of $\RCD$ and $\CAT$ spaces and  the choice $p=2$. The same proof works  for any $p\in(1,\infty)$ and $\Y$ complete as soon as $\X$ is strongly rectifiable in the sense of \cite[Definition 2.18]{GT20} (see also the original definition \cite[Definition 3.1]{GP16}), uniformly locally doubling and supporting a (weak, local 1-1) Poincar\'e inequality.

This line of thought and \cite[Proposition 2.5]{GT20} and its proof also show that, in this case, for $v$ Lipschitz  the conclusions of  \cite[Proposition 3.6]{GT20} hold in a stronger form: one can replace  the approximate $\lims$ in \cite[Definition 3.3]{GT20} with a $\lims$.  Compare with \cite[Definition 2.7]{ZZZ19}.

\subsection{Lipschitz continuity and Zhang-Zhong-Zhu inequality}\label{se:zzz}

In this section  we shall work under the following assumptions:
\begin{itemize}
\item[-] $(\X,\sfd,\mm)$ is an $\RCD(K,N)$ space of essential dimension $d\in\N$.
\item[-] $U\subset\X$ is open, bounded and such that $\mm(\X\setminus U)>0$.
\item[-] $(\Y,\sfd_\Y)$  is a $\CAT(0)$ space.
\item[-] $u\in\ks(U;\Y)\subset L^2(U,\Y)$ is harmonic and $\cont(u)\subset U$ is given by Proposition \ref{prop:boundlinfty}. For convenience, we shall also fix  Borel representatives of $u$ and $\e_2[u]$.
\item[-]  $R>0$ is a fixed parameter, that we shall think of as `maximal radius'.
\item[-] $\hat x\in U$ and $r\in(0,R)$ are fixed so that  the balls  $B:=B_r(\hat x)$,   $2B:=B_{2r}(\hat x)$ are contained in $U$. Also, we define $B'\supset B''\supset B$ as $B':=B_{3r/2}(\hat x)$ and $B'':=B_{4r/3}(\hat x)$.
\end{itemize}
Start noticing that from the bound \eqref{eq:boundest}  and the definition \eqref{eq:defdu} we get
\[
\sfd_u(x,y)\leq \bar C\,\norm \qquad\forall x,y\in B'\qquad\text{ where }\qquad \norm:=\inf_{\o\in\Y}\sqrt{\fint_{2B}\sfd_\Y^2(u(\cdot),\o)\,\d\mm}
\]
for some $C̅=C̅(K^-R^2,N)$ that shall be kept fixed from here on.  We then define $f:^2→$ as
\begin{equation}
\label{eq:deff}
\begin{array}{ll}
-\sfd_u(x,y),&\qquad\text{ if }x,y\in B',\\
- \bar C\,\norm,&\qquad\text{ otherwise}
\end{array}
\right.
\end{equation}
and notice that
\begin{equation}
\label{eq:fbdasotto}
- \bar C\,\norm\leq f(x,y)\leq 0\qquad\forall x,y\in\X.
\end{equation}
The properties of $_u$ (in particular upper semicontinuity and \eqref{eq:dut}) ensure that $f$ has the properties stated at the beginning of Section \ref{se:HLnew}. We then define $f_t$ as in formula \eqref{eq:defft}, i.e.\ we put
\[
\]
Observe that  \eqref{eq:fbdasotto} gives
\begin{equation}
\label{eq:bft22}
-\bar C\,\norm\leq f_t\leq 0\qquad\ on\ \X,\ \forall t>0.
\end{equation}
Recall also that the functions $D^±_t(x):→^+$ have been defined in \eqref{eq:defdpm} and put
\begin{equation}
\label{eq:defdtns}
D^-_t(x),&\qquad\text{ if }K\geq 0,\\
D^+_t(x),&\qquad\text{ if }K< 0.
\end{array}\right.
\end{equation}
Strongly inspired by the ideas in [111],  [110], we see that recalling the limiting property \eqref{eq:dualtilt} and \eqref{eq:liptilt}, we will be in  a good position to prove  the Zhang-Zhong-Zhu inequality if we show that
\begin{equation}
\label{eq:perzzz}
\bd f_t\restr{B''}\leq -K\frac{D_t^2}{t}\mm\qquad\forall  0<t\ll1.
\end{equation}
The Laplacian comparison estimate for the distance and the stability of upper Laplacian bounds under `inf' easily gives that $f_tB”≤C_t$ for some `bad' constant $C_t$, hence by Lemma  \ref{le:lapcomb2} we see that \eqref{eq:perzzz} will follow if we prove that
\[
\tilde\Delta f_t\leq -K\frac{D_t^2}{t}\qquad\mm-a.e.\ on\ B''\  \forall  0<t\ll1.
\]
We are actually able to prove this only at points $x$ for which a minimizer for $f_t(x)$ is in some a priori given full-measure set of `nice' points (those for which the conclusion of Lemma \ref{le:tdF} holds) but it is not clear a priori whether this occurs for $$-a.e.\ $x$. Here is where the variational principle enters into play, in line with the use of Jensen's lemma in the theory of viscosity solutions. We perturb our functions by adding $^2(·,z)/n$ for some well chosen point $z∈$ and $n≫1$ (this does not affect upper Laplacian bounds too much), to be sure that minimizers belong to the given set:
\begin{lemma}[Perturbation lemma]\label{le:partle}
With the same assumptions and notations introduced above, the following holds. Put $\bar T:=\frac{r^2}{72(2+4r^2+\bar C\,\norm)}$ and  let $E\subset B'$ be Borel such that $\mm(B'\setminus E)=0$.

Then for every $t\in(0,\bar T)$ there is $\bar z_t\in B$ such that: for $\mm$-a.e.\ $x\in B''$ and every $n\in\N$, $n>0$
\[
\text{there is a minimizer $T_t(x)$ of }\qquad \X\ni y\quad\mapsto\quad g_{t,n}(x,y,\bar z_t):=f(x,y)+\frac{\sfd^2(x,y)}{2t}+\frac{\sfd^2(y,\bar z_t)}n
\]
and such minimizer belongs to $E\cap B'$.
\end{lemma}
\begin{proof} Let $x\in B''$ and notice that, by the choice of $\bar T$ and direct computation, for $t\in(0,\bar T)$  we have $f(x,\cdot)+\frac{\sfd^2(x,\cdot)}{2t}>c$ outside $B'$ for $c:=4r^2+1$. Thus by  item $(i)$ in Proposition \ref{prop:lapdu}, Theorem \ref{thm:lapbd}   and   Lemma \ref{le:ext} we know that  $\tilde f(x,\cdot):=(f(x,\cdot)+\frac{\sfd^2(x,\cdot)}{2t})\wedge c$ satisfies  $\bd \tilde f(x,\cdot)\leq \frac {C(K^-R^2,N)}t\mm$ and, picking $A:=\cont(u)\cup(\X\setminus U)$, that  $\tilde f(x,\cdot)\in \acm(\X)$.

Now for any $z\in B$ define   $\tilde g_{t,n}(x,y,z):=\tilde f(x,y)+\tfrac{\sfd^2(y,z)}n$. We claim that
\begin{equation}
\label{eq:claimg}
\begin{split}
&\text{$\forall n\in\N,\ n>0,\  x\in B'',\ z\in B,\ t\in(0,\bar T)$ we have:}\\
&\text{if $y$ minimizes $\tilde g_{t,n}(x,\cdot,z)$ then it minimizes $ g_{t,n}(x,\cdot,z)$ and it belongs to $B'$.}
\end{split}
\end{equation}
%Since $\tilde g_{t,n}\leq  g_{t,n}$, this will be proved if we show that for a minimizer $y$ of $\tilde g_{t,n}(x,\cdot,z)$ we have $\tilde g_{t,n}(x,y,z)=  g_{t,n}(x,y,z)$. To see this, notice that
To see this,  let $y$ be a minimizer for $\tilde g_{t,n}(x,\cdot,z)$ and notice that %$x$ is a competitor to get
\[
(-\bar C\,\norm+\tfrac{\sfd^2(x,y)}{2t})\wedge c\leq\tilde g_{t,n}(x,y,z) \leq \tilde g_{t,n}(x,x,z)  \leq f(x,x)+\tfrac1n\sfd^2(x,z)\leq {\rm diam}(B'')^2\leq 4r^2.
\]
In particular, $\tilde f(x,y)\leq 4r^2<c$, hence $\tilde f(x,y)=f(x,y)+\frac{\sfd^2(x,y)}{2t}$ and thus   $\tilde g_{t,n}(x,y,z)=  g_{t,n}(x,y,z)$. Since $\tilde g_{t,n}\leq  g_{t,n}$, we see  that $y$ minimizes  $ g_{t,n}(x,\cdot,z)$. The above also implies   $\sfd^2(x,y)\leq 2t(4r^2+\bar C\,\norm)\leq\frac{r^2}{36}$. Since $x\in B''$ this implies $y\in B'$, so that \eqref{eq:claimg} is proved.

Now for every $x\in B''$ we apply Theorem \ref{prop:varregf} to $\tilde f(x,\cdot)$ to find a Borel map $T(x,\cdot):\X\to \X$ so that for $\mm$-a.e.\ $z\in\X$ the point $T(x,z)$ is the only minimizer of $\tilde g_{t,n}(x,\cdot,z)$ and
\begin{equation}
\label{eq:abscont}
T(x,\cdot)_*\mm\ll\mm\qquad\forall x\in\X.
\end{equation}
By compactness we see that the multivalued map $(x,z)\mapsto K(x,z):=\{\text{minimizers of $\tilde g_{t,n}(x,\cdot,z)$}\}$ satisfies the assumption of the Borel selection theorem \cite[Theorem 6.9.3]{Bogachev07}, thus  there is $\tilde T:\X^2\to\X$ Borel so that $\tilde T(x,z)\in K(x,z)$ for any $x,z$. The uniqueness part of Theorem \ref{prop:varregf} for $\RCD(K,N)$ spaces forces $T(x,\cdot)=\tilde T(x,\cdot)$ $\mm$-a.e., hence \eqref{eq:abscont} gives $\tilde T_*(\mm\times\mm)\ll\mm$.
%The defining properties in   \eqref{eq:defacm} (and the choice $A:=\cont(u)\cup(\X\setminus U)$)  trivially give that
%{\rm Graph}(T)\cap [\cont(u)]^3\quad\subset\quad \{(x,y,z)\in\X^3\ :\ \text{$y$ minimizes }\tilde g_{t,n}(x,\cdot,z)\}
%that together with the fact that for any $x\in\X$ the minimizer of $\tilde g_{t,n}(x,\cdot,z)$ is unique for $\mm$-a.e.\ $z$ (by the uniqueness part of Theorem \ref{prop:varregf} in the case of $\RCD(K,N)$ spaces) imply:
%x_n\to x,\\
%\end{array}\right\}\qquad\Rightarrow\qquad T(x_n,\cdot)\to T(x,\cdot),\quad\mm-a.e..
%It follows that the map $\cont(u)\ni x\mapsto T(x,\cdot)\in L^0(\X;\X)$ is continuous, hence possibly modifying $T$ on a $\mm\times\mm$-negligible set and recalling that $\mm(U\setminus\cont(u))=0$ we can assume that $T:U\times\X\to\X$ is Borel. Thus \eqref{eq:abscont} and Fubini's theorem imply that  $T_*(\mm\restr{U}\times\mm)\ll\mm$.
Since $\mm(B'\setminus E)=0$ by assumption, we have  $(\mm\times\mm)\big((B''\times B)\cap \tilde T^{-1}(B'\setminus E)\big)=0$, thus from Fubini's theorem and  \eqref{eq:claimg} we see that   for $\mm$-a.e.\ $z\in B$ we have:  for $\mm$-a.e.\ $x\in B''$ the point $\tilde T(x,z)$ is in $E\subset B'$ and minimizes $g_{t,n}(x,\cdot,z)$. Since  $n$ ranges over $\N$,
this suffices to conclude.
\end{proof}
%For $\bar T>0$ as in   Lemma \ref{le:partle} above, $t\in(0,\bar T)$ and $\bar z_t\in B$ also as in  Lemma \ref{le:partle} above, we define the function $f_{t,n}:\X\to\R$ as
For $T̅>0$, $t∈(0,T̅)$ and $z̅_t∈B$  as in  Lemma \ref{le:partle} above, we define  $f_t,n:→$ as
\begin{equation}
\label{eq:defftn}
f_{t,n}(x):=\inf_{y\in\X}f(x,y)+\frac{\sfd^2(x,y)}{2t}+\frac1n\sfd^2(y,\bar z_t).
\end{equation}
A first, simple but useful, property of the $f_t,n$'s is:
\begin{proposition} With the same assumptions and notations introduced above the following holds. For $f_{t,n}$ defined as in \eqref{eq:defftn}   we have
\begin{equation}
\label{eq:subopt}
\bd f_{t,n}\restr{B''}\leq \frac {C(K^-R^2,N)}t\mm\qquad\forall t\in(0,\bar T).
\end{equation}
\end{proposition}
\begin{proof} By Lemma \ref{le:partle} and Definitions \eqref{eq:defdu}, \eqref{eq:deff} we know that for $\mm$-a.e.\ $x\in B''$ we have
\[
f_{t,n}(x)=\inf_{y\in B'\cap\cont(u)}f(x,y)+\frac{\sfd^2(x,y)}{2t}+\frac1n\sfd^2(y,\bar z).
\]
On the other hand, by Theorem \ref{thm:lapbd} and  Proposition \ref{prop:lapdu}  we know that
\[
\bd\Big(f(\cdot,y)+\frac{\sfd^2(\cdot,y)}{2t}+\frac1n\sfd^2(y,\bar z)\Big)\restr{B''}\leq \frac {C(K^-R^2,N)}t\mm\qquad\text{ for every $y\in B'$},
\]
%(here the term $\frac1n\sfd^2(y,\bar z)$ is constant - in particular the Laplacian comparison estimates for the distance play no role in handling it).
so that the conclusion  follows from   Lemma \ref{le:inflap2} (using the second part with $E:=\cont(u)$).
\end{proof}
To achieve \eqref{eq:perzzz} we now want to  find suitable upper bounds for $Δ̃f_t,n$. To this aim we  follow [111] and introduce, for fixed $x̅,y̅∈B'$, the auxiliary function $F_x̅,y̅:→$ defined as
\[
F_{\bar x,\bar y}(z):=\left\{
\begin{array}{ll}
\frac1{\sfd_u(\bar x,\bar y)}\Big(\sfd_u^2(z,\bar x)-\sfd_{u,p}^2(z)+\frac14\sfd_u^2(\bar x,\bar y)\Big),&\qquad\text{ if }\sfd_u(\bar x,\bar y)>0\text{ and }z\in B',\\
0,&\qquad\text{ otherwise,}
\end{array}\right.
\]
where $p∈$ is the midpoint of $u(x̅),u(y̅)$ and $_u,p:B'→$ is defined as
\[
\begin{split}
\sfd_{u,p}(x):=&\limi_{x'\to x\atop x'\in\cont(u)}\sfd_\Y(u(x'), p)\qquad\forall x\in B'.
%=&\phantom{\limi\limi\ \,}\ \sfd_\Y(u(x), p)\qquad\forall x\in B'\cap \cont(u).
\end{split}
\]
The relevance of these functions is due to the next two results (analogue of \cite[Proof of Lemma 6.7, Lemma 6.4]{ZZ18}). Notice that the geometry of the target space here enters in a crucial way: in the first of these lemmas we use the kind of `parallelogram inequality'
\begin{equation}
\label{eq:parineq}
(|ps|-|qr|)|qr|\geq \big(|pm_{qr}|^2 -|pq|^2-|m_{qr}q|^2\big)+\big(|sm_{qr}|^2 -|sr|^2-|m_{qr}s|^2\big),
\end{equation}
valid for any $p,q,r,s∈$, where  $m_qr$ is the midpoint of $q,r$ and for brevity we wrote $|pq|$ in place of $_(p,q)$ (see  \cite[Lemma 5.2]{ZZ18} for the proof), while in the second we use   property  \eqref{eq:convdacat3}.
\begin{lemma}
With the same assumptions and notations introduced above, we have
\begin{equation}
\label{eq:fsplit}
f(x,y)\leq f(\bar x,\bar y)+F_{\bar x,\bar y}(x)+F_{\bar y,\bar x}(y),\qquad\forall \bar x,\bar y\in\cont(u),\ \forall x,y\in \X,
\end{equation}
with equality for $x=\bar x$ and $y=\bar y$.
\end{lemma}
\begin{proof}
If $\sfd_u(\bar x,\bar y)=0$ or both $x$ and $y$ are not in $B'$ the claim is obvious. Then we observe that rearranging the terms in \eqref{eq:parineq} we easily get that   \eqref{eq:fsplit} holds for any $x,y,\bar x,\bar y\in\cont(u)$ with $\sfd_u(\bar x,\bar y)>0$ and $x,y\in B'$. The conclusion for $x,y\in B'$ arbitrary follows from the definition of $\sfd_u(x,y)$ and $\sfd_{u,p}(x)$. It remains to deal with the case  $\sfd_u(\bar x,\bar y)>0$  and $x\in B'$, $y\notin B'$: if this happens \eqref{eq:fsplit} reads as
\begin{equation}
\label{eq:perc}
-\bar C\,\norm\leq f(\bar x,\bar y)+F_{\bar x,\bar y}(x).
\end{equation}
To prove this notice that we know the validity of \eqref{eq:fsplit} for $x,\bar x,\bar y$ as in \eqref{eq:perc} and the choice $y:=\bar y$. With this choice we have $F_{\bar y,\bar x}(y)=0$ and thus \eqref{eq:perc} follows from \eqref{eq:fbdasotto}.
\end{proof}
\begin{lemma}\label{le:tdF}
With the same assumptions and notations introduced above, the following holds.

There is $E\subset B'$ Borel with $\mm(B'\setminus E)=0$ such that
\[
\tilde\Delta F_{x,y}(x)\leq 0,\qquad\forall x\in E,\ y\in B'.
\]
\end{lemma}
\begin{proof} For any $p\in\Y$  the identity $\sfd_{u,p}(\cdot)=\sfd_\Y(u(\cdot),p))$ holds on $\cont(u)$, hence $\mm$-a.e.\ on $B'$. Then   \eqref{eq:convdacat3} and Theorem \ref{thm:lapcomp} give $\bd(-\sfd_{u,p}^2)\restr{B'}\leq- 2(d+2)\e_2[u]^2\mm$ and thus  Lemma \ref{le:lapcomb2} implies
\[
\tilde\Delta (-\sfd_{u,p}^2)(x)\leq -2(d+2)\e_2[u]^2(x)\qquad\forall  x\in B'\cap\cont(u)\text{ Lebesgue point of $\e_2[u]^2$ and  $\forall p\in \Y$.}
\]
Similarly, for $x\in\cont(u)$ the identity $\sfd_u(x,y)=\sfd_\Y(u(x),u(y))$ holds for $\mm$-a.e.\ $y\in U$, thus for any such $x$ for which the limiting property  \eqref{eq:convenvar}  holds, since $\sfd_u(x,x)=0$ we have
\[
\tilde\Delta\sfd_u^2(x,\cdot)(x)= \lims_{t\downarrow0}\frac{\h_t(\sfd_u^2(x,\cdot))(x)}{t} = \lims_{t\downarrow0}\frac1t\int \sfd_\Y(u(x),u(y))\,\d\h_t\delta_x(y)\stackrel{\eqref{eq:convenvar} }=2(d+2)\e_2[u]^2(x).
\]

The conclusion  follows  from the subadditivity of $\tilde\Delta$ (which is obvious by definition).
\end{proof}

We then have the following crucial estimate (recall also the definition \eqref{eq:defdtns}):
\begin{proposition}[Laplacian bound for the $f_t$'s]\label{prop:lapft}
With the same assumptions and notations introduced above and for  $\bar T$ as given by Lemma \ref{le:partle}, we have
\[
\bd f_t\restr{B''}\leq -K\frac{D_t^2}{t}\mm\qquad\forall  t\in(0,\bar T).
\]
\end{proposition}
\begin{proof} We apply Lemma \ref{le:partle} with $E\subset B'$ chosen as the set of points for which the conclusions of Lemma \ref{le:tdF} hold and then define the functions $f_{t,n}$ accordingly.  By the very definitions of $f_t$ and $f_{t,n}$ and keeping also in mind that for $x\in B''$ a minimizer for $f_{t,n}$  exists in $B'$ (by Lemma \ref{le:partle}), we have that $f_{t,n}\to f_t$ uniformly on $B''$ as $n\to\infty$.

Hence taking into account the `bad' distributional bound \eqref{eq:subopt}, Lemma \ref{le:lapcomb2} and the stability property \eqref{eq:stablubloc},   to conclude it is sufficient to find $C(K^-R^2,N)>0$  such that
\begin{equation}
\label{eq:optb}
\tilde\Delta f_{t,n}\leq \frac{C(K^-R^2,N)}n-K\frac{D_t^2}{t}\qquad\mm-a.e.\   on\  B''\qquad\forall t\in(0,\bar T),\ n\in\N.
\end{equation}
%We fix $t\in(0,\bar t)$, recall Lemma \ref{le:partle} and  fix $\bar x\in \reg(u)\cap V_2$ such that $T_{t,n}(x)\in \reg(u)\cap V_1$. Notice that, crucially from  Lemma \ref{le:partle}, $\mm$-a.e.\ $\bar x\in V_2$ has this property.
By Lemma \ref{le:partle}  we know  that for $\mm$-a.e.\ $\bar x\in B''\cap E$ there is $\bar y\in E$ minimizer for $f_{t,n}(\bar x)$ (recall the definition \ref{eq:defftn}). Fix such $\bar x,\bar y$ and notice that  for every $x\in B''$ we have
\begin{equation}
\label{eq:bella}
\begin{split}
f_{t,n}(x)&=\inf_{y\in\Y}\Big( f(x,y)+\frac{\sfd^2(x,y)}{2t}+\frac{\sfd^2(y,\bar z)}n\Big),\\
%\text{(by Lemma \ref{le:partle})}\qquad &=\inf_{y\in V_1}\Big( f(x,y)+\frac{\sfd^2(x,y)}{2t}+\frac{\sfd^2(y,\bar z)}n\Big)\\
\text{(by \eqref{eq:fsplit})}\qquad &\leq   f(\bar x,\bar y)+F_{\bar x,\bar y}(x)+\inf_{y\in \Y}\Big(F_{\bar y,\bar x}(y)+\frac{\sfd^2(x,y)}{2t}+\frac{\sfd^2(y,\bar z)}n\Big)\\
&= f(\bar x,\bar y)+F_{\bar x,\bar y}(x)+Q_t\big(F_{\bar y,\bar x}(\cdot)+\tfrac{\sfd^2(\cdot,\bar z)}n\big)(x)
\end{split}
\end{equation}
with equality for $x=\bar x$. By the monotonicity of the heat flow (or the representation formula \eqref{eq:reprht})  it follows that for every $s>0$ we have
\[
\h_sf_{t,n}(\bar x)\leq f(\bar x,\bar y)+\h_sF_{\bar x,\bar y}(\bar x)+\h_s\big(Q_t\big(F_{\bar y,\bar x}(\cdot)+\tfrac{\sfd^2(\cdot,\bar z)}n\big)\big)(\bar x)
\]
hence subtracting \eqref{eq:bella} written for the equality case $x=\bar x$, after passing to the limit  we obtain:
\[
\tilde\Delta f_{t,n}(\bar x)\leq \tilde\Delta F_{\bar x,\bar y}(\bar x)+\tilde\Delta \big(Q_t\big(F_{\bar y,\bar x}(\cdot)+\tfrac{\sfd^2(\cdot,\bar z)}n\big)\big)(\bar x).
\]
Now observe that Lemma \ref{le:tdF}  and the fact that $\bar x\in E$ give $ \tilde\Delta F_{\bar x,\bar y}(\bar x)\leq 0$. Also, by inspecting the equality case $x=\bar x$ in \eqref{eq:bella} we see that the inf in the definition of $Q_t\big(F_{\bar y,\bar x}(\cdot)+\tfrac{\sfd^2(\cdot,\bar z)}n\big)(\bar x)$ is realized in $\bar y$, therefore  Lemma \ref{le:lappunt}, the subadditivity of $\tilde\Delta$ and the definition \eqref{eq:defdtns} give
\[
\begin{split}
\tilde\Delta \big(Q_t\big(F_{\bar y,\bar x}(\cdot)+\tfrac{\sfd^2(\cdot,\bar z)}n\big)\big)(\bar x)%&\leq \tilde\Delta  (F_{\bar y,\bar x}(\cdot)+\tfrac{\sfd^2(\cdot,\bar z)}n)(\bar y)-K\frac{\sfd^2(\bar x,\bar y)}{t}\\
&\leq \tilde\Delta  F_{\bar y,\bar x}(\bar y)+\tfrac1n \tilde\Delta {\sfd^2(\cdot,\bar z_t)}(\bar y)-K\frac{D_t^2(\bar x)}{t}.
\end{split}
\]
Now observe that since  $\bar y\in E$ we have $ \tilde\Delta  F_{\bar y,\bar x}(\bar y)\leq 0$ by Lemma \ref{le:tdF}. To bound the second term we apply Lemma \ref{le:lapcomb2}  with $g\equiv C$ and $ C=C(K^-R^2,N)$ given by Theorem \ref{thm:lapbd}  (recall that $\bar x.\bar y,\bar z_t\in B'\subset B_R(\hat x)$): we obtain  $\tilde\Delta \sfd^2(\cdot,\bar z_t)\leq C$ on $B'$. The conclusion \eqref{eq:optb} follows.
\end{proof}
We are now ready to prove the main result of the paper: % The idea to get Zhang-Zhu inequality is to divide by $t$ the bound \eqref{eq:quasiZZ} and let $t\downarrow0$. If we knew that $-\frac{f_t}t,\frac{D_t^2}{2t^2}$ both  converge to $\frac12\tilt^2(f)$ in $L^1(B)$ we would have more-or-less done, since $\tilt(f)$ is closely related to $\lip(u)$. The problems we encounter in proceeding in this way are that we don't really have integral estimates for $-\frac{f_t}t,\frac{D_t^2}{2t^2}$  (and for this reason the first part of the proof is dedicated to obtain rough uniform bounds) and the fact that we don't really have pointwise convergence to $\frac12\tilt^2(f)$, but only convergence of the $\lims$ as in Proposition xxx (for this reason
%In the course of the proof we shall use  the identity
%|\partial^-f_t|=|\d f_t|\qquad\mm-a.e.,
%valid for any $t>0$ (the descending slope was defined in \eqref{eq:defsl}). There are various ways of seeing this. Start noticing that since $f_t$ is Lipschitz, the slope is an upper gradient and thus the bound $|\d f_t|\leq |\partial^-f_t|$ holds $\mm$-a.e.. Then since $\lip(f_t)\geq |\partial^-f_t|$ is true everywhere, it suffices to show that $\lip(f_t)=|\d f_t|$ $\mm$-a.e.: this follows either by the celebrated results in [26], or via the arguments given in \cite[Prop.3.1]{GP16-2}  (that are based on the fact that  $f_t$ is a multiple of a $\sfd^2$-concave function in conjunction with the $\RCD(K,N)$ condition and the metric Brenier theorem \cite[Theorems 10.3, 10.4]{AmbrosioGigliSavare11}) or by \eqref{eq:radelip} applied  with $\Y=\R$ (recall also item $(ii)$ in Proposition \ref{prop:baseKS}).
\begin{theorem}[Zhang-Zhong-Zhu inequality and Lipschitz continuity of harmonic maps]\label{thm:main}
With the same assumptions and notations introduced above, the following holds.

The map $u:U\to\Y$ satisfies the Zhang-Zhong-Zhu inequality
\begin{equation}
\label{eq:ZZ}
\bd\frac{|\d u|^2}2\geq K|\d u|^2\mm\qquad on\ U
\end{equation}
and  has a representative, still denoted by $u$, that is locally Lipschitz continuous and satisfies
\begin{equation}
\label{eq:main}
\Lip(u\restr B)\leq \frac {C(K^-R^2,N)}r\inf_{\o\in\Y}\,\sqrt{\fint_{2B}\sfd_\Y^2(u(\cdot),\o)\,\d\mm}.
\end{equation}
\end{theorem}
\begin{proof}  \ \\
\st{Rough estimates} Let $\tfrac12 B:=B_{\tfrac r2}(\hat x)$, $\tfrac14 B:=B_{\tfrac r4}(\hat x)$. We claim that for some   $C, T>0$ we have
\begin{equation}
\label{eq:roughclaim}
\big\|\tfrac{f_t}t\big\|_{L^\infty(\frac12B)}+\big\|\tfrac{D_t^2}{t^2}\big\|_{L^\infty(\frac14B)}\leq C \qquad\forall t\in(0, T).
\end{equation}
%To this aim we let $\bar x\in B'''$, $\tilde B:=B_{\frac r{10}}(\bar x)$ and prove the above estimates on the ball $\tilde B$: the arbitrariness of $\bar x$ and the doubling property of $\RCD(K,N)$ spaces then gives the conclusion.
To see this, start noticing that $|\partial^-f_t|\leq\lip(f_t)$ holds everywhere, so by \eqref{eq:radelip} for $\Y=\R$ (or the analogous result in [26]) we get $|\partial^-f_t|\leq|\d f_t|$ $\mm$-a.e.. Then  Theorem \ref{thm:hlvar} and \eqref{eq:tilte2} give
\begin{equation}
\label{eq:l1contr}
\|f_t\|_{L^1( B)}=\|f_t-f_0\|_{L^1( B)}\leq \int_0^t\int _{ B}|\d f_t|^2+C\,\e_2^2[u]\,\d\mm\,\d t\leq tC+\int_0^t\int _{ B}|\d f_t|^2\,\d\mm\,\d t%,\qquad\forall t>0.
\end{equation}
for every $t\in(0,\bar T)$. Now let  $\varphi:=(1-\tfrac{1}{3r}\sfd(\cdot, B))^+$ and notice that
\begin{equation}
\label{eq:anchdop}
\int\varphi^2|\d f_t|^2\,\d\mm=\int\d f_t\cdot\d(f_t\varphi^2)-f_t\varphi\d f_t\cdot\d\varphi\,\d\mm \leq\int\d f_t\cdot\d(f_t\varphi^2)+\tfrac12\varphi^2|\d f_t|^2\,\d\mm+C,
\end{equation}
where in the last inequality we used \eqref{eq:bft22} and the trivial bound $|\d\varphi|\leq \tfrac{1}{3r}\nchi_{2 B}$. Now observe that Proposition \ref{prop:lapft} and the bounds  \eqref{eq:stimadqt2}, \eqref{eq:fbdasotto}   give
\begin{equation}
\label{eq:lapgrezzo}
\bd (-f_t)\restr {B''}\geq -(K^-\bar C\,\norm)\mm\qquad\forall  t\in(0,\bar T)
\end{equation}
and since $\varphi=0$ outside $B''$ and $f_t\varphi^2\leq 0$, by \eqref{eq:bft22} and \eqref{eq:anchdop} we get  the uniform bound
$\int_{ B}|\d f_t|^2\,\d\mm\leq C$ for any  $t\in (0,\bar T).$ We use this estimate in \eqref{eq:l1contr} to deduce $\|f_t\|_{L^1(B)}\leq tC$ and then apply Theorem \ref{thm:ellrt} using again   \eqref{eq:lapgrezzo} to get the first in \eqref{eq:roughclaim}.

For the second, we notice that arguing as for \eqref{eq:claimg} using the uniform bound \eqref{eq:bft22} we see that, possibly picking $\bar T $ smaller, for   $x\in \tfrac14B$ any minimizer $y_t$ for $f_t(x)$ belongs to $\tfrac12B$. Then the claim follows from what already proved and  the estimate \eqref{eq:stimadqt2}.

\st{Core argument} By the continuity in $t$ of $f_t(x)$ (item $(i)$ of Proposition \ref{prop:paseHL}) we see that $\lims_{t\downarrow0}\frac{-f_t}{t}=\inf_{n\in\N}\sup_{t\in \Q\cap(0,\frac1n)}\frac{-f_t}{t}$ pointwise on $\X$. Thus letting, for every $n\in\N$,  $\{t_{n,i}\}_{i\in\N}$ be an enumeration of $\Q\cap(0,\frac1n)$, by Proposition \ref{prop:dualtilt} we have that $\sup_i\frac{-f_{t_{n,i}}}{t_{n,i}}$ converges  to $\tfrac12\tilt(f)^2$ pointwise. Then by the first estimate in \eqref{eq:roughclaim} we see that the convergence is also in $L^1(\tfrac12B)$, hence   by a diagonalization argument we can find $N_n\in\N$ so that $\tilde f_n:=\frac{f_{t_{n,1}}}{t_{n,1}}\wedge \cdots\wedge \frac{f_{t_{n,N_n}}}{t_{n,N_n}}$ converge to $-\tfrac12\tilt(f)^2$ in $L^1(\tfrac12B)$, and then up to pass to a non-relabeled subsequence also pointwise $\mm$-a.e..

Let $A_{n,i}:=\{\frac{f_{t_{n,i}}}{t_{n,i}}=\tilde f_{n}\}\setminus \cup_{j<i}A_{n,j}\subset\X$, notice that for every $n\in\N$ the sets $(A_{n,i})_{i=1,\ldots,N_n}$ form a Borel partition of $\X$ and then put $\tilde g_n:=\sum_i\nchi_{A_{i,n}}\frac{D_{t_{n,i}}^2}{t_{n,i}^2}$. By the last claim in Proposition \ref{prop:dualtilt} and the construction we see that $\tilde g_n\to \tilt(f)^2$ $\mm$-a.e.\ on $B$, so that by the second estimate in \eqref{eq:roughclaim} the convergence is also in $L^1(\frac14B)$.

Now notice that Proposition \ref{prop:lapft} and the last claim in Lemma \ref{le:lapcomb2} (and  induction) give
\begin{equation}
\label{eq:perCacc}
\bd \tilde f_n\leq -K\tilde g_n\mm,\qquad\forall n\in\N,\qquad on \ \tfrac14B,
\end{equation}
so that the $L^1$ convergences, the uniform bounds \eqref{eq:roughclaim} and   the stability property \eqref{eq:stablubloc} imply
\begin{equation}
\label{eq:ZZlip}
\tfrac12\bd\lip^2(u)\geq  K\lip^2(u)\mm
\end{equation}
on $\frac14B$, having also  used \eqref{eq:liptilt}. By the arbitrariness of $B$ and the locality property \eqref{eq:localappr} we infer that \eqref{eq:ZZlip} holds on $U$. We now claim that $\lip^2(u)\in W^{1,2}_{loc}(U)$ and to this aim, by the arbitrariness of $B$  it is sufficient to prove that for any $V\subset\subset\frac14 B$ open we have  $\lip^2(u)\in W^{1,2}(\bar V,\sfd,\mm\restr V)$ (this is a direct consequence of the locality of the concepts of `being Sobolev' and `minimal weak upper gradients', see for instance \cite[Theorem 4.19]{AmbrosioGigliSavare11-2}). Thus let $\varphi\in\Lip_{\bs}(\X)$ be identically 1 on $V$ with support in $\frac14B$ and notice that \eqref{eq:perCacc} and the same computations in \eqref{eq:anchdop} give
\[
\begin{split}
\int\varphi^2|\d \tilde f_n|^2\,\d\mm&\leq \int -K\tilde g_n\varphi^2\tilde f_n +\tfrac12\varphi^2|\d \tilde f_n|^2+2\tilde f_n^2|\d\varphi|^2\,\d\mm.
\end{split}
\]
Since $\sup_n\|\tilde f_n\|_{L^\infty(\frac14B)},\|\tilde g_n\|_{L^\infty(\frac14B)}<\infty$ by \eqref{eq:roughclaim}, this is sufficient to derive a uniform bound on $\int\varphi^2|\d \tilde f_n|^2\,\d\mm$ and thus on $\int_{V}|\d \tilde f_n|^2\,\d\mm$ so that the lower semicontinuity of the Cheeger energy on the space $(\bar V,\sfd,\mm\restr V)$ gives our claim.

We can thus apply Theorem \ref{thm:ellrt} to $f:=\lip^2(u)\in W^{1,2}_{loc}(U)$ using \eqref{eq:ZZlip} and conclude that
\begin{equation}
\label{eq:lipu}
\|\lip(u)\|_{L^\infty(B'')}\leq C\sqrt{\fint_{B'}\lip^2(u)\,\d\mm}\stackrel{\eqref{eq:tilte2}}\leq C\sqrt{\fint_{B'}\e_2^2[u]\,\d\mm}\stackrel{\eqref{eq:invP}}\leq  \frac {\sf C}r\,\norm,
\end{equation}
for some ${\sf C}={\sf C}(K^-R^2,N)$. Now let $(p_n)$ be a countable set dense in the essential image of $u$, put $u_n:=\sfd_\Y(u(\cdot),p_n)$ and notice that - trivially -  we have $\lip(u_n)\leq \lip(u)$ everywhere and  by item $(ii)$ in Proposition \ref{prop:baseKS}  we know that $u_n\in W^{1,2}(U)$. Hence \eqref{eq:lipu} and the simple Lemma \ref{le:lipdf} below ensure that $|\d u_n|\leq \frac {\sf C}r \,\norm$ $\mm$-a.e.\ on $B''$. We can then apply the local Sobolev-to-Lipschitz property (see \cite[Proposition 1.10]{GV21}) and deduce that $u_n$ has a representative $\tilde u_n$ that is locally Lipschitz on $B''$ with local Lipschitz constant uniformly bounded by $\frac {\sf C}r\,\norm$. Since any two points $x,y\in B$ can be joined by a curve lying on $B''$ with length bounded by $3\sfd(x,y)$ (if a geodesic $\gamma$ from $x$ to $y$ is so that $\gamma_t\notin B''$ for some $t$, its length, and thus $\sfd(x,y)$, must be at least  $\tfrac{2r}3$, while a curve passing through the center has length $\leq 2r$), we conclude that $\Lip(\tilde u_n\restr{B})\leq \frac {\sf C}r\,\norm$.

Since this is true for any $n\in\N$, it follows  that for some $N\subset B$ Borel negligible we have
\[
|\sfd_\Y(u(x),p_n)-\sfd_\Y(u(y),p_n)|\leq \frac Cr\sfd(x,y)\,\norm\qquad\forall x,y\in B\setminus N.
\]
Taking the supremum in $n$ we conclude that the restriction of $u$ to $B\setminus N$ is $\frac {\sf C}r\,\norm$-Lipschitz, so that $u:U\to\Y$ has a locally Lipschitz representative satisfying \eqref{eq:main}, as desired.

Finally, \eqref{eq:ZZ} follows from \eqref{eq:ZZlip} taking into account the identity \eqref{eq:radelip}.
\end{proof}
\begin{lemma}\label{le:lipdf}
Let $(\X',\sfd',\mm')$ be a complete separable metric space equipped with a non-negative and non-zero Radon measure finite on bounded sets. Let $U\subset \X'$ be open, $f\in W^{1,2}(U)$ and assume that $\lip(f)\in L^2(U)$. Then $|\d f|\leq \lip(f)$ $\mm$-a.e.\ on $U$.
\end{lemma}
\begin{proof}
Multiplying $f$ by a sequence of Lipschitz cut-off functions with support in $U$ we can, and will, reduce to the case $U=\X'$. Observe that for any curve $\gamma$ and $t\in[0,1]$ for which the metric speed $|\dot\gamma_t|$ of $\gamma$ exists and $\lip f(\gamma_t)$ is finite we have $\lims_{h\to 0}\frac{|f(\gamma_{t+h})-f(\gamma_t)|}{|h|}\leq \lip f(\gamma_t)|\dot\gamma_t|$. Now let $\ppi$ be a test plan and notice  that (by    \cite[Theorem 2.1.21]{GP19}) for $\ppi$-a.e.\ $\gamma$ we have $f\circ\gamma\in W^{1,1}(0,1)$, thus  by what just proved it is easy to see that its distributional derivative is bounded in modulus by  $\lip f(\gamma_t)|\dot\gamma_t|$. By    \cite[Theorem 2.1.21]{GP19} again this is sufficient to  conclude.
\end{proof}

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\def\cprime{$'$} \def\cprime{$'$}
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\end{document}