text:
\section{Introduction and statement of main results}

In this paper, we shall be concerned with the local time of an isotropic
Gaussian random field $\mathbf{T}=\left\{ \mathbf{T}\left( x\right) ,\,x\i
\mathbb{S}^{2}\right\} $ with values in $\mathbb{R}^{d}$ defined on some
probability space $\left( \Omega ,\Im, \mathbb{P}\right)$ by
\begin{equation}  \label{Def:T}
\mathbf{T}\left( x\right) =\left( T_{1}\left( x\right) ,\ldots
,T_{d}\left(x\right) \right) ,\quad x\in \mathbb{S}^{2},
\end{equation}
where $T_{1},\ldots ,T_{d}$ are independent copies of $T_{0}=\left\{T_{0
\left( x\right) ,\text{ }x\in \mathbb{S}^{2}\right\} $, which is a zero-mean
Gaussian random field that satisfies
\begin{equation}  \label{Eq:T0}
\mathbb{E}\left[ T_{0}\left( x\right) T_{0}\left( y\right) \right] =\mathbb{
}\left[ T_{0}\left( gx\right) T_{0}\left( gy\right) \right]
\end{equation
for all $g\in SO\left( 3\right) $ and all $x,y\in \mathbb{S}^{2}$. We call
T_{0}$ the Gaussian field associated with $\mathbf{T}.$

It follows from \eqref{Eq:T0} that the Gaussian field $T_{0}$ is both
2-weakly and strongly isotropic (cf. \cite{MPbook} p.121).
It is well-known (cf. \cite{MPbook}, p.123) that $T_{0}$ has the following
spectral representation
\begin{equation}  \label{rapT}
T_{0}(x) =\sum_{\ell \geq 0}\sum_{m=-\ell }^{\ell }a_{\ell m}Y_{\ell m}( x) ,
\end{equation
where
\begin{equation*}
a_{\ell m}=\int_{\mathbb{S}^{2}}T_{0}\left( x\right) \overline{Y}_{\ell
m}\left( x\right) d\nu \left( x\right) ,
\end{equation*
and $\nu $ is the canonical area measure on the sphere. In the spherical
coordinates $(\vartheta ,\varphi )\in \lbrack 0,\pi ]\times \lbrack 0,2\pi )
, $\nu (dx)=\sin \vartheta d\vartheta d\varphi $. Note that the equality in
\eqref{rapT} holds in both the $L^{2}\left( \mathbb{S}^{2}\times \Omega,d\nu
\left( x\right) \otimes \mathbb{P}\right) $ sense and the $L^{2}\left(
\mathbb{P}\right) $ sense for every fixed $x\in \mathbb{S}^{2}$. The set of
homogenous polynomials $\left\{ Y_{\ell m}:\ell \geq 0,m=-\ell ,...,\ell
\right\} $ are the spherical harmonic functions on $\mathbb{S}^{2},$
representing an orthonormal basis for the space $L^{2}\left( \mathbb{S
^{2},\nu \right) $. In this setting, the random coefficients $\{a_{\ell m},\
\ell \geq 0,m=-\ell ,...,\ell \}$ are Gaussian complex random variables such
that
\begin{equation*}
\begin{split}
\mathbb{E}\left[ a_{\ell m}\right] & =0\text{;} \\
\mathbb{E}\left[ a_{\ell m}\overline{a}_{\ell _{1}m_{1}}\right] &
=\delta_{\ell }^{\ell _{1}}\delta _{m}^{m_{1}}C_{\ell }\text{,}
\end{split
\end{equation*}
where the sequence $\left\{ C_{\ell }\right\} $ of nonnegative numbers is
called the angular power spectrum of $T_{0}$, which fully characterizes the
dependence structures of $T_{0}$. A celebrated theorem of Schoenberg \cit
{schoenberg1942} provides the following expansion for the covariance
function:
\begin{equation*}
\mathbb{E}\left[ T_{0}\left( x\right) T_{0}\left( y\right) \right]
=\sum_{\ell =0}^{+\infty }C_{\ell }\frac{2\ell +1}{4\pi }P_{\ell
}\left(\left\langle x,y\right\rangle \right) ,
\end{equation*}
where for $\ell =0,1,\,2,...,$ $P_{\ell }:[-1,1]\rightarrow \mathbb{R}$
denotes the Legendre polynomial satisfying the normalization condition
P_{\ell }(1)=1$. Thus for every $x\in \mathbb{S}^{2},$
\begin{equation*}
\mathbb{E}\left[ T_{0}\left( x\right) ^{2}\right] =\sum_{\ell
=0}^{+\infty}C_{\ell }\frac{2\ell +1}{4\pi }:=K.
\end{equation*}

For simplicity, we assume in this paper that $K=1.$ Otherwise, we consider
the random field $\{K^{-1/2}T_{0}\left( x\right) ,\text{ }x\in \mathbb{S
^{2}\}$, which does not cause any essential loss of generality.

As shown by Lang and Schwab \cite{lang1}, Lan et al. \cite{LanMarXiao} that
the degree of smoothness of the sample paths of $T_{0}$ can be precisely
characterized by the asymptotic behavior of the angular power spectrum
\{C_{\ell }\}$ at high multipoles $\ell >>0.$ In this paper, we further
illustrate this point by investigating regularity properties of the local
times of the vector-valued random field $\mathbf{T}$ in \eqref{Def:T}.
For this purpose, we recall the following condition from Lan et al. \cit
{LanMarXiao}:

\textbf{Condition (A)} The random field $T_{0}$ is a zero-mean, Gaussian and
isotropic random field indexed by $\mathbb{S}^{2}$, with unit variance and
angular power spectrum satisfying:
\begin{equation*}
C_{\ell }=C_{\ell }(G,\alpha )=G\left( \ell \right) \ell ^{-\alpha }>0,
\end{equation*}
where $\alpha >2$ and there exists a constant $K_{0}\geq 1$ such that for
all $\ell =1,2,...$
\begin{equation*}
K_{0}^{-1}\leq G\left( \ell \right) \leq K_{0}.
\end{equation*}

The regularity of the trajectories of the Gaussian field $T_{0}$ is governed
by $\alpha $. It has been shown in \cite{lang2,LanMarXiao,lang1} that the
sample function of $T_{0}$ is almost surely $k$-times continuous
differentiable if and only if $\alpha >2+2k$. Hence, the field $T_{0}$ is
twice differentiable almost surely (as needed for the Kac-Rice argument in
\cite{AdlerTaylor}, Theorem 12.1.1) if and only if $\alpha >6$.

In this paper we focus on the non-smooth regime $2<\alpha <4$. In this case,
the exact uniform and local moduli of continuity have recently been proved
by Lan et al. \cite{LanMarXiao}. We remark that the regime of $2<\alpha <4$
is indeed the most relevant for many areas of applications; in particular, a
major driving force for the analysis of spherical random fields has been
provided over the last decade by cosmological applications, for instance in
connection to the analysis of Cosmic Microwave Background radiation data
(CMB). Data analysis on CMB maps is currently an active research area, and
the geometry of CMB maps has been deeply investigated (see \cit
{planck2013a, planck2013c, planck2015c}). In this framework there are strong
theoretical and empirical evidence to support the belief that the values of
\alpha$ belong to the non-smooth region $2<\alpha <4$.

Our objective of the present paper is to establish the joint continuity, and
the uniform and local moduli of continuity for the local times of the
vector-valued random field $\mathbf{T}$. Based on these results, we show
that the sample functions of $\mathbf{T}$ and $T_{0}$ are a.s. nowhere
differentiable, and we determine the exact modulus of non-differentiability
of $\mathbf{T}$.

In order to state our main results, we need some notations. In spherical
coordinates $(\vartheta ,\varphi )\in \lbrack 0,\pi ]\times \lbrack 0,2\pi )
, every point $x\in {\mathbb{S}}^{2}$ can be written as $x=(\sin
\vartheta\cos \varphi ,\sin \vartheta \sin \varphi ,\cos \vartheta )$. In
this paper, we always identify the Cartesian and spherical coordinates of
x\in {\mathbb{S}}^{2}$. For any $x\in \mathbb{S}^{2},$ $0<r<\pi ,$ let
D(x,r)=\left\{y\in \mathbb{S}^{2}:d_{\mathbb{S}^{2}}(x,y)<r\right\} $ be an
open disk on $\mathbb{S}^{2},$ where $d_{\mathbb{S}^{2}}(x,y)=\arccos
\left\langle x,y\right\rangle $ denotes the standard geodesic distance on
the sphere, and $\left\langle \cdot ,\cdot \right\rangle $ is the inner
product in $\mathbb{R}^{3}$. Given $(\vartheta ,\varphi )\in \lbrack 0,\pi
]\times \lbrack 0,2\pi)$, denote by $V(\vartheta ,\varphi )=\{(\theta ,\phi
):0\leq \theta \leq\vartheta ,0\leq \phi \leq \varphi \}$ the angular
section originated from the North Pole. It is easy to see that,
respectively, the spherical area of $D(x,r)$ and $V(\vartheta ,\varphi )$ are

\begin{equation}
\nu (D(x,r))=2\pi (1-\cos r)\ \ \ \text{ and }\ \ \ \nu (V(\vartheta
,\varphi ))=\varphi (1-\cos \vartheta ).  \label{Areas}
\end{equation
Our first theorem is concerned with the joint continuity of local times of
the $\mathbf{T}$, see Section \ref{Sec:Joint continuity} below for the
definition of local times and more information.

\begin{theorem}
\label{Th1} Let $\mathbf{T}=\left\{ \mathbf{T}(x),x\in \mathbb{S
^{2}\right\} $ be a Gaussian random field with values in $\mathbb{R}^{d}$
defined in $\left( \ref{Def:T}\right) $. Assume that the associated
isotropic random field $T_{0}$ satisfies Condition (A) with $2<\alpha <4$
and $\beta =4-\left( \alpha -2\right) d>0$. Then for any open set
D\subseteq \mathbb{S}^{2}\ $with $\nu (D)>0$, $\mathbf{T}$ has local times
L\left( \mathbf{t},D\right) $ which is in $\mathbb{R}^{d}\times \mathbb{S
^{2}$ almost surely. Moreover, there is a modification of $L\left( \mathbf{t
,D\right) $ such that it is jointly continuous in the following sense:

\begin{itemize}
\item[(i)] The local time $L\left( \mathbf{t},D(x,r)\right) $ is continuous
in $(\mathbf{t},x,r)$. Moreover, for any two open disks $D_{i}=
\left(x_{i},r_{i}\right) \subseteq \mathbb{S}^{2}$ with $x_{i}\in \mathbb{S
^{2},r_{i}\in \left( 0,\delta \right) ,\ i=1,2,$ and any $\gamma \in
\left(0,\gamma _{0}\right) $ with $\gamma _{0}=\min \left\{ \frac{\beta }
2\left(\alpha -2\right) },1\right\} ,$ it satisfies

\begin{equation*}
\left\vert L\left( \mathbf{t},D_{1}\right) -L\left( \mathbf{s
,D_{2}\right)\right\vert \leq K_{1}\left[ \left\Vert \mathbf{t}-\mathbf{s
\right\Vert^{\gamma }+d_{\mathbb{S}^{2}}\left( x,y\right) ^{\beta
/4}r^{\gamma \left(\alpha -2\right) -\beta /4}\right] r^{\eta }\quad a.s.
\end{equation*}
where $\left\Vert \mathbf{\cdot }\right\Vert $ is the Euclidean distance on
\mathbb{R}^{d}$ and
\begin{equation}  \label{eta}
\eta =\frac{\beta }{2}-\left( \alpha -2\right) \gamma >0;
\end{equation}

\item[(ii)] The local time $L\left( \mathbf{t},V(\vartheta ,\varphi )\right)$
is continuous in $(\mathbf{t},\vartheta ,\varphi )$. Moreover, for any two
angular sections $V_{i}=V(\vartheta _{i},\varphi _{i})\subseteq \mathbb{S
^{2}$ with $(\vartheta _{i},\varphi _{i})\in \lbrack 0,\pi ]\times \lbrack
0,2\pi ),\ i=1,2,$ and any $\gamma \in \left( 0,\gamma _{0}\right),$ it
satisfies
\begin{eqnarray*}
\lefteqn{\left\vert L\left( \mathbf{t},V_{1}\right) -L\left( \mathbf{s
,V_{2}\right) \right\vert} \\
& \leq& K_{2}\left[ \left\vert \varphi _{1}-\varphi _{2}\right\vert \min
\left\{\vartheta _{1}^{2},\vartheta _{2}^{2}\right\} +\min \left\{
\varphi_{1},\varphi _{2}\right\} \left\vert \vartheta
_{1}^{2}-\vartheta_{2}^{2}\right\vert \right]^{\beta /4} \\
&&+K_{2}\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\gamma}\varphi
_{1}^{\eta /2}\vartheta _{1}^{\eta }, \quad a.s.
\end{eqnarray*}
where the constant $\delta \in \left( 0,1\right) $ depending only on
\alpha\ K_{0},$ and $K_{1},K_{2}$ are positive depending only on $\alpha ,\
d $ $K_{0}$ and $\gamma .$
\end{itemize}
\end{theorem}

The next result provides optimal upper and lower bounds for the exact moduli
of continuity for the maximum of local time $L^{\ast }(D)=\sup_{\mathbf{t
\in \mathbb{R}^{d}}L\left( \mathbf{t},D\right) $ \textit{w.r.t} to the
variable $r$ in the set $D=D(x,r)$.

\begin{theorem}
\label{Th2} Under conditions of Theorem \ref{Th1}, there exist positive
constant $K_{3},K_{4}$ depending only on $\alpha ,\ d$ and $K_{0},$ such
that for any $x\in \mathbb{S}^{2},$
\begin{equation}
K_{3}^{-1}\leq \underset{r\rightarrow 0}{\lim \inf }\frac{L^{\ast }\left(
D\left( x,r\right) \right) }{\phi \left( r\right) }\leq \underset
r\rightarrow 0}{\lim \sup }\frac{L^{\ast }\left( D\left( x,r\right) \right)
}{\phi \left( r\right) }\leq K_{3},\ a.s.  \label{ineq:Local Holder of LT}
\end{equation
where the functions $\phi $ is defined by
\begin{equation}
\phi \left( r\right) =\frac{r^{2}}{\left[ \rho _{\alpha }(r/\sqrt{\log
\left\vert \log r\right\vert })\right] ^{d}},  \label{psi1}
\end{equation
with $\rho _{\alpha }(r)=r^{\frac{\alpha }{2}-1}$ for $r\geq 0$.
\end{theorem}

\begin{remark}
The result above further comfirms that the sample functions of $T_{0}$ (and
\mathbf{T)}$ are nowhere differentiable. Actually, in Section \re
{Sec:non-diff},\ we establish the Chung's law of iterated logarithm, which
is essential to the proof of Theorem \ref{Th2} as well.
\end{remark}

The rest of this paper is as follows\emph{: } Section\emph{\ }\re
{preliminaries}\emph{\ }reviews some auxiliary tools for the arguments to
follow. Section \ref{Sec:Joint continuity} and \ref{Sec:Holder condition}
present the proofs of Theorems 1.1 and 1.2., respectively. The Chung's law
of iterated logarithm of $T_{0}$ is established in Section \ref{Sec:non-diff
. Our method for establishing joint continuity and upper bound of H\"{o}lder
conditions for the local times relies on moment estimates on the local
times, where the property of strong local nondeterminsim of $T_{0}$ proved
in Lan et al. \cite{LanMarXiao}. plays an essential role. We also make use
of a chaining argument that is similar to those in \cite{GeHor, Xiao97}.

Throughout this paper, we use $C,K$ to denote a constant whose value may
change in each appearance, and $C_{i,j},K_{i,j}$ to denote the $j$th more
specific positive finite constant in Section $i.$

\medskip

\textbf{Acknowledgement} The authors thank Professor Domenico Marinucci for
stimulating discussions and helpful comments on this paper. Research of X.
Lan is supported by NSFC grants 11501538 and WK0010450002. Research of Y.
Xiao is partially supported by NSF grants DMS-1612885 and DMS-1607089.

\section{Preliminaries}

\label{preliminaries}

In this section, we collect a few technical results which will be
instrumental for most of the proofs to follow. We recall first the following
lemma from \cite{LanMarXiao}, which characterizes the variogram and the
property of strong local nondeterminism of $T_{0}$.

\begin{lemma}
\label{C1} Under Condition (A) with $2<\alpha <4$, there exist positive
constants $K_{2,1}\geq 1,$ $0<\delta <1$ depending only on $\alpha $ and
K_{0},$ such that for any $x,y\in \mathbb{S}^{2},$ if $d_{\mathbb{S}^{2}}(
x,y) <\delta ,$ we have
\begin{equation}  \label{ineq:variogram}
K_{2,1}^{-1}\rho _{\alpha }^{2}\left( d_{\mathbb{S}^{2}}( x,y)\right) \leq
\mathbb{E}\left[ (T_{0}\left( x\right) -T_{0}( y))^{2}\right] \leq
K_{2,1}\rho _{\alpha }^{2}\left( d_{\mathbb{S}^{2}}(x,y) \right) .
\end{equation}
Moreover, there exists a constant $K_{2,2}>0$ depending on $\alpha $ and
K_{0}$ only, such that for all integers $n\geq 1$ and all
x,x_{1},...,x_{n}\in \mathbb{S}^{2},$ we have
\begin{equation}  \label{ineq:SLND}
\mathrm{Var}\left( T_{0}\left( x\right) |T_{0}\left(
x_{1}\right),...,T_{0}\left( x_{n}\right) \right) \geq K_{2,2}\min_{1\leq
k\leq n}\rho_{\alpha }^{2}\left( d_{\mathbb{S}^{2}}\left( x,x_{k}\right)
\right).
\end{equation}
\end{lemma}

For any fixed point $x_{0}\in \mathbb{S}^{2},$ consider the spherical random
field $Z_{0}(x)=T_{0}( x) -T_{0}( x_{0}) ,\ x\in \mathbb{S}^{2}.$ By taking
the same argument as in the proof of Theorem 1 in \cite{LanMarXiao}, we
obtain the following consequence of $\left( \ref{ineq:SLND}\right) $ in
Lemma \ref{C1}.

\begin{corollary}
\label{C2'} Under Condition (A) with $2<\alpha <4$, there exists a constant
K_{2,2}^{\prime }>0,$ such that for all integers $n\geq 1$ and all
x,x_{1},...,x_{n}\in \mathbb{S}^{2},$

\begin{equation*}
\mathrm{Var}\left( Z_{0}\left( x\right) |Z_{0}\left(
x_{1}\right),...,Z_{0}\left( x_{n}\right) \right) \geq K_{2,2}^{\prime
}\min_{0\leq k\leq n}\left\{ \rho _{\alpha }^{2}\left( d_{\mathbb{S
^{2}}(x,x_{k}) \right) \right\} .
\end{equation*}
\end{corollary}

The next auxiliary tool that we will use to prove Theorem \ref{Th1} is the
following lemma, where $\left( \cdot \right) ^{T}$ denotes the transpose of
a matrix or a vector.

\begin{lemma}
\label{Charac-Cov} Let $\mathbf{X}=\left( X_{1},...,X_{n}\right) ^{T}\sim
N\left( \boldsymbol{\mu ,\Theta }\right) $ with a positive definite
covariance matrix $\boldsymbol{\Theta }$ and mean vector $\boldsymbol{\mu
\in }\mathbb{R}^{d}\ \left( d\geq 1\right) ,$ then for all vectors $\mathbf{
}\in \mathbb{R}^{d},$ we have
\begin{equation*}
\left( 2\pi \right) ^{-n/2} \int_{\mathbb{R}^{n}} e^{-i\mathbf{t}^{T
\boldsymbol{\xi }} \mathbb{E}\left[ e^{i\boldsymbol{\xi }^{T}\mathbf{X}
\right] d\boldsymbol{\xi} \leq \left[ \mathrm{Var}\left(
X_{1}\right)\prod\limits_{j=2}^{n}\mathrm{Var}\left(
X_{j}|X_{1},...,X_{j-1}\right) \right] ^{-1/2};
\end{equation*}
Moreover,
\begin{equation*}
\mathrm{Var}\left( X_{1}\right) \prod\limits_{j=2}^{n}\mathrm{Var
\left(X_{j}|X_{1},...,X_{j-1}\right) =\det \left\{ \boldsymbol{\Theta
\right\} .
\end{equation*}
\end{lemma}

In order to prove the lower bounds in Theorems \ref{Th2}, we will also need
to exploit the following two lemmas from Talagrand \cite{Talagrand95}. Let
\{f(x),x\in M\}$ be a centered Gaussian field indexed by $M$ and let
d_{f}(x,y)=\sqrt{\mathbb{E}[(f(x)-f(y))^{2}]}$ be its canonical metric on $M
. For a $d_{f}$-compact manifold $M$, denote by $N_{d_{f}}\left(
M,\varepsilon \right) $ the smallest number of balls of radius $\varepsilon $
in metric $d_{f}$ that are needed to cover $M.$

\begin{lemma}
\label{DudleyLB} If $N_{d_{f}}\left(M,\varepsilon \right) \leq \Psi
\left(\varepsilon \right)$ for all $\varepsilon >0$ and the function $\Psi$
satisfies
\begin{equation*}
\frac{1}{K_{2,3}}\Psi \left( \varepsilon \right) \leq \Psi \left( \frac
\varepsilon }{2}\right) \leq K_{2,3}\Psi \left( \varepsilon \right), \quad
\forall \, \varepsilon >0,
\end{equation*}
where $K_{2,3}>0$ is a finite constant. Then
\begin{equation*}
\mathbb{P}\left\{ \sup_{s,t\in M}\left\vert f(s) -f(t) \right\vert \leq
u\right\} \geq \exp \left( -K_{2,4}\Psi \left(u\right) \right),
\end{equation*}
where $K_{2,4}>0$ is a constant depending only on $K_{2,3}$.
\end{lemma}

\begin{lemma}
\label{DudleyUB} Let $\{f(x), x \in M\} $ be a centered Gaussian field $a.s.$
bounded on a $d_f$-compact set $M$. There exists a universal constant
K_{2,5}>0$ such that for any $u>0,$ we have
\begin{equation*}
\mathbb{P}\left\{ \sup_{x\in M}f(x) \geq K_{2,5}\left( u+\int_{0}^{\overline
d}}\sqrt{\log N_{d}\left( M,\varepsilon \right) }d\varepsilon
\right)\right\} \leq \exp \left( -\frac{u^{2}}{\overline{d}^{2}}\right) ,
\end{equation*}
where $\overline{d}=\sup \left\{ d_{f}( x,y): x,y\in M\right\} $ is the
diameter of $M$ in the metric $d_f$.
\end{lemma}

Based on Lemmas \ref{C1} and \ref{DudleyUB}, we obtain the following result:

\begin{lemma}
\label{LemTupper} Under Condition (A) with $2<\alpha <4$, there exists
positive constants $K_{2,6},K_{2,7}$ depending only on $\alpha $ and $K_{0}$
such that for any $D\left( z,r\right) \subset \mathbb{S}^{2}$ and
0<r<\delta ,$ we have for any $u>K_{2,6}r^{\left( \alpha -2\right) /2},$
\begin{equation}  \label{ineq:Tlower}
\mathbb{P}\left\{ \sup_{x,y\in D\left( z,r\right) }\left\vert
T_{0}\left(x\right) -T_{0}\left( y\right) \right\vert \geq K_{2,7}u\right\}
\geq \exp\left( -\frac{u^{2}}{\left\vert \rho _{\alpha }\left( 2r\right)
\right\vert^{2}}\right) ,
\end{equation}
\end{lemma}

\begin{proof}
Recall Lemma \ref{C1}, we have
\begin{equation*}
\sqrt{K_{2,1}^{-1}}\rho _{\alpha }( x,y) \leq d_{T_{0}}\left(x,y\right) \leq
\sqrt{K_{2,1}}\rho _{\alpha }( x,y) .
\end{equation*}
It follows immediately that, for any $D\left( z,r\right) \subset \mathbb{S
^{2},$ and any $\epsilon \in \left( 0,r\right) ,$
\begin{equation*}
N_{d_{T_{0}}}\left( D\left( z,r\right) ,\epsilon \right) \leq \frac{2\pi
r^{2}}{\pi \left( \epsilon /\sqrt{K_{2,1}}\right) ^{4/\left( \alpha-2\right)
}} \leq 2\left( K_{2,1}\right)^{2/\left( \alpha -2\right) } \frac{r^{2}}
\epsilon ^{4/\left( \alpha -2\right) }},
\end{equation*}
and
\begin{equation*}
\overline{d}=\sup \left\{ d_{T_{0}}\left( x,y\right) :x,y\in
D\left(z,r\right) \right\} \leq \sqrt{K_{2,1}}\rho _{\alpha }\left(
2r\right) ,
\end{equation*}
whence similar to the argument in \cite{LuanXiao12}, we have
\begin{equation*}
\int_{0}^{\overline{d}}\sqrt{\log N_{d_{T_{0}}}\left( D\left(
z,r\right),\epsilon \right) }d\epsilon \leq C_{2,1}r^{\alpha /2-1},
\end{equation*}
for some positive constant $C_{2,1}$ depending on $K_{2,1}.$
Let the constants $K_{2,6}=C_{2,1}$ and $\ K_{2,7}=2K_{2,5}C_{2,1}$, then by
exploiting Lemma \ref{DudleyUB}, we derive $\left( \ref{ineq:Tlower}\right)
.
\end{proof}

Finally, we prove the following approximation for small ball probability,
which is analogy to the argument in \cite{Xiao09}, Theorem 5.1.

\begin{lemma}
\label{Lem:Small Ball P} Under Condition (A) with $2<\alpha <4$, there
exists a positive constant $K_{2,8}$ depending only on $\alpha $ and $K_{0},$
such that for any $\epsilon >0$ and $D\left( z,r\right) \subset \mathbb{S
^{2}$ with $0<r<\delta $, we have
\begin{equation*}
\mathbb{P}\left\{ \sup_{x,y\in D( z,r) }\left\vert T_{0}(x) -T_{0}( y)
\right\vert \leq \epsilon \right\} \geq \exp \left( -K_{2,8}\frac{r^{2}}
\epsilon ^{4/\left( \alpha -2\right) }}\right) ,
\end{equation*}
\end{lemma}

\begin{proof}
We first prove the lower bound of the small ball probability. The canonical
metric $d_{T_{0}}$ on $\mathbb{S}^{2}$ is defined by
\begin{equation*}
d_{T_{0}}\left( x,y\right) =\sqrt{\mathbb{E}\left\vert T_{0}\left( x\right)
-T_{0}\left( y\right) \right\vert ^{2}}.
\end{equation*
By $\left( \ref{ineq:variogram}\right) $\ in Lemma \ref{C1}, we have for any
$x,y\in \mathbb{S}^{2}$ with $d_{\mathbb{S}^{2}}\left( x,y\right) <\delta $,
\begin{equation*}
d_{T_{0}}\left( x,y\right) \leq \sqrt{K_{2,1}}\rho _{\alpha }(x,y).
\end{equation*
In the meantime, recall the metric entropy, it follows immediately that
\begin{equation*}
N_{d_{T_{0}}}\left( D\left( z,r\right) ,\epsilon \right) \leq \frac{2\pi
r^{2}}{\pi \left( \epsilon /\sqrt{K_{2,1}}\right) ^{4/\left( \alpha
-2\right) }}\leq C_{2,2}\frac{r^{2}}{\epsilon ^{4/\left( \alpha -2\right) }}.
\end{equation*
where $C_{2,2}$ is a positive constant depending only on $K_{2,1}$ and
\alpha .$ Hence, Lemma \ref{Lem:Small Ball P}\ is derived by exploiting
Lemma \ref{DudleyLB} with $K_{4,1}=K_{2,4}C_{2,2}.$
\end{proof}

\section{Local times and their joint continuity}

\label{Sec:Joint continuity}

Let us first recall that, for any Borel set $D\subseteq \mathbb{S}^{2}$ and
\omega \in \Omega $, the \emph{occupation measure} of $\mathbf{T}$ on $D$ is
defined by
\begin{equation*}
\mu _{D}\left( I,\omega \right) :=\nu \left\{ x\in D:\mathbf{T}(x,\omega)
\in I\right\}
\end{equation*}
for all Borel sets $I\subset \mathbb{R}^{d}$, where
$\nu $ is the canonical area measure on the unit sphere. If $\mu
_{D}(\cdot,\omega )$ is absolutely continuous \emph{w.r.t} the Lebesgue
measure $\lambda _{d}$ on $\mathbb{R}^{d}$, then we say that $\mathbf{T
(\cdot,\omega )$ has \emph{local times} on $D$, and define a local time
L\left( \mathbf{t},D,\omega \right) $ as the Radon--Nikod\'{y}m derivative
of $\mu_{D}$ with respect to $\lambda _{d}$, i.e.,
\begin{equation}  \label{def:Local time}
L\left( \mathbf{t},D,\omega \right) =\frac{d\mu _{D}(\cdot ,\omega )}
d\lambda _{d}}(\mathbf{t}),\qquad \forall \mathbf{t}\in \mathbb{R}^{d}.
\end{equation}
As usual, we will from now on omit $\omega $ from the notation for the local
times.

We refer to Geman and Horowitz \cite{GeHor} for a comprehensive survey on
local times of random fields. In particular, Theorems 6.3 and 6.4 in \cit
{GeHor} show that $L\left( \mathbf{t},D\right) $ satisfies the following
occupation density identity: For every Borel set $D\subseteq \mathbb{S}^{2}$
and for every measurable function $f:\mathbb{R}\rightarrow \mathbb{R}_{+},$
\begin{equation}  \label{eq:LT-occup density}
\int_{D}f\left( \mathbf{T}( x) \right) d\nu ( x) =\int_{\mathbb{R}^{d}}f(
\mathbf{t}) L( \mathbf{t},D) d\mathbf{t}.
\end{equation}

We split the proof of Theorem \ref{Th1} into two parts.

\begin{proof}[\textbf{Proof of Theorem \protect\ref{Th1}: Existence}]
The Fourier transform of the occupation measure $\mu _{D}$ is
\begin{equation*}
\widehat{\mu }_{D}\left( \boldsymbol{\xi }\right) =\int_{D}e^{i\boldsymbol
\xi }^{T}\mathbf{T}(x)}d\nu (x),\quad \forall \boldsymbol{\xi \in \mathbb{R}
^{d}.
\end{equation*
Now applying Fubini's theorem and the independence of components of $\mathbf
T}$, we derive
\begin{equation*}
\begin{split}
\mathbb{E}\left[ \int_{\mathbb{R}^{d}}\left\vert \widehat{\mu }\left( \xi
\right) \right\vert ^{2}d\xi \right] & =\int_{D}d\nu \left( x\right)
\int_{D}d\nu \left( y\right) \int_{\mathbb{R}^{d}}\mathbb{E}\left[ e^{
\boldsymbol{\xi }^{T}\left( \mathbf{T}\left( x\right) -\mathbf{T}\left(
y\right) \right) }\right] d\boldsymbol{\xi } \\
& =\int \int_{D\times D}\left( \mathbb{E}\left[ \big|{T_{0}}\left( x\right)
{T_{0}}\left( y\right) \big|^{2}\right] \right) ^{-d/2}d\nu \left( x\right)
d\nu \left( y\right) \\
& \leq K_{2,1}^{d/2}\int \int_{D\times D}\left[ d_{\mathbb{S}^{2}}(x,y
\right] ^{(1-\alpha /2)d}d\nu (x)d\nu (y) \\
& \leq K_{2,1}^{d/2}\int_{0}^{2\pi }\int_{0}^{\pi }\vartheta ^{(1-\alpha
/2)d}\sin \vartheta d\vartheta d\phi <\infty .
\end{split
\end{equation*
In the above, the first inequality follows from $\left( \ref{ineq:variogram
\right) $ in Lemma \ref{C1}\ and the last inequality follows from the
condition that $\alpha \in \left( 2,4\right) $ and $(\alpha -2)d<4$. Hence,
by the Plancherel theorem (see i.e., \cite{Rudin}, Ch. 9), we see that
\mathbf{T}$ a.s. has local times which can be represented in the $L^{2}
\mathbb{R}^{d})$-sense as
\begin{equation}
L\left( \mathbf{t},D\right) =\frac{1}{2\pi }\int_{\mathbb{R}^{d}}e^{-
\boldsymbol{\xi }^{T}\mathbf{t}}\int_{D}e^{i\boldsymbol{\xi }^{T}\mathbf{T
\left( x\right) }d\nu \left( x\right) d\boldsymbol{\xi }.  \label{repLT}
\end{equation
The proof of existence is thus completed.
\end{proof}

An immediate consequence of the above proof of the existence of local times
is the following:

\begin{corollary}
Under the same conditions as in Theorem \ref{Th1}, almost surely the range
\mathbf{T}\left( \mathbb{S}^{2}\right) $ has positive Lebesgue measure.
\end{corollary}

In order to prove the joint continuity of local times, we first prove the
moment estimates in Lemmas \ref{L1} and \ref{L2}, by extending the argument
in Xiao \cite{Xiao97} to the spherical random field $\mathbf{T}$. The new
ingredient for the proofs is the property of strong local nondeterminism in
Lemma \ref{C1}. \medskip

\begin{lemma}
\label{L1} Under conditions of Theorem \ref{Th1}, there exists a positive
constant $K_{3,1}$ depending only on $\alpha ,\ d$ and $K_{0},$ such that
for any open set $D\subset \mathbb{S}^{2}$ with $\nu \left( D\right) >0$,
\mathbf{t}\in \mathbb{R}^{d}$, and integers $n\geq 1,$ it holds that
\begin{equation*}
\mathbb{E}\left[ L\left( \mathbf{t},D\right) ^{n}\right] \leq
K_{3,1}^{n}\left( \left( n-1\right) !\right) ^{d\left( \alpha
-2\right)/4}\nu ( D) ^{( n-1) \beta /4+1}.
\end{equation*}
where $\beta $ is defined in Theorem \ref{Th1}.
\end{lemma}

\begin{proof}
It follows from $( \ref{repLT}) $ (see also \cite{GeHor,Xiao09}) that for
all $\mathbf{t}\in \mathbb{R}^{d}$ and integers $n\geq 1,$
\begin{equation*}
\mathbb{E}\left[ L\left( \mathbf{t},D\right) ^{n}\right] =\frac{1}
\left(2\pi \right) ^{n}} \int_{D^{n}} \int_{\mathbb{R}^{nd}}e^{-
\sum_{j=1}^{n}\mathbf{t}^{T}\boldsymbol{\xi }_{j}}\mathbb{E}\left[
e^{i\sum_{j=1}^{n} \boldsymbol{\xi }_{j}^{T}\mathbf{T}\left( x_{j}\right)
\right] \prod\limits_{j=1}^{n}\left( d\nu \left( x_{j}\right) `d\boldsymbol
\xi }_{j}\right) ,
\end{equation*}
where $\boldsymbol{\xi }_{j}\in \mathbb{R}^{d}$ for $j=1,...,n$. By the
positive definiteness of covariance matrix of $T_{0}\left(
x_{1}\right),...,T_{0}\left( x_{n}\right) ,$ we have
\begin{eqnarray*}
\lefteqn{\int_{\mathbb{R}^{nd}}\mathbb{E}\left[ e^{i\sum_{j=1}^{n
\boldsymbol{\xi }_{j}^{T}\left[ \mathbf{T}( x_{j}) -\mathbf{t}\right]
\right] \prod \limits_{j=1}^{n} d\boldsymbol{\xi }_{j}} \\
&=&\prod\limits_{k=1}^{d} \int_{\mathbb{R}^{n}}\mathbb{E}\left[
e^{i\sum_{j=1}^{n}\xi _{j,k}\left[ T_{k}\left(x_{j}\right) -t_{k}\right]
\right] \prod\limits_{j=1}^{n}d\xi _{j,k} \\
&\leq &\frac{( 2\pi ) ^{nd/2}}{\left[ \det \left( \mathrm{Cov}\left( T_{0}(
x_{1}) ,...,T_{0}( x_{n}) \right)\right) \right] ^{d/2}} \\
&\leq &( 2\pi ) ^{nd/2}\left[ \mathrm{Var}( T_{0}(x_{1}) )
\prod\limits_{j=2}^{n}\mathrm{Var}( T_{0}(x_{j}) |T_{0}( x_{1}) ,...,T_{0}(
x_{j-1})) \right] ^{-d/2}.
\end{eqnarray*}
in view of Lemma \ref{Charac-Cov}. Recall the assumption of unit variance of
$T_{0}$ and the inequality $\left( \ref{ineq:SLND}\right) $ in Lemma \ref{C1
, we obtain that
\begin{equation}  \label{ELeq}
\mathbb{E}\left[ L( \mathbf{t},D) ^{n}\right] \leq C_{3,1}^{n-1}\left[
\int_{D^{n}}\prod\limits_{j=2}^{n}\frac{1}{\min_{1\leq i\leq
j-1}\rho_{\alpha }^{d}\left( x_{i},x_{j}\right) }d\boldsymbol{\nu }\right]
\end{equation}
where the constant $C_{3,1}>0$ depends only on $K_{2,2},$ and $d\boldsymbol
\nu }=d\nu \left( x_{1}\right) \cdots d\nu \left( x_{n}\right) .$ Now let
j\in \left\{ 2,...,n\right\} $ fixed, and define the following sets that are
disjoint except on the boundaries,
\begin{equation}  \label{GammaD}
\Gamma _{i} =\left\{ x\in D: d_{\mathbb{S}^{2}}( x,x_{i}) =\min \left\{ d_
\mathbb{S}^{2}}( x,x_{i^{\prime }}) , i^{\prime}=1,...,j-1\right\} \right\} .
\end{equation}
Observing $D=\cup _{i=1}^{j-1}\Gamma _{i}$, we have
\begin{eqnarray}
\lefteqn{\int_{D}\frac{1}{\min_{1\leq i\leq j-1}\rho _{\alpha
}^{d}(x_{i},x_{j}) }d\nu ( x_{j}) }  \notag \\
&=&\sum_{i=1}^{j-1}\int_{\Gamma _{j}}\frac{1}{\rho _{\alpha
}^{d}\left(x_{i},x\right) }d\nu \left( x\right) \leq
\sum_{i=1}^{j-1}\int_{0}^{2\pi}\int_{0}^{r_{j}( \phi ) }\frac{\theta }{\rho
_{\alpha}^{d}( \theta ) }d\theta d\phi  \notag \\
&=&\sum_{i=1}^{j-1}\frac{2}{\beta }\int_{0}^{2\pi }\left[ r_{i}\left( \phi
\right) \right] ^{\beta /2}d\phi \leq C_{3,2}\sum_{i=1}^{j-1}\left
\int_{0}^{2\pi }\frac{1}{2}\left[ r_{i}\left( \phi \right) \right] ^{2}d\phi
\right] ^{\beta /4}  \label{ineq:mineq2} \\
&\leq & C_{3,2}\left( j-1\right) \left( \frac{1}{\left( j-1\right)
\sum_{i=1}^{j-1}\nu \left( \Gamma _{i}\right) \right) ^{\beta /4} \leq
C_{3,2}\left( j-1\right) \left( \frac{\nu \left( D\right) }{\left(j-1\right)
}\right) ^{\beta /4}  \label{ineq:mineq3}
\end{eqnarray}
where we have used Jensen's inequality above and the constant $C_{3,2}>0$
depends on $\alpha $ and $d$. Moreover, the inequality in $( \re
{ineq:mineq2}) $ holds if and only if $\beta =4-( \alpha-2) d>0.$ It is
readily seen that
\begin{eqnarray*}
\mathbb{E}\left[ L\left( \mathbf{t},D\right) ^{n}\right] &\leq&
C_{3,1}^{n-1}\int_{D}\left[ \int_{D^{n-1}}\prod\limits_{j=2}^{n}\frac{d\nu (
x_{1}) \cdots d\nu \left( x_{n-1}\right) }{\min_{1\leq i\leq j-1}\rho
_{\alpha }^{d}( x_{i},x_{j}) }\right] d\nu (x_{n}) \\
&\leq & K_{3,1}^{n-1}\left[ \left( n-1\right) !\right] ^{( \alpha-2) d/4}\nu
( D) ^{( n-1) \beta /4+1},
\end{eqnarray*
in view of the two results $\left( \ref{ELeq}\right) $ and (\ref{ineq:mineq3
), where the constant $K_{3,1}>0$ depends on $\alpha $ and $d$. Hence, the
lemma is proved.
\end{proof}

Recall $\eta =\beta /2-( \alpha -2) \gamma $ defined in $\left(\ref{eta
\right) .$ Obviously, $\eta <\beta /2<2$ for any $\gamma \in (0,1) .$ Now we
have the following moment estimation:

\begin{lemma}
\label{L2} Under conditions of Theorem \ref{Th1}, there exists a positive
constant $K_{3,2}$ depending on $\alpha,$ $d\ K_{0\text{ }}$ and $\gamma ,$
such that for any open set $D\subset \mathbb{S}^{2}$ with $\nu
\left(D\right) >0$, and any $\mathbf{s},\mathbf{t}\in \mathbb{R}^{d}$, all
even integers $n\geq 2,\,0<\gamma <1$ satisfying $\eta >0,$ we have
\begin{equation*}
\mathbb{E}\left\{ \left[ L( \mathbf{t},D) -L( \mathbf{s},D) \right]
^{n}\right\} \leq K_{3,2}^{n}( n!) ^{2-\eta/2}\left\Vert \mathbf{t}-\mathbf{
}\right\Vert ^{n\gamma }\nu (D) ^{( n-1) \eta /2+1},.
\end{equation*}
\end{lemma}

\begin{proof}
Recall $( \ref{repLT}) ,$ we have
\begin{eqnarray*}
\lefteqn{\mathbb{E}\left\{ \left[ L\left( \mathbf{t},D\right) -L\left(
\mathbf{s},D\right) \right] ^{n}\right\}} \\
&=&\frac{1}{\left( 2\pi \right) ^{n}} \int_{D^{n}}\int_{\mathbb{R}^{nd}
\mathbb{E} \left[ e^{i\sum_{j=1}^{n}\boldsymbol{\xi }_{j}^{T}\mathbf{T}
\left( x_{j}\right) }\right] \prod\limits_{j=1}^{n}\left[ e^{-i\mathbf{t}^{T
\boldsymbol{\xi }_{j}}-e^{-i\mathbf{s}^{T}\boldsymbol{\xi }_{j}}\right] d\nu
\left( x_{j}\right) d\boldsymbol{\xi }_{j}.
\end{eqnarray*}
By the fact that, for any $\gamma \in ( 0,1) ,$ we have
\begin{equation*}
\left\vert e^{-i\mathbf{t}^{T}\boldsymbol{\xi }_{j}}-e^{-i\mathbf{s}^{T
\boldsymbol{\xi }_{j}}\right\vert = \left\vert e^{-i\left( \mathbf{t}
\mathbf{s}\right) ^{T}\boldsymbol{\xi }_{j}}-1\right\vert \leq
2^{1-\gamma}\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\gamma }\left\Vert
\boldsymbol{\xi }_{j}\right\Vert ^{\gamma },
\end{equation*}
with $j=1,...,n,$ and hence,
\begin{eqnarray*}
\lefteqn{\mathbb{E}\left\{ \left[ L( \mathbf{t},D) -L( \mathbf{s},D) \right]
^{n}\right\} } \\
&\leq& \frac{2^{n\left( 1-\gamma \right)}\left\Vert \mathbf{t}-\mathbf{s
\right\Vert ^{n\gamma }}{\left( 2\pi \right) ^{n}} \int_{D^{n}}\int_{\mathbb
R}^{nd}} e^{-\frac{1}{2}\mathrm{Var}\left( \sum_{j=1}^{n}\boldsymbol{\xi
_{j}^{T}\mathbf{T} \left( x_{j}\right) \right)}
\prod\limits_{j=1}^{n}\left\Vert \boldsymbol{\xi }_{j}\right\Vert ^{\gamma}
\boldsymbol{\xi }_{j}d\nu ( \mathbf{x}) .
\end{eqnarray*}
Since $\left\vert a+b\right\vert ^{\gamma }\leq \left\vert
a\right\vert^{\gamma }+\left\vert b\right\vert ^{\gamma }$ for any real
numbers $a,b$ and $0<\gamma <1,$ we have $\left\Vert \boldsymbol{\xi
_{j}\right\Vert^{\gamma }\leq \sum_{k=1}^{d}\left\vert \xi _{j,k}\right\vert
^{\gamma }$, which leads to
\begin{eqnarray*}
\lefteqn{\int_{\mathbb{R}^{nd}} e^{-\frac{1}{2} \mathrm{Var}( \sum_{j=1}^{n
\boldsymbol{\xi }_{j}^{T}\mathbf{T}( x_{j})
)}\prod\limits_{j=1}^{n}\left\Vert \boldsymbol{\xi }_{j}\right\Vert
^{\gamma}d\boldsymbol{\xi }_{j} } \\
&\leq &\sum_{\mathbf{k}\in \left\{ 1,...,d\right\} ^{n}} \int_{\mathbb{R
^{nd}} e^{-\frac{1}{2}\mathrm{Var}\left( \sum_{j=1}^{n}\boldsymbol{\xi
_{j}^{T}\mathbf{T}\left( x_{j}\right)
\right)}\prod\limits_{j=1}^{n}\left\vert \xi _{j,k_{j}}\right\vert ^{\gamma
}d\boldsymbol{\xi }_{j},
\end{eqnarray*}
with $\mathbf{k=}\left( k_{1},...,k_{n}\right) $ and $k_{j}\in
\left\{1,...,d\right\} $ for $j=1,...,n.$ That is
\begin{equation}  \label{IntJ}
\left\vert \mathbb{E}\left[ L( \mathbf{t},D) -L( \mathbf{s},D) \right]
^{n}\right\vert \leq \left( 2^{\gamma }\pi \right)^{-n}\left\Vert \mathbf{t}
\mathbf{s}\right\Vert ^{n\gamma }\sum_{\mathbf{k}\in \left\{ 1,...,d\right\}
^{n}}\int_{D^{n}}J_{\mathbf{k}}( \mathbf{x}) d\boldsymbol{\nu }
\end{equation}
where $d\boldsymbol{\nu }=d\nu _{1}...d\nu _{n}$ and $J_{\mathbf{k}}$ is the
integral
\begin{equation*}
J_{\mathbf{k}}\left( \mathbf{x}\right) =\int_{\mathbb{R}^{nd}}e^{-\frac{1}{2
\mathrm{Var}\left( \sum_{j=1}^{n}\boldsymbol{\xi }_{j}^{T}\mathbf{T}\left(
x_{j}\right) \right)}\prod\limits_{j=1}^{n}\left\vert \xi
_{j,k_{j}}\right\vert ^{\gamma }d\boldsymbol{\xi }_{j}\ ,
\end{equation*}
for each fixed point $\mathbf{k}$ in the discrete space $\left\{
1,...,d\right\} ^{n}.$ By a generalized H\"{o}lder's inequality and Lemma2.4
in \cite{Xiao97} (see also \cite{Cuzick82}), we see that $J_{\mathbf{k
}\left( \mathbf{x}\right) $ is bounded by
\begin{eqnarray}
\lefteqn{\prod\limits_{j=1}^{n}\left[ \int_{\mathbb{R}^{nd}} e^{-\frac{1}{2
\mathrm{Var}( \sum_{j=1}^{n}\sum_{k=1}^{d}\xi_{j,k}T_{k}( x_{j}) )
}\left\vert \xi _{j,k_{j}}\right\vert^{n\gamma }d\boldsymbol{\xi }\right]
^{1/n} }  \notag \\
&=&\frac{\left( 2\pi \right) ^{nd-1} \int_{\mathbb{R}} \left\vert
v\right\vert ^{n\gamma } \exp \left\{ -\frac{v^{2}}{2}\right\} d\nu
\prod\limits_{j=1}^{n}\left( \widetilde{\sigma }_{j}^{2}\right)^{-\gamma /2
}{\left[ \det \mathrm{Cov}\left( T_{k}\left( x_{j}\right),1\leq k\leq
d,1\leq j\leq n\right) \right] ^{1/2}}  \notag \\
&\leq &\frac{\left( 2\pi \right) ^{nd-1}2^{\frac{n\gamma +1}{2}}\Gamma
\left( \frac{n\gamma +1}{2}\right) }{\left[ \det \mathrm{Cov
\left(T_{0}\left( x_{1}\right) ,...,T_{0}\left( x_{n}\right) \right) \right]
^{d/2}}\prod\limits_{j=1}^{n}\left( \widetilde{\sigma }_{j}^{2}\right)
^{-\gamma/2},  \label{ineq:Jk1}
\end{eqnarray}
where $\widetilde{\sigma }_{j}^{2}$ is the conditional variance of
T_{k_{j}}\left( x_{j}\right) $ given $T_{l}\left( x_{i}\right) $ ($l\neq
k_{j}$ or $l=k_{j}$ but $i\neq j$) and $\Gamma \left( \cdot \right) $ is the
Gamma function. Now we define a permutation $\pi $ of $\left\{1,...,n\righ
\} $ such that $\pi \left( 1\right) =1,$ and
\begin{equation*}
d_{\mathbb{S}^{2}}\left( x_{\pi ( j) },x_{\pi ( j-1)}\right) =\min \left\{
d_{\mathbb{S}^{2}}\left( x_{i},x_{\pi \left(j-1\right) }\right) ,i\in
\left\{ 1,...,n\right\} \backslash \left\{ \pi( 1) ,...,\pi ( j-1) \right\}
\right\} ,
\end{equation*}
Then by $\left( \ref{ineq:SLND}\right) $\ in Lemma \ref{C1}, we see that
K_{2,2}^{\gamma n/2}\prod\limits_{j=1}^{n}\frac{1}{\widetilde{\sigma
_{j}^{\gamma }}$ is bounded by
\begin{eqnarray*}
\lefteqn{ \prod\limits_{j=1}^{n}\frac{1}{\min \left\{ \left[ \rho _{\alpha
}\left( d_{\mathbb{S}^{2}}\left( x_{\pi \left( j\right) },x_{i}\right)
\right) \right] ^{\gamma }: i\neq \pi \left( j\right) \right\} }} \\
&\leq &\prod\limits_{j=1}^{n} \frac{1}{\left[ \min \left\{ \frac{1}{2
\rho_{\alpha }\left( d_{\mathbb{S}^{2}}( x_{\pi ( j) },x_{\pi( j-1) })
\right) ,\rho _{\alpha }\left( d_{\mathbb{S}^{2}}( x_{\pi ( j) },x_{\pi (
j+1) })\right) \right\} \right] ^{\gamma }} \\
&\leq &\prod\limits_{j=1}^{n}\frac{4}{\left[ \rho _{\alpha }\left( d_
\mathbb{S}^{2}}( x_{\pi ( j) },x_{\pi ( j-1)}) \right) \right] ^{2\gamma }
\leq \prod\limits_{j=1}^{n}\frac{4}{\left[ \min_{1\leq i\leq j-1}\rho
_{\alpha }\left( d_{\mathbb{S}^{2}}(x_{\pi ( j) },x_{\pi ( i) }) \right)
\right]^{2\gamma }}
\end{eqnarray*}
For the sake of notation's simplicity, we denote by $\pi ( j) =j$ for each
j=1,...,n.$ By $\left( \ref{ineq:Jk1}\right) $\ and Lemma \ref{Charac-Cov},
we derive that
\begin{equation}  \label{ineq:Jk2}
J_{\mathbf{k}}\left( \mathbf{x}\right) \leq \frac{C_{3,2}^{n}\left(n-
\right) !}{\Pi _{j=2}^{n}\left[ \min_{1\leq i\leq j-1}\rho _{\alpha} \left(
d_{\mathbb{S}^{2}}( x_{j},x_{i}) \right) \right]^{d+2\gamma }}
\end{equation}
where $C_{3,2}$ is a positive constant depending on $K_{2,2}$ and $d.$
Therefore, by $( \ref{IntJ}) $ and $( \ref{ineq:Jk2}) $ above, we have
\begin{equation}  \label{ineq:Ik1}
\mathbb{E}\left\{ \left[ L( \mathbf{t},D) -L( \mathbf{s},D) \right]
^{n}\right\} \leq \left( \frac{dC_{3,2}}{2^{\gamma }\pi }\right) ^{n}
\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{n\gamma }n!\sum_{\mathbf{k}\in
\{ 1,...,d\} ^{n}}I_{\mathbf{k}},
\end{equation}
where
\begin{equation}  \label{def:Ik}
I_{\mathbf{k}}=:\int_{D^{n}}\Pi _{j=2}^{n}\left[ \min_{1\leq i\leq
j-1}\rho_{\alpha }\left( d_{\mathbb{S}^{2}}\left( x_{j},x_{i}\right) \right)
\right]^{-\left( d+2\gamma \right) }d\boldsymbol{\nu }.
\end{equation}
Similar to the argument in the proof of Lemma \ref{L1}, for any fixed $j\in
\left\{ 2,...,n\right\} ,$ we define the domains $\Gamma _{i},\ i=1,...,j-1,$
same as in $\left( \ref{GammaD}\right) ,$ then we can obtain
\begin{eqnarray}
A_{j} &:=&\int_{D}\frac{1}{\left[ \min_{1\leq i\leq j-1}\rho _{\alpha}\left(
d_{\mathbb{S}^{2}}\left( x_{j},x_{i}\right) \right) \right]^{d+2\gamma }
d\nu \left( x_{j}\right)  \notag \\
&=&\sum_{i=1}^{j-1}\int_{\Gamma _{i}}\frac{1}{\left[ \min_{1\leq i\leq
j-1}\rho _{\alpha }\left( d_{\mathbb{S}^{2}}( x_{j},x_{i})\right) \right]
^{d+2\gamma }}d\nu \left( x_{j}\right)  \notag \\
&\leq &\sum_{i=1}^{j-1}\int_{0}^{2\pi }\int_{0}^{r_{i}\left( \phi \right)
\frac{\theta }{\left[ \rho _{\alpha }\left( \theta \right) \right
^{d+2\gamma }}d\theta d\phi =\sum_{i=1}^{j-1}\eta ^{-1}\int_{0}^{2\pi }\left
r_{i}\left( \phi \right) \right] ^{\eta }d\phi  \notag \\
&\leq &C_{3,3}\sum_{i=1}^{j-1}\left[ \int_{0}^{2\pi }\frac{1}{2}\left
r_{i}\left( \phi \right) \right] ^{2}d\phi \right] ^{\eta/2}
=C_{3,3}\sum_{i=1}^{j-1}\left[ \nu \left( \Gamma _{i}\right) \right]^{\eta
/2}  \label{intmin} \\
&\leq &C_{3,3}\left( j-1\right) \left( \frac{1}{\left( j-1\right) }
\sum_{i=1}^{j-1}\nu \left( \Gamma _{i}\right) \right) ^{\eta /2} \leq
C_{3,3}\left( j-1\right) \left( \frac{\nu \left( D\right) }{\left(j-1\right)
}\right) ^{\eta /2}.  \label{ineq:Aj}
\end{eqnarray
where constant $\eta $ is the one defined in $\left( \ref{eta}\right) $ and
C_{3,3}=\eta ^{-1}\left( 2\pi \right) ^{1-\eta /2}$. The inequality in
\left( \ref{intmin}\right) $\ holds if and only if $0<\eta <2$, which is
always true for all $\alpha \in \left( 2,4\right) ,$ $\gamma \in
\left(0,1\right) ,\ d\in \mathbb{N}^{+}$ satisfying $\left( \alpha -2\right)
\left( d+2\gamma \right) <4.$ Hence, by the definition of $I_{\mathbf{k}}$
in $\left( \ref{def:Ik}\right) $ and the inequality $( \ref{ineq:Aj}) $
above, we have
\begin{eqnarray}
I_{\mathbf{k}} &=&\int_{D}\left[ \int_{D^{n-1}}\Pi _{j=2}^{n}A_{_{j}}d\nu
\left( x_{n}\right) \cdots d\nu \left( x_{2}\right) \right] d\nu
\left(x_{1}\right)  \label{mineq2} \\
&\leq &( 2\pi ) ^{n\eta /2}C_{3,3}^{n}\left[ ( n-1) !\right] ^{1-\eta /2}\nu
( D) ^{( n-1) \eta /2+1}.  \notag
\end{eqnarray}
Thus, by $\left( \ref{ineq:Ik1}\right) $ and $\left( \ref{mineq2}\right) $
above, we immediately obtain
\begin{eqnarray*}
\mathbb{E}\left\{ \left[ L\left( \mathbf{t},D\right) -L\left( \mathbf{s
,D\right) \right] ^{n}\right\} &\leq &\left( n-1\right) !\left\Vert \mathbf{
}-\mathbf{s}\right\Vert ^{n\gamma }\sum_{\mathbf{k}\in
\left\{1,...,p\right\} ^{n}}C_{3,4}^{n}r^{( n-1) \eta +2} \\
&\leq &K_{3,2}^{n}\left[ ( n-1) !\right] ^{2-\eta /2}\left\Vert \mathbf{t}
\mathbf{s}\right\Vert ^{n\gamma }\nu ( D) ^{(n-1) \eta /2+1}
\end{eqnarray*}
where the constants $C_{3,4}=C_{3,2}\left( 2\pi \right) ^{\eta /2}C_{3,3}$
and $K_{3,2}=dC_{3,4}.$ The proof is then completed.
\end{proof}

\begin{proof}[Proof of Theorem \textbf{\protect\ref{Th1}: }Joint Continuity]

It follows immediately from Kolmogorov's continuity theorem and Lemmas \re
{L1} and \ref{L2} that, for any two sets $D_{1},D_{2}\subset \mathbb{S}^{2}$
with $\nu \left( D_{i}\right) >0,i=1,2,$ and any $\mathbf{s},\mathbf{t}\in
\mathbb{R}^{d}$, we have
\begin{equation}  \label{ineq:JC1}
\left\vert L( \mathbf{t},D_{1}) -L( \mathbf{s},D_{1})\right\vert \leq
C_{3,5}\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\gamma}\nu ( D_{1})
^{\eta /2},
\end{equation}
and
\begin{equation}  \label{ineq:JC2}
\left\vert L( \mathbf{s},D_{1}) -L( \mathbf{s},D_{2}) \right\vert \leq
C_{3,6}\nu ( \Delta D) ^{\left[ 1-( \alpha -2) d/4\right] },
\end{equation}
where $C_{3,5},C_{3,6}$ are positive constants depending only on
\alpha,A,d, $ and $C_{3,5}$depends on $\gamma $ as well. Moreover, the set
\Delta D=\left\{ D_{1}\cup D_{2}\right\} \backslash \left\{ D_{1}\cap
D_{2}\right\} $, that is, the union of $D_{1}$ and $D_{2},$ excluding their
intersection. Therefore, by $\left( \ref{ineq:JC1}\right) ,\left( \re
{ineq:JC2}\right) $ together with the following inequality
\begin{equation*}
\left\vert L( \mathbf{t},D_{1}) -L( \mathbf{s},D_{2})\right\vert \leq
\left\vert L\left( \mathbf{t},D_{1}\right) -L\left( \mathbf{s},D_{1}\right)
\right\vert +\left\vert L( \mathbf{s},D_{2})-L( \mathbf{s},D_{2})
\right\vert ,
\end{equation*}
we have

\begin{enumerate}
\item[(i)] If replacing $D$ with $D\left( x,r\right) ,$ then for any
r_{1},r_{2}\in \left( 0,\delta \right) ,$ $x\in \mathbb{S}^{2},$
\begin{eqnarray*}
\lefteqn{ \left\vert L( \mathbf{t},D( x,r_{1}) ) -L(\mathbf{s},D( x,r_{2}) )
\right\vert } \\
&\leq &\pi C_{3,5}\left\Vert \mathbf{t}-\mathbf{s}\right\Vert
^{\gamma}r_{1}^{\eta }+\pi C_{3,6}\left\vert r_{1}^{2}-r_{2}^{2}\right\vert
^{\left[1-\left( \alpha -2\right) d/4\right] }
\end{eqnarray*}
and for any $r\in \left( 0,\delta \right) ,$ $x,y\in \mathbb{S}^{2}\ $with
d_{\mathbb{S}^{2}}\left( x,y\right) <2r,$
\begin{eqnarray*}
\lefteqn {\left\vert L\left( \mathbf{t},D\left( x,r\right) \right) -L\left(
\mathbf{s},D\left( y,r\right) \right) \right\vert } \\
&\leq &\pi C_{3,5}\left\Vert \mathbf{t}-\mathbf{s}\right\Vert
^{\gamma}r^{\eta }+6C_{3,6}\left[ d_{\mathbb{S}^{2}}( x,y) r\right] ^{\left[
1-( \alpha -2) d/4\right] },
\end{eqnarray*}
where we have used the fact that
\begin{eqnarray*}
\nu \left( D_{1}\cap D_{2}\right) &=&4\left( \frac{\pi }{2}-\arcsin \frac{d_
\mathbb{S}^{2}}\left( x,y\right) }{r}\right) (1-\cos r) \\
&&-d_{\mathbb{S}^{2}}\left( x,y\right) \sqrt{r^{2}-d_{\mathbb{S}^{2}}(x,y)
^{2}/4} +o\left( d_{\mathbb{S}^{2}}\left( x,y\right) r\right)
\end{eqnarray*}
and
\begin{eqnarray*}
\nu ( \Delta D) &=& 2\pi (1-\cos r)-\nu \left( D_{1}\cap D_{2}\right) \\
&\leq & 6d_{\mathbb{S}^{2}}( x,y) r
\end{eqnarray*}
Here we have denoted by $o( \cdot ) $ the higher order terms.

\item[(ii)] If replacing $D$ with the angular section $V(\vartheta
,\varphi)=\{(\theta ,\phi ):0\leq \theta \leq \vartheta ,0\leq \phi \leq
\varphi \}$ with $(\vartheta ,\varphi )\in \lbrack 0,\pi ]\times \lbrack
0,2\pi )$, then for any
\begin{eqnarray*}
\lefteqn{\left\vert L\left( \mathbf{t},V(\vartheta _{1},\varphi _{1})\right)
-L\left( \mathbf{s},V(\vartheta _{2},\varphi _{2})\right) \right\vert} \\
& \leq & \pi C_{3,5}\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\gamma
}\varphi_{1}^{\eta /2}\vartheta _{1}^{\eta } \\
&&+\pi C_{3,6}\left[ \left\vert \varphi _{1}-\varphi _{2}\right\vert \min
\left\{ \vartheta _{1}^{2},\vartheta _{2}^{2}\right\}+\min \left\{
\varphi_{1},\varphi _{2}\right\} \left\vert \vartheta
_{1}^{2}-\vartheta_{2}^{2}\right\vert \right] ^{\beta /4}
\end{eqnarray*}
The joint continuity of local times $L\left( \mathbf{t},D\right) $ \emph
w.r.t.} $\mathbf{t,}D$ in different cases is then obtained and hence the
proof of Theorem \ref{Th1} is completed.
\end{enumerate}
\end{proof}

\section{H\"{o}lder conditions of local times}

\label{Sec:Holder condition}

In order to prove Theorem \ref{Th2}, we need the following lemma which is
readily seen in view of Lemmas \ref{L1} and \ref{L2} :

\begin{lemma}
\label{Lemma:EL+r.v.} Under conditions of Theorem \ref{Th1}, there exist
positive constants $K_{4,1}$ $K_{4,2}$ depending on $\alpha ,\ K_{0},\ d$
and $K_{4,2}$ depends on $\gamma $ as well, such that for any open disk
D=D\left( x,r\right) \subset \mathbb{S}^{2}$ with $r\in \left(
0,\delta\right) $, any $\mathbf{t}\in \mathbb{R}^{d}$, $x_{0}\in \mathbb{S
^{2}$ and all even integers $n\geq 2,$ and $0<\gamma <1,$
\begin{equation*}
\mathbb{E}\left\{ \left[ L( \mathbf{t}+\mathbf{T}( x_{0}),D) \right]
^{n}\right\} \leq K_{4,1}^{n}( n!) ^{\frac{d}{4}( \alpha -2) }r^{\frac{n}{2
\left[ 4-( \alpha -2) d\right] },
\end{equation*}
\begin{equation*}
\mathbb{E}\left\{ \left[ L( \mathbf{t}+\mathbf{T}( x_{0},D) -L( \mathbf{s}
\mathbf{T}( x_{0}) ,D) \right]^{n}\right\} \leq K_{4,2}^{n}( n!) ^{2-\eta
/2}\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{n\gamma }r^{n\eta },
\end{equation*}
where $\eta $ is the constant defined in $( \ref{eta}) $.
\end{lemma}

\begin{proof}
For any points $x_{1},...,x_{n}\in \mathbb{S}^{2},$ let $\mathbf{Z}(x_{j})
\mathbf{T}( x_{j}) -\mathbf{T}( x_{0}) ,$ $j=1,...,n$, we have
\begin{eqnarray*}
\lefteqn{ \det \mathrm{Cov}\left[ Z_{0}\left( x_{1}\right) ,...,Z_{0}(x_{n})
\right] } \\
&=&\mathrm{Var}( Z_{0}( x_{1}) ) \prod\limits_{j=2}^{n}\mathrm{Var}( Z_{0}(
x_{j}) |Z_{0}( x_{1}),...,Z_{0}( x_{j-1}) ) \\
&\geq & K_{2,2}( K_{2,2}^{\prime }) ^{( n-1) /2}\rho_{\alpha }( x_{1},x_{0})
\prod\limits_{j=2}^{n}\min_{0\leq i\leq j-1}\rho _{\alpha }( x_{j},x_{i}) .
\end{eqnarray*}
in view of Corollary \ref{C2'}. Let $L_{\mathbf{Z}}( \mathbf{t},D) $ be the
local time of $\mathbf{Z}$ at $\mathbf{t}$ in $D,$ and recall (\ref{ELeq})
and (\ref{ineq:mineq3}), then we can obtain that
\begin{eqnarray*}
\lefteqn{\mathbb{E}\left\{ \left[ L_{\mathbf{Z}}( \mathbf{t},D) \right
^{n}\right\} } \\
&\leq& \left( K_{2,2}^{^{\prime }}/K_{2,2}\right) ^{-1/2}(2\pi ) ^{-nd/2}(
K_{2,2}^{\prime }) ^{-nd/2} \\
&&\times \int_{D}\left[ \int_{D^{n-1}}\frac{d\nu ( x_{1}) \cdots d\nu (
x_{n-1}) }{\Pi _{j=2}^{n}\min_{0\leq i\leq j-1} \left[ \rho_{\alpha }( d_
\mathbb{S}^{2}}( x_{i},x_{j}) ) \right]^{d}}\right] \frac{d\nu ( x_{1}) }
\left[ \rho _{\alpha }(x_{1},x_{0}) \right] ^{d}} \\
&\leq & ( C_{4,1}) ^{n}n!\left( \frac{r^{2}}{n!}\right) ^{n\beta /4},
\end{eqnarray*
where $C_{4,1}$ is a positive constant depending on $K_{3,1},K_{2,2}$ and
K_{2,2}^{^{\prime }}.$ Likewise, recall (\ref{ineq:Ik1}) and (\ref{mineq2}),
we have
\begin{eqnarray*}
\lefteqn{\mathbb{E}\left[ L_{\mathbf{Z}}( \mathbf{t},D) -L_{\mathbf{Z}}(
\mathbf{s},D) \right] ^{n}} \\
&\leq& C_{4,2}\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{n\gamma } \\
&& \times \int_{D} \left[ \int_{D^{n-1}}\frac{d\nu \left( x_{n}\right)
\cdots d\nu \left( x_{2}\right) }{\Pi _{j=2}^{n}\min_{0\leq i\leq j-1} \left[
\rho_{\alpha }\left( d_{\mathbb{S}^{2}}\left( x_{i},x_{j}\right) \right)
\right]^{d+2\gamma }}\right] \frac{d\nu \left( x_{1}\right) }{\left[ \rho
_{\alpha}\left( x_{1},x_{0}\right) \right] ^{d+2\gamma }} \\
&\leq & ( C_{4,3}) ^{n}( n!) ^{2-\eta /2}\left\Vert \mathbf{t}-\mathbf{s
\right\Vert ^{n\gamma }r^{n\eta },
\end{eqnarray*}
where $C_{4,2}$ and $C_{4,3}$ are positive constants depending on
K_{2,2},K_{2,2}^{\prime },K_{3,2}$ and $\gamma ,d.$ The results in Lemma \re
{Lemma:EL+r.v.}\ is then derived by the fact that
\begin{equation*}
L_{\mathbf{Z}}( \mathbf{t},D) =L( \mathbf{t+T}(x_{0}) ,D) .
\end{equation*}
\end{proof}

Based on Lemma \ref{Lemma:EL+r.v.}, we now follow the similar line as in the
proof of Lemma 2.7 in \cite{Xiao97}, and obtain the results below:

\begin{lemma}
\label{Lemma:PrLT+r.v.} Assume conditions of Theorem \ref{Th1} hold, there
exists a positive constant $K_{4,3},$ $K_{4,4}$ depending on $d,$ $\alpha
,K_{0}$ and $K_{4,4}$ depends on $\gamma $ as well, such that for any open
disk $D=D( x,r) \subset \mathbb{S}^{2}$ with $r\in \left(0,\delta \right) $,
any $u>0,$ $\mathbf{t}\in \mathbb{\mathbb{R}}^{d}$, $x_{0}\in \mathbb{S}^{2}
,
\begin{equation}  \label{Pr:LT+r.v.}
\mathbb{P}\left\{ L( \mathbf{t}+\mathbf{T}( x_{0}) ,D)\geq \frac{K_{4,3}r^{2
}{\left[ \rho _{\alpha }( ur) \right] ^{d}}\right\} \leq e^{-1/u^{2}}.
\end{equation}
and
\begin{equation}  \label{Pr:LTincrem+r.v.}
\mathbb{P}\left\{ \left\vert L( \mathbf{t}+\mathbf{T}(x_{0}) ,D) -L( \mathbf
s}+\mathbf{T}( x_{0}),D) \right\vert \geq \frac{K_{4,4}r^{2}\left\Vert
\mathbf{t}-\mathbf{s}\right\Vert ^{\gamma }}{\left[ \rho _{\alpha }( ur)
\right]^{d}u^{4\gamma }}\right\} \leq e^{-u^{-2}}.
\end{equation}
\end{lemma}

\begin{proof}
Let
\begin{equation*}
\Lambda =\frac{L( \mathbf{t}+\mathbf{T}( x_{0}) ,D) }{r^{2}},\ u_{n}=\frac{
}{\sqrt{n}}
\end{equation*}
with $n\in\mathbb{N}^{+}.$ Then by Chebyshev's inequality and Lemma \re
{Lemma:EL+r.v.}, we have
\begin{eqnarray*}
\mathbb{P}\left\{ \Lambda \geq \frac{K_{4,3}}{\left[ \rho _{\alpha
}\left(u_{n}r\right) \right] ^{d}}\right\} &\leq &\frac{\mathbb{E}\left[
\Lambda^{n}\right] \rho _{\alpha }^{n}( u_{n}r) }{( K_{3,4})^{n}}( n!) ^
\frac{d}{4}( \alpha -2) }r^{\frac{n}{2}\beta } \\
&\leq &\left( \frac{K_{4,1}}{K_{4,3}}\right) ^{n}\left( n!\right) ^{\frac{d}
4}\left( \alpha -2\right) }u_{n}^{\frac{n}{2}d\left( \alpha -2\right) }. \\
&\leq &\exp \left\{ -\left( \frac{\left( \alpha -2\right) d}{8}-\ln \frac
K_{4,1}}{K_{4,3}}\right) n\right\} \\
&\leq &\exp \left\{ -2/u_{n}^{2}\right\} ,
\end{eqnarray*}
where the last inequality follows from Stirling's formula, and the constants
satisfy $K_{4,3}=K_{4,1}\exp \left\{ 2-\frac{\left( \alpha -2\right) d}{8
\right\} .$ Now for any $u>0$ small enough, there exists $n\in\mathbb{N
^{+}, $ such that $u_{n+1}\leq u<u_{n},$ and
\begin{eqnarray*}
\mathbb{P}\left\{ \Lambda \geq \frac{K_{4,3}}{\left[ \rho _{\alpha
}\left(ur\right) \right] ^{d}}\right\} &\leq & \mathbb{P}\left\{ \Lambda
\geq \frac{K_{4,3}}{\left[ \rho _{\alpha }\left( u_{n}r\right) \right] ^{d}
\right\} \\
&\leq& \exp \left\{ -2/u_{n}^{2}\right\} \leq \exp \left\{ -u^{-2}\right\} .
\end{eqnarray*}
Hence inequality $\left( \ref{Pr:LT+r.v.}\right) $ is derived.

The proof of estimation $\left( \ref{Pr:LTincrem+r.v.}\right) $ is similar
to the argument above and we omit it here. Thus the proof of Lemma \re
{Lemma:PrLT+r.v.} is completed.
\end{proof}

Now we sketch a proof for Theorem \ref{Th2}.

\begin{proof}[\textbf{Proof of Theorem \protect\ref{Th2}}]
The proof for the upper bounds in the inequalies $\left( \ref{ineq:Local
Holder of LT}\right) $ is based on a chaining argument and quite similar to
the proof of Theorem 1.1 and 1.2 in \cite{Xiao97} section 3 by replacing the
notations $X(t),$ $B(\tau ,2^{-n})$ with ones of $\mathbf{T}\left( x\right) ,
$ $D\left( x_{0},2^{-n}\right) $ for some $x_{0}\in \mathbb{S}^{2},$ and
Lemma 2.7, 3.1 in \cite{Xiao97} with Lemmas \ref{LemTupper} and \re
{Lemma:PrLT+r.v.} in this paper.

For the lower bounds in the inequities $(\ref{ineq:Local Holder of LT}),$ we
first let $\mathbf{I}$ be the closure of the set $\mathbf{T}(D)\mathbf{=
\left\{ \mathbf{T}(x),x\in D\right\} $ for any disk $D=D(z,r)\subset \mathbb
S}^{2}.$ Recall the definition of local time $\left( \ref{def:Local time
\right) $ or formula $\left( \ref{eq:LT-occup density}\right) ,$ we have for
any $\omega \in \Omega ,$
\begin{eqnarray}
\nu (D) &=&\mu _{D}(I,\omega )=\int_{\mathbf{I}}L(\mathbf{t},D,\omega )
\mathbf{t}  \notag \\
&\leq &L^{\ast }(D,\omega )\cdot \frac{\pi ^{d/2}}{\Gamma (d/2+1)}\left\vert
\sup_{x,y\in D}\left\Vert \mathbf{T}(x)-\mathbf{T}(y)\right\Vert \right\vert
^{d}.  \label{ineq:LT-mod}
\end{eqnarray
That is, for any $z\in \mathbb{S}^{2},$
\begin{eqnarray*}
\liminf_{r\rightarrow 0}\frac{L^{\ast }(D(z,r))}{\phi _{1}\left( r\right) }
&\geq &C_{4,4}\cdot \liminf_{r\rightarrow 0}\left\{ \frac{\rho _{\alpha }(r
\sqrt{\log \log r^{-1}})}{\sup_{x,y\in D(z,r)}\left\Vert \mathbf{T}\left(
x\right) -\mathbf{T}(y)\right\Vert }\right\} ^{d} \\
&\geq &C_{4,4}\left\{ \lim_{r\rightarrow 0}\sup_{x,y\in D(z,r)}\frac
\left\Vert \mathbf{T}(x)-\mathbf{T}(y)\right\Vert }{\rho _{\alpha }(r/\sqrt
\log \log r^{-1}})}\right\} ^{-d} \\
&=&C_{4,4}K_{5,3}^{-d},
\end{eqnarray*
in view of Proposition \ref{Prop:Loc non-dif} in Section \ref{Sec:non-diff},
which derives the lower bounds in the equalities $\left( \ref{ineq:Local
Holder of LT}\right) $ in Theorem \ref{Th2}. The positive constant $C_{4,4}$
depends only on $d.$ The proof is then completed.
\end{proof}

\section{Modulus of non-differentiability \label{Sec:non-diff}}

In this section, we establish local and global Chung's law of the iterated
logarithm, or we say, the mudulus of non-differentiability of the random
field $\mathbf{T}$, which are the essential of proving the lower bound of
maximum local time in Theorem \ref{Th2}.

Before giving the main results, we first introduce the following
band-limited random field. For any two integers $1\leq L<U\leq \infty ,$ we
define a random field as follows:
\begin{equation*}
T_{0}^{L,U}( x) =\sum_{\ell =L}^{U}\sum_{m=-\ell }^{\ell }a_{\ell m}Y_{\ell
m}( x) ,\ x\in \mathbb{S}^{2}.
\end{equation*}
Observe that $T_{0}^{L,U}\left( x\right) $ and $T_{0}^{L^{\prime
},U^{\prime}}\left( x\right) $ are independent for $L<U<L^{\prime
}<U^{\prime }$ in view of the orthogonality properties of the Fourier
components of the field $T_{0}(x)$ and the assumption of Gaussianity.

Meantime, let $T_{0}^{\Delta }$ be the random field defined by
\begin{equation*}
T_{0}^{\Delta }=T_{0}-T_{0}^{L,U}.
\end{equation*}
For any $r$ small, take $L=\left[ r^{-1}( B( r) )^{-\kappa _{1}}\right] $
and $U=\left[ r^{-1}( B( r) )^{1-\kappa _{1}}\right] ,$ where $B( r) $ is a
function such as $( \log \log r^{-1}) ^{\kappa _{2}}$ or $\left( \log
r^{-1}\right)^{\kappa _{2}}$ and the constants $\kappa _{1},\kappa _{2}$ are
to be determined. Here $[ \cdot ] $ denotes integer part as usual. Then we
have the following approximation for $T_{0}^{\Delta }:$

\begin{lemma}
\label{LemTtail} Under the same condition as in Theorem \ref{Th1}, there
exist positive constants $K_{5,1}$ and $K_{5,2}$ depending only on $\alpha $
and $K_{0},$ such that for any $0<r<\delta ,$ $0<\kappa _{1}\leq \frac
\alpha }{2}-1$ and
\begin{equation*}
u > K_{5,1}( B( r) ) ^{-\kappa _{1}( 2-\frac{\alpha }{2}) }\sqrt{\log B( r)
\ r^{\frac{\alpha }{2}-1},
\end{equation*}
we have
\begin{equation*}
\mathbb{P}\left\{ \sup_{x,y\in D( z,r) }\left\vert T_{0}^{\Delta}( x)
-T_{0}^{\Delta }( y) \right\vert \geq u\right\} \leq \exp \left( -\frac{1}
K_{5,2}}\frac{\left( B\left( r\right) \right) ^{\kappa _{1}( 4-\alpha )
}u^{2}}{r^{\alpha -2}}\right) .
\end{equation*}
\end{lemma}

\begin{proof}
Like in many other arguments in this paper, we start by introducing a
suitable Gaussian metric $d_{T_{\Delta }}$ defined on $D( z,r)\subset
\mathbb{S}^{2}$ by
\begin{equation*}
d_{T_{0}^{\Delta }}( x,y) :=\left[ \mathbb{E}\left\vert T_{0}^{\Delta }( x)
-T_{0}^{\Delta }( y) \right\vert ^{2}\right] ^{1/2}.
\end{equation*}
Once again, due to the fact that $d_{\mathbf{T}^{\Delta }}( x,y)\leq \sqrt
K_{2,1}}\rho _{\alpha }( x,y) ,$ a simple metric entropy argument yields
\begin{equation*}
N_{d_{T_{0}^{\Delta }}}( D( z,r) ,\varepsilon ) \leq C_{5,1}\frac{r^{2}}
\epsilon ^{4/( \alpha -2) }},
\end{equation*}
with $C_{5,1}$ depending on $K_{2,1}.$ More precisely, recall
\begin{equation*}
d_{T_{0}^{\Delta }}^{2}\left( x,y\right) =\left(
\sum_{\ell=0}^{L-1}+\sum_{\ell =U+1}^{\infty }\right) \frac{2\ell +1}{4\pi
C_{\ell}\left\{ 1-P_{\ell }(\cos \theta )\right\} .
\end{equation*}
where $\theta =d_{\mathbb{S}^{2}}\left( x,y\right) .$ Let $0<\theta <r,$ by
Lemma \ref{C1'} in the Appendix, we obtain
\begin{eqnarray*}
d_{\mathbf{T}^{\Delta }}^{2}( x,y) &\leq & K_{A}(L^{4-\alpha }\theta
^{2}+U^{2-\alpha }) \\
&\leq & K_{A}\left[ B( r) ^{-\kappa _{1}( 4-\alpha )}+B( r) ^{-( 1-\kappa
_{1}) ( \alpha -2) }\right] r^{\alpha -2} \\
&\leq & K_{A}B( r) ^{-\kappa _{1}( 4-\alpha )}r^{\alpha -2}:=\left\vert h(
r) \right\vert ^{2}r^{\alpha -2},
\end{eqnarray*}
where
\begin{equation*}
h( r) =:\sqrt{K_{A}}B( r) ^{-\frac{\kappa _{1}}{2}( 4-\alpha ) }.
\end{equation*}
Hence, if we let
\begin{equation*}
\overline{d}:=\sup \left\{ d_{T_{0}^{\Delta }}( x,y) :x,y\in D( z,r)
\right\} ,
\end{equation*}
which obviously have $\overline{d}\leq h( r) r^{\frac{\alpha }{2}-1}$, then
\begin{equation*}
\begin{split}
& \int_{0}^{\overline{d}}\sqrt{\log N_{d_{T_{0}^{\Delta }}}( D(z,r)
,\epsilon ) }\ d\epsilon \leq \int_{0}^{h( r) r^{\frac{\alpha }{2}-1}}\sqrt
\log C_{4,1}\frac{r^{2}}{\epsilon ^{4/(\alpha -2) }}}\ d\epsilon \\
& \leq \frac{2}{\sqrt{\alpha -2}}\sqrt{K_{2,1}}r^{\frac{\alpha }{2}-1}\int_
\sqrt{\log \frac{\sqrt{K_{2,1}}}{h\left( r\right) }}}^{+\infty }ud\left(
-e^{-u^{2}}\right) \\
& \leq \frac{4h\left( r\right) }{\sqrt{\alpha -2}}\sqrt{\log \frac{\sqrt
K_{2,1}}}{h\left( r\right) }}\ r^{\frac{\alpha }{2}-1}.
\end{split
\end{equation*}
By exploiting Lemma \ref{DudleyUB}, we immediately derive that, for any
\begin{equation*}
u > C_{5,2}( B( r) ) ^{-\kappa _{1}( 2-\alpha /2) }\sqrt{\log B( r) }\
r^{\alpha /2-1}
\end{equation*}
with $C_{5,2}>0$ depending on $K_{2,5},K_{A},\alpha ,\kappa _{1},$ it holds
that
\begin{equation*}
\mathbb{P}\left\{ \sup_{x,y\in D( z,r) }\left\vert T_{0}^{\Delta}( x)
-T_{0}^{\Delta }( y) \right\vert \geq u\right\} \leq \exp \left( -\frac{u^{2
}{4K_{2,5}^{2}\left\vert h( r)\right\vert ^{2}r^{\alpha -2}}\right)
\end{equation*
Letting $K_{5,1}=2K_{2,5}C_{5,2}$ and $K_{5,2}=4K_{2,5}^{2}K_{A},$ the proof
is then completed.
\end{proof}

Now let us focus on the Chung's law of the iterated logarithm.

\begin{proposition}
\label{Prop:Loc non-dif} Under the conditions of Theorem \ref{Th1}, there
exists positive constants $K_{4,2}$ such that for any $z\in \mathbb{S}^{2}$
with
\begin{equation}  \label{eq:loc non-diff}
\lim_{r\rightarrow 0}\sup_{x,y\in D( z,r) }\frac{\left\Vert \mathbf{T}( x)
\mathbf{T}( y) \right\Vert }{\rho_{\alpha }\left( r/\sqrt{\log \log r^{-1}
\right) } =K_{5,3}.
\end{equation}
\end{proposition}

\begin{proof}
Due to Lemma 7.1.1 in Marcus and Rosen \cite{MRbook} and the fact that
\left\Vert \mathbf{T}( x) -\mathbf{T}( y) \right\Vert =\sqrt{d
|T_{0}(x)-T_{0}(y)|,$ we only need to prove the upper and lower bounds of
the following form: there exist positive and finite constants $C_{5,3}$ and
C_{5,4}$ such that
\begin{equation}  \label{ineq:Llocmod}
\lim_{r\rightarrow 0}\sup_{x,y\in D( z,r) } \frac{|T_{0}(x)-T_{0}(y)|}{\rho
_{\alpha }\left( r/\sqrt{\log \log r^{-1}}\right) } \geq C_{5,3},\ \
\hbox{
a.s.}
\end{equation}
and
\begin{equation}  \label{ineq:Ulocmod}
\lim_{r\rightarrow 0}\sup_{x,y\in D( z,r) }\frac{|T_{0}(x)-T_{0}(y)|}{\rho
_{\alpha }\left( r/\sqrt{\log \log r^{-1}}\right) } \leq C_{5,4},\ \
\hbox{
a.s. }
\end{equation
which implies $( \ref{eq:loc non-diff}) $\ with $K_{5,3}\in \lbrack
C_{5,3},C_{5,4}].$

Recall $\left( \ref{ineq:LT-mod}\right) $ and the upper bound of $\left( \re
{ineq:Local Holder of LT}\right) $ in Theorem \ref{Th2}$,$ we have for any
z\in \mathbb{S}^{2},$
\begin{eqnarray*}
&&\liminf_{r\to 0}\sup_{x,y\in D( z,r) }\frac{\left\Vert \mathbf{T}( x)
\mathbf{T}( y)\right\Vert }{\rho _{\alpha }\left( r/\sqrt{\log \log r^{-1}
\right) } \\
&\geq &C_{4,4}\left[ \limsup_{r\to 0}\frac{L^{\ast}( D( z,r) ) }{\phi _{1}(
r) }\right]^{-1} \geq \frac{C_{4,4}}{K_{3}}.
\end{eqnarray*
where $\phi _{1}$ is defined in $\left( \ref{psi1}\right) .$ One can verify
(cf. Lemma 7.1.6 in \cite{MRbook}) that this implies $\left( \re
{ineq:Llocmod}\right) .$

Now let us focus on the Proof of $\left( \ref{ineq:Ulocmod}\right) $. Let
B( r) =\left( \log \log r^{-1}\right) ^{\kappa _{2}}$ for any $r\in \left(
0,\delta \right) $ and $\kappa _{2}>0$ to be determined. Now we choose
r_{k}=( 2\log k) ^{-k},k=1,2,\cdots ,$ and $L_{k}=\left[ B\left(
r_{k}\right) ^{-\kappa _{1}}r_{k}^{-1}\right] $ as well as $U_{k}=\left[
B\left( r_{k}\right) ^{1-\beta }r_{k}^{-1}\right] ,$ where $\beta \in (0
\frac{\alpha }{2}-1]$ and $\left[ \cdot \right] $ denotes the integer part
as before. Obviously $L_{k}<U_{k}<L_{k+1}<U_{k+1}$ for any $k\in \mathbb{N
^{+}.$ We would like to prove that for some constant $C_{5,5}>0,$
\begin{equation}  \label{eq:UTmain}
\dsum\limits_{k=1}^{\infty }\mathbb{P}\left\{ \sup_{x,y\in D(z,r_{k})
}\left\vert T_{0}^{L_{k},U_{k}}( x)-T_{0}^{L_{k},U_{k}}( y) \right\vert \leq
C_{5,5}\rho _{\alpha}\left( \frac{r_{k}}{\sqrt{\log \log r_{k}^{-1}}}\right)
\right\} =\infty .
\end{equation
\label{UTmain} Thus, due to the independence of $T_{0}^{L_{k},U_{k}}$ for
different $k^{\prime }s,$ we have
\begin{equation}
\limsup_{k\to\infty}\sup_{x,y\in D(z,r_{k}) }\left\vert T_{0}^{L_{k},U_{k}}(
x)-T_{0}^{L_{k},U_{k}}( y) \right\vert \leq C_{5,5}\rho _{\alpha}\left(
\frac{r_{k}}{\sqrt{\log \log r_{k}^{-1}}}\right) ,\ a.s.
\end{equation}
in view of the Borel-Cantelli Lemma. The equation $\left( \ref{eq:UTmain
\right) $ can be derived by the fact that
\begin{equation*}
d_{T_{0}^{L_{k},U_{K}}}=\sqrt{\mathbb{E}\left\vert T_{0}^{L_{k},U_{k}}(x)
-T_{0}^{L_{k},U_{k}}( y) \right\vert ^{2}} \leq d_{T_{0}},
\end{equation*
and thus, similar to the argument as in the proof of Lemma \ref{Lem:Small
Ball P}, we have for some constant $C_{4,5}>0,$
\begin{eqnarray*}
&&\mathbb{P}\left\{ \sup_{x,y\in D( z,r_{k}) }\left\vert
T_{0}^{L_{k},U_{k}}( x) -T_{0}^{L_{k},U_{k}}( y) \right\vert \leq
C_{5,5}\rho _{\alpha }\left( \frac{r_{k}}{\sqrt{\log \log r_{k}^{-1}}
\right) \right\} \\
&\geq & \exp \left( -\frac{2K_{2,8}}{C_{5,5}}\log k\right) =k^{-1/2},
\end{eqnarray*}
where we have taken $C_{5,5}=4\log eK_{2,8}.$

On the other hand, recall $T_{0}^{\Delta _{k}}=T_{0}-T_{0}^{L_{k},U_{k}}.$
\textit{Let } $B\left( r_{k}\right) ^{\kappa _{1}\left( 4-\alpha \right)}=(
\log \log r_{k}^{-1}) ^{\alpha /2},$ then it is readily seen that for any
k\in \mathbb{N}^{+},$
\begin{equation*}
\rho _{\alpha }\left( r_{k}/\sqrt{\log \log r_{k}^{-1}}\right) > (B( r_{k})
) ^{-\kappa _{1}( 2-\alpha /2) }\sqrt{\log B( r_{k}) }r_{k}^{( \alpha -2)
/2},
\end{equation*}
and thus, by Lemma \ref{LemTtail}, we have for any constant $C_{5,6}>0,$
\begin{eqnarray*}
&&\mathbb{P}\left\{ \sup_{x,y\in D( z,r_{k}) }\left\vert T_{0}^{\Delta
_{k}}( x) -T_{0}^{\Delta _{k}}( y)\right\vert >C_{5,6}\rho _{\alpha }\left(
r_{k}/\sqrt{\log \log r_{k}^{-1}}\right) \right\} \\
&\leq & \exp \left( -\frac{C_{5,6}^{2}}{K_{5,2}}\frac{\left(
B\left(r_{k}\right) \right) ^{\kappa _{1}\left( 4-\alpha \right) }}{\left(
\log\log r_{k}^{-1}\right) ^{\alpha /2-1}}\right) \leq \exp \left( -\frac
C_{5,6}^{2}}{K_{5,2}}\log \log r_{k}^{-1}\right) \\
&\leq &\exp \left( -\frac{C_{5,6}^{2}}{K_{5,2}}\log k\right) =k^{-2},
\end{eqnarray*}
where we have chosen $C_{5,6}^{2}\log e=2K_{5,3}$. \textit{\ Therefore, we
have }
\begin{equation*}
\dsum\limits_{k=1}^{\infty }\mathbb{P}\left\{ \sup_{x,y\in D(z,r_{k})
}\left\vert T_{0}^{\Delta _{k}}( x) -T_{0}^{\Delta_{k}}( y) \right\vert >
C_{5,6}\rho _{\alpha }\left( \frac{r_{k}}{\sqrt{\log \log r_{k}^{-1}}
\right) \right\} <\infty ,
\end{equation*}
and again by the Borel-Cantelli Lemma, we have
\begin{equation}  \label{UTtail}
\limsup_{r\to0}\sup_{x,y\in D(z,r_{k}) }\left\vert T_{0}^{\Delta _{k}}( x)
-T_{0}^{\Delta_{k}}( y) \right\vert \leq C_{5,6}\rho _{\alpha }\left( \frac
r_{k}}{\sqrt{\log \log r_{k}^{-1}}}\right) ,\ a.s.
\end{equation}
The inequality $\left( \ref{ineq:Ulocmod}\right) $ is then derived in view
of $( \ref{UTmain}) $ and $( \ref{UTtail}) .$
\end{proof}

\section{Appendix}

\label{appendix}

\begin{lemma}
\label{C1'} There exists a positive constant $K_{A}$ depending on $K_{0}$
and $\alpha $, such that for any $\theta >0$ small, and positive integers $L
\frac{K_{A}}{\theta },$ we have
\begin{equation*}
\sum_{\ell =1}^{L}\frac{2\ell +1}{4\pi }C_{\ell }\left\{ 1-P_{\ell
}(\cos\theta )\right\} \leq K_{A}L^{4-\alpha }\theta ^{2}\text{ ,}
\end{equation*}
and, for any $U>1,$
\begin{equation*}
\sum_{\ell =U}^{\infty }\frac{2\ell +1}{4\pi }C_{\ell }\left\{
1-P_{\ell}(\cos \theta )\right\} \leq K_{A}U^{2-\alpha }.
\end{equation*}
\end{lemma}

\begin{proof}
Recall first that, from Condition (A), there exists a positive constant
K_{0}$ such that for all $\ell =1,2,...$
\begin{equation*}
K_{0}^{-1}\ell ^{-\alpha +1}\leq \frac{2\ell +1}{4\pi }C_{\ell } \leq
K_{0}\ell ^{-\alpha +1}.
\end{equation*}
We recall also the following\ Hilb's asymptotics results (see \cite{szego
,page 195, Theorem 8.21.6): for $K_{A}>0,$ we have uniformly
\begin{equation*}
P_{\ell }(\cos \theta ) =\left\{ \frac{\theta }{\sin \theta
\right\}^{1/2}J_{0}((\ell +\frac{1}{2})\theta )+\delta _{\ell }(\theta ),
\end{equation*}
where
\begin{equation*}
\delta _{\ell }(\theta )<<\left\{
\begin{array}{cl}
\theta ^{2}O(1) & \text{ for }0<\theta <\frac{K_{A}}{\ell } \\
\theta ^{1/2}O(\ell ^{-3/2}) & \text{ for }\theta >\frac{K_{A}}{\ell
\end{array}
\right. ;
\end{equation*}
Moreover,
\begin{equation*}
\lim_{u\rightarrow 0}\frac{\left\{ 1-J_{0}(K_{A}u)\right\} }{K_{A}^{2}u^{2}}
\frac{1}{2}\text{.}
\end{equation*}
Thus, by using the fact that
\begin{equation*}
\frac{\theta }{\sin \theta }-1=\frac{\theta ^{2}}{6}+O(\theta ^{3})\text{
,as }\theta \rightarrow 0\text{,}
\end{equation*}
we obtain that
\begin{eqnarray*}
\sum_{\ell =1}^{L}\frac{2\ell +1}{4\pi }C_{\ell }\left\{ 1-P_{\ell
}(\cos\theta )\right\} &\leq & K_{0}\sum_{\ell =1}^{L}\ell ^{1-\alpha
}\left( \frac{\ell ^{2}}{K_{A}^{2}}-\frac{1}{6}\right) \theta ^{2} \\
&\leq & \frac{K_{0}}{\left( 4-\alpha \right) K_{A}^{2}}L^{4-\alpha
}\theta^{2}.
\end{eqnarray*}
On the other hand, recall that $1-P_{\ell }(\cos \theta )\leq 2$ uniformly
for all $\theta ,$ whence we have, for any $U>1,$
\begin{equation*}
\sum_{\ell =U}^{\infty }\frac{2\ell +1}{4\pi }C_{\ell }\left\{
1-P_{\ell}(\cos \theta )\right\} \leq K_{1,0}\sum_{\ell =U}^{\infty }\ell
^{1-\alpha}\leq \frac{2K_{1,0}}{\alpha -2}U^{2-\alpha }.
\end{equation*}
Let $K_{A}=\max \left\{ \sqrt[3]{\frac{K_{0}}{\left( 4-\alpha \right) }}
\frac{2K_{0}}{\alpha -2}\right\} ,$ the proof is then completed.
\end{proof}

\bigskip

redpajama_set_name: RedPajamaArXiv


The Black Women in Sport Foundation (BWSF) recently announced the winners of its inaugural essay contest for Women's History Month. The contest was created as an inter-generational history project to continue sharing the rich legacy of African-American women achievements with the younger generation. Youth paid tribute to the late Delores Andy, Venus Williams and Serena Williams with their essays.
One of the winners from the category of fifth- to eighth-graders included: Kierra Adams, (first place), Northeast Frankford Boys & Girls Club of Philadelphia.
The winners were selected based on the scores received from independent judges Linda Carroll-Pitts, Ora Lambright and Jerika Johnstone. The contest was open to participants of BWSF programs and special events. BWSF received entries from youth at the Northeast Frankford Boys & Girls Club of Philadelphia; Masterman School; Beverly Hills Middle School; Tanner Duckrey School; McKinley School; and SCOP Programs Girls Track.


Heng Yin1,2, Xavier C. Fretté3, Lars P. Christensen3, Kai Grevsen1.
1Department of Food Science, Faculty of Science and Technology, Aarhus University, Kirstinebjergvej 10, Aarslev DK-5792, Denmark.
2Liaoning Provincial Key Laboratory of Carbohydrates, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People's Republic of China.
Greek oregano is commonly used as a spice and in traditional medicine in Eurasia. The plant is rich in secondary metabolites, such as volatile organic compounds (VOC) and polyphenols. Chitosan oligosaccharides (COS) are used as a plant elicitor. The objectives of this study were to determine the effects of COS on the growth and content of secondary metabolites in Greek oregano. Four COS treatments (50, 200, 500, and 1000 ppm) were used in a field experiment. The 200 and 500 ppm COS treatments promoted plant height growth, whereas 50 and 200 ppm COS upregulated the content of polyphenols significantly (38 and 29%, respectively). The COS treatments induced H2O2 generation in Greek oregano leaves; thus, the effect of H2O2 treatment was studied to investigate the possible role of H2O2 in growth and polyphenol production. A low concentration of H2O2 also promoted plant height growth, but only tendencies to higher polyphenol content were seen.


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On Wednesday 5th October 2016, TV3 launched The Pat Kenny Show.
The current affairs programme hosted by broadcasters Pat Kenny and Colette Fitzpatrick. The programme goes out live with an audience of up to 150 people at the Virgin Media TV3 HD Studio in Ballymount.
53 Degrees Design was responsible for the set construction including all built-in controllable RGB LEDs.


The Counselors of Real Estate (CRE) is an international organisation of high profile property professionals which include principals of prominent real estate, financial, legal and accounting firms as well as recognised leaders of government and academia.
Membership is extended by invitation and candidates may independently apply if they confirm a minimum of 10 years of real estate experience and 3 years of property advisory services. A stringent process of selection ensures that individuals awarded with the "CRE" credential have attained the highest levels of expertise and achievement in their real estate speciality, particularly in the provision of sophisticated advice and high-level problem solving.
The organisation is characterised by its selective, powerful, highly functioning network of leading property professionals worldwide. Members who hold the CRE credential are senior real estate decision makers; they are a diverse, seasoned group encompassing a variety of backgrounds – developers, investors, appraisers, attorneys, financial experts, portfolio and pension fund managers, architects, etc.
In addition to a strong American base, The Counselors has expanded its global reach, with a vibrant, international presence reflecting 20 countries. Jonathan is on the Board and is Vice Chair of the CRE European Chapter. With so many of the world's most pressing issues rooted in real property, The Counselors takes seriously its involvement in matters relating to transportation, affordable housing, education, the environment, and more recently, economies in turmoil.
Jonathan became a CRE in 2005.
The Landauer/White Award is presented each year to an individual who has furthered the ideals of The Counselors of Real Estate, demonstrated outstanding professionalism in real estate and been a notable influence civically. Find out more about the award here.
Jonathan received this award in 2009.


The Kush God is back at it with his release of the K.O.N.Y. mixtape, for those of who haven't yet experienced the smooth Harlem flow of the new JETS member this is your chance. I've been listening to Smoke Dza for a while and he totally embodies the Harlem king pin persona mixed with the cool of a pot head, and smoke doesn't fall short of giving you all he is on this mixtape. The K.O.N.Y. mixtape has more features than his other works but hey how can u not look forward to a mixtape that has features of people like Wiz Khalifa and Joey Bada$$! Any way this mixtape is a solid project but it is still missing the special something that has been evident on other Smoke Dza projects. K.O.N.Y. is super far from bad but not great either it's just all RIIIIIIIIGHT!
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text:
Review: Mercedes-Benz S-Class
Mercedes-Benz has launched the seventh generation of its flagship S-Class, demonstrating its worldview of a car that merges the concepts of uber luxury and next-generation technology
It's not incorrect to say that Mercedes-Benz is a brand built around the S-Class. Recently, it unveiled a EUR730m Factory 56 in Sindelfingen to begin production of its seventh-generation S-Class model which has been its flagship offering since the seventies. When you're ready to invest nearly three-quarters of a billion euros, future intent becomes crystal clear.
The new S-Class features an aluminium hybrid bodyshell. As a stately saloon, it has considerable road presence. The long-wheelbase model is now 34mm longer than the outgoing model, 55mm wider and 12mm taller.
As Britta Seeger, a board member at Daimler AG and head of marketing and sales at Mercedes-Benz recently told Gulf Business, Asian markets prefer the long-wheelbase version as nine out of 10 of its customers there are chauffeured. In the Western and American markets, however, customers prefer to drive themselves and hence opt for a shorter wheelbase variant.
For those in the driver's seat, there's much to be had with the six-cylinder 2.9-litre in-line engine that is being offered at launch with the S450 and S500.
With a power output of 367hp and 435hp respectively, they also produce 500Nm and 520Nm of torque, which result in an average 0-100kph time of 5.1 seconds and 4.9 seconds – helped of course, by the nine-speed automatic gearbox.
Both models get a 22hp boost under acceleration due to the EQ Boost thanks to an integrated starter generator. Another upcoming hybrid version of the S-Class will also be able to deliver 100km of electric range. Mercedes added that a V8 version of the car with a 48V mild-hybrid version will follow shortly. Seeger confirmed that plans are afoot, expectedly, to launch an AMG version of the S-Class, and an eagerly anticipated top-drawer Maybach variant too.
A six-cylinder 2.9-litre in-line engine is being offered at launch with the S450 and S500
"Since we launched in 2015, the S-Class Maybach has been a huge success with more than 50,000 units sold since then. We will launch the Maybach this year, with additional distinctive features that will be available on the to-be-launched S-Class Maybach," says Seeger.
While the Maybach version of the S-Class will make its appearance later in the year, the existing S-Class we now have doesn't skimp on luxury. For example, it features a new 4D 30-speaker 1,750 watt Burmester high-end audio system. The fourth-dimension effect comes by way of the bass, which apart from the 400 watt subwoofer, is also fed through the pores of the seat through built-in vibrators.
Each of those seats, by the way, are controlled by 18 individual motors and have 10 different massage options – it beats business class seats in an aircraft by a long way.
But for a passenger within the car, one of the biggest highlights will be the revamped user interface – the MBUX. Mercedes says that it has increased the computing power of MBUX by 50 per cent compared to the outgoing model. There were reportedly over 30 million lines of code written by hundreds of engineers as they fine-tuned the processes of the car that go far beyond changing the cabin's ambient lighting or turning down the AC.
Mercedes has eliminated 27 analogue controls from the car and replaced them with gesture controls and swipes – a gesture can be used to open the sunroof, or with only a swipe, the driver can transfer the entire infotainment display from the front console to the rear-seat passenger's screen.
The long-wheelbase model is now 34mm longer than the outgoing model, 55mm wider and 12mm taller
One of Mercedes' biggest strengths with the MBUX was to make the system compatible with 27 languages – Arabic included – activated with a straightforward "Hey Mercedes". It has a natural language understanding feature that allows it to be properly conversational and pick up dialects and pronunciations. No more bloopers of the infotainment displaying the nearest public washroom on the screen, when all you asked was to turn up the music volume.
The MBUX goes further too. By way of an MBUX Smart Home function which uses WLAN and sensors, you can ask Mercedes to check the temperature of your home and whether you left the electric kettle at home on, all while seated in your car.
In China, the MBUX is also being made compatible with payment gateways to allow customers to, for example, order food or groceries, buy movie tickets and even make restaurant reservations from their MBUX itself. The decision to offer unique features to the Chinese market is a conscious one – considering that China is S-Class' biggest market. "The strongest market for the S-Class are, in order, China, US, South Korea and Germany. In China, we find the youngest customer segment [with an average age of] around 40. Many of them are first-car buyers, and are into new technology," explains Seeger.
She adds that in South Korea, the S-Class is particularly strong with women, with every fourth S-Class customer in the country being a woman.
The other top markets for the S-Class – the US and Western Europe, especially Germany – show steadfast customer loyalty, with between 70-80 per cent of the car's buyers there previously owning a Mercedes.
As noteworthy as Factory 56 is, so is Daimler's Immendingen test track facility that was built in 2018 with an investment of EUR200m.
Spread across 520 hectares, it is here that Mercedes has tested not only some of its most significant safety features (the body of the S-Class automatically rises by 8cm when it detects an imminent side crash, besides deploying rear seat passenger airbags), but is also where it has been testing its autonomous driving capabilities.
Seeger confirmed that they have received regulatory approvals to activate the new S-Class' Level 3 autonomous driving capabilities on German roads starting next year. This will allow autonomous driving on public roads at speeds of up to 60kph.
The car is equipped with the technology of up to Level 4 autonomous driving, which allows it to self-park in public garages. However, she cautioned that getting necessary regulatory approvals from governments around the world can be a tedious task and might delay the rollout of Level 3 or Level 4 technologies in several markets.
But Seeger says that for highest Level 5 of complete autonomous driving, Mercedes has plans beyond its passenger car fleet. "We have a very clear strategy and logic for the use cases [of Level 5]. One use of Level 5 is less on the passenger car side in crowded city traffic, but a higher use case in our trucks for hub-to-hub delivery."
A fully autonomous S-Class isn't a near-term priority, but a fully-electric flagship sibling for the S-Class definitely is. The new EQS with a promised range of around 700km will make an appearance in 2021.
Will the S-Class then finally meet its ultimate competition, and that too from a sibling? "We'll give a lot of S-Class DNA to the EQS, but in addition we will also show a new interpretation on the interiors and exteriors and include additional features exclusive to the EQS," Seeger notes, quickly adding, "the S-Class and EQS will not cannibalise each other".
"The S-Class stands for the core of Mercedes-Benz – it is the tradition and heart of Mercedes-Benz." The new-generation model makes that abundantly clear.
Auto Lifestyle Luxury Mercedes-Benz S-Class UAE
Varun Godinho October 10, 2020
Dubai, Energy
Covid-19 PCR tests mandatory for those visiting inpatients at medical facilities in Abu Dhabi

redpajama_set_name: RedPajamaCommonCrawl


This page contains a directory for all types of weapons in PlayerUnknown's BattleGrounds (PUBG) for PC, PS4, and XBOX One.
This page contains a database directory for all types of weapons in PlayerUnknown's BattleGrounds (PUBG) for PC, PS4, and XBOX One, including weapons descriptions, stats, ammunition, map availability, recommended attachments and upgrades, usage and strategies in the game. Click on a weapon type to go to its directory page for detailed guides on each weapon under the category.
(AR) A very useful weapon that can be used at any distance. Its Power and Accuracy is high, and you can also use it to snipe enemies by adding a scope.
(SMG) SMGs are useful whenever you are moving around near enemy lanes. You can also use handgun bullets for some SMGs, which comes pretty handy whenever you're saving your rarer ammo.
(LMG) LMGs can only be obtained through air drops. The Light Machine Gun is a very powerful weapon, hence its scarcity in the battlefield.
(SG) Shotguns are very useful in close range combat. This type of weapon is good to pair with Assault Rifles.
(SR) Sniper Rifles are used primarily for long range combat. Those who use SRs typically hide in high vantage points to get a better look around the map. This type of weapon has high accuracy and power.
(DMR) DMRs are like Sniper Rifles but with lower accuracy and power. However, this type of weapon is much more versatile than SRs because it can also be used in mid-range distance.
Bow The Bow is a weapon that needs a lot of practice to used and master. It requires a lot of skill due to the fact that it has slower travel time and lower accuracy.
Pistol Can be equipped separately from the main weapon. Because it is easy to obtain, it is one of the first types of weapons that players can master in the game.
Melee Weapons These are used to eliminate enemies in very close range. These weapons do not have ammo, making it a perfect weapon of choice for those who are running out of ammo.
Grenades Weapons that are thrown towards enemies.
Attachments / Add-On Optional add-on to main weapons like scopes and silencers.
Ammunition Various ammunition to be loaded in main and sub weapons.


Burial with full military honors is set for Sunday afternoon in a cemetery outside the village of Schuylerville.
Streetman was 20 years old when he was killed guarding the Kum River Line near the town of Taejon.
They both went missing in November, 1950.


text:
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After settling in Green Hills, Sonic is eager to prove he has what it takes to be a true hero. His test comes when Dr. Robotnik returns, this time with a new partner, Knuckles, in search for an emerald that has the power to destroy civilizations. Sonic teams up with his own sidekick, Tails, and together they embark on a globe-trotting journey to find the emerald before it falls into the wrong hands.
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text:
Asus launches Zenfone 7 and 7 Pro with the triple flip camera setup
By Sriram
image credit: timesofindia.indiatimes.com
The much awaited Asus 7 and 7 pro has been officially launched by Asus with a lot of different and unique features. The next generation device from Asus is still carrying the flip camera technology, with a triple camera setup this time. The device has been officially launched for Europe and will be available in the market from September 1, 2020. However, the company has made no official announcement about its global availability. But if the trend is to be followed, the phone will be launched in other parts of the world within the next month or two.
The company has launched two variants of the phone called the Zenfone 7 and 7 Pro. Both the mobile phones come with a 6.67 inch screen which is a notchless screen capable of 90Hz refresh rate and is a complete OLED panel. Both the smartphone will have the latest Snapdragon 865 chipset, which will provide exceptional performance and gaming capabilities. However, the difference can be noticed in both the models as the process is clocked at 2.4GHz vs. 3.1GHz in both 7 and 7Pro models of Asus Zenfone. Zenfone 7 will come in a 6 and 8 GB RAM variant and a storage of 128 GB, while the 7 Pro model will come in only 8GB variant and a 256GB storage capacity.
Both the device will get a side-mounted fingerprint sensor and a 5,000 mAh battery along with the capability to expand the storage up to 2TB. The major feature of the phone is the all new triple flip camera setup which will have the 64-megapixel main sensor, 12-megapixel ultrawide, and an 8-megapixel telephoto camera with 3x optical zoom. The same camera can be flipped and used as a selfie camera as well. The cost of the phone will be $750 for the Zenfone 7 model and $955 for the 7Pro model.


September 21, 2011. ""Characterization of Vibrio cholerae extracellular serine protease B, VesB," Shilpa Gadwal, Ph.D. student in Microbiology and Immunology and "Muscle Specific Effects of Calorie Restriction on Skeletal Muscle Glucose Uptake & Signaling," Donel Sequea, MD/PhD student in Molecular and Integrative Physiology, both at the University of Michigan.
October 12, 2011″Urogynecological Biomechanics: And The Word Of The Day Is…," Steven Abramowitch, Ph.D. Assistant Professor in the Departments of Bioengineering and Obstetrics, Gynecology, and Reproductive Sciences at the University of Pittsburgh.
"Structural and functional studies of the glmS ribozyme". Daniel Klein, PhD., UMBC alumnus, now Senior Research Scientist at Merck & Co.
"Painful Channels: the Role of Electrophysiology in Target Selection and Drug Screening". Rebecca Klein, PhD., UMBC alumnus, now Senior Research Scientist at Merck & Co.
"Building Better CARs." Avery Posey, PhD., alumnus of the UMBC MARC and Meyerhoff programs, now a postdoc in the Department of Pathology at the University of Pennsylvania.
"Temperature Matters! How body temperature can affect cancer therapy." Elizabeth Repasky, PhD., Professor, Department of Immunology at the Roswell Park Cancer Institute.


Adorable Mason Jar Project with Sugar Cookies Jar Recipe. Fits hoop sizes per shape of design for 5x7, 6x9, 8x11.
All formats also included. A pdf printable recipe card is also included with the complete recipe. Tutorial is also included. Also a printable tag to print for the jar or stitch and attach to the jar with the wet mix and preparation is also included. Super cute design!


text:
BLACK SABBATH "Logo" /Patch/
Band: BLACK SABBATH
Product Code: BSL-PATCH
BLACK SABBATH "Devil Twins" /Mug/
Official merch. Official Black Sabbath 16 oz ceramic mug with wide handle in a decorated gift box. Packed in a matching cardboard display box. 100% Officially Licensed Product. Microwave and dishwasher safe! Size: 10 cm x 8,5 cm. Black Sabbath/Singatures Network/Rock Express, 2008 (BSMUG01). M..
BLACK SABBATH "13 – Flame Circle" /TS/
Official merch. Barcode: 5023209710599 Bravado, 34192059CT. Gildan. Ring Spun. Made in Bangladesh. Printed in UK. 100% cotton...
BLACK SABBATH "13 - Fire" /TS/
BLACK SABBATH "Classic Albums - Paranoid" /DVD/
The fourth full-length DVD by the legendary British Heavy/Doom Metal band. The 40th anniversary of the "Paranoid" album celebrated with new DVD! The making of Black Sabbath's "Paranoid" is the latest in the Classic Albums DVD series put out by Eagle Vision. The latest additi..
BLACK SABBATH "The Last Supper" /DVD; Live/
The third DVD by the legendary British Hard Rock/Heavy Metal band. This is the first-ever live DVD from legendary Metal superstars, which includes all of Sabbath's biggest hits performed live. The "Last Supper" features all of the original members of Black Sabbath on their sold-out 19..
BLACK SABBATH "Black Sabbath" /LP/
Very rare compilation album which contains tracks from first two official albums of legendary British Heavy Metal band. Rare edition with titles in English and Russian. Melodia, 1970/1990 (C90 29145 002). Unofficial release. Made in USSR. Used: very good. Tracklist: 1.Black Sabbath..
BLACK SABBATH "Live At Last" /LP; Live/
The first live album by legendary British Heavy/Doom Metal band. Despite its wide distribution, it was released without any permission or knowledge from the band, and is thus considered "unofficial" in some quarters; however, the release was legal. In the US it was available only as an im..
BLACK SABBATH "Sabbath Bloody Sabbath" /LP/
The fifth studio album by the British Heavy Metal legend, released in December 1973. With this album, the band expanded upon their slow, crunching style of music by strings, keyboards and more complex orchestral arrangements. For the first time in their career, the band began to receive favorable r..
BLACK SABBATH "Vol. 4" /GLP/
The fourth full-length album by legendary British Heavy/Doom Metal band. "Vol. 4" is the point in Black Sabbath's career where the band's legendary drug consumption really starts to make itself felt. And it isn't just in the lyrics, most of which are about the blurry line betw..
BLACK SABBATH "13" /CD/
Nineteenth full-length album by the legendary British Heavy Metal band. The album was released on 10 June in Europe and 11 June in North America, via Vertigo Records and Republic Records in the US, and via Vertigo Records worldwide. It is the first studio album released by Black Sabbath since..
BLACK SABBATH "13" /Ltd. Deluxe 3D Digipack 2CD Set/
Special Edition release of the nineteenth full-length album by the legendary British Heavy Metal band. The album was released on 10 June in Europe and 11 June in North America, via Vertigo Records and Republic Records in the US, and via Vertigo Records worldwide. It is the first studio album releas..
BLACK SABBATH "Sabbath Bloody Sabbath" /Digipack CD/
Remastered re-release of the fifth full-length studio album by legendary British Hard Rock/Heavy Metal band. With this album, the band expanded upon their slow, crunching style of music by strings, keyboards and more complex orchestral arrangements. For the first time in their career, the band bega..
BLACK SABBATH "Sabotage" /CD/
Third remastered re-release of the sixth full-length studio album by legendary British Hard Rock/Heavy Metal band. Sanctuary Records, 1975/2004 (SMRCD036) Made in UK. Tracklist: 1. Hole In The Sky 2. Don't Start (Too Late) 3. Symptom Of The Universe 4. Megalomania 5. The Thrill Of It ..
BLACK SABBATH "Born Again" /CD/
The eleventh full-length studio album by legendary British Heavy Metal band. The first and only album the band recorded with lead vocalist Ian Gillan, best known for his work with Deep Purple. It was also the last Black Sabbath album for nine years to feature original bassist Geezer Butler and the ..
BLACK SABBATH "Dehumanizer" /CD/
The sixteenth full-length studio album by legendary British Heavy/Doom Metal band. One of Sabbath's heaviest albums and the first studio album in over a decade to feature vocalist Ronnie James Dio and drummer Vinny Appice, and their first in nine years to feature original bassist Geezer Butler...


Can you tell us about how you first came to work for Parkwood Leisure?
I was working at Sainsbury but I wanted to do something in the fitness industry. I saw the opportunity on line so applied, and was successful.
What attracted you to Parkwood Leisure in particular?
I was attracted to the Multiskilled Apprenticeship, as it covered a number of areas including working as a lifeguard and a fitness instructor. These were all areas that appealed to me, along with doing a job that's more active.
What do you remember of your first days in the role?
In my interview the manager was very welcoming, also the staff were all very friendly and started talking to me right away.
What career development opportunities have been available to you at Parkwood Leisure?
I have been promoted to Trainee Duty Manager and I looking to progress to becoming a Duty Manager.
Because I did the multi skilled role I cover a bit of everything in the building. I like to help out where needed, covering swim teaching, gym and reception - this means every day is different. In my new role I am responsible for assisting the Duty Manager running the shift, and making sure the pool is covered and all the set ups are done on time.
Would you recommend the Parkwood Leisure Apprenticeship programme to others?
Yes because its not just a job to earn money, you also gain experience and qualifications, leading to a career with opportunities to develop.


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Since 1974, Marmot has produced high-quality technical clothing and equipment for mountaineers, outdoor enthusiasts and extreme Alpinists.
Dramatic business growth and manual distribution processes led to space and operational constraints. Marmot wanted to evaluate potential automation with integration into a new SAP warehouse management system and needed a design to sustain future growth and potential acquisition of new brands.
Evaluated four scenarios with varying levels of mechanization to meet design requirements, including conveyable and non-conveyable product storage and processing.
Selected technologies included: a 3-level pick module, conveyor-fed processing from picking through shipping, a light-directed put wall for small items, and a shipping sortation system.
Led equipment procurement and turnkey implementation of the new 300,000 sq. ft. facility in Reno, NV.
Integrated a new warehouse control system (WCS) to interface with the SAP EWM host system for enhanced functional capabilities and efficiencies.
The project was completed within a tight timeframe.
New operations modal and automation enabled greater efficiencies and positioned the facility for enhanced future automation when growth exceeds the planned horizon projected volumes.
Operations reduced spending as a percent of sales from 3.1% to 2.2% and reduced labor as a percent of sales from 2.2% to 1.6% over the planning horizon.
The WCS interface reduced the need for EWM host system modifications, resulting in lower programming and engineering costs, easier future upgrades, quicker implementation, and less complex security and training.Contingency planning mitigated risk with regard to the SAP integration.