text:
\begin{document}

\begin{abstract}
 We prove that for a certain class of $n$ dimensional rank one  locally symmetric spaces, if $f \in L^p$, $1\leq p \leq 2$, then the Riesz means of order $z$ of $f$ converge to $f$ almost everywhere, for $\operatorname{Re}z> (n-1)(1/p-1/2).$
\end{abstract}

\dedicatory{To the memory of Professor Michel Marias.}

\title[Riesz means]{Riesz means on locally symmetric spaces}
\author{
	Effie Papageorgiou
}
\title{Riesz means on   locally symmetric spaces}
\curraddr{Department of Mathematics, Aristotle University of Thessaloniki,
Thessaloniki 54.124, Greece }
\subjclass[2020]{42B15, 42B08, 22E30, 22E40}
\keywords{Riesz means, symmetric spaces, locally symmetric spaces, almost everywhere convergence}
\maketitle

\section{Introduction and statement of the results}

In the present work we deal with the almost everywhere convergence of the Riesz means. We extend the results on rank one symmetric spaces in \cite{GIUMA} to a class of rank one locally symmetric spaces. Recall that non-compact rank one symmetric spaces are the real, complex and quaternionic hyperbolic spaces,  and the octonionic hyperbolic plane.

Let $G$ be a semi-simple, non-compact, connected Lie group with finite
center and let $K$ be a maximal compact subgroup of $G$. Consider the
symmetric space of non-compact type $X=G/K$. Let $\dim X=n$. Denote by $\mathfrak{g}$ and $\mathfrak{k}$ the Lie algebras of $G$ and $K$, respectively. We have the Cartan decomposition $\mathfrak{g}=\mathfrak{p} \oplus  \mathfrak{k}$. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$ and let $\mathfrak{a}^*$ be its dual. If $\mathfrak{a} \cong \mathbb{R}$, then we say that $X$ has rank one.

Consider the multiplier
\begin{equation}  \label{mult}
s_{R}^z(\lambda)=\left( 1-\frac{\rho^2+\lambda^2}{R}\right)^z_{+}, \; \lambda \in \mathfrak{a^*},
\end{equation}
where $\rho $ is the half sum of positive roots, counted with their
multiplicities, $R\geq \rho^2$ and $z\in \mathbb{C}$ with $\operatorname{Re}z\geq 0$. Denote by $\kappa_{R}^z$ the inverse spherical Fourier transform of
$s_{R}^z$ in the sense of distributions and consider the Riesz means on $X$ to be the convolution operator $S_R^z$:
\begin{equation}
S_R^z(f)(x)=\int_{G}\kappa _R^z(y^{-1}x)f(y)dy,\quad f\in C_{0}(X).  \label{operatorX}
\end{equation}

Let $\Gamma $ be a discrete and torsion free subgroup of $G$ and consider the locally symmetric space $M=\Gamma \backslash X=\Gamma
\backslash G/K.$ Then $M$, equipped with the projection of the canonical
Riemannian structure of $X$, becomes a Riemannian manifold.

To define Riesz means on $M$, we first observe that if $f\in
C_{0}^{\infty }(M)$, then the function $S_{R}^{z }f$ defined by (\ref{operatorX}) is right $K$-invariant and left $\Gamma $-invariant. So
$S_{R}^{z }$ can be considered as an operator acting on functions on $M,$ which we shall denote by $\widehat{S}_{R}^{z }$.
Let
\[\
P_s(x, y)=\sum_{ \gamma \in \Gamma}e^{-sd(x,\gamma y)}, \quad s>0,
\]
denote the Poincar\'{e} series.
The critical exponent $\delta(\Gamma)$ is defined by
\begin{equation*}
\delta (\Gamma)=\inf \{ s>0: P_s(x,y)< +\infty \}.
\end{equation*}
Note that  $\delta(\Gamma)\leq 2\rho$, \cite{Weber}.

The counting function $N_{\Gamma}$ of $\Gamma$  is given by
\begin{equation}\label{counting}
N_{\Gamma}(x,y,A)=\# \{ \gamma \in \Gamma:\;d(x,\gamma y)\leq A \}, \quad x,y\in X, \; A>0,
\end{equation}
where  $\# E$ denotes the cardinality of the set $E$.

	We say that a rank one locally symmetric space $M = \Gamma \backslash G/K$ belongs in the \textit{class (R)} if

	(i) $\delta(\Gamma)< \rho$, and

	(ii) the counting function  satisfies the following uniform asymptotics: there is a constant $c > 0$, such that for all $x, y \in X,$
	\begin{equation}\label{asymptotics}
	\lim_{A\rightarrow +\infty}
	\frac{N_{\Gamma} (x, y, A)} {e^{\delta(\Gamma)A}} = c.
	\end{equation}

Our main result is the following.

\begin{theorem}\label{Thm}
	Let $1\leq p \leq 2$. If $M \in (R)$ and $\operatorname{Re}z>(n-1)(\frac{1}{p}-\frac{1}{2})$, then
	\[\
	\lim\limits_{R\rightarrow +\infty}\widehat{S}_R^zf(x)=f(x), \; \text{a.e., for } f\in L^p(M).
	\]
\end{theorem}

Theorem \ref{Thm} extends the results of \cite{GIUMA} in the class (R) of rank one locally symmetric spaces. Note that (\ref{asymptotics}) holds for a rich class of groups, e.g. when  $\Gamma$ is convex co-compact (including uniform lattices and Schottky groups with no parabolic element) or geometrically finite of divergence type with at least one parabolic element. For more details on the class of groups satisfying (\ref{asymptotics}), see \cite{FOMAMA} and the references therein.

The Riesz means have already been extensively studied in the case of $\textbf{R}^n$ (\cite{CHRIST8,CHRIST7,DAVCHANG,STEIN29}). The case of Lie groups and Riemannian manifolds of nonnegative curvature is treated in \cite{ALEXOLO} and the case of elliptic differential operators on compact manifolds in \cite{Berard,ChristSogge,GIUTRA,HORM,Sogge}. For the case of rank one noncompact symmetric spaces see \cite{GIUMA} and for the case of $SL(3, \mathbb{H})/Sp(3)$ see \cite{zhu}.

This paper is organized as follows. In Section 2 we present the necessary
preliminaries, and in Section 3 we prove Theorem \ref{Thm}. For the proof of Theorem \ref{Thm}, we study the mapping properties of the maximal operator associated with Riesz means. For the local part, we prove that the maximal operator is dominated by the Hardy-Littlewood maximal operator. For the part at infinity, we use a summation argument and the properties implied by (\ref{asymptotics}).

Throughout this article, the different constants will always be denoted by the same letter $c$.

\section{Preliminaries}
In this section we recall some basic facts about symmetric spaces. For more
details see for example \cite{ANprelim, FOMAMA, HEL}.
\subsection{Symmetric spaces}
Let $G$ be a semisimple Lie group, connected, noncompact, with finite center
and let $K$ be a maximal compact subgroup of $G$. We denote by $X$ the noncompact symmetric space $G/K$. In the sequel we
assume that $\operatorname{dim}X=n$. Denote by $\mathfrak{g}$ and $\mathfrak{k}$ the Lie
algebras of $G$ and $K$. Let also $\mathfrak{p}$ be the subspace of $
\mathfrak{g}$ which is orthogonal to $\mathfrak{k}$ with respect to the
Killing form. The Killing form induces a $K$-invariant scalar product on $
\mathfrak{p}$ and hence a $G$-invariant metric on $G/K$. Denote  by $d(.,.)$ the Riemannian
distance and by $dx$ the associated Riemannian measure on $X$.

Fix $\mathfrak{a}
$ a maximal abelian subspace of $\mathfrak{p}$ and denote by $\mathfrak{a}
^{\ast }$ the real dual of $\mathfrak{a}$. If $\operatorname{dim}\mathfrak{a}=l$, we
say that $X$ has rank $l$. We also say that $\alpha \in \mathfrak{a}^{\ast }$ is a root vector, if
\begin{equation*}
\mathfrak{g}^{\alpha }=\left\{ X\in \mathfrak{g}:[H,X]=\alpha (H)X,\text{
	for all }H\in \mathfrak{a}\right\} \neq \left\{ 0\right\} .
\end{equation*}
Denote by $\rho $ the half sum of positive roots, counted with their
multiplicities.

From now on, we assume that $\operatorname{rank}X=1$. Let $A$ be the analytic subgroup of $G$ with Lie
algebra $\mathfrak{a}\cong \mathbb{R}$. Let $\mathfrak{a}_{+}\subset \mathfrak{a}$ be a positive Weyl chamber and let $
\overline{\mathfrak{a}_{+}}$ be its closure. We have the Cartan decomposition
\begin{equation}
G=K\exp \overline{\mathfrak{a}_{+}}K.  \label{kak}
\end{equation}
Then, each element $x \in G $ is written uniquely as $x =k_1(\exp H)k_2$.  Denote by $x_0 = eK$ a base point of $X$. If we choose points $x,y$ in the homogeneous space $X$, there are isometries $g,h \in G $ such that $x = gx_0$ and $y = hx_0$. Because of the Cartan decomposition (\ref{kak}) of the Lie group $G$, there are $k, k^{\prime}\in K$ and a unique $H \in \overline{\mathfrak{a}_{+}}$ with $g^{-1}h=k \exp H k^{\prime}$ . Then,
\[\
d(x, y)=\|H\|,
\]
where $d(x,y)$ is the distance on $X$, \cite{Weber}.

Normalize the Haar measure $dk$ of $K$ such that $\int_{K}dk=1$. Then, from
the Cartan decomposition, it follows that
\begin{equation*}
\int_{G}f(g)dg=\int_{K}dk_{1}\int_{{\mathfrak{a}_{+}}}{\delta (H)}
dH\int_{K}f(k_{1}\exp (H)k_{2})dk_{2},
\end{equation*}
where the modular function $\delta (H)$ satisfies the estimate \begin{equation}\label{modular}\delta(H)\leq ce^{2\rho \|H\|}.\end{equation} We identify functions on $X = G/K$ with functions on $G$ which are $K$-invariant on the right, and hence bi-$K$-invariant functions on $G$, with functions on $X$, $K$-invariants on the left. Note that if $f$ is $K$-bi-invariant, then
\begin{equation}
\int_{G}f\left( g\right) dg=\int_{X}f\left( x\right) dx=c\int_{{\mathfrak{a}_{+}}}f(\exp H)\delta(H)dH.  \label{rightinv}
\end{equation}

\subsection{The spherical Fourier transform}

Denote by $S(K\backslash G/K)$ the Schwartz space of $K$-bi-invariant
functions on $G$. The spherical Fourier transform $\mathcal{H}$ is defined
by
\begin{equation*}
\ \mathcal{H}f(\lambda )=\int_{G}f(x)\varphi _{\lambda }(x)\;dx,\quad
\lambda \in \mathfrak{a^{\ast }},\quad f\in S(K\backslash G/K),
\end{equation*}
where $\varphi _{\lambda }$ are the elementary spherical functions on $G$.
Let $S(\mathfrak{a^{\ast }})$ be the usual Schwartz space on $\mathfrak{
	a^{\ast}}$. Denote by $W$ the Weyl group
associated to the root system of $(\mathfrak{g},\mathfrak{a})$ and let $S(\mathfrak{a^{\ast }})^{W}$ be the subspace
of $W$-invariants in $S(\mathfrak{a^{\ast }})$. Then, by a
celebrated theorem of Harish-Chandra, $\mathcal{H}$ is an isomorphism
between $S(K\backslash G/K)$ and $S(\mathfrak{a^{\ast }})^W$. Its inverse
is given by
\begin{equation*}
\ (\mathcal{H}^{-1}f)(x)=c\int_{\mathfrak{a\ast }}f(\lambda )\varphi
_{-\lambda }(x)\frac{d\lambda }{|\mathbf{c}(\lambda )|^{2}},\quad x\in
G,\quad f\in S(\mathfrak{a^{\ast }})^{W},
\end{equation*}
where $\mathbf{c}(\lambda )$ is the Harish-Chandra function.

\subsection{The counting function}  Set
  \begin{equation}\label{countingset}
\Gamma_A(x, y) =\{ \gamma \in \Gamma: d(x,\gamma y) \leq  A\}, \quad x, y \in X, \; A > 0,
  \end{equation}and recall that the counting function of $\Gamma$ is
$N_{\Gamma}(x,y,A)=\# \Gamma_A(x, y)$.

In our case, the counting function $N_{\Gamma} (x, y, A)$ satisfies
\begin{equation}\label{asymptotics1}
\lim_{A\rightarrow +\infty}
\frac{N_{\Gamma} (x, y, A)} {e^{\delta(\Gamma)A}} = c.
\end{equation}
Note that (\ref{asymptotics1}) implies that we have
 \begin{equation}\label{asymptotics2}
 N_{\Gamma}(x,y,A)\leq c e^{\delta(\Gamma)A}.
 \end{equation}

 In \cite[Lemma 6]{FOMAMA}, it is proved the following result.
\begin{lemma}
	Assume that $\Gamma$ satisfies (\ref{asymptotics1}). If $s> \delta(\Gamma)$, then there are positive constants $A$ and $c$ such that
	\begin{equation}\label{poincaresum}
	\sum_{ \gamma \in \Gamma}e^{-sd(x,\gamma y)}\leq c\sum_{ \gamma \in \Gamma_A(x,y)}e^{-sd(x,\gamma y)},
	\end{equation}
 for all $x, y \in X$.
\end{lemma}

Combining (\ref{asymptotics2}) and (\ref{poincaresum}) for $s>\delta(\Gamma)$, we have
\begin{equation}\label{uniform}
\sum_{ \gamma \in \Gamma}e^{-sd(x,\gamma y)}\leq c N_{\Gamma}(x,y,A)\leq ce^{\delta(\Gamma)A}\leq c.
\end{equation}
\section{Proof of Theorem \ref{Thm}}
In this section we shall study the boundedness of the maximal operator
\begin{equation}\label{maximal}
\widehat{S}_{\ast}^zf(x)=\sup\limits_{R>0}|\widehat{S}_R^zf(x)|, \quad \text{for }f\in L^p(M), \quad 1\leq p\leq 2.
\end{equation}

Our aim is to prove the $L^p(M)-(L^p+L^r)(M)$ boundedness of $\widehat{S}_*^z$ in Theorem \ref{thmmapping}, which is our main result.
Then,  Theorem \ref{Thm} follows by intepolation and well-known measure theoretic arguments, see for example \cite[Theorem 2.1.14]{GRAF}.

The Riesz means kernel $\kappa_R^z$ on $X$ is given by
\begin{equation*}
\kappa_R^z(\exp H)=\mathcal{H}^{-1}\left(\left(1-\frac{\lambda^2+\rho^2}{R}\right)^z_{+}\right)\left(\exp H\right), \quad \lambda \in \mathfrak{a^{\ast }},\;H \in \overline{\mathfrak{a}	_{+}}.
\end{equation*}
 Using this formula and the expression of the inverse spherical Fourier transform $\mathcal{H}^{-1}$ in the case of rank one symmetric spaces, Giulini and Mauceri in \cite{GIUMA} obtained the following estimate of $\kappa_R^z$:
\begin{equation}\label{estimate}
|\kappa_R^z(\exp H)| \leq c(z)R^{n/2}(1+\sqrt{R}\|H\|)^{-\operatorname{Re}z-(n+1)/2}(1+\|H\|)^{(n-1)/2}e^{-\rho \|H\|}.
\end{equation}

 Recall now that the Riesz means operator on $M$ is initially defined as a convolution operator on $G$,
 \begin{equation}\label{33}
 (\widehat{S}_R^zf)(x)=\int_G\kappa_R^z(y^{-1}x)f(y)dy, \quad f\in C_0^{\infty}(M).
 \end{equation}
Set $\kappa_R^z(x,y) =\kappa_R^z(y^{-1}x) $ and
\begin{equation}\label{serieskernel}
\widehat{\kappa}_R^z(\tilde{x},\tilde{y})=\sum_{\gamma \in \Gamma}\kappa_R^z(x,\gamma y).
\end{equation}
We shall first prove the following result.
\begin{proposition}
	If $M\in (R)$, then the series (\ref{serieskernel}) converges and the Riesz means operator $\widehat{S}^z_R$ on $M$ is given by
	\begin{equation}\label{35}
	(\widehat{S}_R^z f) (\tilde{x}) =\int_M\widehat{\kappa}^z_R(\tilde{x},\tilde{y}) f (\tilde{y})d\tilde{y}.
	\end{equation}
\end{proposition}
\begin{proof}
	Use the Cartan decomposition and write $(\gamma y)^{-1}x =k_{\gamma} \exp H_{\gamma} k_{\gamma}^{\prime}$. Then, since $\kappa_{R}^z$ is $K$-bi-invariant, $\kappa_{R}^z((\gamma y)^{-1}x) =\kappa_{R}^z(\exp H_\gamma )$. The distance on $M$ is defined by
	\begin{equation}\label{distM}
	d_M(\tilde{x},\tilde{y})=\inf\limits_{\gamma \in \Gamma} d(x,\gamma y)=\|H_{\gamma}\|,
	\end{equation}
	where $\tilde{x}=\pi(x)$ and $\pi:X\rightarrow M$ denotes the covering map.

	Recall that $\delta(\Gamma)<\rho$. Then, estimate (\ref{estimate})  implies that  for $0<\epsilon <\rho-\delta(\Gamma)$,
	\begin{align}\label{(3)}
|\widehat{\kappa}_{R}^{z}(\tilde{x},\tilde{y})|&\leq c(z)R^{n/2}\sum_{ \gamma \in \Gamma}(1+\sqrt{R}\|H_{\gamma}\|)^{-\operatorname{Re}z-(n+1)/2}(1+\|H_{\gamma}\|)^{(n-1)/2}e^{-\rho \|H_{\gamma}\|} \notag \\
&\leq c(z)R^{n/2}\sum_{ \gamma \in \Gamma} (1+\sqrt{R}\|H_{\gamma}\|)^{-\operatorname{Re}z-(n+1)/2}e^{-(\rho-\epsilon) \|H_{\gamma}\|} \notag \\
&\leq c(z)R^{n/2} (1+\sqrt{R}d_M(\tilde{x}, \tilde{y}))^{-\operatorname{Re}z-(n+1)/2}\sum_{ \gamma \in \Gamma}e^{-(\rho-\epsilon) \|H_{\gamma}\|} \notag \\
&\leq c(z)R^{n/2} (1+\sqrt{R}d_M(\tilde{x}, \tilde{y}))^{-\operatorname{Re}z-(n+1)/2}P_{\rho-\epsilon}(x,y)<+\infty.
	\end{align}
	Thus, it remains to prove (\ref{35}). Since $\kappa_{R}^z$ and $f$ are right-$K$-invariant, from (\ref{33}) we get that
	\begin{equation*}
	(\widehat{S}_R^zf)(x)=\int_{X}\kappa_R^z(x,y)f(y)dy.
	\end{equation*}
	 Now, since $f$ is left $\Gamma$-invariant, by Weyl’s formula we find that
	 \begin{align*}
	 (\widehat{S}_R^zf)(x)&=\int_{X}\kappa_R^z(x,y)f(y)dy=\int_{\Gamma \backslash X}\left( \sum_{ \gamma \in \Gamma} \kappa_R^z(x,\gamma y)f(\gamma y)\right)d\tilde{y}\\&=\int_M \widehat{\kappa}_R^z(\tilde{x},\tilde{y})f(\tilde{y})d\tilde{y}.
	 \end{align*}
\end{proof}

 Next, we proceed as in \cite{GIUMA}, and we prove the following lemma.
\begin{lemma}
	\label{S*L2}  If $f\in L^2(M)$ and $z\in \mathbb{C}$ with $\operatorname{Re}z>0$,
	then $\|\widehat{S}_*^zf\|_2\leq c(z)\|f\|_2$.
\end{lemma}
\begin{proof}
Let $\widehat{H}_R$ be the heat semigroup on $M$. By \cite[Lemma 6]{LOMAjga}, it holds $\|\widehat{H}_R\|_{L^2(M)\rightarrow L^2(M)} =e^{-\lambda_0 R} \leq 1$, where $ \lambda_0 \geq 0$ is the bottom of the $L^2$-spectrum of the Laplacian on $M$. Thus, by \cite[Chapter III, MAXIMAL THEOREM]{STEINred}, the heat maximal operator $f\rightarrow \widehat{H}_{\ast}f:=\sup\limits_{R>0}|\widehat{H}_Rf|$ is bounded on $L^2 (M)$. Thus, it suffices to prove the boundedness of $(\widehat{S}^z-\widehat{H})_{\ast}$. Using the spectral theorem for $\Delta_M$ and the Mellin transform we have
	\begin{equation}\label{mellin}
	(\widehat{S}^z_R-\widehat{H}_R)f=\int_{\mathbb{R}}c(z,s)R^{-is}(-\Delta_M	)^{is}fds,
	\end{equation}
	where $|c(z,s)|\leq c(z)(1+|s|)^{-(\operatorname{Re}z+1)}$ \cite{GIUMA}. So, the integral in (\ref{mellin}) converges. Since $L^2(M)$ is a complete Banach lattice, from \cite{COW}, we can write
	\[\
	(\widehat{S}^z-\widehat{H})_{\ast}f=\sup_{R>0}|(\widehat{S}_R^z-\widehat{H}_R)f|\leq c(z) \int_{\mathbb{R}} (1+|s|)^{-(\operatorname{Re}z+1)}|(-\Delta_M)^{is}f|ds.
	\]
	Thus, since $\|(-\Delta_M)^{is}\|_{L^2(M)\rightarrow L^2(M)}\leq 1$ by the spectral theorem, we obtain
	\[\
	\|(\widehat{S}^z-\widehat{H})_{\ast}f\|_{L^2(M)}\leq c(z)\|f\|_{L^2(M)}.
	\]
\end{proof}
Our main result is the following.

\begin{theorem}\label{thmmapping}
Let $\operatorname{Re}z> (n - 1)/2$. There exists $q > 2$ such that if $1 < p \leq q^{\prime}$, then $\widehat{S}_{\ast}^z$ maps $L^p (M)$ continuously into $(L^p + L^r)(M)$ for every $r \in [qp^{\prime} /(p^{\prime}-q), + \infty]$. Moreover, $\widehat{S}_{\ast}^z$ maps $L^1(M)$ continuously into $(L^{1,w}+L^r)(M)$ for every $r\in[q, +\infty]$.
\end{theorem}
\begin{proof}
	We have
	\begin{align*}
	\widehat{S}_R^zf(\tilde{x})&=\int_{M}\widehat{\kappa}^z_R(\tilde{x}, \tilde{y})f(\tilde{y})d\tilde{y}\\
	&=\int_{B(\tilde{x},1)}\widehat{\kappa}^z_R(\tilde{x}, \tilde{y})f(\tilde{y})d\tilde{y}+\int_{B(\tilde{x},1)^c}\widehat{\kappa}^z_R(\tilde{x}, \tilde{y})f(\tilde{y})d\tilde{y}:=I_1+I_2.
	\end{align*}
	\textit{Estimates for $I_1$.} 	Let ${N}_{\psi}$ denote the maximal operator
	\begin{equation}\label{psi} \mathcal{N}_{\psi}f(\tilde{x})=\sup\limits_{t\in (0,1)}\psi(t)\int\limits_{B_M(\tilde{x},t)} |f (\tilde{y})|d\tilde{y}, \quad \psi(t)=t^{-n}, \quad t>0.
	\end{equation}
	Using (\ref{uniform}), (\ref{(3)}) and (\ref{psi}), we have
	\begin{align}
	|I_1|&\leq \int_{B(\tilde{x},1)}|\widehat{\kappa}_{R}^z(\tilde{x}, \tilde{y})||f(\tilde{y})|d\tilde{y} \notag \\&\leq c(z)R^{n/2} \int_{B(\tilde{x},1)}\frac{1}{(1+\sqrt{R}d_M(\tilde{x}, \tilde{y}))^{\operatorname{Re}z+(n+1)/2}}|f(\tilde{y})|d\tilde{y}\notag\\
	&= c(z)\int_{B(\tilde{x},1)}d_M(\tilde{x}, \tilde{y})^{-n} \frac{(\sqrt{R}d_M(\tilde{x}, \tilde{y}))^n}{(1+\sqrt{R}d_M(\tilde{x}, \tilde{y}))^{\operatorname{Re}z+(n+1)/2}}|f(\tilde{y})|d\tilde{y} \notag \\
	&= c(z) \sum\limits_{\nu=-\infty}^0\int_{2^{\nu-1}<d_M(\tilde{x},\tilde{y})\leq  2^{\nu}}d_M(\tilde{x}, \tilde{y})^{-n} \frac{(\sqrt{R}d_M(\tilde{x}, \tilde{y}))^n}{(1+\sqrt{R}d_M(\tilde{x}, \tilde{y}))^{\operatorname{Re}z+(n+1)/2}}|f(\tilde{y})|d\tilde{y} \notag \\
	&\leq c(z) \sum\limits_{\nu=-\infty}^0\int_{2^{\nu-1}<d_M(\tilde{x},\tilde{y})\leq  2^{\nu}} 2^{-n(\nu-1)}\frac{(\sqrt{R}2^{\nu})^{n}}{(1+\sqrt{R}2^{\nu-1})^{\operatorname{Re}z+(n+1)/2}}|f(\tilde{y})|d\tilde{y}\notag \\
	&\leq c(z) \sum\limits_{\nu=-\infty}^0 \frac{(\sqrt{R}2^{\nu})^{n}}{(1+\sqrt{R}2^{\nu-1})^{\operatorname{Re}z+(n+1)/2}}\psi(2^{\nu})\int_{d_M(\tilde{x},\tilde{y})\leq 2^{\nu}}|f(\tilde{y})|d\tilde{y}\notag\\
	&\leq c(z) \mathcal{N}_{\psi}f(\tilde{x})\sum\limits_{\nu=-\infty}^0 \frac{(\sqrt{R}2^{\nu})^{n}}{(1+\sqrt{R}2^{\nu-1})^{\operatorname{Re}z+(n+1)/2}}.\label{seira}
	\end{align}
	Let $2^k<\sqrt{R}\leq 2^{k+1}$, for some $k\in \mathbb{Z}$. Thus, $\sqrt{R}=2^{k+\epsilon}$, for some $\epsilon \in (0,1]$. Then, note that
	\begin{align}
	\sum\limits_{\nu=-\infty}^0\frac{(\sqrt{R}2^{\nu})^{n}}{(1+\sqrt{R}2^{\nu-1})^{\operatorname{Re}z+(n+1)/2}}&=\sum\limits_{\nu=-\infty}^0\frac{(2^{k+\nu+\epsilon})^n}{(1+2^{k+\nu-1+\epsilon})^{\operatorname{Re}z+(n+1)/2}} \notag \\
	&\leq \sum\limits_{\ell=-\infty}^{+\infty}\frac{(2^{\ell+\epsilon})^n}{(1+2^{\ell-1+\epsilon})^{\operatorname{Re}z+(n+1)/2}}\notag\\
	& \leq c\sum\limits_{\ell=-\infty}^{0}(2^{\ell})^n+c\sum\limits_{\ell=0}^{+\infty}(2^{\ell})^{-\operatorname{Re}z+(n-1)/2}<c, \label{newsum}
	\end{align}
	provided that $\operatorname{Re}z>(n-1)/2$. Combining (\ref{seira}) and (\ref{newsum}), it follows that
	\begin{equation}\label{local}
	|I_1|\leq c(z)\mathcal{N}_{\psi}f(\tilde{x}).
	\end{equation}

	Since $\psi(t)$ behaves like $|B_M(\tilde{x},t)|^{-1}$, as $t\rightarrow 0^+$, it follows that $|I_1|$ is dominated by the Hardy-Littlewood maximal function over the balls of radius at most $1$. A standard covering lemma shows that the maximal operator $\mathcal{N}_{\psi}$ is of weak type $1-1$ and is bounded on $L^p (M)$ for every $1<p\leq \infty$.

	\textit{Estimates for $I_2$.} Using (\ref{(3)}), we have
	\begin{align}
|I_2|&\leq \int_{B(\tilde{x},1)^c}|\kappa_{R}^z(\tilde{x}, \tilde{y})||f(\tilde{y})|d\tilde{y} \notag \\
& \leq c(z) \int_{B(\tilde{x},1)^c}\sum_{ \gamma \in \Gamma} R^{n/2} (1+\sqrt{R}\|H_{\gamma}\|)^{-\operatorname{Re}z-(n+1)/2}e^{-(\rho-\epsilon)\|H_{\gamma}\|}|f(\tilde{y})|d\tilde{y}. \label{inf1}
	\end{align}
Note that since $d_M(\tilde{x}, \tilde{y})>1$, it follows that $\|H_{\gamma}\|>1$ for every $\gamma \in \Gamma$. Thus, taking into account that $R\geq \|\rho\|^2$, we have
\begin{equation}\label{inf2} R^{n/2} (1+\sqrt{R}\|H_{\gamma}\|)^{-\operatorname{Re}z-(n+1)/2}\leq (\sqrt{R})^{-\operatorname{Re}z+(n-1)/2}\leq c_{\rho},
\end{equation}
provided that $\operatorname{Re}z>(n-1)/2$.

Let $T$ be the integral operator on $M$ with kernel
\begin{equation}\label{kernelpoin}\widehat{\kappa}(\tilde{x}, \tilde{y})=P_{\rho-\epsilon}(x,y).\end{equation} Then, by (\ref{inf1}) and (\ref{inf2}), it follows that
	\begin{equation*}\label{I2}
|I_2| \leq c(z) T(|f|)(\tilde{x}).
\end{equation*}
 We claim that $\widehat{\kappa}\in L^q(M)$, for every $q>2$. Then, by Young's inequality,  the operator $T$
 maps $L^p (M)$, $1 \leq p \leq  q^{\prime}$, continuously into $ L^r(M)$ for every $r \in [qp^{\prime} /(p^{\prime}-q), + \infty]$.

Using (\ref{poincaresum}), (\ref{asymptotics1}) and the fact that $\Gamma_A(x,y)$ is a finite set, we have
\begin{align}
\left( \sum_{ \gamma \in \Gamma}e^{-(\rho-\epsilon )d(x,\gamma y)} \right)^q
&\leq \left(c\sum_{ \gamma \in \Gamma_A}e^{-(\rho-\epsilon)d(x,\gamma y)} \right)^q \notag\\
&\leq c(\#\Gamma_A(x,y))^{q-1}\sum_{ \gamma \in \Gamma}e^{-q(\rho-\epsilon)d(x,\gamma y)} \notag\\
&\leq cN(x,y,A)^{q-1}\sum_{ \gamma \in \Gamma}e^{-q(\rho-\epsilon)d(x,\gamma y)} \notag\\
&\leq ce^{(q-1)\delta(\Gamma)}\sum_{ \gamma \in \Gamma}e^{-q(\rho-\epsilon)d(x,\gamma y)}.\label{43}
\end{align}

Using Weyl's formula, (\ref{modular})  and (\ref{rightinv}), we have that \begin{align}
\int_M \sum_{ \gamma \in \Gamma}e^{-q(\rho-\epsilon)d(x,\gamma y)} dy \notag&=\int_X e^{-q(\rho-\epsilon)d(x,y)}dy\\&=c\int_0^{\infty}e^{-r((q-2)\rho-q\epsilon)}dr<+\infty,\label{44}
\end{align}
for $\epsilon<\frac{(q-2)\rho}{q}$. Thus, combining (\ref{43}) and (\ref{44}), we obtain that for every $q>2$,
\begin{equation}\label{integr}
\int_M \left( \sum_{ \gamma \in \Gamma}e^{-(\rho-\epsilon )d(x,\gamma y)} \right)^q dy\leq c.\end{equation}

By (\ref{kernelpoin}) and (\ref{integr}), it follows that the claim that $\widehat{\kappa}\in L^q(M)$, $q>2$, is valid. Thus, the proof of Theorem \ref{thmmapping} is complete.
\end{proof}
Using complex interpolation, we have the following result.
\begin{theorem}\label{interpol}  	Let $1\leq p \leq 2$. If $M \in (R)$ and $\operatorname{Re}z>(n-1)(\frac{1}{p}-\frac{1}{2})$, then for every $r >pq/(2-p+pq-q)$, we have
	\[
	\|\widehat{S}_{\ast}^zf\|_{(L^p+L^r)(M)}\leq c(z)\|f\|_{L^p(M)}
	\]
for every $f\in L^p(M)$.
\end{theorem}

As a corollary of Theorem \ref{interpol} and standard measure-theoretic arguments, Theorem \ref{Thm} follows.

\section*{Acknowledgement}
	The author would like to thank Professor Anestis Fotiadis for his valuable
	remarks.


TITLE: Restricting the DeRham cohomology class of a submanifold to a coordinate neighborhood.
QUESTION [1 upvotes]: Suppose $M$ is an $n$-manifold and $A$ a $k$-dimensional submanifold, both compact and oriented.  Let the deRham cohomology class of $A$ be denoted $[\phi_A]$.  The class is defined by $$\int_M\phi_A\wedge\omega=\int_A\omega$$ for every closed $k$-form $\omega$ on $M$.
If $U$ is a coordinate neighborhood does it follow that $$\int_U\phi_A\wedge\omega=\int_{A\cap U}\omega$$ for every closed $k$-form $\omega$ on $M$?
Here we take restrictions of forms as necessary so that the statements make sense.

REPLY [2 votes]: Note that $\phi_A$ is defined up to an exact form form $d\alpha$ and
$$\int_U (\phi_A + d\alpha) \wedge \omega = \int_U \phi_A \wedge \omega + \int_U d(\alpha\wedge w)$$
and
$$\int_U d(\alpha\wedge \omega) = \int_{\partial U} \alpha \wedge \omega$$
So the LHS of your equation is not well defined if $\partial U \neq \emptyset$.
If you consider that $\omega$ is of compact support in $U$, then you can extend $\omega$ by $0$ to $M$, and everything reduces to your first equation.


text:
\section{Wonderful Models}\label{wonderful}
\subsection{The Geometric Definition of Building Sets and Nested Sets}
    \label{some}
     In this section we recall from \cite{DCP1}, \cite{DCP2}
     the  definitions of building set and nested set of subspaces.
    Let $V$ be a real or complex finite dimensional vector space

    endowed with an Euclidean or Hermitian non-degenerate product
    and let  $\A$ be a central subspace   arrangement in  $V$.
    For every $A\in V$, we will  denote by  $A^\perp$ its
orthogonal.
We denote   by ${\mathcal C}^{}_{\A}$ the closure  under the sum of $\A^{}$ and
by $\A^{\perp}$ the arrangement of subspaces in \(V\)
\[\A^\perp=\{A^\perp \:|\: A\in \A\}.\]

\begin{definition}
The  collection of subspaces $\G \subset {\mathcal C}^{}_{\A}$ is called  {\em building set associated to \(\A\)} if \({\mathcal C}^{}_{\A}= {\mathcal C}^{}_{\G}\) and  every element $C$ of ${\mathcal C}^{}_{\A}$ is the direct sum
 $C= G_1\oplus G_2\oplus \ldots\oplus G_k$ of  the maximal elements
$G_1,G_2,\ldots,G_k$  of
$\G^{}$ contained in $C$ (this is called the \(\G\)-decomposition of \(C\)).

\end{definition}

Given a subspace  arrangement  \(\A\), there are several building sets associated to it. Among these there always are a maximum and a minimum (with respect to inclusion). The maximum is ${\mathcal C}^{}_{\A}$, the minimum is the building set of {\em irreducibles} that is defined as follows.

\begin{definition}
 Given a subspace $U\in\Cc_\A$, a \textbf{decomposition of} $U$ in $\mathbf{\Cc_\A}$ is a collection
$\{U_1,\ldots,U_k\}$ ($k>1$) of non zero subspaces in $\Cc_\A$ such that
\begin{enumerate}
 \item $U=U_1\oplus\cdots\oplus U_k;$
 \item for every subspace $A\in\Cc_\A$ such that $A\subset U$, we have $A\cap U_1,\ldots,A\cap U_k \in \Cc_\A$ and
$A=\left(A\cap U_1\right)\oplus\cdots\oplus \left(A\cap U_k\right)$.
\end{enumerate}
\end{definition}
\begin{definition}
 A nonzero subspace $F\in\Cc_\A$ which does not admit a decomposition  is called \textbf{irreducible} and the set of
irreducible subspaces is denoted by $\mathbf{\Fc_\A}$.
\end{definition}
\begin{remark}
\label{rootirreducibles}
As an example, let us consider  a root system \(\Phi\) in \(V\) (real or complexified vector space) and its associated
 root  arrangement (i.e.  $\A^\perp$ is the hyperplane arrangement provided by the hyperplanes orthogonal to   the roots in  \(\Phi\)). In this case  the  building set of irreducibles is the set  whose elements are  the  subspaces spanned by  the irreducible root subsystems of \(\Phi\) (see \cite{YuzBasi}).

\end{remark}

 \begin{definition}
Let $\G$ be a building set associated to \(\A\).
A subset $\MS\subset \G$ is called
($\G$-)nested, if
 given any subset $\{U_1,\ldots ,U_h \}\subseteq \MS$ (with \(h>1\)) of pairwise non comparable
	elements, we have that $U_1 + \cdots + U_h\notin \G$.
\end{definition}

\subsection{The Example of the Root System \(A_{n-1}\)}
\label{secexamplean}

 Let \(V=\R^n\) or \(\C^n\) and let us consider  the real or complexified root arrangement of type \(A_{n-1}\).  We think of it  as an {\em essential} arrangement, i.e. we consider the hyperplanes defined by the equations \(x_i-x_j=0\) in the quotient space \(V/<(1,1,...,1)>\).

Let us denote by  $\Fc_{A_{n-1}}$ the building set of irreducibles associated to this arrangement. According to Remark \ref{rootirreducibles}, it is made  by all the subspaces in \(V\) spanned by the irreducible root subsystems.  Therefore there  is a bijective correspondence between the elements of $\Fc_{A_{n-1}}$ and the subsets of $\{1,\cdots,n\}$ of cardinality at
least two:  if the orthogonal of \(A\in \Fc_{A_{n-1}}\) is the subspace  described by the equation  \(x_{i_1}=x_{i_2}= \cdots =x_{i_k}\) then we represent \(A\) by the set \(\{i_1,i_2,\ldots, i_k\}\).

As a consequence,   a $\Fc_{A_{n-1}}$-nested set  \(\Sc\) is  represented by a set  (which we still call \(\Sc\)) of subsets of $\{1,\cdots,n\}$ with the property that any of its elements has  cardinality \(\geq 2\) and if \(I\) and \(J\) belong to \(\Sc\) than either \(I\cap J=\emptyset\) or one of the two sets is included into the other.

As an example we can consider  the  $\Fc_{{A_{8}}}$-nested set represented  by the  three sets \(\{1,5\}\), \(\{2,4,6\}\), \(\{2,4,6,7,8\}\). This means that the elements of the nested set are the  three subspaces whose orthogonal subspaces in  \(V/<(1,1,...,1)>\) are described respectively by the equations \(x_{1}=x_{5}\), \(x_{2}=x_{4} =x_{6}\) and
\(x_{2}=x_{4}= x_{6}=x_{7}=x_{8}\).

We observe that we  can  represent a $\Fc_{A_{n-1}}$-nested set   \(\MS\)
 by an oriented   forest on \(n\) leaves  in the following way. We consider the set \({\tilde \MS}=\MS\cup\{1\}\cup \{2\}\cup\cdots \cup \{n\}\).
Then the forest  coincides  with the Hasse diagram of  \({\tilde \MS}\) viewed as a poset by the  inclusion relation:  the roots of the trees correspond to  the maximal elements of \(\MS\), and the orientation goes from the roots to the leaves, that are the vertices \(\{1\},\{2\},\ldots, \{n\} \).

Let us now focus on the maximal building set \(\Cc_{A_{n-1}}\) associated with the root arrangement of type \(A_{n-1}\). It is made by all the subspaces that can be obtained as the span of a set of roots. Using the same notation as before, these subspaces can be put in bijective correspondence with the partitions of $\{1,\cdots,n\}$ such that at least one part has cardinality \(\geq 2\).

For instance,
\[   \{1,4\},\{2\},\{3,5,9\},\{6\},\{7,8\}\]
corresponds to the subspace whose orthogonal is described by the equations \(x_{1}=x_{4}\) and  \(x_{3}=x_{5} =x_{9}\) and
\(x_{7}=x_{8}\).

The \(\Cc_{A_{n-1}}\)-nested sets  are given by chains of subspaces in \(\Cc_{A_{n-1}}\) (with respect to   inclusion). In terms of partitions, this corresponds to give chains of the above described partitions of $\{1,\cdots,n\}$ (with respect to the refinement relation).

One can find in   \cite{GaiffiServenti} a description of the maximal model \(Y_{\Cc_{{A_{n-1}}}}\) and in \cite{GaiffiServenti2} a description of all the \(S_n\)-invariant building sets associated with the root system \(A_{n-1}\).

\subsection{The construction of wonderful models and their cohomology}
  \label{constru}
In this section we  recall  from   \cite{DCP1} the  construction and the main properties of the De Concini-Procesi models.

 The  interest in these models  was at first motivated by an  approach to Drinfeld's construction of special
solutions for Khniznik-Zamolodchikov equation (see \cite{drinfeld}).  Moreover, in \cite{DCP1} it was shown, using the cohomology description of these models, that  the rational homotopy type of the complement of a complex  subspace arrangement  depends only on the intersection lattice.

 Then real and complex  De Concini-Procesi models
turned out to play a key role in several   fields of mathematical research:  subspace and toric  arrangements,  toric varieties (see for instance \cite{DCP3}, \cite{feichtneryuz}, \cite{rains}),   tropical geometry (see \cite{feichtnersturmfels}),
moduli spaces  and       configuration spaces (see for instance \cite{etihenkamrai}, \cite{LTV}), box splines, vector partition functions  and  index theory (see  \cite{DCP4}, \cite{cavazzanimoci}),  discrete  geometry    (see \cite{feichtner}).

 Let us recall how they are defined. We will  focus on the case when \(\KK=\R\) or \(\KK=\C\).
 Let \(\A\) be a subspace arrangement in the real or complex space  \(V\)
 endowed with a non-degenerate euclidean or Hermitian product
  and let \(\emme(\A^\perp)\) be the  complement in \(V\) of the arrangement \(\A^\perp\).
  Let $\G$ be a building set associated to \(\A\) (we can suppose that it contains $V$).
 Then one considers   the map
\begin{displaymath}
	 i\: :\: \pp( \emme (\A^\perp)  ) \rightarrow  \pp(V) \times \prod
_{D\in
	\G-\{V\}}^{}\pp(V /D^\perp)
\end{displaymath}
where in the first coordinate we have the inclusion and the map from
$\emme (\A^\perp) $ to $\pp(V /D^\perp )$
 is the restriction of the
canonical projection ${\displaystyle (V-D^\perp) \rightarrow
\pp(V /D^\perp)}$.
\begin{definition}
The  (compact) wonderful  model
$ Y_{\G}$ is  obtained
by  taking the closure of the image of $ i$.
\end{definition}
De Concini and Procesi in \cite{DCP1} proved  that
  the complement $ { \mathcal D}$ of  $\pp( \emme (\A^\perp) ) $ in $
Y_{\G}$ is a divisor with normal crossings whose irreducible components are in bijective correspondence with the elements of \(\G-\{V\}\) and are denoted by \(\Dc_G\)
($G\in
	\G-\{V\}$).

If we denote by $\upi$ the
projection of $
Y_{\G}$ onto
	the first component \(  \pp(V)\),  then ${ \mathcal  D}_G$		can  be characterized as the unique
irreducible component such that
	$ \upi({ \mathcal  D}_G)=\pp(G^\perp)$.

	A complete characterization of the boundary is then provided by the
		observation that, if we consider  a collection \(
	{ \mathcal T} \) of subspaces in \( \G \) containing \(V\),  then  \[
	{ \mathcal D}_{{ \mathcal T}}= \bigcap_{A\in { \mathcal
	T}-\{V\}}{\mathcal D}_{A}\] is non empty if and only if \( {\mathcal T} \)
	is \(\G\)-nested, and in this case \(	{\mathcal D}_{{\mathcal T}}\)  is a smooth irreducible subvariety obtained as a normal crossing intersection.
Sometimes we will denote $Y_{\G}$ by
 ${ \mathcal D}_\MS$ with $\MS=\{V\}$.

The integer cohomology ring of the models \(Y_\G\)  in the complex case was studied  in  \cite{DCP1}, where  a  presentation by generators and relations  was provided. The cohomology is   torsion free, and in  \cite{YuzBasi}  Yuzvinski explicitly described  some $\Z$-bases (see also  \cite{GaiffiBlowups}).  We  briefly  recall these results (for a description of the cohomology ring in the real case see \cite{rains}).

Let \(\G\) be a building set containing \(V\) and let us consider a $\G$-nested set $\MS$.
We  take a
subset ${\cal H}\subset \G$  and an  element $B \in \G$
with the property that $A\subsetneq B$ for all $A\in {\cal H}$
and  we put  $\MS_B=\{A\in \MS\: :\:
A\subsetneq B\}$. As in \cite{DCP1}, we define the non negative integer
$	d_{{\cal H},B}^\MS$:
\begin{definition}
\begin{displaymath}
	d_{{\cal H},B}^\MS=dim B-dim
	\left( \sum _{A\in {\cal H}\cup \MS_B}^{}A \right)
\end{displaymath}
\end{definition}
Then we consider the polynomial ring $\Z[c_A]$ where the variables
$c_A$ are  indexed by the elements of $\G$.
\begin{definition} Given \(\G\), $\MS$, ${\cal H}$  and  $B$ as before, we define the following polynomial in $\Z[c_A]$:
\begin{displaymath}
	P_{{\cal H},B}^\MS=\left(\prod _{A\in {\cal H}}^{}c_A \right) \left(\sum _{B\subset
	C}^{}c_C \right)^{d_{{\cal H},B}^\MS}
\end{displaymath}
\end{definition}
Let us now call by $I_\MS$  the ideal in $\Z[c_A]$ generated by these
polynomials, for fixed $\MS$ and varying ${\cal H}, B$.
\begin{theorem}[{see \cite[Section 5.2]{DCP1}}]
Let $\G$  and   $\MS$ be as before, and let us consider the complex model \(Y_\G\). The natural map $\phi \: : \Z[c_A] \mapsto H^{\ast}(\Dc_\MS,\Z)$, defined by sending
$c_A$ to the cohomology class
$[\Dc_A]$ associated to the divisor $\Dc_A$ (restricted to $\Dc_S$), induces an
isomorphism  between ${\displaystyle \Z[c_A]/I_\MS}$ and $H^{\ast}(\Dc_\MS,\Z)$.

In particular,  in the case
$\MS=\{V\}$, we obtain
\begin{displaymath}
\Z[c_A]/I_{\{V\}}	\simeq H^{\ast}(Y_{\G},\Z)
\end{displaymath}
\label{coom}
\end{theorem}

\begin{definition}

 Let $\G$  and   $\MS$ be as before. A function $f\: :\: \G \: \mapsto \N $ is
called $\G,\MS$-admissible if it is $f=0$ or if $f\neq 0$, $supp\, f\cup \MS$ is
$\G$-nested and, for every $A\in supp\, f$,
$f(A)<d_{(supp\, f)_A,A}^{\MS}$.

\end{definition}

Now, given a $\G,\MS$-admissible function $f$, we can consider  in
$H^{\ast}({ \mathcal D}_\MS,\Z)\simeq \Z[c_A]/I_{\MS}$ the monomial
${\displaystyle m_f=\prod _{A\in \G}^{}c_A^{f(A)}}$. We will call
``$\G,\MS$-admissible'' such monomials.

\begin{theorem}[{see \cite[Section 3]{YuzBasi} and \cite[Section 2]{GaiffiBlowups}}]\label{base coomologia}
The set ${\cal B}_{\G,\MS}$ of $\G,\MS$-admissible monomials is a $\Z$-basis for
$ H^{\ast}({ \mathcal D}_\MS,\Z) $.
\end{theorem}

\subsection{A More General Construction}
\label{secgeneralblowup}

The construction of De Concini-Procesi models  can be viewed as a  special case of  other more general constructions that, starting from a `good' stratified variety, produce models by blowing up a suitable subset of  strata. Among these there are  the models  described by MacPherson and Procesi in \cite{procesimacpherson} and by  Li in \cite{li}.

 In Li's paper one can also find a  comparison among several  constructions of wonderful compactifications by  Fulton-Machperson (\cite{fultonmacpherson}),  Ulyanov (\cite{Ulyanov}), Kuperberg-Thurston (\cite{Kuperbergthurston}), Hu (\cite{Hu}). A further interesting survey including tropical compactifications can be found in Denham's paper \cite{denham}.

We recall here some basic facts adopting the language  and the notation of Li's paper.

\begin{definition}\label{def:simple} A simple arrangement of subvarieties (or `simple stratification') of a nonsingular variety \(Y\) is a finite set \(\Lambda = \{\Lambda_i\}\) of nonsingular closed subvarieties \(\Lambda_i\) properly contained in \(Y\) satisfying the following conditions:\\
(i) the  intersection of \(\Lambda_i\) and \(\Lambda_j\)  is nonsingular and the tangent bundles satisfy \(T(\Lambda_i\cap \Lambda_j) = T(\Lambda_i)_{|(\Lambda_i\cap \Lambda_j)}\cap T(\Lambda_j)_{|(\Lambda_i\cap \Lambda_j)}\),\\
(ii) \(\Lambda_i \cap \Lambda_j\) either is equal to some stratum in \(\Lambda\)  or is empty.

\end{definition}

\begin{definition}
Let \(\Lambda\) be an arrangement of subvarieties of  \(Y\).  A subset \(\G' \subseteq \Lambda\) is called a building set of \(\Lambda\)  if \(\forall \Lambda_i \in \Lambda - \G'\)  the minimal elements in \(\{G  \in \G' \: : \: G \supseteq \Lambda_i\}\)  intersect transversally and the intersection is \(\Lambda_i\).

\end{definition}

Then, if one has a simple stratification  \(\Lambda\) of a nonsingular variety \(Y\) and a  building set \(\G'\), one can construct a wonderful model \(Y_{\G'}\)   by  considering  (by  analogy with \cite{DCP1})  the closure of the image of the  locally closed embedding
\[\left( Y-\bigcup_{\Lambda_i\in \Lambda}\Lambda_i \right ) \rightarrow \prod_{G\in \G'}Bl_GY\] where \(Bl_GY\) is the blowup of \(Y\) along \(G\).

It turns out that

\begin{theorem}[{see \cite[Theorem 1.3]{li}}]
\label{teo:listabuilding}
If one arranges the elements \(G_1,G_2,...,G_N\)  of \(\G'\) in such a way that for every \(1\leq i \leq N\) the set \(\{G_1,G_2,\ldots , G_i\}\) is building, then \(Y_{\G'}\) is isomorphic to the variety  \[Bl_{{\widetilde G_N}}Bl_{{\widetilde G_{N-1}}}\
\cdots Bl_{{\widetilde G_2}}Bl_{G_1}Y\]
where \({\widetilde G_i}\) denotes the dominant transform of $G_i$ in $Bl_{{\widetilde G_{i-1}}}
\cdots Bl_{{\widetilde G_2}}Bl_{G_1}Y$.
\end{theorem}

\begin{remark}
\label{rem:ordinescoppiamenti}
As remarked by Procesi-MacPherson in \cite[Section 2.4]{procesimacpherson} it is
always possible to choose a linear ordering on the set $\G'$ such that every initial segment is building. We can do this by ordering $\G'$ in such a way that we always blow up first the strata of smaller  dimension.
\end{remark}

We show two examples that will be crucial in the following sections.

\begin{example}
In the case of    subspace arrangements, the De Concini-Procesi construction
and the above construction produce the same models  (the only warning is that in the preceding sections a building set \(\G\) was described in a dual way,  so the building set of subvarieties \(\G'\) is made by  the  orthogonals of the subspaces in  \(\G\)).

\end{example}

\begin{example}\label{es:boundary}
Given a  De Concini-Procesi   model \(Y_\G\), we notice that its boundary strata   give rise to a simple arrangement of subvarieties, and that the set of all strata is a building set. So it is possibile to  obtain a `model of the model \(Y_\G\)'.

The boundary strata of these `models of models' are indexed by the nested sets of the building set of all the strata of \(Y_\G\).  More precisely, according to the definition given in \cite[Section 4]{procesimacpherson}, a nested set \(\Sc\) in this sense is a collection of \(\G\)-nested sets containing $\{V\}$ linearly ordered by inclusion
(we will come back to this, using a more combinatorial definition,  in Section \ref{sec:buildingcombinatorial}).

\end{example}

config: arxiv


TITLE: Compactness of a subset of $W^{2, \infty}(\mathbb R^d, \mathbb R^d)$ in $L^2(\mathbb R^d, \mathbb R^d)$
QUESTION [0 upvotes]: i would like to know if the following subset in $W^{2, \infty}(\mathbb R^d, \mathbb R^d)$
$$
\mathcal B:=\{b\in W^{2, \infty}(\mathbb R^d, \mathbb R^d):||b||_{L^{\infty}(\mathbb R^d, \mathbb R^d)}\leq M,\ ||\operatorname{div}b\|_{L^{\infty}(\mathbb R^d)}\leq M\}\quad\text{for a given $M>0$}
$$
is compact in $L^2(\mathbb R^d, \mathbb R^d)$.
I think that this is not true because the only hypothesis on $\operatorname{div}$ is not enough to guarantee the equi-boundedness of derivatives of $b$. Indeed i think that if i have a subsequence $b_n\in W^{1, \infty}(\mathbb R^d, \mathbb R^d)$ such that $||\operatorname{div}b_n||_{L^{\infty}(\mathbb R^d)}\leq M$ and $\|b_n\|_{L^{\infty}(\mathbb R^d, \mathbb R^d)}\leq M$ then this does not imply that $\|b_n\|_{W^{1, \infty}(\mathbb R^d, \mathbb R^d)}\leq M$. If this was true, i would have that $\mathcal B\subset\subset W^{1, \infty}\subset\subset L^2$ on compact subsets. Could you help me/give me some useful suggestion?
Thank you

REPLY [1 votes]: You cannot have compactness for function spaces on all of $\mathbb R^n$. Take $\phi$ a non-zero smooth function, supported on $(0,1)$. Then define the sequence $\phi_n(x):=\phi(x+n)$. It will not have a converging subsequence.


TITLE: Fast way to calculate Fraction
QUESTION [1 upvotes]: I want to solve this problem not by using common denominator. Is there any innovative and fast way ?.
$$
{1 \over 6} + {4 \over 21} + {5 \over 84} + {7 \over 228} + {9 \over 532} =\
{\large ?}
$$

REPLY [3 votes]: You can avoid converting to the single common denominator $1596$ here by grouping terms in a way that creates some nice cancellations:
$$\begin{align}
{1\over6}+{4\over21}+{5\over84}+{7\over228}+{9\over532}&={1\over6}+\left({4\over21}+{5\over4\cdot21} \right)+\left({7\over3\cdot76}+{9\over7\cdot76} \right)\\
&={1\over6}+{16+5\over4\cdot21}+{49+27\over21\cdot76}\\
&={1\over6}+{21\over4\cdot21}+{76\over21\cdot76}\\
&={1\over6}+{1\over4}+{1\over21}\\
\end{align}$$
From here it's fairly easy to get
$${1\over6}+{1\over4}+{1\over21}={5\over4\cdot3}+{1\over7\cdot3}
={35+4\over28\cdot3}
={13\over28}$$
Whether this is innovative or fast is hard to say.  But I think it beats computing
$${1\cdot266+4\cdot76+5\cdot19+7\cdot3+9\cdot7\over3\cdot4\cdot7\cdot19}$$


TITLE: How to prove this limit of two sequences
QUESTION [2 upvotes]: suppose there exists a $k>0$ and two sequences $a_n$ and $b_n$ where $n$ from $1$ to $\infty$ such that two sequences $(a_n=b_n)$ for all $n>k$. I want to show that if $(a_n)$ converge, $(b_n)$ also converges.
Since $a_n$ is convergent, so there exists a $N>0$ such that for all $\epsilon$, $|a_n-L| < \epsilon$ for all $n>N$. For this prove, I don't know if I can just pick $N\geq k$ and conclude that $|b_n-L| < \epsilon$ for all $n>N\geq k$ because $(a_n=b_n)$ for all $n>k$
How do I continue this proof to make it correct? Thank you

REPLY [1 votes]: If $N$ satisfies the condition of convergence of $a_n$, then so does any larger number, and you can replace $N$ by $\max(N,K)$, which will satisfy the condition of convergence for $b_n$.