text:
\chapter{Non-local Dirichlet spaces with heat kernel estimates}\label{Ch-non-loc}

Let $(X,d,\mu)$ be a metric measure space. We assume that $B(x,r):=\{y\in X\mid d(x,y)<r\}$ has compact closure for any $x\in X$ and any $r\in(0,\infty)$, and that $\mu$ is Ahlfors $d_H$-regular, i.e.\ there exist $c_{1},c_{2},d_{H}\in(0,\infty)$ such that $c_{1}r^{d_{H}}\leq\mu\bigl(B(x,r)\bigr)\leq c_{2}r^{d_{H}}$ for any $r\in\bigl(0,+\infty\bigr)$. Furthermore, we assume that on $(X,\mu)$ there is a heat kernel $p_t(x,y)$.
If the metric measure space $(X,d,\mu)$ satisfies a certain chain condition and all its metric balls are relatively compact, it is known from~\cite[Theorem 4.1]{GK08} that the sub-Gaussian heat kernel estimates~\eqref{eq:subGauss-upper} will force the associated Dirichlet form to be local. If the Dirichlet form is non-local, the mentioned result tells us that the only possible heat kernel estimates are
\def\Phihke#1{\left(1+#1\right)^{-d_H-d_W}}
\begin{equation}\label{eq:HKE-non-loc}
c_{5}t^{-\frac{d_{H}}{d_{W}}}
\Phihke{ c_{6} \frac{d(x,y)}{t^{1/d_{W}}}}
\le
p_{t}(x,y)
\le
c_{3}t^{-\frac{d_{H}}{d_{W}}}
\Phihke{ c_{4} \frac{d(x,y)}{t^{1/d_{W}}}},
\end{equation}
with $0<d_W\leq d_H+1$. We refer to~\cite{GK08,GHL:TAMS2003,GHL14,GrigHuHu17,GrigKajino17,GrigLiu15,BeGriPittVess14} for further details and results in this direction. The present chapter is devoted to the study of the spaces $\mathbf{B}^{p,\alpha}(X)$ in this framework of non-local Dirichlet spaces with heat kernel estimates~\eqref{eq:HKE-non-loc}. To compare both situations more easily, we will follow the same structure as the previous chapter about the local case.

\section{Metric characterization of  Besov spaces}\label{S:MCB_nl}
We start by relating our Besov space $\mathbf{B}^{p,\alpha}(X)$ with other Besov type spaces that have been considered in the literature in the non-local context; see e.g.~\cite{Gri,Str03}.

For $\alpha\in [0,\infty)$ and $p\in [1,\infty)$, recall the Besov  seminorm
\[
\| f \|_{p,\alpha}= \sup_{t >0} t^{-\alpha} \left( \int_X \int_X |f(x)-f(y) |^p p_t (x,y) d\mu(x) d\mu(y) \right)^{1/p}
\]
and the Besov space
\[
\mathbf{B}^{p,\alpha}(X)=\{ f \in L^p(X,\mu)\, :\,
 \| f \|_{p,\alpha} <+\infty \}.
\]

Recall also the metric Besov type seminorm defined in \eqref{eq:Besov-seminorm-r}. That is, for $f\in L^p(X,\mu)$,
\[
N^{\alpha}_{p}(f,r):=\frac{1}{r^{\alpha+d_{H}/p}}\biggl(\iint_{\Delta_r}|f(x)-f(y)|^{p}\,d\mu(x)\,d\mu(y)\biggr)^{1/p}.
\]
Here, as in the previous chapters, for $r>0$ the set $\Delta_r$ denotes the collection of all
$(x,y)\in X\times X$ with $d(x,y)<r$.
For any $q\in [1,\infty)$ such that $p\le q$, define
\[
N_{p,q}^{\alpha}(f):=\brak{\int_0^\infty \brak{N_p^{\alpha}(f,r)}^{q}\, \frac{dr}{r}}^{1/q},
\]
and for $q=\infty$, define
\[
N_{p,\infty}^{\alpha}(f):=\sup_{r>0} N_p^{\alpha}(f,r).
\]
The Besov space is then defined by
\[
\mathfrak{B}^{\alpha}_{p,q}(X)=\{u\in L^p(X)\, :\, N_{p,q}^{\alpha}(f)<\infty\}.
\]
We determine first which of the these spaces coincides with $\bm{B}^{p,\alpha}(X)$ when $p\in[1,\infty)$ and $\alpha\in[0,1/p)$.

\begin{theorem}\label{Besov non-local}
Let $p \ge 1$ and $ 0\le \alpha < 1/p$. We have $\mathfrak{B}^{\alpha d_W}_{p,\infty}(X)=\mathbf{B}^{p,\alpha}(X)$ and there exist two constants $c,C>0$ (depending on $p,\alpha, d_H, d_W$) such that
\[
c\,N_{p,\infty}^{\alpha d_W}(f) \le \| f \|_{p,\alpha} \le C\,N_{p,\infty}^{\alpha d_W}(f).\footnote{Here and after, the letters $c,C$ denote constants independent of important parameters which may change at each circumstance.}
\]

\end{theorem}

\begin{proof}
We follow the proof for Theorem \ref{Besov characterization} by  first proving the lower bound. By the upper bound in \eqref{eq:HKE-non-loc}, one has for $r,t>0$
\begin{align*}
& \int_X \int_X |f(x)-f(y) |^p p_t (x,y) d\mu(x) d\mu(y)
\\ \ge &
\frac{c}{t^{d_H/d_W}}\int_X \int_{B(y,r)}\frac{|f(x)-f(y) |^p}{\brak{1+\frac{d(x,y)}{t^{1/d_W}}}^{d_H+d_W}} d\mu(x) d\mu(y)
\\ \ge &
\frac{c}{t^{d_H/d_W}}\frac{1}{\brak{1+\frac{r}{t^{1/d_W}}}^{d_H+d_W}}  \int_X \int_{B(y,r)} |f(x)-f(y) |^p d\mu(x) d\mu(y)
\\ \ge &
\frac{c}{t^{d_H/d_W}} \frac{ r^{\alpha d_W p+d_{H}} }{\brak{1+\frac{r}{t^{1/d_W}}}^{d_H+d_W}} N^{\alpha d_W}_{p}(f,r)^p
\end{align*}
Choosing $r=t^{1/d_W}$, it easily follows that
\[
\| f \|_{p,\alpha} \ge c \sup_{t>0} N^{\alpha d_W}_{p}(f,t^{1/d_W})=c\,N_{p,\infty}^{\alpha d_W}(f).
\]
We now turn to the upper bound. We set $A(t)$ and $B(t)$ as in~\eqref{E:A(t)}, respectively~\eqref{E:B(t)},

so that $ \int_X \int_X |f(x)-f(y) |^p p_t (x,y) d\mu(x) d\mu(y) =A(t)+B(t)$.
By \eqref{eq:HKE-non-loc} and the inequality
$|f(x)-f(y)|^{p}\leq 2^{p-1}(|f(x)|^{p}+|f(y)|^{p})$,
\begin{align}
A(t)\notag
&\leq
\frac{c_{3}}{t^{d_{H}/d_{W}}}\int_{X}\int_{X\setminus B(y,r)} \biggl(1+c_{4}\frac{d(x,y)}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}} \cdot 2^{p}|f(y)|^{p}\,d\mu(x)\,d\mu(y)
\notag\\ &=
\frac{2^{p}c_{3}}{t^{d_{H}/d_{W}}}\sum_{k=1}^{\infty}\int_{X}\int_{B(y,2^{k}r)\setminus B(y,2^{k-1}r)}\biggl(1+c_{4}\frac{d(x,y)}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}}|f(y)|^{p}\,d\mu(x)\,d\mu(y)\notag\\
&\leq
\frac{2^{p}c_{3}}{t^{d_{H}/d_{W}}}\sum_{k=1}^{\infty}\int_{X}\mu\bigl(B(y,2^{k}r)\bigr)\biggl(1+c_{4}\frac{2^{(k-1)}r}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}}|f(y)|^{p}\,d\mu(y)
\notag\\&\leq
\frac{2^{p}c_{3}}{t^{d_{H}/d_{W}}}\sum_{k=1}^{\infty}c_{2}r^{d_{H}}2^{kd_{H}}\|f\|_{L^{p}}^{p}\biggl(1+c_{4}\frac{2^{(k-1)}r}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}}
\notag\\ &=
C \|f\|_{L^{p}}^{p} \sum_{k=1}^{\infty} \Bigl(\frac{2^{k-1}r}{t^{1/d_W}}\Bigr)^{d_{H}}\biggl(1+\frac{c_{4}}{2}\frac{2^{k}r}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}}
\notag\\ &\leq
C\|f\|_{L^{p}}^{p} \sum_{k=1}^{\infty}\int_{\frac{2^{k-1} r}{t^{1/d_{W}}}}^{\frac{2^k r}{t^{1/d_{W}}}} s^{d_{H}}\biggl(1+\frac{c_{4}}{2}s\biggr)^{-d_H-d_{W}}\frac{ds}{s}
\notag\\ &\le
C\|f\|_{L^{p}}^{p} \frac{t^{\alpha p}}{r^{\alpha d_W p}} \int_{0}^{\infty}s^{\alpha d_W p+d_{H}-1} \biggl(1+\frac{c_{4}}{2}s\biggr)^{-d_H-d_{W}}\,ds
\notag\\ &\leq
C \frac{t^{\alpha p}}{r^{\alpha d_W p}} \|f\|_{L^{p}}^{p},
\label{eq:nonLocalHKBesov-norms-upper-proof1}
\end{align}
where the last integral is bounded as long as $\alpha \in [0,1/p)$.

On the other hand, for $B(t)$, by \eqref{eq:HKE-non-loc} we have
\begin{align}
B(t)\notag
&\leq
\frac{c_{3}}{t^{d_{H}/d_{W}}}\int_{X}\int_{B(y,r)} \biggl(1+c_{4}\frac{d(x,y)}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}} |f(x)-f(y)|^{p}\,d\mu(x)\,d\mu(y)\notag
\\&\leq
\frac{c_{3}}{t^{d_{H}/d_{W}}}\sum_{k=1}^{\infty}\int_{X}\int_{B(y,2^{1-k}r)\setminus B(y,2^{-k}r)} \biggl(1+c_{4}\frac{d(x,y)}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}} |f(x)-f(y)|^{p}\,d\mu(x)\,d\mu(y)\notag
\\&\leq
\frac{c_{3}}{t^{d_{H}/d_{W}}}\sum_{k=1}^{\infty}\int_{X}\int_{B(y,2^{1-k}r)} \biggl(1+c_{4}\frac{2^{-k}r}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}} |f(x)-f(y)|^{p}\,d\mu(x)\,d\mu(y)\notag
\\&\leq
c_{3} \sum_{k=1}^{\infty}\frac{(2^{1-k}r)^{\alpha d_W p+d_{H}}}{t^{d_{H}/d_{W}}} \biggl(1+c_{4}\frac{2^{-k}r}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}} \frac{1}{(2^{1-k}r)^{\alpha d_W p+d_{H}}}\int_{X}\int_{B(y,2^{1-k}r)}|f(x)-f(y)|^{p}\,d\mu(x)\,d\mu(y)\notag
\\&\leq
C t^{\alpha p}\sup_{s\in(0,r]}N^{\alpha d_W}_{p}(f,s)^{p}\sum_{k=1}^{\infty}\Bigl(\frac{2^{-k}r}{t^{1/d_W}}\Bigr)^{d_H+\alpha d_W p} \biggl(1+c_{4}\frac{2^{-k}r}{t^{1/d_{W}}}\biggr)^{-d_H-d_{W}} \notag
\\&\leq
C t^{\alpha p}\sup_{s\in(0,r]}N^{\alpha d_W}_{p}(f,s)^{p},
\label{eq:nonLocalHKBesov-norms-upper-proof2}
\end{align}
where the last series converges for all $\alpha\in [0,1/p)$.

Combining \eqref{eq:nonLocalHKBesov-norms-upper-proof1} and \eqref{eq:nonLocalHKBesov-norms-upper-proof2}, we conclude that by taking $r\to \infty$,
\[
\sup_{t>0} \frac{1}{t^{\alpha p}} \int_X \int_X |f(x)-f(y) |^p p_t (x,y) d\mu(x) d\mu(y) \le C \sup_{s>0}N^{\alpha d_W}_{p}(f,s)^{p},
\]
and hence $\| f \|_{p,\alpha}\le CN_{p,\infty}^{\alpha d_W}(f)$.
The proof is thus complete.
\end{proof}

We can also establish several inclusions that cover the case when $\alpha\in[1/p,\infty)$.

\begin{proposition}\label{P:Inclusions_Besov_nl}
Let $p\in[ 1,\infty)$.  We have
\begin{enumerate}
\item if $\alpha \in [0,1/p]$, then $\mathfrak{B}^{\alpha d_W}_{p,p}(X) \subset \mathbf{B}^{p,\alpha }(X)$.
\item  if $\alpha \in [1/p, \infty)$, then $\mathbf{B}^{p,\alpha}(X) \subset \mathfrak{B}^{\alpha d_W}_{p,p}(X)$.
\end{enumerate}
In particular, $\mathbf{B}^{p,1/p}(X)=\mathfrak{B}^{d_W/p}_{p,p}(X)$.
\end{proposition}
\begin{proof}
We first show $(1)$. Fixing a decreasing geometric sequence $\{r_k\}_{k\in \Z}$, we can write
\[
\sum_{k\in \Z}  N_p^{\alpha}(f,r_k)^p \simeq \int_0^{\infty} N_p^{\alpha}(f,r)^p \frac{dr}{r}.
\]
Using the upper bound in \eqref{eq:HKE-non-loc}, for any $\alpha \in [0,1/p]$ we have that
\begin{align*}
&\int_X \int_X |f(x)-f(y) |^p p_t (x,y) d\mu(x) d\mu(y)
\\ &=
\sum_{k\in \Z} \int_X \int_{B(x,r_k)\setminus B(x,r_{k+1})} |f(x)-f(y) |^p p_t (x,y) d\mu(x) d\mu(y)
\\ &\le
\sum_{k\in \Z}  c_{3}t^{-\frac{d_{H}}{ d_{W}}}
\brak{1+c_4\frac{r_{k+1}}{t^{1/d_W}}}^{-d_H-d_W}\int_X \int_{B(x,r_k)\setminus B(x,r_{k+1})} |f(x)-f(y) |^p d\mu(x) d\mu(y)
 \\ &=
\sum_{k\in \Z}  c_{3} t\brak{t^{1/d_W}+c_4 r_{k+1}}^{-d_H-d_W}\int_X \int_{B(x,r_k)\setminus B(x,r_{k+1})} |f(x)-f(y) |^p d\mu(x) d\mu(y)
 \\ &\le
 \sum_{k\in \Z}  c_{3} t \brak{t^{1/d_W}}^{-d_W+\alpha p d_W}\brak{c_4 r_{k+1}}^{-d_H-\alpha p d_W}\int_X \int_{B(x,r_k)\setminus B(x,r_{k+1})} |f(x)-f(y) |^p d\mu(x) d\mu(y)
 \\ &\le
C\sum_{k\in \Z}  t^{\alpha p} N_p^{\alpha d_W}(f,r_{k+1})^p
\le
C t^{\alpha p} N_{p,p}^{\alpha d_W}(f)^p.
\end{align*}
This implies
\[
\| f \|_{p,\alpha } \le C N_{p,p}^{\alpha d_W}(f),
\]
and hence $\mathfrak{B}^{\alpha d_W}_{p,p}(X) \subset \mathbf{B}^{p,\alpha }(X)$.

Next we prove $(2)$. Notice that for any $\delta\in (0,1)$ and $d_W'=d_W/\delta$, there exist two constants $c,C$ such that
\[
p_t(x,y) \ge
c \int_{t^{1/\delta}}^{\infty} \frac{t}{s^{1+\delta}} s^{-d_H/d_W'} \exp\brak{-C\brak{\frac{d(x,y)^{d_W'}}{s}}^{1/(d_W'-1)}}ds.
\]
See for instance \cite[Section 5.4]{Gri}. Following the proof in Theorem~\ref{Besov characterization}, we obtain for $\alpha\ge 1/p $,
\begin{align*}
&  \int_X \int_X |f(x)-f(y) |^p p_s (x,y) d\mu(x) d\mu(y)
\\ &  \ge
c \int_{t^{1/\delta}}^{\infty} \frac{t}{s^{1+\delta}} s^{-d_H/d_W'}  \int_X \int_{B\brak{y,s^{1/d_W'}}} |f(x)-f(y) |^p d\mu(x) d\mu(y)  ds
\\ &  \ge
c \int_{t^{1/\delta d_W'}}^{\infty} \frac{t}{r^{\delta d_W'-\alpha d_W p}} N_p^{\alpha d_W}(f,r)^p \frac{dr}{r}
\\&  \ge
c t^{\alpha p} \int_{t^{1/d_W}}^{\infty} N_p^{\alpha d_W}(f,r)^p \frac{dr}{r},
\end{align*}
where we get the second inequality by changing the variable. It follows that
\[
\| f \|_{p,\alpha}\ge c\, \sup_{t>0} \brak{\int_{t^{1/d_W}}^{\infty} N_p^{\alpha d_W}(f,r)^p \frac{dr}{r}}^{1/p} = c\,N_{p,p}^{\alpha d_W}(f),
\]
and we conclude $(2)$.

\end{proof}

We finish this section by giving an alternative, equivalent expression for the seminorm $\|f\|_{1,1}$ that will turn useful later on.

\begin{lemma}\label{lem:BesovEquivSN}
Let $f\in \bfB^{1,1}(X)$. Then the seminorm $\|f\|_{1,1}$ is equivalent to
\[
W_{1,1}(f):=\int_X\int_X \frac{|f(x)-f(y)|}{d(x,y)^{d_H+d_W}} d\mu(x) d\mu(y).
\]
\end{lemma}
\begin{proof}
We first show that $\|f\|_{1,1} \le CW_{1,1}(f) $, which follows directly from the heat kernel estimate \eqref{eq:HKE-non-loc}. Indeed, for every $t>0$,
\begin{align*}
&\frac1t \int_X\int_X |f(x)-f(y)|p_t(x,y) d\mu(x) d\mu(y)
\\ &\le
\frac{c_3}{t} \int_X\int_X t^{-\frac{d_H}{d_W}} \brak{1+c_4\frac{d(x,y)}{t^{1/d_W}}}^{-d_H-d_W} |f(x)-f(y)|d\mu(x) d\mu(y)
\\ &=
\frac{c_3}{t}\int_X\int_X \frac{t}{\brak{t^{1/d_W}+c_4d(x,y)}^{d_H+d_W}} |f(x)-f(y)|d\mu(x) d\mu(y)
\le CW_{1,1}(f).
\end{align*}
Next we show the reverse inequality. For every $t>0$,
\begin{align*}
&\int_X\int_X \frac{c_5}{\brak{t^{1/d_W}+c_6d(x,y)}^{d_H+d_W}} |f(x)-f(y)|d\mu(x) d\mu(y)
\\ &\le
\frac{1}{t}\int_X\int_X  |f(x)-f(y)|p_t(x,y) d\mu(x) d\mu(y) \le \|f\|_{1,1}.
\end{align*}
By Lebesgue's dominated convergence theorem, we have $W_{1,1}(f) \le C \|f\|_{1,1}$. This completes the proof.
\end{proof}

\section{Coarea type estimates}\label{S:coarea_nl}
From Theorem~\ref{Besov non-local} we can derive as in the local case several estimates involving the fractional perimeter in the non-local setting, which in this case include the case when $d_W<1$.
As in Chapter~\ref{LDsGHKU}, given a function $u\in L^1(X,\mu)$ we denote $E_s(u)=\{x\in X,\;u(x)>s\}$, for $E\subset K$ denote by $(E)_r$ the $r$-neighborhood of $E$ and by $\partial E$ its topological boundary.

\begin{theorem}\label{T:non-local_coarea}
Let $X$ be Ahlfors $d_H$-regular with a heat kernel that satisfies the upper estimate in~\eqref{eq:HKE-non-loc}. For $u\in L^1(X,\mu)$, assume that there is $0<\alpha\leq \min\{d_H/d_W,1\}$ and $R>0$ such that
\begin{equation}\label{extended-inner-boundary-estimate-non-loc}
h_R(u,s) = \sup_{r \in(0,R]} \frac1{r^{\alpha d_W}} \mu \bigl\{ x\in E_s(u): d(x, X\setminus E_s(u)) < r  \bigr\}
\end{equation}
is in $L^1 (\mathbb{R},ds)$. Then, $u \in \mathbf{B}^{1,\alpha}(X)$ and there exist constants $C_1,C_2>0$ independent of $u$ such that
\begin{equation}\label{E:non-local_coarea}
\| u \|_{1, \alpha} \le C_1 R^{-\alpha}\|u\|_{L^1(X,\mu)}+  C_2 \int_{\mathbb{R}} h(u,s) ds.
\end{equation}
In particular, for any set $E\subset X$ we have
\begin{equation}\label{E:non-local_coarea_1E}
\| \bm{1}_E \|_{1, \alpha} \le C_1 R^{-\alpha}\mu(E)+  C_2 \sup_{r \in(0,R]} \frac1{r^{\alpha d_W}} \mu \bigl\{ x\in E\,\colon\,d(x, X\setminus E) < r  \bigr\}.
\end{equation}
\end{theorem}
\begin{proof}
This is analogous to the local case and we only give a sketch of the proof. Applying Theorem~\ref{T:non-local_coarea} with $f=\bm{1}_{E_s(u)}$ and $p=1$ we have that, for any $R>0$ and $s\in\mathbb{R}$,
\begin{multline}\label{E:non-local-coarea01}
t^{-\alpha}\int_{X\times X}|\bm{1}_{E_s(u)}(y)-\bm{1}_{E_s(u)}(x)|\,p_t(x,y)\,d\mu(y)d\mu(x)\\
\leq C_1 R^{-\alpha d_W}\|\bm{1}_{E_s(u)}\|_{L^1(X,\mu)}+C_2\sup_{r\in(0,R]}N_1^{\alpha d_W}(\bm{1}_{E_s(u)},r).
\end{multline}
Moreover, in view of definition~\eqref{eq:Besov-seminorm-r} and applying the Ahlfors $d_H$-regularity of the space $X$ we get
\begin{align*}
N_1^{\alpha d_W}(\bm{1}_{E_s(u)},r)&=\frac{1}{r^{\alpha d_W+d_H}}\mu\otimes\mu\big(\{(x,y)\in E_s(u)\times (X\setminus E_s(u))\,\colon\,d(x,y)<r\}\big)\\
&\leq \frac{1}{r^{\alpha d_W+d_H}}\int_{\{x\in E_s(u)\,\colon\,d(x,X\setminus E_s(u)\}}\mu\big(B(y,r)\big)\,d\mu(x)\\
&\leq \frac{1}{r^{\alpha p}}\mu\big(\{x\in E_s(u)\,\colon\,d(x,X\setminus E_s(u)\}\big)=h_R(u,s).
\end{align*}
Plugging this into~\eqref{E:non-local-coarea01}, integrating both sides of over $\mathbb{R}$ and noting that $\int_{\mathbb{R}}\|\bm{1}_{E_s(u)}\|_{L^1(X,\mu)}=\|u\|_{L^1(X,\mu)}$ finally yields~\eqref{E:non-local_coarea}. The last assertion follows from the fact that $E_s(\bm{1}_E)=E$ if $s\in [0,1)$ and zero otherwise.
\end{proof}
\begin{remark}
The previous theorem could be stated with $0<\alpha<1$. However, if $\alpha>d_H/d_W$, there will be no function $h_R(u,s)$ satisfying the required conditions. Analogous reasons explain this restriction for $\alpha$ in all subsequent statements.
\end{remark}
We can now derive the same corollaries as in the local case, since these do not involve heat kernel estimates. We restate them for completeness and refer for proof to the previous chapter.
\begin{corollary}\label{C:non-local-ahlfors-coarea-1E-ulc}
Let $X$ be a uniformly locally connected space with a heat kernel that satisfies the upper heat kernel estimate~\eqref{eq:HKE-non-loc}. If $E \subset X$ has finite measure and for some $0<\alpha\leq \min\{\frac{d_H}{d_W},1\}$
\begin{equation*}
\sup_{r\in(0,R]} \frac1{r^{\alpha d_W}} \mu\bigl\{ x\in E: d(x,\partial E) <r\bigr\}<\infty,
\end{equation*}
then $\bm{1}_E \in \mathbf{B}^{1,\alpha}(X) $ and
\begin{equation*}
\| \bm{1}_E \|_{1, \alpha} \le C_1 R^{-\alpha d_W} \mu(E) + C_2  \sup_{r\in(0,R]} \frac1{r^{\alpha d_W}} \mu\bigl\{ x\in E: d(x,\partial E) <r\bigr\}.
\end{equation*}
\end{corollary}

\begin{corollary}\label{C:non-local-JKLM}
Let $X$ be a uniformly locally connected space with a heat kernel that satisfies the upper estimate~\eqref{eq:HKE-non-loc} and let $u\in L^1(X,\mu)$. Assume that $\partial E_s(u)$ is finite and $s\mapsto  | \partial E_s(u) |$ is in $L^1(\mathbb{R})$, where $| \partial E_s(u) |$ denotes the cardinality of $\partial E_s(u)$. Then, if $d_H\leq d_W$, we have $u \in \mathbf{B}^{1,\frac{d_H}{d_W}}(X)$ and there exists a constant $C>0$ independent of $u$ such that
\begin{equation*}\label{E:non-local-JKLM}
 \| u \|_{1, d_H/d_W} \le C \int_{\mathbb{R}} | \partial E_s(u) | ds.
\end{equation*}
In particular, if $E \subset X$ is a set of finite measure whose boundary is finite, then $\bm{1}_E \in \mathbf{B}^{1,\frac{d_H}{d_W}}(X)$ and
\[
\| \bm{1}_E \|_{1, d_H/d_W} \le C  | \partial E |.
\]
\end{corollary}
In the same fashion, the analogous of Proposition~\ref{P:BesovLB} will hold when the level sets $E_s(u)$ are uniformly porous at large scales.
\section{Sets of finite perimeter and fractional content of boundaries}\label{S:FP_nl}

\subsection{Characterizations of the sets of finite perimeter}
In the previous sections we have provided the non-local counterpart of all ingredients that lead to the characterization result provided by Theorem~\ref{T:FP_char} in the local case. With the corresponding restriction on the parameter $\alpha$, the same result will hold in this setting.

\begin{theorem}\label{T:FP_char_non-local}
Under the assumptions of Theorem~\ref{Besov non-local}, for a bounded measurable set $E \subset X$  and $0<\alpha \leq\min\{1,d_H/d_W\}$

 we consider the following properties:
\begin{enumerate}
\item\label{FP_char1_nl} $\bm{1}_E \in \mathbf{B}^{1,\alpha}(X)$;
\item\label{FP_char2_nl} $\displaystyle \sup_{r>0} \frac{1}{r^{\alpha d_W+d_{H}}} (\mu \otimes \mu) \big(\{ (x,y) \in E \times E^c\, :\,
 d(x,y) < r\}\big) <+\infty$;
\item\label{FP_char3_nl} $\displaystyle \limsup_{r \to 0^+} \frac{1}{r^{\alpha d_W+d_{H}}} (\mu \otimes \mu) \big(\{ (x,y) \in E \times E^c\,
:\, d(x,y) < r\}\big) <+\infty$;
\item\label{FP_char4_nl} $\displaystyle \limsup_{r\to 0^+}\frac{1}{r^{\alpha d_W}}\mu\big(\{x\in E~\colon~d(x,E^c)<r\}\big)<+\infty$;
\item\label{FP_char5_nl} $\displaystyle \limsup_{r\to 0^+}\int_{\{x\in E\colon d(x,E^c)<r\}}\int_{B(x,r)\cap E^c}\frac{1}{d(x,y)^{d_H+\alpha d_W}}d\mu(y)d\mu(x)<+\infty$;
\item\label{FP_char6_nl} $\displaystyle \int_E \int_{E^c} \frac{1}{d(x,y)^{d_H+\alpha d_W}}d\mu(y)d\mu(x) <+\infty$.
\end{enumerate}
Then, the following relations hold:
\begin{enumerate}[label=(\roman*)]
\item \eqref{FP_char1_nl}~$\Leftrightarrow$~\eqref{FP_char2_nl}~$\Leftrightarrow$~\eqref{FP_char3_nl};

moreover there exist constants $c,C >0$ independent of $E$ such that
\begin{multline*}
c \sup_{r>0} \frac{1}{r^{\alpha d_W+d_{H}}} (\mu \otimes \mu) \big(\{ (x,y) \in E \times E^c\,
:\,
 d(x,y) < r\} \big) \\
 \le  \| \bm{1}_E \|_{1,\alpha} \le C \sup_{r>0} \frac{1}{r^{\alpha d_W+d_{H}}} (\mu \otimes \mu) \big(\{ (x,y) \in E \times E^c\,
 :\, d(x,y) < r\}\big);
\end{multline*}
\item \eqref{FP_char4_nl}~$\Rightarrow$~\eqref{FP_char5_nl}~$\Rightarrow$~\eqref{FP_char1_nl};
\item \eqref{FP_char6_nl}~$\Rightarrow$~\eqref{FP_char1_nl};
\item If in addition $E$ is porous, c.f.~\eqref{D:porous}, then \eqref{FP_char1_nl}~$\Rightarrow$~\eqref{FP_char4_nl};
\end{enumerate}
\end{theorem}
The proofs follow verbatim, replacing Theorem~\ref{Besov characterization} by Theorem~\ref{Besov non-local} and Theorem~\ref{ahlfors-coarea} by Theorem~\ref{T:non-local_coarea}. Notice that Proposition~\ref{P:local_geom_cond} is does not actually involve any heat kernel estimates. Assuming uniformly locally connectedness on the space, which includes the case of $X$ being geodesic, we obtain as in Corollary~\ref{C:ahlfors-coarea-1E-ulc} the corresponding characterization of sets with finite $\alpha$-perimeter in terms of inner Minkowski contents.

\subsection{Further results}
\begin{theorem}\label{thm-HKE-E-Ec1-non-loc}
Assume that $p_t(x,y)$ satisfies the upper heat kernel estimate~\eqref{eq:HKE-non-loc} and $d_H<d_W$. If there exists $0<\alpha<d_H/d_W$ such that for all $\varepsilon>0$,
\begin{equation}\label{E:nbd_estimateU-non-loc}
\mu\otimes\mu\{(x,y)\in E^c\times E\, :\, d(x,y)<\varepsilon\}\leqslant c_7\left(\varepsilon^{d_H+\alpha d_W}+\varepsilon^{2d_H }\right),
\end{equation}
then for all $t>0$ it holds that
\begin{equation}\label{eq:PtE-upper-non-loc}
\left(t^{-\alpha}+t^{-{{d_H}}/d_{W}}\right)\| P_t \mathbf{1}_E -\mathbf{1}_E\|_{L^1(X,\mu)} \leqslant c_{11}<\infty,
\end{equation}
where $c_{11}$ is a constant given by~\eqref{eq:NLPtE-upperc11} that depends on $\delta,d_H$ and $d_W$.
\end{theorem}
\begin{proof}
In view of~\eqref{eq:HKE-non-loc}, $p_t(x,y)>s$ implies

\begin{equation}\label{E:def_F(t,s)}
d(x,y)<t^{1/d_W}c_4^{-1}\Big[\Big(c_3^{-1}t^{\frac{d_H}{d_W}}s\Big)^{-\frac{1}{d_W+d_H}}-1\Big]=:F(t,s).
\end{equation}
Moreover, $F(t,s)>0$ if and only if
$s<c_3t^{-\frac{d_H}{d_W}}$ and

the corresponding first term in the expression~\eqref{e-u-dd'} can now be bounded by
\begin{multline*}
\int_0^{c_3t^{-d_H/d_W}} c_7c_4^{-d_H+\alpha d_W}t^{\frac{d_H+\alpha d_W}{d_W}}\Big[\Big(c_3^{-1}t^{\frac{d_H}{d_W}}s\Big)^{-\frac{1}{d_W+d_H}}-1\Big]^{d_H+\alpha d_W}ds\\
=	c_7c_3c_4^{-(d_H+\alpha d_W)}t^\alpha\int_0^1\Big(u^{-\frac{1}{d_W+d_H}}-1\Big)^{d_H+\alpha d_W}du
\end{multline*}

which is finite since by assumption $0<\delta<d_H<d_W$. Thus,
\begin{equation*}
c_7\int_0^{c_3t^{-d_H/d_W}} (F(t,s))^{d_H+\alpha d_W}ds\leq c_7c_3c_4^{-(d_H+\alpha d_W)}t^{\alpha}\frac{d_W+d_H}{d_W(1-\alpha)}
\end{equation*}
and, since $d_H<d_W$, the same estimate for the second term in~\eqref{e-u-dd'} with $\alpha=d_H/d_W$ yields
\begin{equation*}
c_7\int_0^{c_3t^{-d_H/d_W}} (F(t,s))^{d_H+d_H}ds\leq c_7c_3c_4^{-2d_H}t^{\frac{d_H}{d_W}}\frac{d_W+d_H}{d_W-d_H}
\end{equation*}
so that~\eqref{eq:PtE-upper-non-loc} holds with
\begin{equation}\label{eq:NLPtE-upperc11}
c_{11}=c_7c_3c_4^{-d_H}\max\Big\{c_4^{\alpha d_W}\frac{d_W+d_H}{d_W(1-\alpha)},c_4^{-d_H}\frac{d_W+d_H}{d_W-d_H}\Big\}.
\end{equation}
\end{proof}

\begin{theorem}\label{thm-HKE-E-Ec2-non-loc}
Assume that $p_t(x,y)$ satisfies the lower heat kernel estimate in~\eqref{eq:HKE-non-loc}. If there exist $\alpha>0$ and $\varepsilon>0$ such that
\begin{equation}\label{E:nbd_estimateL-non-loc}
\mu\otimes\mu\{(x,y)\in E^c\times E
\, :\,
d(x,y)<\varepsilon\}\geq c_8\varepsilon^{d_H+\alpha d_W},
\end{equation}
then, for $t=\varepsilon^{d_W}$, it holds that
\begin{equation}\label{eq:PtE-lower-non-loc}
t^{-\alpha}\| P_t \mathbf{1}_E -\mathbf{1}_E \|_{L^1(X,\mu)} \geqslant 2 c_{5}c_8e^{-c_{6}} >0.
\end{equation}
\end{theorem}
\begin{corollary}
If there exists $\alpha>0$ and $\varepsilon_0>0$ such that~\eqref{E:nbd_estimateL-non-loc} holds for any $0<\varepsilon<\varepsilon_0$, then~\eqref{eq:PtE-lower-non-loc} holds for any $0<t<\varepsilon^{d_W}$.
\end{corollary}

\begin{proof}[Proof of Theorem~\ref{thm-HKE-E-Ec2-non-loc}.]
The lower estimate in~\eqref{eq:HKE-non-loc} implies that
\begin{equation*}
p_t(x,y)\geq c_5t^{-\frac{d_H}{d_W}}c_6^{-(d_H+d_W)}
\end{equation*}
and hence assumption~\eqref{E:nbd_estimateL-non-loc} yields
\begin{align*}
\|\bm{1}_{E^c} P_t\bm{1}_{E}\|_{L^1(X,\mu)} &\geq \int_{A_{E,t^{1/d_W}}}p_t(x,y)\,\mu(dx)\,\mu(dy) \geq c_5 c_6^{-(d_H+d_W)} t^{-\frac{d_H}{d_W}}\mu\otimes\mu (A_{E,t^{1/d_W}})\\
& \geq c_5c_6^{-(d_H+d_W)}t^{-\frac{d_H}{d_W}}c_8t^{\frac{(d_H+\alpha d_W)}{d_W}}=c_5c_8c_6^{-(d_H+d_W)}t^{\alpha}.
\end{align*}
\end{proof}

Under weaker assumptions, the following localized version of Theorem~\ref{thm-HKE-E-Ec1-non-loc} is available.
\begin{theorem}\label{thm-HKE-E-Ec1-non-loc-ball}
Let $E\subset B\subset X$ and assume that $p_t(x,y)$ satisfies the upper heat kernel estimate~\eqref{eq:HKE-non-loc}. If there exists $0<\alpha<1$ such that for all $\varepsilon>0$,
\begin{equation}\label{E:nbd_estimateU-non-loc-ball}
\mu\otimes\mu\{(x,y)\in (B\cap E^c)\times E\, :\, d(x,y)<\varepsilon\}\leqslant c_7\varepsilon^{d_H+\alpha d_W},
\end{equation}
then for all $t>0$ it holds that
\begin{equation}\label{eq:PtE-upper-non-loc-ball}
t^{-\alpha}\| P_t \mathbf{1}_E -\mathbf{1}_E\|_{L^1(B,\mu)} \leqslant c_{11}<\infty,
\end{equation}
where $c_{11}$ is a constant given by~\eqref{eq:NLPtE-upperc11-ball} that depends on $\alpha,d_H$ and $d_W$.
\end{theorem}
\begin{proof}
Following the proof of Theorem~\ref{thm-HKE-E-Ec1-ball} verbatim we have

\begin{equation*}
\|  \mathbf{1}_{E^c} P_t \mathbf{1}_E \|_{L^1(B,\mu)}\leq\int_0^{c_3t^{-d_H/d_W}}c_7F(t,s)^{d_H+\alpha d_W}ds.
\end{equation*}
Since $0<\alpha<1$ by assumption, the same computations as in the proof of Theorem~\ref{thm-HKE-E-Ec1-non-loc} imply that~\eqref{eq:PtE-upper-non-loc-ball} holds with
\begin{equation}\label{eq:NLPtE-upperc11-ball}
c_{11}=c_7c_3c_4^{-(d_H+\alpha d_W)}\frac{d_W+d_H}{d_W(1-\alpha)}.
\end{equation}
\end{proof}

\section{Critical exponent for $\mathbf{B}^{1,\alpha}(X)$}\label{S:criticalE_nl}
The observation that the space $\bfB^{1,\alpha}(X)$ is trivial for any $\alpha>1$ is the starting point of the discussion that will be carried out in this section, where we compute
\begin{equation}\label{E:critical_a_non-local}
\alpha^*=\inf \{ \alpha >0\,\colon\, \mathbf{B}^{1,\alpha}(X) \text{ is trivial} \}.
\end{equation}
\begin{proposition}\label{prop:NLTrivialB}
If $f\in \bfB^{1,\alpha}(X)$ with $\alpha>1$, then $f=0$.
\end{proposition}
\begin{proof}
Notice first that for any $f\in\bfB^{1,\alpha}(X)$,
\[
 \|P_t(|f-f(\cdot)|)\|_{L^1(X,\mu)} \le t^{\alpha} \|f\|_{1,\al}.
\]
Now assume that $f$ is bounded (in fact, one can consider $f_N$ defined in the proof of Proposition \ref{prop:BesovCEp}). Then,
\begin{align*}
\|P_t(|f-f(\cdot)|)\|_{L^1(X,\mu)} &\le \int_X P_t(|f-f(x)|^2)(x) d\mu(x)\\
& \le 2\|f\|_{L^{\infty}(X)}  \|P_t(|f-f(\cdot)|)\|_{L^1(X,\mu)} \le 2t^{\alpha}\|f\|_{L^{\infty}(X)}  \|f\|_{1,\al}.
\end{align*}
On the one hand, since $\al>1$, we have
\[
\lim_{t\to 0}\frac1t \int_X P_t(|f-f(x)|^2)(x) d\mu(x)= 0.
\]
On the other hand, as in Lemma \ref{lem:energyasbesov} and Proposition \ref{prop:energyasbesov} we know that
\[
\frac1t\int_X P_t(|f-f(x)|^2)(x) d\mu(x)=2\Ecal_t(f),
\]
where $\Ecal_t(f)$ is decreasing. Hence, $\|P_t(|f-f(\cdot)|)\|_{L^1(X,\mu)}=0$ for every $t>0$ and therefore $P_t f(x)=f(x)$ for every $t>0$. The upper bound in \eqref{eq:HKE-non-loc} yields
\[
|f(x)|=|P_t f(x)| \le \int_X p_t(x,y) |f(y)| d\mu(y) \le \frac{C}{t^{d_H/d_W}} \N{f}_{L^1(X,\mu)}
\]
and letting $t\to\infty$, we obtain that $f=0$.
\end{proof}

In order to compute the critical parameter $\alpha^*$, we shall make the following assumption on the underlying metric measure space $(X,d, \mu)$:
\begin{assumption}\label{assumption non local}
There exists a constant $C>0$,  such that for every $f \in \mathfrak{B}^{1}_{1,\infty}(X)$
\[
\sup_{\varepsilon >0} \frac{1}{\varepsilon^{d_H+1}}
 \int_{\Delta_\varepsilon}|f(y)-f(x)|\, d\mu(x)d\mu(y) \le C \limsup_{\varepsilon\to 0^+}\frac{1}{\varepsilon^{d_H+1}}
 \int_{\Delta_\varepsilon} |f(y)-f(x)| \, d\mu(x)d\mu(y),
\]
where $\Delta_\varepsilon=\{ (x,y) \in X \times X, d(x,y) < \varepsilon \}$.
\end{assumption}

This property of the metric space $(X,d,\mu)$ is for instance satisfied if $(X,d)$ satisfies the assumptions of section \ref{weak BE metric space}, see remark \ref{chaining metric}.

One has then the following theorem:

\begin{theorem}\label{T:non-local_critical_alpha}
Under Assumption~\ref{assumption non local} and if $d_W \le 1$, then $\alpha^*=1$ and $\mathbf{B}^{1,\alpha^*}(X)=\mathfrak{B}^{d_W}_{1,1}(X)$. If $d_W  > 1$, then $\alpha^*=\frac{1}{d_W}$ and $\mathbf{B}^{1,\alpha^*}(X)=\mathfrak{B}^{1}_{1,\infty}(X)$.
\end{theorem}
\begin{proof}
If $d_W\le1$, it follows from Proposition \ref{prop:NLTrivialB} that $\bfB^{1,\alpha}(X)$ is trivial for any $\alpha>1$. It remains to justify that $\bfB^{1,\alpha}(X)$ or equivalently $\mathfrak{B}^{\alpha d_W}_{1,\infty}(X)$ is not trivial if $\alpha <1$. This follows from the fact 4 below since one has then $  \mathfrak{B}^1_{1,\infty}(X) \subset \mathfrak{B}^{\alpha d_W}_{1,\infty}(X) $.

\medskip
If $d_W>1$, the conclusion follows by considering the following four facts.

{\bf Fact 1.} $\bfB^{1,\alpha}(X)$ is trivial for $\alpha>1$. This immediately follows from Proposition \ref{prop:NLTrivialB}.

\medskip
{\bf Fact 2.} $\bfB^{1,1}(X)$ is trivial. Let $f\in \bfB^{1,1}(X)$, then for any $\varepsilon>0$, Lemma \ref{lem:BesovEquivSN} yields
\begin{align*}
\frac{1}{\varepsilon^{d_H+1}} \int_{\Delta_\varepsilon} |f(y)-f(x)|\, d\mu(x)d\mu(y)

\le\varepsilon^{d_W-1} \int_{\Delta_\varepsilon} \frac{|f(y)-f(x)|}{d(x,y)^{d_H+d_W}}\, d\mu(x)d\mu(y)
\le
C \varepsilon^{d_W-1} \|f\|_{1,1}.
\end{align*}
Thus,
\[
 \limsup_{\varepsilon\to 0^+}\frac{1}{\varepsilon^{d_H+1}}
 \int_{\Delta_\varepsilon} |f(y)-f(x)| \, d\mu(x)d\mu(y)=0.
 \]
 From assumption \eqref{assumption non local}, this implies that $f$ is constant and thus 0.
\medskip

{\bf Fact 3.} $\bfB^{1,\alpha}(X)$ is trivial for any $\frac1{d_W}<\alpha<1$.
By Theorem \ref{Besov non-local}, we have $\bfB^{1,\alpha}(X) =\mathfrak B_{1,\infty}^{\alpha d_W}(X)$. Thus it suffices to show that $\mathfrak B_{1,\infty}^{\beta}(X)$ is trivial for $1<\beta<d_W$. For any $\varepsilon>0$,
\begin{align*}
\frac{1}{\varepsilon^{d_H+1}} \int_{\Delta_\varepsilon} |f(y)-f(x)|\, d\mu(x)d\mu(y)
\le
\frac{\varepsilon^{\beta-1} }{\varepsilon^{d_H+\beta}} \int_{\Delta_\varepsilon} |f(y)-f(x)|\, d\mu(x)d\mu(y)
\le
C \varepsilon^{\beta-1}N_{1,\infty}^{\beta}(f).
\end{align*}
Taking the limsup when $\varepsilon$ goes to zero, again  we have that $f$ is a constant.

\medskip

{\bf Fact 4.} It remains to justify that $\bfB^{1,\frac1{d_W}}(X)$ is not trivial, or equivalently, $\mathfrak B_{1,\infty}^{1}(X)$ is not trivial.

Fix $x_0\in X$ and $R>0$. Set
\[
\phi_{x_0,R}(x)=\max\{0,R-d(x_0,x)\}.
\]
Then $\phi_{x_0,R}$ is non-negative, $L^1$-integrable and supported in the ball $B(x_0,R)$ (see also Section 4.1 for these functions).
For any $r>0$, we have
\begin{align*}
&\iint_{\Delta_r} |\phi_{x_0,R}(x)-\phi_{x_0,R}(y)|\, d\mu(x) d\mu(y)
\\ \le&
2\int_{B(x_0,R)}\int_{B(x_0,R)^c \cap B(y,r)} |R-d(x_0,y)| \,d\mu(x) d\mu(y)

\\ &
+\int_{B(x_0,R)}\int_{B(x_0,R)\cap B(y,r)} |d(x_0,x)-d(x_0,y)| \,d\mu(x) d\mu(y)
\\ \le&
\,C r^{d_H+1} R^{d_H}.
\end{align*}
The second inequality above holds since $x\in B(x_0,R)^c$ and $y\in B(x_0,R)$ imply $R-d(x_0,y)\le d(x,y)\le r$. Therefore it easily follows from the definition that $\phi_{x_0,R}\in \mathfrak B_{1,\infty}^1(X)$ with semi-norm estimate depending on $R$.
\end{proof}

\section{Sobolev and isoperimetric inequalities}\label{S:Sobo_iso_nl}

\subsection{Discussion on Sobolev inequalities for non-local Dirichlet spaces}
The standard Poincar\'e type inequalities do not make sense for
non-local Dirichlet forms (although there are some forms of Poincar\'e type inequalities for Dirichlet forms, such as \cite{OS-CT,Chen2017}); however, the following global Sobolev inequality
makes sense: do there exist constants $C>0$ and $\kappa\ge 1$ such that
for every $u\in\mathcal{F}$,
\[
\left(\int_X|u|^{2\kappa}\, d\mu\right)^{\frac{1}{2\kappa}}
  \le C\l \sqrt{\mathcal{E}(u,u)}
\]
We now explore one condition under which the above inequality will hold.
Recall that $\mathcal{E}$ is \emph{transient} if there is some almost-everywhere
positive $f\in L^1(X)$ such that
\[
\int_0^\infty P_tf\, dt
\]
is finite almost
everywhere in $X$. Here $P_tf$ is the heat extension of $f$, namely,
\[
P_tf(x)=\int_X p_t(x,y)\, f(y)\, d\mu(y).
\]
Thus transience of the form $\mathcal{E}$ implies that $P_tf$ decays fast
as $t\to \infty$ almost everywhere in $X$. For the rest of this subsection
we will assume that $\mathcal{E}$ is a transient regular Dirichlet form on
$L^2(X)$, and that $(\mathcal{E},\mathcal{F})$ is strongly continuous.
By completing $(\mathcal{E},\mathcal{F})$ with respect to the norm
$u\mapsto\Vert u\Vert_{L^2(X)}+\sqrt{\mathcal{E}(u,u)}$ if necessary, we may
assume also that this pair is a Hilbert space (see~\cite[Theorem~1.5.2,
Lemma~1.5.5]{FOT}). We now have the following version of Sobolev
inequality:

\begin{lemma}[Theorem~1.5.3 in \cite{FOT} or Theorem 2.1.5 in \cite{ChenFukushima} ]\label{lem:g-Sobolev}
For a Dirichlet form $(\mathcal{E},\mathcal{F})$,
  the strongly continuous   semigroup $P_t$ is

transient if and only if

there exists a
bounded function $g\in L^1(X)$ such that $g$ is positive almost everywhere
in $X$ such that for all $u\in\mathcal{F}$ we have
\[
\int_X|u|\, g\, d\mu\le \sqrt{\mathcal{E}(u,u)}.
\]
\end{lemma}

While the conclusion of the above lemma looks remarkably like the
desired Sobolev inequality, the problem here is that we do not have
good control over $g$, and in general we would like to have control
over the $L^q(X,\mu)$-norm of $u$ for some $q>0$, not just over the
$L^1(X, g\, d\mu)$-norm.

For a compact set $K\subset X$ we set the variational capacity of $K$ to be
the number
\[
\text{Cap}_0(K):=\inf_u \mathcal{E}(u,u),
\]
where the infimum is over all $u\in\mathcal{F}\cap C_0(X)$ (where
$C_0(X)$ is the collection of all compactly supported continuous functions on $X$)
with $u(x)\ge 1$ for all $x\in K$, see~\cite[Section~2.4]{FOT}.
The following capacitary type inequality holds.

\begin{lemma}[{\cite[Lemma~2.4.1]{FOT}}]
For $u\in\mathcal{F}\cap C_0(X)$ and $t>0$ let
$K_t$ denote the set $\{x\in X\, :\, |u(x)|\ge t\}$.
Then we have that
\[
\int_0^\infty 2t\, \text{\rm Cap}_0(K_t)\, dt\le 4\, \mathcal{E}(u,u).
\]
\end{lemma}

The above lemma indicates that there are plenty of compact sets in $X$
with finite capacity. Let $K\subset X$ be a compact set such that
$\text{Cap}_0(K)$ is finite. Then we can find a sequence
$u_k\in\mathcal{F}\cap C_0(X)$ with $u_k\ge 1$ on $K$ and
$\lim_k\mathcal{E}(u_k,u_k)=\text{Cap}_0(K)<\infty$.
Since truncations of a function in $\mathcal{F}$ do not increase its
Dirichlet energy, we know that
$\mathcal{E}(\widehat{u_k},\widehat{u_k})\le \mathcal{E}(u_k,u_k)$
where $\widehat{u_k}=\max\{0,\min\{u_k,1\}\}$. Thus without loss of generality
we may assume that $0\le u_k\le 1$ on $X$. Therefore $\{u_k\}_k$ forms
a locally bounded sequence in $L^2(X)$, and by multiplying each
$u_k$ by a fixed $\eta\in\mathcal{F}\cap C_0(X)$ with $0\le \eta\le 1$ on $X$
and $\eta=1$ on the ball $B(x_0,n)$ for some fixed $x_0\in X$ and
$n\in\mathbb{N}$, we get a bounded sequence in $\mathcal{F}$, which,
as $\mathcal{F}$ is a Hilbert space, gives a
convex combination subsequence that converges to
some function $u\in\mathcal{F}$ (we do this for each positive integer $n$
and employ a Cantor-type diagonalization argument to do so) with
$\mathcal{E}(u,u)\le \lim_k\mathcal{E}(u_k,u_k)=\text{Cap}_0(K)$.
Employing a further subsequence argument, we can ensure that
$u\ge 1$ in $K$; however, $u$ may no longer be continuous on $X$,
see also~\cite[Lemma~2.1.1]{FOT}.
Such a function $u$ is called a \emph{$0$-th order equilibrium potential}
of $K$ in~\cite{FOT}.

Given the above notion of capacity, the following theorem identifies a property
on $\mathcal{E}$ under which we have the desired Sobolev inequality.

\begin{theorem}[{\cite[Theorem~2.4.1]{FOT}}]\label{T:isoperim-nonlocal}
If there is some $\kappa\ge 1$ and $\Theta>0$ such that
\begin{equation}\label{eq:isoperim-nonlocal}
\mu(K)^{1/\kappa}\le \Theta\, \text{\rm Cap}_0(K)
\end{equation}
for each compact $K\subset X$, then there is some $C>0$ with
$0<C^2\le (4\kappa)^\kappa \Theta$ such that
\[
\left(\int_X|u|^{2\kappa}\, d\mu\right)^{1/2\kappa}\le C\, \sqrt{\mathcal{E}(u,u)}.
\]
\end{theorem}

The inequality~\eqref{eq:isoperim-nonlocal} is an analog of the
\emph{isoperimetric inequality} adapted to the non-local Dirichlet form.
The optimal constant $\Theta$ is called the isoperimetric constant of $\mathcal{E}$,
and is a non-local analog of the Cheeger constant.
Indeed, \cite[Theorem~2.4.1]{FOT} claims even more, that the support of
a Sobolev type inequality is equivalent to this isoperimetric inequality.
Should $\mathcal{E}$ support such an inequality, then the total capacity of
a compact set $K$ given by
\[
\text{Cap}_1(K):=\inf_u \int_X u^2\, d\mu+\mathcal{E}(u,u)
\]
with infimum over all $u\in \mathcal{F}\cap C_0(X)$ with $u\ge 1$ on $K$
has the same class of null capacity sets as $\text{Cap}_0$.
We refer the interested reader to~\cite[Theorem~2.4.3]{FOT} for
examples of Sobolev type inequality for measures in Euclidean spaces.

We point out here that this notion of isoperimetric inequality is weaker
than the traditional one where $\text{Cap}_0(K)$ is replaced with the
relative $1$-capacity of $K$, namely infimum of the numbers
$\int_X|\nabla u|\, d\mu$ over all $u\in \mathcal{F}\cap C_0(X)$ with
$u\ge 1$ on $K$. In the case of the non-local Dirichlet form the
quantity $\int_X|\nabla u|\, d\mu$ does not make usual sense (although  see \cite{HRT} for a version of such an integral for arbitrary Dirichlet forms).
On the other hand, the results of~\cite[Theorem~2.1.1]{FOT} tell us
that $\text{Cap}_0$ has an extension to a Choquet capacity on $X$.

Fix $u\in\mathcal{F}\cap C_0(X)$, and let $t>0$. For $\varepsilon>0$
we set
\[
u_t^\varepsilon=\frac{\min\{\varepsilon, \max\{u-t,0\}\}}{\varepsilon}.
\]
Then by the Markov property of $\mathcal{E}$ we know that
$\mathcal{E}(u_t^\varepsilon,u_t^\varepsilon)\le \mathcal{E}(u,u)/\varepsilon^2$.
Observe that $u_t^\varepsilon\to \chi_{K_t}$ in $L^1(X)$ as $\varepsilon\to 0^+$.
Suppose we know that
$\liminf_{\varepsilon\to 0^+}\mathcal{E}(u_t^\varepsilon,u_t^\varepsilon)<\infty$.
Then
\[
\text{Cap}_0(K_t)\le
  \liminf_{\varepsilon\to 0^+}\mathcal{E}(u_t^\varepsilon,u_t^\varepsilon)<\infty.
\]
If $\mathcal{E}$ is strongly local and supports a $2$-Poincar\'e inequality,
then using the fact that distance functions
belong to $\mathcal{F}$ we can obtain a comparison between
$\text{Cap}_0(K_t)$
and the Minkowski co-dimension $1$ content of $\partial K_t$ for almost
every $t>0$, see for example the arguments in~\cite{CJKS, JKY}.
For strongly local Dirichlet forms we have shown in Chapter~4 that
the correct BV class is $\mathbf{B}^{1,1/2}(X)$.
For non-local Dirichlet forms (and indeed even for forms that are not
strongly local) it is not clear which of the Besov classes
$\mathbf{B}^{1,\alpha}(X)$ should be the correct analog of BV functions,
see Theorem~\ref{T:FP_char_non-local}. From this theorem, we see that
if $E\subset X$ is bounded and measurable such that
$\bm{1}_E\in \mathbf{B}^{1,\alpha}(X)$, then by choosing $r>2\text{diam}(E)$,
\[
\frac{c}{r^{\alpha d_W+d_H}}\mu(E)\, \mu(B(x_0,r/2))
\le \Vert \bm{1}_E\Vert_{1,\alpha}
\]
where $x_0\in X$ such that $B(x_0,r/2)\cap E$ is empty and
$B(x_0,r/2)\subset\bigcup_{x\in E}B(x,r)$. One can look on this as
the Besov space analog of the isoperimetric inequality.

\subsection{Besov spaces embeddings}

Note that by combining the results of Part I, Chapter 2,  with the results of this chapter, one immediately obtains the following corollaries. The proofs are similar to the ones in Section  \ref{section Sobolev local}, so will be omitted for concision.

\begin{corollary}
Let $(X,\mu,\mathcal{E},\mathcal{F})$ be a symmetric non-local Dirichlet space with  heat kernel estimates \eqref{eq:HKE-non-loc}. Let $0<\delta < d_H $. Let $1 \le p < \min \left\{  \frac{d_H}{\delta} , \frac{d_W}{\delta} \right\}$.   There exists a constant $C_{p,\delta} >0$ such that for every $f \in \mathbf{B}^{p,\delta/d_W}(X)$,
\[
\sup_{s \ge 0} s \mu \left( \{ x \in X, | f(x) | \ge s \} \right)^{\frac{1}{q}} \le C_{p,\delta} \sup_{r>0} \frac{1}{r^{\delta+d_{H}/p}}\biggl(\iint_{\Delta_r}|f(x)-f(y)|^{p}\,d\mu(x)\,d\mu(y)\biggr)^{1/p}
\]
where $q=\frac{p d_H}{ d_H -p \delta}$. Furthermore, for every $0<\delta <d_H $, there exists a constant $C_{\emph{iso},\delta}$ such that for every measurable $E \subset X$, $\mu(E) <+\infty$,
\begin{align}\label{isoperimetric intro3}
\mu(E)^{\frac{d_H-\delta}{d_H}} \le C_{\emph{iso},\delta} \sup_{r>0} \frac{1}{r^{\delta+d_{H}}} (\mu \otimes \mu) \left\{ (x,y) \in E \times E^c
\, :\,
 d(x,y) \le r\right\}
\end{align}
\end{corollary}

\begin{corollary}
Let $(X,\mu,\mathcal{E},\mathcal{F})$ be a symmetric non-local Dirichlet space with  heat kernel estimates \eqref{eq:HKE-non-loc} and $d_W \le d_H$. Let $0<\delta < d_H $. Let  $p=\frac{d_W}{\delta}$ and assume $p \ge 1$.   There exists a constant $C_{p,\delta} >0$ such that for every $f \in \mathbf{B}^{p,\delta/d_W}(X) $,
\[
\sup_{s \ge 0} s \mu \left( \{ x \in X, | f(x) | \ge s \} \right)^{\frac{1}{q}} \le C_{p,\delta} \brak{\int_0^\infty \brak{N_p^{d_W/p}(f,r)}^{p}\, \frac{dr}{r}}^{1/p},
\]
where $q=\frac{p d_H}{ d_H -p \delta}$ and
\[
N^{d_W/p}_{p}(f,r):=\frac{1}{r^{d_W/p+d_{H}/p}}\biggl(\iint_{\Delta_r}|f(x)-f(y)|^{p}\,d\mu(x)\,d\mu(y)\biggr)^{1/p}
\]

\end{corollary}


TITLE: Function and Fourier transform vanish on an interval
QUESTION [11 upvotes]: I'm no expert on these things (and this may not be cutting edge research level; it's really motivated by this MSE question), but it seems that there are non-zero measures (and also functions (?), I would think) on $\mathbb R$ such that both the function and its Fourier transform vanish on an interval. This is usually mentioned in the context of Beurling's work on spectral gaps.

(1) Is there a nice snappy way to see this? (2) How regular can such an $f$ be? Could it be a Schwartz function?

REPLY [8 votes]: Over at MSE (see linked question), user1952009 (= reuns on MO) has given a very elegant construction of a Schwartz function with this property. I am reproducing his/her answer here.
Let $g,h\in C_0^{\infty}$ with support contained in $[1/3,1/2]$, and write $\widehat{h}_n=\int_0^1 h(x)e^{-2\pi inx}\, dx$ for the discrete Fourier coefficients of $h$, viewed as a $1$-periodic function $h_p$. Let
$$
f(x)=\sum_{n=-\infty}^{\infty} \widehat{h}_n g(x-n) .
$$
Then
$$
\widehat{f}(\xi)= \sum \widehat{h}_n e^{2\pi in\xi}\widehat{g}(\xi) = h_p(\xi)\widehat{g}(\xi) .
$$
Thus $f$ is as desired; this calculation also confirms that $f\in\mathcal S$.


TITLE: Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of prime numbers, then $\lim_{n \to \infty}\frac{p_{n+1}}{p_n} = 1$?
QUESTION [9 upvotes]: The Bertrand postulate is that $p_{n+1} < 2p_{n}$ but it is far to be enough to conclude.
I made a program in $R$, to see the convergence of this sequence for the first $500$ prime numbers and it seems to converge to 1. Is this true, and how to prove it ?

REPLY [3 votes]: Here's a different proof of that fact, which doesn't make a reference to the prime number theorem assuming the limit exists. We'll use the fact that the series
$$\sum_{n=1}^\infty \frac{1}{p_n}$$
diverges, which has a nice elementary proof due to Erdos. Now suppose that the limit is more than $1$. Then $p_{n+1}/p_n>1+a$ for some $a>1$ and all $n\geq N$ for some $N$. Thus
$$\sum_{n=1}^\infty \frac{1}{p_n}\leq \sum_{n=1}^{N-1} \frac{1}{p_n} + \frac{1}{p_n}\sum_{i=0}^\infty \frac{1}{(1+a)^i}$$
which converges.
It would be interesting to give an elementary (not using the prime number theorem) proof that the limit exists.


TITLE: Similar Matrices in a field
QUESTION [2 upvotes]: Let $p$ be a odd prime integer. And $A$ and $B$ be matrices from the set  $GL(l,p)$. Further, suppose that $A$ and $B$ are similar. That is, there exists an invertible matrix $H$ such that $B = HAH^{-1}$. Finally let
$$ S = \{ H \in  GL(l,p):  B = HAH^{-1}\bmod~p \}.$$
I am trying to determine the number of elements of the set $S$.
My approach is to solve $B = HAH^{-1}$ for the unknown elements of $H$. Notice that will result in having a homogeneous linear system with $l^2$ unknowns, where $l$ is the number of rows and columns of $H$.  Experimentally, I observed that the number of free variables when solving this system is $l$. Would this imply that the number elements of the set $S$ is $(p-1)^l$?
Any help with this will be highly appreciated. You may contact me directly at  j1 dot feeb "@" G mail  dot   com.  Thanks in advance.

REPLY [0 votes]: The following is taken from http://www.westminster.edu/staff/offnerde/CountingEigenvalues.pdf.
For any matrix $A\in M_{l}(\mathbb{F})$, let $S(A)  \subseteq M_{l}(\mathbb{F})$ be the set of matrices similar to $A$, and let $C(A)$ denote the subgroup of $GL_l(\mathbb{F})$ of matrices $P$ such that $PA = AP$.
Lemma: For Any $J \in M_{l}(\mathbb{F})$, $|S(J)|\ = \frac{|G_{l}(\mathbb{F})|}{|C(J)|}$.
Proof: Fix a matrix $J \in M_{l}(\mathbb{F})$. We wish to find the cardinality of
$$ S(J) = \{{AJA^{-1}: A \in G_{l}(\mathbb{F})}\}.$$
For any matrix $A$, $B$ $\in G_{l}(\mathbb{F})$, $AJA^{-1} = BJB^{-1}$ if and only if
$B^{-1}AJ = JB^{-1}A$, i.e. if $B^{-1}A \in C(J)$. Now $B^{-1}A \in C(J)$ if and only if $A$ and $B$ are in the same coset of $C(J)$ in $G_{l}(\mathbb{F})$ and thus $S(J)$ has the same cardinality as the number of cosets of $C(J)$ in $G_{l}(\mathbb{F})$, which is equal to $|G_{l}(\mathbb{F})|/|C(J)|$ by Lagrange's Theorem.


TITLE: Uniqueness of Thin QR Factorization.
QUESTION [4 upvotes]: Let $A \in \mathbb C^{m \times n}$, have linearly independent columns. Show: If $A=QR$, where $Q \in \mathbb C^{m \times n}$ satisfies $Q^*Q=I_n$ and $R$ is upper triangular with positive diagonal
elements, then $Q$ and $R$ are unique.
$Q^*$ is tranpose of $Q$
Is a exercise of Numerical Matrix Analysis of Ipsen
How do I  can prove this? I need help
$A=Q_1R_1=Q_2R_2$ then $R_1=Q_1^*Q_2R_2$, so I need to prove that $Q_1^*Q_2=I_n$

REPLY [7 votes]: Let $A=Q_1R_1=Q_2R_2$ then we have $Q_2^{-1}Q_1=R_2 R_1^{-1}$ but $Q_2^{-1}Q_1$ is unitary matrix so it's diagonalizable and their eigenvalues belong to the unit circle of the complex plane and it's a triangular matrix with positive diagonal elements (eigenvalues) so these eigenvalues are equal to $1$ and then this matrix is similar to $I_n$ hence it's equal to $I_n$. Conclude.


TITLE: Do equivariant crepant resolutions always exist?
QUESTION [6 upvotes]: Let $X_t$ be a family of algebraic varieties (my interest is Calabi-Yau varieties, but I don't think that's important) over $\mathbb{C}$, smooth for $t \neq 0$, on which a group $G$ acts fibre-wise.  Suppose further that $X_0$ admits at least one crepant resolution.  Does there always exist an equivariant crepant resolution?  If not, are there conditions under which such exists?

REPLY [6 votes]: Consider $\mathbb Z/2$ acting on $\{xy-zw=t\}$ by $x\leftrightarrow y$. This swaps the two small resolutions of the central fibre (the 3-fold ordinary double point $xy=zw$ in $\mathbb C^4$). So there can't be an equivariant small resolution.
(A formal proof might go along these lines: $\mathbb Z/2$ does act on the blow up of the ODP, swapping the two rulings of the $\mathbb P^1\times\mathbb P^1$ exceptional divisor. The small resolutions are contractions of this blow up. If $\mathbb Z/2$ acted on one of them, it would act on its $H^2$. Pulling back, its action on $H^2(\mathbb P^1\times\mathbb P^1)\cong\mathbb Z\oplus\mathbb Z$ would be the identity on the contracted $\mathbb Z$ summand, contradicting the fact that it swaps the summands.)
However if your (finite?) group action permutes the singular loci with no stabilisers then there would surely be a crepant resolution.


TITLE: Determine logaritmic function based on samples of unique visitors
QUESTION [3 upvotes]: I need to estimate the unique visitors for any given number of days. I have the unique visitors per day, day, week and month.
 1 day :  20000 visitors
 7 days:  85000 visitors
30 days: 200000 visitors

How can I determine a logaritmic function that has these 3 points?

REPLY [1 votes]: We should expect the unique visitors to grow quickly the first days.
The visitors coming back after a long period is relatively small, but there will always be a group of new visitors. So in the long run the numbers should be almost linear.
This would suggest a logarithmic function like:
$$
a*log(x+1) + bx
$$

Determining a and b this function based on samples is fairly hard. However the function can be approached for $x > 1$ with:
$$
\sqrt{ax^2+bx}
$$

a and b can be calculated using Quadatic regression.
With 3 samples and $y=0$ on $x=0$ the function would not exactly pass through all 3 points, but it'd be a best fit. Taken that we don't really care about $x<1$, the best is to introduce c having the error biggest on $t=0$ and becoming less relevant for a bigger x.
$$
\sqrt{ax^2+bx+c}
$$
We calculate a, b and c using the discriminant.
$$
a = (x3*(y2-y1) + x2*(y1-y3) + x1*(y3-y2)) / ((x1-x2) * (x1-x3) * (x2-x3))
$$
$$
b = (x3^2(y1-y2) + x1^2(y2-y3) + x2^2*(y3-y1) )/ ((x1-x2) * (x1-x3) * (x2-x3))
$$
$$
c = (x3*(x2*(x2-x3)*y1 + x1*(x3-x1)*y2) + x1*(x1-x2)*x2*y3) / ((x1-x2) * (x1-x3) * (x2-x3))
$$
This answer is courtesy of Nico Schoonderwoerd aka @klup


TITLE: Bound on Derivatives
QUESTION [3 upvotes]: Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function with the following properties.

$\phi(x) = 1$ if $|x| \leq 1$
$\phi(x) = 0$ if $|x| \geq 2$
$0 \leq \phi \leq 1$
$\phi$ is radial.

Let $\phi_k(x) = \phi(\frac{x}{k})$. How do I show that for each multiindex $\alpha$, there exists a constant $C_\alpha$ such that $|D^\alpha \phi_k| \leq \frac{C_\alpha}{k^{\textrm{deg }\alpha}}$ uniformly in $k$?

REPLY [2 votes]: For $\alpha\in\mathbb N^n$ and $x\in\mathbb{R}^n$, we have $D^{\alpha}\phi_k(x) = \dfrac{x^{\alpha}}{k^{\mathrm{deg}\alpha}}D^{\alpha}\phi \left(\frac xk\right)$. Since the support of $\phi$ is contained in the ball $\bar{B}(0,2)$, we have $\lVert D^{\alpha}\phi_k\rVert \leq \dfrac{2^{\mathrm{deg}\alpha}}{k^{\mathrm{deg}\alpha}}\cdot \lVert D^{\alpha}\phi\rVert$, where $\displaystyle\lVert f\rVert =\sup_{x\in \mathbb R} |f(x)| $. We get the result with $C_{\alpha} = 2^{\mathrm{deg}\alpha}\cdot \lVert D^{\alpha}\phi\rVert$.


TITLE: Topological relation between Complex numbers and R^2
QUESTION [2 upvotes]: Topologically,are the space of complex numbers and the Euclidean space $\mathbb R^2$ (with the standard topologies) the same?

REPLY [5 votes]: The map $f:\mathbb C\to\mathbb R^2$ given by
$$f(x+iy)=(x,y)$$
is a homeomorphism.


TITLE: Using the digits $2$, $3$, $4$, $5$ and $6$, find the sum of all $5$-digit numbers that can be formed such that no two digits are same?
QUESTION [1 upvotes]: Using the digits $2$, $3$, $4$, $5$ and $6$, find the sum of all $5$-digit numbers that can be formed such that no two digits are same?
Why each of the number would be in any place $4!$ times and not $5!$ times?
And how to find the sum? Please can someone explain. There are many questions based on this concept. But I am not able to understand how to find sum of all the $5$-digit numbers.

REPLY [0 votes]: Here is another way to look at it.  As noted elsewhere, there are 120 different numbers.  Since each place (1s, 10s, 100s, etc.) holds 2, 3, 4, 5, and 6 the exact same number of times, the average for each place is $120 \times 44444 = 5333280$.


TITLE: Why is there a duality between spaces and commutative algebras?
QUESTION [46 upvotes]: 1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety its algebra of regular functions to $\mathbb{A}^1$, and to each finitely generated reduced algebra its (ringed) space of algebra homomorphisms to $\mathbb{C}$. This is almost a tautology, but in some sense a deep one.
2) The category of compact Hausdorff spaces is equivalent to the opposite category of $\mathrm{C}^*$-algebras. The equivalence associates to a compact Hausdorff space its algebra of continuous functions to $\mathbb{C}$ (equipped with the sup norm), and to each $\mathrm{C}^*$-algebra its space of $\mathrm{C}^*$-algebra homomorphisms to $\mathbb{C}$. This is due to Gelfand.
3) The category of totally-disconnected compact Hausdorff spaces is equivalent to the opposite category of Boolean algebras. The equivalence associates to a totally-disconnected compact Hausdorf space its algebra of continuous functions to $\{0,1\}$, and to each Boolean algebra its space of Boolean algebra homomorphisms to $\{0,1\}$. This is due to Stone.
4) The $\infty$-category of simply-connected rational spaces of finite type is equivalent to the opposite $\infty$-category of simply-coconnected coconncetive $\mathbb{E}_\infty$-algebras of finite type over $\mathbb{Q}$. The equivalence associates to a rational space $X$ its cochain complex $C^*(X,\mathbb{Q})$, and to each $\mathbb{E}_\infty$-algebra over $\mathbb{Q}$ its space of $\mathbb{Q}$-linear $\mathbb{E}_\infty$-algebra maps to $\mathbb{Q}$. This is an $\infty$-categorical reformulation due to Lurie of a classical theorem of Sullivan.
5) The $\infty$-category of pro-$p$-finite spaces is equivalent to the opposite category of solvable $\mathbb{E}_\infty$-algebras over $\overline{\mathbb{F}}_p$. The equivalence associates to a pro-$p$-finite space $\{X_i\}_{i \in I}$ the corresponding colimit of cochain complexes $colim_i C^*(X_i,\overline{\mathbb{F}}_p)$. If $X$ is a $p$-finite space then $X$ can be reconstructed as the space of $\overline{\mathbb{F}}_p$-linear $\mathbb{E}_\infty$-algebra maps from $C^*(X,\overline{\mathbb{F}}_p)$ to $\overline{\mathbb{F}}_p$. Here $\overline{\mathbb{F}}_p$ is the algebraic closure of $\mathbb{F}_p$. The theorem in this form is due to Lurie, but a the original version of the theorem (working with $p$-complete spaces) is due to Mandell.

Indeed, if $R$ is a commutative-ring type object in some category of geometric objects, and $X$ is a geometric object carrying a sufficient supply of maps to $R$, then it is worthwhile to look at the ring of functions from $X$ to $R$, as this object will carry a lot of information on $X$. In the other direction, it is often useful to study commutative-ring objects by trying to realize them as rings of functions on some geometric object, their "spectrum". However, even with this in mind, it is still quite striking that this approach yields so many anti-equivalences between full subcategories of geometric objects and full subcategories of commutative-algebraic objects. It is like the geometer/topologist/homotopy-theorist is walking on one continent, and the commutative-algebraist is walking on another, and yet they keep stumbling upon the exact same species of categories (only in reverse).
What is going on here? Is there any conceptual explanation to this phenomenon? Is there any general argument which suggests, even heuristically, that the above equivalences are to be expected?

REPLY [5 votes]: I have not yet considered the new answer in full detail, so apologies for not addressing it. I'm coming back to this question after a while, so I thought I'd share some observations which came up during that time, and which might interest someone (at least, those someones who thought the original question was of any interest).
1) Example (3) above is actually a particular case of Example (5) with $p=2$, in the sense that it is obtained by restricting from 2-pro-finite spaces to pro-finite sets. Indeed, recall the notion of "solvable" which appears in Lurie's $p$-adic homotopy theory paper. In the discrete case, the data of a solvable $\overline{\mathbb{F}}_2$-algebra is essentially equivalent to the data of a Boolean algebra. In fact, example (3) admits a version for every prime $p$, replacing Boolean algebras with algebras which satisfy the relation $a^p=a$ for every $a$. These categories of "$p$-Boolian algebras" are in fact all equivalent to the opposide category of pro-finite sets (or totally disconnected compact Hausdorff spaces).
2) In every context where there is an interesting notion of geometric objects and commutative algebra objects which are connected by some duality, there is usually also a related category of "abelian group objects" underlying the situation. In particular, the commutative algebras are commutative algebra objects in this category, and there is typically an adjunction between geomtric objects and abelian group objects which associates to an abelian group its "underlying space" and to a geometric obejct the free abelian group generated from it. This later abelian group should be dual in some since to the underlying abelian group of the algebra of functions on the space. The adjunction between geometric objects and abelian group objects then induces a comonad on the category of abelian group objects. In that case one can ask two types of questions: (i) to what extent is this adjunction comonadic? in other words, how close is it to exhibiting spaces as coalgebras over this comonad? and (ii) how close is this comonad to being the cocommutative coalgebras comonad? When both answers are sufficiently positive we get that spaces look a lot like cocommutative coalgebras in abelian groups. Since abelian groups often have something that is close to self duality, this makes spaces close to the opposite of commutative algebras.
3) If one is working in the $\infty$-categorical setting, and $\mathcal{C}$ is some $\infty$-category of geometric objects, then one has a natural choice for the $\infty$-category $\mathcal{A}$ of abelian group objects, namely, the stabilization $Sp(\mathcal{C})$ of $\mathcal{C}$. Alternatively, one can tensor the stabilization with a field, such as $\mathbb{Q}$ or $\overline{\mathbb{F}}_p$, the choices underlying examples (4) and (5) above. If $\mathcal{C}$ is furthermore presentable then we will have an adjunction $\mathcal{C} \leftrightarrows \mathcal{A}$ where we can consider the left functor as an analogue of the free abelian group functor. If $\mathcal{C}$ carries a symmetric monoidal structure which preserves colimits in each variable separately then $\mathcal{A}$ will inherit the same type of structure and the left functor $\mathcal{C} \to \mathcal{A}$ will be monoidal.
4) Let $K: \mathcal{A} \to \mathcal{A}$ be the comonad corresponding to the adjunction $\mathcal{C} \leftrightarrows \mathcal{A}$. Since $\mathcal{A}$ is monoidal it makes since to ask if $K$ is the coalgebra comonad of some operad enriched in $\mathcal{A}$. This question can be tackled using Goodwillie calculus and obstruction theory, see work of Gijs Heuts (e.g., https://arxiv.org/abs/1510.03304). In particular, these obstructions vanish when working over a field of characteristic $0$, so if we take $\mathcal{A}$ to be the stabilization tensored with $\mathbb{Q}$, then the comonad $K$ will always come from some operad in $\mathcal{A}$. The spaces of $n$-ary operations of this operad (considered as $\Sigma_n$-objects in $\mathcal{A}$) are given by the Goodwillie derivatives of $K$ of $\mathcal{A}$. For this operad to be the commutative operad we need that the derivatives are all the unit object. This is likely to happen for completely abstract reasons when the monoidal structure on $\mathcal{C}$ is the Cartesian one, i.e., when $\mathcal{C}$ is Cartesian closed (a typical property of categories of geometric objects). Here is a cool heuristic explanation of this phenomenon that I learned from Tomer Schlank: the right adjoint $\mathcal{A} \to \mathcal{C}$ preserves Cartesian products and under our assumptions the left adjoint sends Cartesian products to tensor products in $\mathcal{A}$. Since $\mathcal{A}$ is stable products and coproducts in $\mathcal{A}$ coincide, and are often known simply as direct sums. One may then conclude that the comonad $K: \mathcal{A} \to \mathcal{A}$ sends direct sums to tensor products. In other words, in the analogy between functors and functions underlying the Goodwille calculus, $K$ is analogous to a function $\mathbb{R} \to \mathbb{R}$ which sends addition to multiplication. In other words, $K$ is analogous to an exponential function. As such, its derivatives at $0$ are the sequence of powers of some number $a$. There is a natural family of operads which have the same property. If $A \in \mathcal{A}$ is a cocommutative bialgebra object then the theory of commutaive $A$-algebras is controlled by an operad whose $\Sigma_n$-object of $n$-ary operations is $A^{\otimes n}$. It seems likely (though I don't have a proof of this), then for every Cartesian closed $\infty$-category $\mathcal{C}$ there is a cocommutative bialgebra object in $\mathcal{A} = Sp(\mathcal{C}) \otimes \mathbb{Q}$ such that the comonad associated to the adjunction $\mathcal{C} \leftrightarrows \mathcal{A}$ is the comonad of cocommutative $A$-coalgebras. For example, when $\mathcal{C}$ is the $\infty$-category of spaces this $A$ is the unit object of $\mathcal{A}$ and one gets the commutative operad.
Edit: A closer look at 1 shows that this $A$ is actually always the unit, see Lemma B.3. On the other hand, the situation is more complicated than I thought, so remark (4) should really be taken loosely.


TITLE: If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
QUESTION [2 upvotes]: Suppose $G_1$ and $G_2$ are two finite groups and let $H\leq G_1\times G_2$.
Is it true that $H=P_1\times P_2$ with $P_i\leq G_i$?
I think so for since $H\subseteq G_1\times G_2$ we must have $H=P_1\times P_2$ with $P_i\subseteq G_i$. We have $P_i\neq \phi$, otherwise $H=\phi$. Indeed, $(1_{G_1}, 1_{G_2})\in H$ for $H\leq G_1\times G_2$.  Now, if $a, b\in P_1$ then $$(ab, 1_{G_2})=(a, 1_{G_2})\cdot (b, 1_{G_2})\in H=P_1\times P_2$$ so that $ab\in P_1$. A similar reason would work for the inverse.
I read somewhere this was not true, but I can't see what is wrong in the above argument.
Obs: The group structure on $G_1\times G_2$ is given pointwise.

REPLY [1 votes]: Andrea Mori has given you a counterexample, so I'll try to comment on your attempted argument.
It looks like you have shown that if $P_1 \leq G_1$ and $P_2 \leq G_2$, then $P_1 \times P_2 \leq G_1 \times G_2$.  Let's see what happens when we try to prove the converse.
Suppose $H \leq G_1 \times G_2$.  We aim to find $P_1 \leq G_1$ and $P_2 \leq G_2$ such that $H = P_1 \times P_2$.  Then certainly $P_1$ must contain the set $S_1 = \{x \in G_1 : (x,y) \in H \text{ for some } y \in G_2\}$ and $P_2$ must contain $S_2 = \{y \in G_2 : (x,y) \in H \text{ for some } x \in G_1\}$.  In fact, one can show these sets $S_1, S_2$ are subgroups of $G_1$ and $G_2$, respectively.  We shouldn't take anything extraneous in $P_1$ and $P_2$, so let's just try letting $P_1$ and $P_2$ being these subgroups, i.e.,
\begin{align*}
P_1 &= \{x \in G_1 : (x,y) \in H \text{ for some } y \in G_2\}\\
P_2 &= \{y \in G_2 : (x,y) \in H \text{ for some } x \in G_1\} \, .
\end{align*}
Certainly $H \subseteq P_1 \times P_2$: given $(x,y) \in H$, then $x \in P_1$ and $y \in P_2$ by definition, so $(x,y) \in P_1 \times P_2$.  But is it true that $H \supseteq P_1 \times P_2$?  Given $(x,y) \in P_1 \times P_2$, then there exist $v \in G_2$ and $u \in G_1$ such that $(x,v) \in H$ and $(u,y) \in H$.  But there is no reason that $(x,y) \in H$, and in general this is false.  For instance, in the case of the diagonal subgroup $\Delta=\{(g,g)\mid g\in G\}$ from Andrea Mori's answer, we have $P_1 = P_2 = G$, so $P_1 \times P_2 = G \times G$, which is much bigger than $\Delta$.
Here's a way to think about this geometrically.  The diagonal subgroup is like the line $y = x$ in the plane.  If we take $P_1$ and $P_2$ to be the projections of the line onto the $x$- and $y$-axes, does taking $P_1 \times P_2$ give us the line back?  No, of course not: we get the entire plane.