New criteria for Hunt's hypothesis (H) of L\'evy processes
Wei Sun
Department of Mathematics and Statistics
Concordia University
Montreal
H3G 1M8
Canada
wei.sun@concordia.ca
\noindent Novel necessary and sufficient conditions are presented for the validity of Hunt's hypothesis (H) of L\'evy processes. As applications, new examples of L\'evy processes satisfying (H) are obtained. Moreover, it is shown that a general class of pure jump subordinators can be decomposed into the summation of two independent subordinators satisfying (H). \vskip 0.5cm \noindent {Keywords: Hunt's hypothesis (H), Getoor's conjecture, L\'{evy processes, subordinators. \vskip 0.5cm \noindent AMS Subject Classification: 60J45, 60G51.
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\begin{document}

\title{New criteria for Hunt's hypothesis (H) of L\'evy processes}
  \author{
Wei Sun
            \\ Department of Mathematics and Statistics\\
             Concordia University\\
             Montreal, H3G 1M8, Canada\\
             wei.sun@concordia.ca}
   \date{}
\maketitle

\begin{abstract}
\noindent Novel necessary and sufficient conditions are presented
for the validity of Hunt's hypothesis (H) of L\'evy processes. As
applications, new examples of L\'evy processes satisfying (H) are
obtained. Moreover, it is shown that a general class of pure jump
subordinators can be decomposed into the summation of two
independent subordinators satisfying (H).

\vskip 0.5cm \noindent {\bf Keywords:} Hunt's hypothesis (H),
Getoor's conjecture,  L\'{e}vy processes, subordinators. \vskip
0.5cm \noindent AMS Subject Classification: 60J45, 60G51.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[short title]{Introduction}

A L\'evy process $X$ on $\mathbf{R}^n$ is said to satisfy Hunt's
hypothesis (H) if every semipolar set of $X$ is polar. More than
forty years ago, Professor R.K. Getoor raised the problem that for
which L\'evy processes semipolar sets are always polar. He
conjectured that essentially all L\'evy processes satisfy (H).
Getoor's conjecture is the major open problem in the field of
potential theory for L\'evy processes (cf. e.g. \cite[page
70]{B}). In this paper, we will present new criteria for the
validity of (H).

Let us start with a brief introduction to Hunt's hypothesis (H).
For simplicity, we consider here only (H) for L\'evy processes;
however, we should point out that (H) plays a crucial role in the
potential theory of (dual) Markov processes. We refer the readers
to \cite[Chapter VI]{BG68} for a systematic introduction to (H)
for Markov processes.

Let $(\Omega,{\cal F},P)$ be a probability space and
$X=(X_t)_{t\ge 0}$ be a L\'{e}vy process on $\mathbf{R}^n$  with
L\'{e}vy-Khintchine exponent $\psi$, i.e.,
\begin{eqnarray*}
E[\exp\{i\langle z,X_t\rangle\}]=\exp\{-t\psi(z)\},\  z\in
\mathbf{R}^n,t\ge 0.
\end{eqnarray*}
Hereafter $E$ denotes the expectation {with respect to} $P$ and
$\langle\cdot,\cdot\rangle$ denotes the Euclidean inner product of
$\mathbf{R}^n$. The classical L\'{e}vy-Khintchine formula tells us
that
\begin{eqnarray*}
\psi(z)=i\langle a,z\rangle+\frac{1}{2}\langle
z,Qz\rangle+\int_{\mathbf{R}^n} \left(1-e^{i\langle
z,x\rangle}+i\langle z,x\rangle 1_{\{|x|<1\}}\right)\mu(dx),
\end{eqnarray*}
where $a\in \mathbf{R}^n,Q$ is a symmetric nonnegative definite
$n\times n$ matrix, and $\mu$ is a measure (called the L\'evy
measure) on $\mathbf{R}^n\backslash\{0\}$ satisfying
$\int_{\mathbf{R}^n\backslash\{0\}} (1\wedge
|x|^2)\mu(dx)<\infty$. We use Re$(\psi)$ and Im$(\psi)$ to denote
respectively the real and imaginary parts of $\psi$, and use also
$(a,Q,\mu)$ to denote $\psi$. For $x\in \mathbf{R}^n$, we denote
by $P^x$ the law of $x+X$ under $P$. In particular, $P^0=P$.

Denote by ${\cal B}^n$ the family of all nearly Borel measurable
subsets of $\mathbf{R}^n$. For $D\subset E$, we denote the first
hitting time of $D$ by
$$
\sigma_D:=\inf\{t>0:X_t\in D\}.
$$
A set $D\subset E$ is called polar  if there exists a set $C\in
{\cal B}^n$ such that $D\subset C$ and $P^x(\sigma_C<\infty)=0$
for every $x\in \mathbf{R}^n$. $D$ is called a thin set if there
exists a set $C\in {\cal B}^n$ such that $D\subset C$ and
{$P^x(\sigma_C=0)=0$} for every $x\in \mathbf{R}^n$. $D$ is called
semipolar if $D\subset\bigcup_{n=1}^{\infty}D_n$ for some thin
sets $\{D_n\}_{n=1}^{\infty}$. $X$ is said to satisfy Hunt's
hypothesis (H) if every semipolar set of $X$ is polar.

To appreciate the importance of (H), we recall below some
important principles of potential theory that are equivalent to
(H). We refer the readers to \cite[Proposition 1.1]{HSZ} for a
summary of the proofs. For $\alpha>0$, a finite $\alpha$-excessive
function $f$ on $\mathbf{R}^n$ is called a regular potential
provided that $E^x\{e^{-\alpha T_n}f(X_{T_n})\}\rightarrow
E^x\{e^{-\alpha T}f(X_{T})\}$  for $x\in E$ whenever $\{T_n\}$ is
an increasing sequence of stopping times with limit $T$. Denote by
$(U^{\alpha})_{\alpha>0}$ the resolvent operators for $X$.

\begin{itemize}
\item {\bf Bounded maximum principle:} If $\nu$ is a finite
measure with compact support $K$ such that $U^{\alpha}\nu$ is
bounded, then $\sup\{U^{\alpha}\nu(x):x\in
E\}=\sup\{U^{\alpha}\nu(x):x\in K\}$.

\item {\bf Bounded energy principle:} If $\nu$ is a finite measure
with compact support such that  $U^{\alpha}\nu$ is bounded, then
$\nu$ does not charge semipolar sets.

\item {\bf Bounded regularity principle:} If $\nu$ is a finite
measure with compact support such that $U^{\alpha}\nu$ is bounded,
then $U^{\alpha}\nu$ is regular.

\item {\bf Bounded positivity principle:} If $\nu$ is a finite
signed measure such that  $U^{\alpha}\nu$ is bounded, then $\nu
U^{\alpha}\nu\geq 0$, where $\nu
U^{\alpha}\nu:=\int_EU^{\alpha}\nu(x)\nu(dx)$.

\end{itemize}

Hunt's hypothesis (H) is also equivalent to some other important
properties of Markov processes. For example, Blumenthal and Getoor
\cite[Proposition (4.1)]{BG70} and Glover \cite[Theorem
(2.2)]{G83} showed that (H) holds if and only if the fine and
cofine topologies differ by polar sets; Fitzsimmons and Kanda
\cite{FK} showed that (H) is equivalent to the dichotomy of
capacity.

In spite of its importance, Hunt's hypothesis (H) has been
verified only in special situations. Some forty years ago, Getoor
conjectured that essentially all L\'evy processes satisfy (H). To
motivate this paper, we recall below the results obtained so far
for Getoor's conjecture that are closely related to our results.

Blumenthal and Getoor \cite{BG70} showed that all  stable
processes with index $\alpha\in (0,2)$ on the line satisfy (H).
Kanda \cite{Ka76} and Forst \cite{F75} proved independently that
(H) holds if $X$ has bounded continuous transition densities (with
respect to the Lebesgue measure $dx$) and the L\'{e}vy-Khintchine
exponent $\psi$ satisfies \begin{equation}\label{Kanda}|\mbox{Im}
(\psi)|\leq M(1+\mbox{Re}(\psi)) \end{equation}
 for some constant
$M>0$. Rao \cite{R77} gave a short proof of the Kanda-Forst
theorem under the weaker condition that $X$ has resolvent
densities.
 In particular, for $n\ge 1$, all
stable processes with index $\alpha\neq 1$ satisfy (H). Kanda
\cite{Ka78} proved that (H) holds for stable processes on
$\mathbf{R}^n$ with index $\alpha= 1$ if we assume that the linear
term vanishes.  Glover and Rao \cite{GR86} proved that
$\alpha$-subordinates of general Hunt processes satisfy (H) (cf.
Proposition \ref{GR} below).
%Rao  \cite{R87}(1987) derived some of the above-mentional
%results, using some general energy considerations developed in
%Pop-Stojanovic and Rao \cite{PR81}(1981). By using the results in
%\cite{R87},
{Rao \cite{R88} proved that if all 1-excessive functions of $X$
are lower semicontinuous and \begin{equation}\label{Rao}|{\rm
Im}(\psi)|\leq (1+{\rm Re}(\psi))f(1+{\rm Re}(\psi)),
\end{equation} where $f$ is an increasing function on $[1,\infty)$
such that $\int_N^{\infty}(\lambda
f(\lambda))^{-1}d\lambda=\infty$ for every $N\geq 1$, then $X$
satisfies (H).}

Hu and Sun \cite{HS11} showed that if $Q$ is non-degenerate then
$X$ satisfies (H); if $Q$ is degenerate then, under the assumption
that $\mu({\mathbf{R}^n\backslash \sqrt{Q}\mathbf{R}^n})<\infty$,
$X$ satisfies (H) if and only if the equation
$$
\sqrt{Q}y=-a-\int_{\{x\in {\mathbf{R}^n\backslash
\sqrt{Q}\mathbf{R}^n}:\,|x|<1\}}x\mu(dx)
$$
has at least one solution $y\in \mathbf{R}^n$. They also showed
that if $X$ is a subordinator satisfying (H) then its drift
coefficient must be 0.  Recently Hu, Sun and Zhang \cite{HSZ} gave
a comparison result on L\'evy processes which implies that big
jumps have no effect on the validity of (H). Also, they gave a new
necessary and sufficient condition for (H) (cf. Proposition
\ref{thmnb} below) and obtained an extended Kanda-Forst-Rao
theorem.

In the next section of this paper, we will present novel necessary
and sufficient conditions for the validity of (H) for L\'evy
processes. It is a bit surprising to us that these criteria were
not known until now. As applications of these  delicate criteria,
we give in Section 3 new examples of L\'evy processes satisfying
(H) and show in Section 4 that a general class of pure jump
subordinators can be decomposed into the summation of two
independent subordinators satisfying (H) (see Theorem \ref{sub1}
below).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[short title]{New criteria for Hunt's hypothesis (H)}\setcounter{equation}{0}

Let $X$ be a L\'evy process on $\mathbf{R}^n$ with
L\'{e}vy-Khintchine exponent $\psi$. From now on till the end of
this paper we assume that all 1-excessive functions are lower
semicontinuous, equivalently, $X$ has resolvent densities. We
refer the readers to \cite[Theorem 2.1]{Haw} for more
characterizations of this weak assumption.

We Define
\begin{equation}\label{ABAB}
A:=1+{\rm Re}(\psi),\ \ B:=|1+\psi|.
\end{equation}
For a finite (positive) measure $\nu$ on $\mathbf{R}^n$, we denote
$$ \hat\nu(z):=\int_{\mathbf{R}^n}e^{i\langle z,x\rangle}\nu(dx).
$$
$\nu$ is said to have finite 1-energy if
\begin{equation}\label{energy}
\int_{\mathbf{R}^n}\frac{A(z)}{B^{2}(z)}|\hat \nu(z)|^2dz<\infty.
\end{equation}
Let $\nu$ be a finite measure  on $\mathbf{R}^n$ of finite
1-energy. For $\lambda>0$, define
$$
c(\lambda):=\int_{\mathbf{R}^n}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^2dz.
$$
Then, we have (see \cite[Theorem 1 and the proof of Theorem
2]{R88})
\begin{equation}\label{ld23}
\lim_{\lambda\rightarrow\infty}c(\lambda)=\lim_{\lambda\rightarrow\infty}\int_{\mathbf{R}^n}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^21_{\{A(z)\le\lambda\}}dz<\infty.
\end{equation}
The remarkable result of Rao (\cite[Theorem 1]{R88}) tells us that
whether $X$ satisfies (H) depends on if the limit in (\ref{ld23})
equals 0.

 By \cite[Theorems 4.3 and 5.1]{HSZ}, we get
the following proposition.
\begin{prop}\label{thmnb} Let
 $f$ be an increasing
function on $[1,\infty)$ such that $ \int_N^{\infty}(\lambda
f(\lambda))^{-1}d\lambda=\infty $ for some $N\ge 1$. Then (H)
holds for $X$ if and only if
\begin{eqnarray}\label{thm2.4-1}
\lim_{\lambda\rightarrow\infty}\int_{\{B(z)>A(z)f(A(z))\}}\frac{\lambda}{\lambda^2+B^2(z)}|\hat
\nu(z)|^2dz=0
\end{eqnarray}
for any finite measure $\nu$ of finite 1-energy.
\end{prop}

For $\varsigma>1$, we define
$$N^{\varsigma}_x:=\varsigma^{(\varsigma^x)}\ \ {\rm for}\
x\ge 0.$$ To simplify notations, we use $N$ to denote
$N^{\varsigma}$ whenever there is no confusion caused. Throughout
this paper, we use $\log$ to denote $\log_e$. \vskip 0.5cm
\begin{thm}\label{slight2} Let $\nu$ be a finite measure  on $\mathbf{R}^n$ of finite 1-energy. Then, the following condition
is fulfilled for any $\delta>0$.
\begin{eqnarray*}
{Condition\ (C^{\delta})}:\ \int_{\mathbf{R}^n}
\frac{1}{B(z)\log(2+B(z))[\log\log(2+B(z))]^{1+\delta}}|\hat
\nu(z)|^2dz<\infty.
\end{eqnarray*}
\end{thm}

\noindent \emph{Proof.} We fix a $\delta>0$ and let $\nu$ be an
arbitrary finite measure on $\mathbf{R}^n$ of finite 1-energy.
Denote $$ F:=\{z\in\mathbf{R}^n:B(z)\ge 4A(z)\}.
$$
To prove the theorem, it suffices to show that
$$
\int_{F} \frac{1}{B(z)\log(B(z))[\log\log(B(z))]^{1+\delta}}|\hat
\nu(z)|^2dz<\infty.
$$

Set $N_k=2^{(2^k)}$ for $k\ge 1$. Define $f^{\delta}(\lambda)=1$
when $1\le\lambda<4$ and
$$
f^{\delta}(\lambda)=k^{1+\delta}(\log(N_k)-\log(N_{k-1})),\ \ {\rm
when}\ N_{k-1}\le \lambda<N_k,\ k\ge 2.
$$
Then, we obtain by (\ref{ld23}) that
\begin{eqnarray}\label{wei22}
\infty&>&\int_1^{\infty}\frac{d\lambda}{\lambda
f^{\delta}(\lambda)}\int_{\mathbf{R}^n}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^21_{\{A(z)\le\lambda\}}dz\nonumber\\
&=&\int_{\mathbf{R}^n}|\hat\nu(z)|^2dz\int_{A(z)}^{\infty}\frac{d\lambda}{f^{\delta}(\lambda)
(\lambda^2+B^2(z))}\nonumber\\
&=&\int_{\mathbf{R}^n}|\hat\nu(z)|^2dz\int_{1}^{\infty}\frac{A(z)d\eta}{f^{\delta}(A(z)\eta)
(A^2(z)\eta^2+B^2(z))}\nonumber\\
&=&\int_{\mathbf{R}^n}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\int_{1}^{\infty}\frac{(\frac{B(z)}{A(z)})^2d\eta}{f^{\delta}(A(z)\eta)
(\eta^2+(\frac{B(z)}{A(z)})^2)}\nonumber\\
&\ge&\int_{\mathbf{R}^n}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\sum_{k=2}^{\infty}\int_{\{1\vee\frac{N_{k-1}}{A(z)}\le
\eta<
\frac{N_k}{A(z)}\}}\frac{(\frac{B(z)}{A(z)})^2d\eta}{k^{1+\delta}(\log(N_k)-\log(N_{k-1}))
(\eta^2+(\frac{B(z)}{A(z)})^2)}\nonumber\\
&=&\int_{\mathbf{R}^n}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=2}^{\infty}\frac{\arctan\frac{N_k}{B(z)}-
\arctan\frac{N_{k-1}\vee A(z)}{B(z)}}{k^{1+\delta}(\log(N_k)-\log(N_{k-1}))}\nonumber\\
&\ge&\sum_{k_0=2}^{\infty}\int_{\{N_{k_0-1}\le B(z)<
N_{k_0}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=2}^{\infty}\frac{\arctan\frac{N_k}{B(z)}-
\arctan\frac{N_{k-1}\vee A(z)}{B(z)}}{k^{1+\delta}(\log(N_k)-\log(N_{k-1}))}\nonumber\\
&\ge&\sum_{k_0=2}^{\infty}\int_{\{z\in F:\,N_{k_0-1}\le B(z)<
N_{k_0}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan\frac{N_{k_0}}{B(z)}-
\arctan(\frac{N_{k_0-1}}{B(z)}\vee\frac{1}{4})}{k_0^{1+\delta}(\log(N_{k_0})-\log(N_{k_0-1}))}\nonumber\\
&=&\sum_{k_0=2}^{\infty}\left\{\int_{\{z\in F:\,N_{k_0-1}\le B(z)<
\frac{N_{k_0}}{2}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan\frac{N_{k_0}}{B(z)}-
\arctan(\frac{N_{k_0-1}}{B(z)}\vee\frac{1}{4})}{k_0^{1+\delta}(\log(N_{k_0})-\log(N_{k_0-1}))}\right.\nonumber\\
&&+\left.\int_{\{z\in F:\,\frac{N_{k_0}}{2}\le B(z)<
N_{k_0}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan\frac{N_{k_0}}{B(z)}-
\arctan(\frac{N_{k_0-1}}{B(z)}\vee\frac{1}{4})}{k_0^{1+\delta}(\log(N_{k_0})-\log(N_{k_0-1}))}\right\}\nonumber\\
&\ge&\sum_{k_0=2}^{\infty}\left\{\int_{\{z\in F:\,N_{k_0-1}\le
B(z)<
\frac{N_{k_0}}{2}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan2-
\arctan1}{k_0^{1+\delta}(\log(N_{k_0})-\log(N_{k_0-1}))}\right.\nonumber\\
&&+\left.\int_{\{z\in F:\,\frac{N_{k_0}}{2}\le B(z)<
N_{k_0}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan1-
\arctan\frac{1}{2}}{k_0^{1+\delta}(\log(N_{k_0})-\log(N_{k_0-1}))}\right\}\nonumber\\
&\ge&\sum_{k_0=2}^{\infty}\int_{\{z\in F:\,N_{k_0-1}\le B(z)<
{N_{k_0}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan1-
\arctan\frac{1}{2}}{k_0^{1+\delta}\log(N_{k_0-1})}\nonumber\\
&\ge&\frac{\arctan1-
\arctan\frac{1}{2}}{(5/\log2)^{1+\delta}}\sum_{k_0=2}^{\infty}\int_{\{z\in
F:\,N_{k_0-1}\le B(z)<
{N_{k_0}}\}}\frac{1}{B(z)\log(B(z))[\log\log(B(z))]^{1+\delta}}|\hat\nu(z)|^2dz\nonumber\\
&=&\frac{\arctan1-
\arctan\frac{1}{2}}{(5/\log2)^{1+\delta}}\int_{F}\frac{1}{B(z)\log(B(z))[\log\log(B(z))]^{1+\delta}}|\hat\nu(z)|^2dz.\nonumber
\end{eqnarray}
Therefore,  Condition ($C^{\delta}$) is fulfilled. \hfill\fbox

\vskip 0.5cm
\begin{thm}\label{nesu} (i) $X$ satisfies (H)  if the following condition holds:
\begin{eqnarray}\label{zeta}
& &{ Condition\ (C^{\log})}:\ { For\ any\ finite\
measure}\ \nu\ { on}\ \mathbf{R}^n\ { of\ finite\ 1{\textrm{-}}energy,\ there}\nonumber\\
& &{exist\ a\ constant}\ \varsigma>1\  and\ a\ sequence\
\ \{y_k\uparrow\infty\}\ { such\ that}\ y_1>1\ and\nonumber\\
& &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{k=1}^{\infty}\int_{\{y_k\le
B(z)< (y_k)^{\varsigma}\}} \frac{1}{B(z)\log(B(z))}|\hat
\nu(z)|^2dz<\infty.
\end{eqnarray}

(ii) Suppose $X$ satisfies (H). Then, for\ any\ finite\ measure
$\nu$ on $\mathbf{R}^n$  of\ finite\ 1-energy and any
$\varsigma>1$, there exists a sequence\ $\{y_k\uparrow\infty\}$
such\ that $y_1>1$ and (\ref{zeta})\ {holds}.

\end{thm}

\noindent \emph{Proof.}\ \ \textbf{Assertion (i):}

Let $\nu$ be an arbitrary finite measure on $\mathbf{R}^n$ of
finite 1-energy. We choose a constant $\varsigma$ and a sequence
$\{y_k\}$ described as in Condition ${\rm (C^{\log})}$. We assume
without loss of generality that $y_1>\varsigma$ and
$(y_k)^{\varsigma}<y_{k+1}$ for $k\in \mathbf{N}$. Set
$$
x_k=\log_{\varsigma}(\log_{\varsigma}{y_k})\ \ {\rm for}\
k\in\mathbf{N}.
$$ Then $x_k+1<x_{k+1}$ for
each $k\in\mathbf{N}$.

By (\ref{energy}), (\ref{ld23}) and the dominated convergence
theorem, we find that
\begin{equation}\label{op1}
\lim_{\lambda\rightarrow\infty}\int_{\{B(z)\ge
N_{x_1}\}}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^2dz=\lim_{\lambda\rightarrow\infty}c(\lambda)<\infty.
\end{equation}
We will show below that
$\lim_{\lambda\rightarrow\infty}\int_{\{B(z)\ge
N_{x_1}\}}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^2dz=0$.

For $\lambda>1$, we define
$$
g(\lambda):=\sum_{k=1}^{\infty}\frac{1_{\{N_{x_k+\frac{1}{2}}\le\lambda<
N_{x_k+\frac{3}{4}}\}}}{\log\lambda}.
$$
Then, we have
\begin{eqnarray}\label{op2}
& &\int_{N_{x_1}}^{\infty}\frac{g(\lambda)d\lambda}{\lambda
}\int_{\{B(z)\ge
N_{x_1}\}}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^2dz\nonumber\\
&=&\int_{\{B(z)\ge
N_{x_1}\}}|\hat\nu(z)|^2dz\int_{N_{x_1}}^{\infty}\frac{g(\lambda)d\lambda}{
\lambda^2+B^2(z)}\nonumber\\
&=&\int_{\{B(z)\ge N_{x_1}\}}|\hat\nu(z)|^2dz\int_{
\frac{N_{x_1}}{A(z)}}^{\infty}\frac{g(A(z)\eta)A(z)d\eta}{
A^2(z)\eta^2+B^2(z)}\nonumber\\
&=&\int_{\{B(z)\ge
N_{x_1}\}}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\int_{
\frac{N_{x_1}}{A(z)}}^{\infty}\frac{g(A(z)\eta)(\frac{B(z)}{A(z)})^2d\eta}{
\eta^2+(\frac{B(z)}{A(z)})^2}\nonumber\\
&\le&\int_{\{B(z)\ge
N_{x_1}\}}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\int_{\{\frac{N_{x_k+\frac{1}{2}}}{A(z)}\le
\eta<
\frac{N_{x_k+\frac{3}{4}}}{A(z)}\}}\frac{(\frac{B(z)}{A(z)})^2d\eta}{\log(N_{x_k+\frac{1}{2}})
(\eta^2+(\frac{B(z)}{A(z)})^2)}\nonumber\\
&=&\int_{\{B(z)\ge
N_{x_1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{\log(N_{x_k+\frac{1}{2}})}\nonumber\\
&=&\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{\log(N_{x_k+\frac{1}{2}})}\nonumber\\
& &+\sum_{l=1}^{\infty}\int_{\{N_{(x_l)+1}\le B(z)<
N_{x_{(l+1)}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{\log(N_{x_k+\frac{1}{2}})}\nonumber\\
  &=&\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\left\{\sum_{k=1}^{l-1}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{\log(N_{x_k+\frac{1}{2}})}\right.\nonumber\\
& &\ \ \ \
\left.+\sum_{k=l}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{\log(N_{x_k+\frac{1}{2}})}\right\}\nonumber\\
&&+\sum_{l=1}^{\infty}\int_{\{N_{(x_l)+1}\le B(z)<
N_{x_{(l+1)}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\left\{\sum_{k=1}^{l}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{\log(N_{x_k+\frac{1}{2}})}\right.\nonumber\\
& &\ \ \ \
\left.+\sum_{k=l+1}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{\log(N_{x_k+\frac{1}{2}})}\right\}\nonumber\\
&\le&\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\left\{\frac{N_{x_{(l-1)}+\frac{3}{4}}}{B(z)}\sum_{k=1}^{l-1}\frac{1}{\log(N_{x_k+\frac{1}{2}})}\right.\nonumber\\
& &\ \ \ \
\left.+\frac{\pi}{2\log(N_{x_l+\frac{1}{2}})}+\sum_{k=l+1}^{\infty}\frac{\frac{B(z)}{N_{x_k+\frac{1}{2}}}}
{\log(N_{x_k+\frac{1}{2}})}\right\}\nonumber\\
&&+\sum_{l=1}^{\infty}\int_{\{N_{(x_l)+1}\le B(z)<
N_{x_{(l+1)}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\left\{\frac{N_{(x_{l})+\frac{3}{4}}}{B(z)}
\sum_{k=1}^{l}\frac{1}{\log(N_{x_k+\frac{1}{2}})}\right.\nonumber\\
& &\ \ \ \
\left.+\sum_{k=l+1}^{\infty}\frac{\frac{B(z)}{N_{x_k+\frac{1}{2}}}}{\log(N_{x_k+\frac{1}{2}})}\right\}\nonumber\\
&\le&\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz
\left(\frac{1}{\varsigma^{x_1+\frac{1}{2}}(\log\varsigma)(1-\frac{1}{\varsigma})(B(z))^{1-\varsigma^{-\frac{1}{4}}}}\right.\nonumber\\
& &\ \ \ \ \left.+\frac{\pi \varsigma^{\frac{1}{2}}}{2\log
(B(z))}+\frac{1}{\varsigma^{x_1+\frac{1}{2}}(\log\varsigma)(1-\frac{1}{\varsigma})(B(z))^{\varsigma^{\frac{1}{2}}-1}}\right)\nonumber\\
&&+\sum_{l=1}^{\infty}\int_{\{N_{(x_l)+1}\le B(z)<
N_{x_{(l+1)}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz
\left(\frac{1}{\varsigma^{x_1+\frac{1}{2}}(\log\varsigma)(1-\frac{1}{\varsigma})(B(z))^{1-\varsigma^{-\frac{1}{4}}}}\right.\nonumber\\
& &\ \ \ \ \left.+\frac{1}{\varsigma^{x_1+\frac{1}{2}}(\log\varsigma)(1-\frac{1}{\varsigma})(B(z))^{\varsigma^{\frac{1}{2}}-1}}\right)\nonumber\\
&\le&D_1\int_{\mathbf{R}^n}
\frac{1}{B(z)\log(2+B(z))[\log\log(2+B(z))]^{1+\delta}}|\hat
\nu(z)|^2dz\nonumber\\
&&+ \frac{\pi
\varsigma^{\frac{1}{2}}}{2}\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le
B(z)< N_{(x_{l})+1}\}}\frac{1}{B(z)\log(B(z))}|\hat\nu(z)|^2dz,
\end{eqnarray}
where $D_1>0$ and $\delta>0$ are constants depending only on
$\varsigma$. The fact that $\frac{\pi}{2}-\arctan x\le\frac{1}{x}$
for $x>0$ has been used above to get the second inequality.

Since $\int_{N_{x_1}}^{\infty}\frac{g(\lambda)d\lambda}{\lambda
}=\infty$, we obtain by (\ref{zeta})-(\ref{op2}) and  Theorem
\ref{slight2} that $\lim_{\lambda\rightarrow\infty}\int_{\{B(z)\ge
N_{x_1}\}}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^2dz=0$. The
proof of Assertion (i) is complete by Proposition \ref{thmnb}.

\noindent \textbf{Assertion (ii):}

Suppose that $X$ satisfies (H). Let $\nu$ be an arbitrary finite
measure on $\mathbf{R}^n$ of finite 1-energy and $\varsigma>1$ be
a constant. By \cite[Theorem 5.1]{HSZ},
$\lim_{\lambda\rightarrow\infty}c(\lambda)=0$. We choose an
increasing sequence of positive numbers $\{x_k\}$ such that
\begin{equation}\label{ld24}
c(\lambda)\le\frac{1}{2^k},\ \ {\rm if}\ \lambda\ge x_k.
\end{equation}
We assume without loss of generality that $x_k+1<x_{k+1},\
k\in\mathbf{N}$. Set
$$
y_k=N^{\varsigma}_{x_k}\ \ {\rm for}\ k\in\mathbf{N}. $$Denote $$
F:=\{z\in\mathbf{R}^n:B(z)\ge 2A(z)\}.
$$
To prove (\ref{zeta}), it suffices to show that
$$ \sum_{k=1}^{\infty}\int_{\{z\in F:N_{x_k}\le B(z)< N_{x_k+1}\}}
\frac{1}{B(z)\log(B(z))}|\hat \nu(z)|^2dz<\infty.
$$

For $\lambda>1$, we define $ f(\lambda)=\log\lambda$. Set $$
\Lambda:=\bigcup_{k=1}^{\infty}\{\lambda:N_{x_{k}}\le
\lambda<N_{x_k+1}\}.
$$
By (\ref{ld24}), we get
\begin{eqnarray}
\infty&>&\int_{\Lambda}\frac{d\lambda}{\lambda
f(\lambda)}\int_{\mathbf{R}^n}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^21_{\{A(z)\le\lambda\}}dz\nonumber\\
&=&\int_{\mathbf{R}^n}|\hat\nu(z)|^2dz\int_{\Lambda\cap\{A(z)\le\lambda\}}\frac{d\lambda}{f(\lambda)
(\lambda^2+B^2(z))}\nonumber\\
&=&\int_{\mathbf{R}^n}|\hat\nu(z)|^2dz\int_{\frac{\Lambda}{A(z)}\cap\{1\le\eta\}}\frac{A(z)d\eta}{f(A(z)\eta)
(A^2(z)\eta^2+B^2(z))}\nonumber\\
&=&\int_{\mathbf{R}^n}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\int_{\frac{\Lambda}{A(z)}\cap\{1\le\eta\}}\frac{(\frac{B(z)}{A(z)})^2d\eta}{f(A(z)\eta)
(\eta^2+(\frac{B(z)}{A(z)})^2)}\nonumber\\
&\ge&\int_{\mathbf{R}^n}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}
\int_{\{1\vee\frac{N_{x_{k}}}{A(z)}\le \eta<
\frac{N_{x_{k}+1}}{A(z)}\}}\frac{(\frac{B(z)}{A(z)})^2d\eta}{\log(N_{x_{k}+1})
(\eta^2+(\frac{B(z)}{A(z)})^2)}\nonumber\\
&=&\int_{\mathbf{R}^n}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\frac{\arctan\frac{N_{x_{k}+1}}{B(z)}-
\arctan\frac{N_{{x_{k}}}\vee A(z)}{B(z)}}{\log(N_{x_{k}+1})}\nonumber\\
&\ge&\sum_{k=1}^{\infty}\int_{\{z\in F:\,N_{x_{k}}\le B(z)<
N_{x_{k}+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan\frac{N_{x_{k}+1}}{B(z)}-
\arctan(\frac{N_{x_{k}}}{B(z)}\vee\frac{1}{2})}{\log(N_{x_{k}+1})}\nonumber\\
&=&\sum_{k=1}^{\infty}\left\{\int_{\{z\in F:\,N_{x_{k}}\le B(z)<
\frac{N_{x_{k}+1}}{2}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan\frac{N_{x_{k}+1}}{B(z)}-
\arctan(\frac{N_{x_{k}}}{B(z)}\vee\frac{1}{2})}{\log(N_{x_{k}+1})}\right.\nonumber\\
&&+\left.\int_{\{z\in F:\,\frac{N_{x_{k}+1}}{2}\le B(z)<
N_{x_{k}+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan\frac{N_{x_{k}+1}}{B(z)}-
\arctan(\frac{N_{x_{k}}}{B(z)}\vee\frac{1}{2})}{\log(N_{x_{k}+1})}\right.\nonumber\\
&\ge&\sum_{k=1}^{\infty}\left\{\int_{\{z\in F:\,N_{x_{k}}\le B(z)<
\frac{N_{x_{k}+1}}{2}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan2-
\arctan1}{\log(N_{x_{k}+1})}\right.\nonumber\\
&&+\left.\int_{\{z\in F:\,\frac{N_{x_{k}+1}}{2}\le B(z)<
N_{x_{k}+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan1-
\arctan\frac{1}{2}}{\log(N_{x_{k}+1})}\right\}\nonumber\\
&\ge&\sum_{k=1}^{\infty}\int_{\{z\in F:\,N_{x_{k}}\le B(z)<
{N_{x_{k}+1}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\frac{\arctan1-
\arctan\frac{1}{2}}{\log(N_{x_{k}+1})}\nonumber\\
&\ge&\frac{\arctan1- \arctan\frac{1}{2}}{\varsigma}
\sum_{k=1}^{\infty}\int_{\{z\in F:\,N_{x_{k}}\le B(z)<
N_{x_{k}+1}\}}\frac{1}{B(z)\log(B(z))}|\hat \nu(z)|^2dz.\nonumber
\end{eqnarray}
The proof of Assertion (ii) is complete.\hfill\fbox

\vskip 0.5cm
\begin{thm}\label{add1} (i)
$X$ satisfies (H)  if the following condition holds:
\begin{eqnarray}\label{new345}
& &{ Condition\ (C^{\log\log})}:\ { For\ any\ finite\
measure}\ \nu\ { on}\ \mathbf{R}^n\ { of\ finite\ 1{\textrm{-}}energy,}\nonumber\\
& &\ \ \ \ \ {there\ exist\ a\ constant}\ \varsigma>1\ { and\ a\
sequence\ of\ positive\ numbers}\
\{x_k\}\nonumber\\
& &\ \ \ \ \ { such\ that}\ N^{\varsigma}_{x_1}>e,\
x_k+1<x_{k+1},\ k\in
\mathbf{N},\ \sum_{k=1}^{\infty}\frac{1}{x_k}=\infty,\ { and}\nonumber\\
& &\ \ \ \ \ \ \sum_{k=1}^{\infty}\int_{\{N^\varsigma_{x_k}\le
B(z)< N^\varsigma_{x_k+1}\}} \frac{1}{B(z)\log
(B(z))[\log\log(B(z))]}|\hat \nu(z)|^2dz<\infty.
\end{eqnarray}

(ii) Suppose $X$ satisfies (H). Then, for any finite measure $\nu$
on $\mathbf{R}^n$ of finite 1-energy and any $\varsigma>1$, there
exists a sequence of positive numbers $\{x_k\}$ such that $
N^{\varsigma}_{x_1}>e$, $x_k+1<x_{k+1}$, $k\in \mathbf{N}$, $
\sum_{k=1}^{\infty}\frac{1}{x_k}=\infty$, and (\ref{new345})\
{holds}.

\end{thm}

The proof of Theorem \ref{add1} is similar to, but more delicate
than that of Theorem \ref{nesu}. We put it in the Appendix.



\vskip 0.5cm We define
\begin{eqnarray*}
& &{\rm  Condition\ (C^0)}:\ {\rm For\ any\ finite\
measure}\ \nu\ {\rm on}\ \mathbf{R}^n\ {\rm of\ finite\ 1{\textrm{-}}energy,}\\
& &\ \ \ \ \ \int_{\mathbf{R}^n} \frac{1}{B(z)\log
(2+B(z))[\log\log(2+B(z))]}|\hat \nu(z)|^2dz<\infty,
\end{eqnarray*}
and
\begin{eqnarray*}
& &{\rm Condition\ (C^{B/A}):}\ {\rm There\ exists\ a\ constant}\ C>0\ {\rm such\ that}\\
& &\ \ \ \ \ B(z)\le CA(z)\log (2+B(z))[\log\log(2+B(z))],\ \
\forall z\in \mathbf{R}^n.
\end{eqnarray*}

\vskip 0.5cm
\begin{cor}\label{slight} Condition ($C^{B/A}$)
$\Rightarrow$ Condition ($C^0$) $\Rightarrow$ (H).
\end{cor}

\noindent \emph{Proof.} This is a direct consequence of Theorem
\ref{add1}.\hfill\fbox



\section[short title]{Application 1: New examples of L\'evy processes satisfying (H)}\setcounter{equation}{0}

Theorems \ref{nesu} and \ref{add1} provide novel necessary and
sufficient conditions for  the validity of Hunt's hypothesis (H)
for L\'evy processes. Different from the classical Kanda-Forst
condition (\ref{Kanda}) and Rao's condition (\ref{Rao}), our
Conditions (${\rm C^{\log}}$) and (${\rm C^{\log\log}}$) require
only that Im$(\psi)$ is partially well-controlled by $1+{\rm
Re}(\psi)$. These weaker conditions are fulfilled by more general
L\'evy processes and reveal the more essential reason for the
validity of (H).

In this section, we will present new examples of L\'evy processes
satisfying (H). First, we give the following consequence of
Theorem \ref{nesu}.

%The purpose of these preliminary examples is to exhibit the
%strength of the criteria obtained in Section 2. Further examples
%should be expected. In fact, based on Theorems \ref{nesu} and
%\ref{add1}, there are two ways to further explore Getoor's
%conjecture. One the one hand, we can construct more general
%examples of L\'evy processes satisfying Condition (${\rm
%C^{\log}}$), or Condition (${\rm C^{\log\log}}$), and hence
%satisfy (H). One the other hand, we can consider if there exists a
%(maybe pathological) L\'evy process which does not satisfy
%Condition (${\rm C^{\log}}$) and hence does not satisfy (H).



\begin{prop}\label{partial}
$X$ satisfies (H)  if the following conditions hold:

(i) $ \frac{1}{c}|z|^{\alpha_1}\le A(z)\le B(z)\le
c|z|^{\alpha_2}$ for $|z|\ge 1$, where $0<\alpha_1<\alpha_2\le 2$
and $c>1$ are constants.

(ii) There\ exist\ $\varsigma>1$, $\kappa>0$, and\ a\ sequence\
$\{z_k\}$ such\ that $z_1>1$,
$c^{\frac{\varsigma+1}{\alpha_1}}z_k^{\frac{\varsigma\alpha_2}{\alpha_1}}<z_{k+1}$,
$k\in \mathbf{N}$, and
$$
B(z)\le \kappa A(z)\log(B(z)),\ \ {\rm for}\ z_k\le |z|<
c^{\frac{\varsigma+1}{\alpha_1}}z_k^{\frac{\varsigma\alpha_2}{\alpha_1}},\
k\in\mathbf{N}.
$$
\end{prop}

\noindent \emph{Proof.} Suppose Conditions (i) and (ii) hold. Then
$X$ has bounded continuous transition densities by Hartman and
Wintner \cite{Hart}. Let $\nu$ be a finite measure  on
$\mathbf{R}^n$ of finite 1-energy. Set $$ y_k=cz_k^{\alpha_2},\ \
k\in\mathbf{N}.
$$
Then,
\begin{eqnarray*}
& &\sum_{k=1}^{\infty}\int_{\{y_k\le B(z)< (y_k)^{\varsigma}\}}
\frac{1}{B(z)\log(B(z))}|\hat \nu(z)|^2dz\\&\le&
\sum_{k=1}^{\infty}\int_{\{z_k\le |z|<
c^{\frac{\varsigma+1}{\alpha_1}}z_k^{\frac{\varsigma\alpha_2}{\alpha_1}}\}}
\frac{\kappa A(z)}{B^{2}(z)}|\hat
\nu(z)|^2dz+\int_{\{z:\,|z|<1,\,B(z)\ge c\}}\frac{1}{B(z)\log(B(z))}|\hat \nu(z)|^2dz\\
&\le&\int_{\mathbf{R}^n}\frac{\kappa A(z)}{B^{2}(z)}|\hat
\nu(z)|^2dz+\frac{(\nu(\mathbf{R}^n))^2}{c\log c}\int_{\{|z|<1\}}dz\\
 &<&\infty.
\end{eqnarray*}
Hence Condition (${\rm C^{\log}}$) is fulfilled and the proof is
complete by Theorem \ref{nesu}(i).

\begin{rem}
Blumenthal and Getoor introduced in \cite{BG61} different indexes
for L\'evy processes on $\mathbf{R}^n$. In particular, they
defined
$$
\beta=\inf\left\{\alpha\ge 0:
\frac{|\psi(z)|}{|z|^{\alpha}}\rightarrow 0\ {\rm as}\
|z|\rightarrow\infty\right\},
$$
and
$$
\beta''=\sup\left\{\alpha\ge 0: \frac{{\rm
Re}\psi(z)}{|z|^{\alpha}}\rightarrow \infty\ {\rm as}\
|z|\rightarrow\infty\right\}.
$$
It is well known that all  stable processes with index $\alpha\in
(0,2]$ satisfy (H) (cf. Section 1 for the references). The main
unknown case of Getoor's conjecture is whether Hunt's hypothesis
(H) holds for L\'evy processes with indexes $\beta''<\beta$. Our
Proposition \ref{partial} provides a sufficient condition for the
validity of (H) in this case.

Note that in Condition (ii) of Proposition \ref{partial} $z_{k+1}$
may be chosen to be much bigger than $z_k$ for each
$k\in\mathbf{N}$. Hence Proposition \ref{partial} can be used to
construct a class of L\'evy processes with indexes $\beta''<\beta$
and satisfying (H). As a concrete example, we will give in Example
\ref{example1} below a class of subordinators satisfying (H).
Recall that by virtue of the following remarkable result of Glover
and Rao, whenever we obtain a new class of subordinators
satisfying (H), we obtain a new class of time-changed Markov
processes satisfying (H).

\begin{prop}\label{GR} (Glover and Rao \cite{GR86}) Let $(X_t)_{t\ge 0}$ be a standard Markov process on a locally compact space with a
countable base and $(T_t)_{t\ge 0}$ be an independent subordinator
satisfying Hunt's hypothesis (H). Then $(X_{T_t})_{t\ge 0}$
satisfies (H).
\end{prop}

\end{rem}

\begin{exa}\label{example1}
Let $0<\alpha_1<\alpha_2<1$, $c_1>1$, $\varsigma>1$ and
$0<\kappa_1\le c_1$. Denote $$
c:=c_1\left(\frac{2}{\alpha_2}+\frac{1}{1-\alpha_2}+8\right).
$$
We choose a sequence $\{z_k\}$ satisfying $z_1>1$,
$c^{\frac{\varsigma+1}{\alpha_1}}z_k^{\frac{\varsigma\alpha_2}{\alpha_1}}<z_{k+1}$,
$k\in \mathbf{N}$.

Let $X$ be a pure jump subordinator with L\'evy measure
$\mu(dx):=\rho(x)dx$. Suppose $\rho$ satisfies the following
conditions:

(i) $\frac{1}{c_1x^{1+\alpha_1}}\le\rho(x)\le
\frac{c_1}{x^{1+\alpha_2}}$ for $0<x\le 1$; and $\rho(x)=0$,
otherwise.

(ii) $\rho(x)\ge \frac{\kappa_1}{x^{1+\alpha_2}}$ for
$\frac{1}{2c^{\frac{\varsigma+1}{\alpha_1}}z_k^{\frac{\varsigma\alpha_2}{\alpha_1}}}\le
x< \frac{1}{z_{k}}$, $k\in \mathbf{N}$.

\noindent Then $X$ satisfies (H).

In fact, for $z\in \mathbf{R}$ with $|z|\ge1$, we have
\begin{eqnarray}\label{june1}
A(z)&=&1+{\rm Re}\psi(z)\nonumber\\
&=&1+\int_0^{\infty}(1-\cos(zx))\mu(dx)\nonumber\\
&\ge&\int_{0}^{\frac{1}{|z|}}(1-\cos(zx))
\frac{1}{c_1x^{1+\alpha_1}}dx\nonumber\\
&\ge&\int_{0}^{\frac{1}{|z|}}
\frac{|z|^2x^2}{4c_1x^{1+\alpha_1}}dx\nonumber\\
&\ge&\frac{1}{8c_1}|z|^{\alpha_1},
\end{eqnarray}
and
\begin{eqnarray}\label{3322}
B(z)&=&|1+\psi(z)|\nonumber\\
&\le&A(z)+|{\rm Im}\psi(z)|\nonumber\\
&\le&1+\int_0^{\infty}(1-\cos(zx))\mu(dx)+\int_0^{\infty}|\sin(zx)|\mu(dx)\nonumber\\
&\le&1+\int_{0}^{\frac{1}{|z|}}
\left(\frac{|z|^2x^2}{2}+|z|x\right)\frac{c_1}{x^{1+\alpha_2}}dx+2\int_{\frac{1}{|z|}}^{1}\frac{c_1}{x^{1+\alpha_2}}dx\nonumber\\
&\le&c_1\left(\frac{2}{\alpha_2}+\frac{1}{1-\alpha_2}+\frac{1}{2}\right)|z|^{\alpha_2}.
\end{eqnarray}
For $z_k\le
|z|<c^{\frac{\varsigma+1}{\alpha_1}}z_k^{\frac{\varsigma\alpha_2}{\alpha_1}}$,
$k\in \mathbf{N}$, we have
\begin{eqnarray*}
A(z) &\ge&\int_{\frac{1}{2|z|}}^{\frac{1}{|z|}}(1-\cos(zx))
\frac{\kappa_1}{x^{1+\alpha_2}}dx\nonumber\\
&\ge&\int_{\frac{1}{2|z|}}^{\frac{1}{|z|}}
\frac{\kappa_1|z|^2x^2}{4x^{1+\alpha_2}}dx\nonumber\\
&\ge&\frac{\kappa_1}{16}|z|^{\alpha_2}.
\end{eqnarray*}
Hence Conditions (i) and (ii) of Proposition \ref{partial} are
fulfilled and therefore $X$ satisfies (H).
\end{exa}
 \vskip 0.2cm By
Corollary \ref{slight}, we obtain the following proposition.

\begin{prop}\label{pro-1} Let $X$ be a L\'evy process on $
\mathbf{R}$. Suppose that
\begin{equation}\label{logd}
\liminf_{|z|\to
\infty}\frac{|\psi(z)|}{|z|(\log\log|z|)^{\delta}}>0
\end{equation}
for some constant $\delta>0$. Then $X$ satisfies (H).
\end{prop}

\begin{exa}
Let $X$ be a L\'{e}vy process  on $\mathbf{R}$ with
L\'{e}vy-Khintchine exponent $(a,Q,\mu)$. Suppose that there exist
constants $\delta>0$ and $c>0$ such that
$$
d\mu\geq \frac{c[\log(-\log|x|)]^{\delta}}{x^2}dx \ \ { on}\ \
\left\{x\in \mathbf{R}: 0<|x|<\frac{1}{e}\right\}.$$ Similar to
(\ref{june1}), we can show that (\ref{logd}) holds. Therefore, $X$
satisfies (H) by Proposition \ref{pro-1}. Note that in this
example it does not matter if $a$ or $Q$ equals 0.
\end{exa}

\section[short title]{Application 2: Decomposition of subordinator into two independent ones satisfying (H)}\setcounter{equation}{0}

To the best of our knowledge, which subordinators satisfy Hunt's
hypothesis (H) is still unknown in general. In this section, we
will apply Theorem \ref{nesu} to obtain an affirmative result
(Theorem \ref{sub1}) on (H) of subordinators. The proof of this
new result illustrates the strength of the delicate criteria given
in Section 2.

\begin{thm}\label{sub1} Let $X$ be a pure jump subordinator. Suppose that the
L\'evy measure $\mu$ of $X$ has density $\rho$ which satisfies the
following condition:

There exist constants $c>1$ and $0<\alpha_1<\alpha_2<1$ such that
$\frac{1}{cx^{1+\alpha_1}}\le\rho(x)\le \frac{c}{x^{1+\alpha_2}}$
for $x\in (0,1]$.

Then, there exist two independent pure jump subordinators $X^1$
and $X^2$ such that $X=X^1+X^2$ and both $X^1$ and $X^2$ satisfy
(H).

\end{thm}

\noindent \emph{Proof.}\ \  We assume without loss of generality
that $\rho(x)=0$ for $x\not\in (0,1]$. For $i=1,2$, set
\begin{eqnarray}\label{2345}
\rho^i(x)=\left\{
  \begin{array}{ll}
   \frac{1}{2cx^{1+\alpha_1}}+\rho_i(x),&\ {\rm if}\ x\in (0,1],\\
    0,&\ {\rm otherwise},
  \end{array}
\right.
\end{eqnarray}
where $\rho_1$ and $\rho_2$ are two non-negative functions defined
on $(0,1]$ satisfying
\begin{equation}\label{2346}\rho_1(x)+\rho_2(x)=\rho(x)-\frac{1}{cx^{1+\alpha_1}}, \ \
x\in (0,1].\end{equation} We will suitably define $\rho_1$,
$\rho_2$ below to ensure that the pure jump subordinators $X^1$
and $X^2$ with L\'evy densities $\rho^1$ and $\rho^2$,
respectively, satisfying the requirements of the desired
decomposition.

We fix a constant $\varsigma>1$ and set $\varepsilon_0=1$,
$\varepsilon_1=1/2$,
$\rho_1(x)=\rho(x)-\frac{1}{cx^{1+\alpha_1}}$,
 $\rho_2(x)=0$ for $x\in (\varepsilon_1,1]$. We
will define $\varepsilon_n,\rho_1(x),\rho_2(x)$, $x\in
(\varepsilon_n,\varepsilon_{n-1}]$ by induction. Suppose
$\varepsilon_1,\dots,\varepsilon_n,\rho_1(x),\rho_2(x)$, $x\in
(\varepsilon_n,1]$, $n\ge 1$, have been defined. First, we
consider the case that $n$ is an odd number. Define
$d\mu^{(n)}=(\rho_1(x)1_{(\varepsilon_{n},1](x)})dx$. By the
Riemann-Lebesgue lemma,
$$
\lim_{z\rightarrow\infty}\int_{\mathbf{R}}\sin(zx)\mu^{(n)}(dx)=0.
$$
Hence there exists a constant, which is denoted by $z_{n+1}$, such
that
\begin{equation}\label{Lebes0}
\left|\int_{\mathbf{R}}\sin(zx)\mu^{(n)}(dx)\right|\le 1,\ \ {\rm
for}\ |z|\ge z_{n+1}.
\end{equation}
We assume without loss of generality that
$z_{n+1}>\frac{1}{\varepsilon_{n}}$.

Similar to (\ref{june1}) and (\ref{3322}), we can show that for
$z\in \mathbf{R}$ with $|z|\ge 1$,
\begin{equation}\label{ZA0}
1+\int_0^{\infty}(1-\cos(zx))\frac{1}{2cx^{1+\alpha_1}}dx\ge
\frac{1}{16c}|z|^{\alpha_1}
\end{equation}
and
\begin{eqnarray}\label{BB}
& &1+\int_0^{\infty}(1-\cos(zx))\frac{c}{x^{1+\alpha_2}}dx+\int_0^{\infty}|\sin(zx)|\frac{c}{x^{1+\alpha_2}}dx\nonumber\\
&\le&1+\int_{0}^{\frac{1}{|z|}}
\left(\frac{|z|^2x^2}{2}+|z|x\right)\frac{c}{x^{1+\alpha_2}}dx+2\int_{\frac{1}{|z|}}^{1}\frac{c}{x^{1+\alpha_2}}dx\nonumber\\
&\le&c\left(\frac{2}{\alpha_2}+\frac{1}{1-\alpha_2}+\frac{1}{2}\right)|z|^{\alpha_2}.
\end{eqnarray}
Define
\begin{equation}\label{ec1}
c_1:=c\left(\frac{2}{\alpha_2}+\frac{1}{1-\alpha_2}+\frac{1}{2}\right).\end{equation}
Set
\begin{equation}\label{dcv}
z_{n+1}'=\left[16c(c_1z^{\alpha_2}_{n+1})^{\zeta}\right]^{1/\alpha_1}
\end{equation}
and define
\begin{equation}\label{ssa}\varepsilon_{n+1}=1/(z_{n+1}')^{1/(1-\alpha_2)}.
\end{equation}
For $x\in (\varepsilon_{n+1},\varepsilon_n]$, we define
$\rho_1(x)=0$ and $\rho_2(x)=\rho(x)-\frac{1}{cx^{1+\alpha_1}}$.
For the case that $n$ is an even number, we can similarly define
$\varepsilon_{n+1},\rho_1(x),\rho_2(x)$, $x\in
(\varepsilon_{n+1},\varepsilon_{n}]$ but with the places of
$\rho_1$ and $\rho_2$ switched.

Now for $i=1,2$ we let $\rho^i$ be defined as above and let $X^i$
be a pure jump subordinator with the L\'evy density $\rho^i$. By
(\ref{2345}) and (\ref{2346}), we can assume without loss of
generality that $X^1$ and $X^2$ are independent and $X=X^1+X^2$.
We will show that $X^1$ satisfies (H). The proof for $X^2$ is
similar so we omit it. We use $\psi_{X^1}$ to denote the
L\'{e}vy-Khintchine exponent of $X_1$ and use $A_{X^1}$,
$B_{X^1}$, etc. to denote the corresponding quantities of $X^1$
(cf. (\ref{ABAB})). Let $n\ge 1$ be an odd number. For $z_{n+1}\le
|z|\le z_{n+1}'$, we obtain by (\ref{Lebes0}), (\ref{ZA0}) and
(\ref{ssa}) that
\begin{eqnarray}\label{cvbx0}
B_{X^1}(z)&=&|1+\psi_{X^1}(z)|\nonumber\\
&\le&A_{X^1}(z)+|{\rm Im}\psi_{X^1}(z)|\nonumber\\
&\le&A_{X^1}(z)+\int_0^{\varepsilon_{n+1}}\frac{c|z|x}{x^{1+\alpha_2}}dx+1+\left|\int_{0}^{\infty}\sin(zx)\frac{1}{2cx^{1+\alpha_1}}dx\right|\nonumber\\
&\le&A_{X^1}(z)+\frac{c}{1-\alpha_2}+1+\int_{0}^{\frac{1}{|z|}}
\frac{zx}{2cx^{1+\alpha_1}}dx+\int_{\frac{1}{|z|}}^{1}\frac{1}{2cx^{1+\alpha_1}}dx\nonumber\\
&\le&c^1A_{X^1}(z),
\end{eqnarray}
where $c^1>1$ is a constant independent of $z$ and $n$.

Let $\nu$ be a finite measure  on $\mathbf{R}$ of finite 1-energy
w.r.t. $X^1$. Set $y_k=c_1z_{k+1}^{\alpha_2},\ \ k\in\mathbf{N}$.
Then, we obtain by (\ref{ZA0})-(\ref{dcv}) and (\ref{cvbx0}) that
\begin{eqnarray*}
& &\sum_{k\ {\rm odd}}^{\infty}\int_{\{y_k\le B_{X^1}(z)<
(y_k)^{\varsigma}\}} \frac{1}{B_{X^1}(z)\log(B_{X^1}(z))}|\hat
\nu(z)|^2dz\\
&\le& \sum_{k\ {\rm odd}}^{\infty}\int_{\{z_{k+1}\le
|z|<z_{k+1}'\}}\frac{1}{B_{X^1}(z)\log(B_{X^1}(z))}|\hat
\nu(z)|^2dz\\
& &+\int_{\{z:\,|z|<1,\,B_{X^1}(z)\ge c_1\}}\frac{1}{B_{X^1}(z)\log(B_{X^1}(z))}|\hat \nu(z)|^2dz\\
&\le& \sum_{k\ {\rm odd}}^{\infty}\int_{\{z_{k+1}\le
|z|<z_{k+1}'\}} \frac{c^1A_{X^1}(z)}{B_{X^1}^{2}(z)}|\hat
\nu(z)|^2dz\nonumber\\
& &+\int_{\{z:\,|z|<1,\,B_{X^1}(z)\ge c_1\}}\frac{1}{B_{X^1}(z)\log(B_{X^1}(z))}|\hat \nu(z)|^2dz\\
&\le&\int_{\mathbf{R}}\frac{c^1A_{X^1}(z)}{B_{X^1}^{2}(z)}|\hat
\nu(z)|^2dz+\frac{(\nu(\mathbf{R}))^2}{c_1\log c_1}\int_{\{|z|<1\}}dz\\
 &<&\infty.
\end{eqnarray*}
Hence Condition (${\rm C^{\log}}$) is fulfilled and the proof is
complete by Theorem \ref{nesu}.\hfill\fbox

\begin{rem}
Theorem \ref{sub1} leads us to consider the following question on
 Hunt's
hypothesis (H): \vskip 0.2cm

Suppose that $X^1$ and $X^2$ are two independent L\'evy processes
on $\mathbf{R}^n$ such that both of them satisfy (H). Does
$X^1+X^2$ satisfy (H)?
\end{rem}

So far we have not been able to resolve this question. We will
present below a related result which is on the validity of (H) for
perturbed L\'evy processes.

\begin{prop}\label{prop11} Let $X_1$ and $X_2$ be two
independent L\'evy processes on $\mathbf{R}^n$ with
L\'evy-Khintchine exponents $\psi_1$ and $\psi_2$, respectively.
Suppose that $X_2$ satisfies (H) and there exists a constant $c>0$
such that
\begin{equation}\label{1}
|\psi_1|\le c(1+{\rm Re}(\psi_2)). \end{equation} Then, $X_1+X_2$
satisfies (H). \end{prop}

\noindent{\bf Proof.} Denote by $\psi$ the L\'evy-Khintchine
exponent of $X:=X_1+X_2$. Then $A(z)=1+{\rm Re}\psi_1(z)+{\rm
Re}\psi_2(z)$ and $B(z)=|1+\psi_1(z)+\psi_2(z)|$ for
$z\in\mathbf{R}^n$. Let $\nu$ be a finite measure of finite
1-energy. We assume without loss of generality that $c>1$ and
define $f\equiv 10c$. We will show that (\ref{thm2.4-1}) holds and
hence $X$ satisfies (H) by Proposition \ref{thmnb}.

Note that (\ref{1}) implies that
\begin{equation}\label{4}({\rm Im}\psi_1(z))^2\le c^2A^2(z).\end{equation}
Suppose $B(z)>A(z)f(A(z))$. Then,
\begin{equation}\label{5}({\rm Im}\psi_2(z)+{\rm Im}\psi_1(z))^2> (100c^2-1)A^2(z).\end{equation}
By (\ref{4}) and (\ref{5}), we get $$({\rm Im}\psi_2(z)+{\rm
Im}\psi_1(z))^2> \left(100-\frac{1}{c^2}\right)({\rm
Im}\psi_1(z))^2,$$ which implies that $$\label{7}({\rm
Im}\psi_2(z))^2+({\rm Im}\psi_1(z))^2>
\left(50-\frac{1}{2c^2}\right)({\rm Im}\psi_1(z))^2$$ and hence
\begin{equation}\label{71}({\rm Im}\psi_2(z))^2> \left(49-\frac{1}{2c^2}\right)({\rm Im}\psi_1(z))^2.\end{equation}
By (\ref{71}), we get
\begin{eqnarray}\label{9}
({\rm Im}\psi_2(z)+{\rm Im}\psi_1(z))^2&\ge&\frac{({\rm Im}\psi_2(z))^2}{4}-({\rm Im}\psi_1(z))^2\nonumber\\
&>&\left(\frac{1}{4}-\frac{1}{49-\frac{1}{2c^2}}\right){\rm
Im}\psi_2(z))^2.
\end{eqnarray}

By (\ref{9}), we get
\begin{equation}\label{10}\int_{\{B(z)>A(z)f(A(z))\}}\frac{\lambda}{\lambda^2+({\rm
Im}\psi(z))^2}|\hat \nu(z)|^2dz\le
c'\int_{\mathbf{R}^n}\frac{\lambda}{\lambda^2+({\rm
Im}\psi_2(z))^2}|\hat \nu(z)|^2dz\end{equation} for some constant
$c'>0$. By (\ref{1}) and the assumption that $\nu$ has finite
1-energy, we have
\begin{equation}\label{3}\int_{\mathbf{R}^n}\frac{1+{\rm
Re}\psi_2(z)}{|1+\psi_2(z)|^2}|\hat
\nu(z)|^2dz<\infty.\end{equation}  Therefore, the limit in
(\ref{thm2.4-1}) equals 0 by (\ref{10}), (\ref{3}) and the
assumption that $X_2$ satisfies (H). The proof is complete.
\hfill\fbox

\begin{rem} In Proposition \ref{prop11}, if we assume that $X_2$ satisfies the Kanda-Forst condition (\ref{Kanda}) or Rao's condition
(\ref{Rao}), then the result is obtained by the corresponding
theorems of Kanda-Forst and Rao. But our Proposition \ref{prop11}
can be applied to general L\'evy processes $X_2$ satisfying (H),
e.g., the $X_2$ satisfying Conditions (i) and (ii) of Proposition
\ref{partial}.
\end{rem}

\begin{exa}

Let $X_1$ and $X_2$ be two independent L\'evy processes on
$\mathbf{R}^n$ with L\'evy-Khintchine exponents $\psi_1$ and
$\psi_2$, respectively. Suppose that ${\rm Re}\psi_2(z)\ge
c_1|z|^{\alpha}$ for $|z|\ge1$, where $0<\alpha\le 2,\ c_1>0$ are
constants, and $X_2$ satisfies (H) (for example, $X_2$ is a
subordinator as given in Example \ref{example1}). Suppose
$|\psi_1(z)|\le c_2|z|^{\alpha}$ for $|z|\ge1$, where $c_2>0$ is a
constant. Then, $X_1+X_2$ satisfies (H) by Proposition
\ref{prop11}.

\end{exa}

\section[short title]{Appendix}\setcounter{equation}{0}

\noindent {\bf Proof of Theorem \ref{add1}.}

\noindent \textbf{Assertion (i):}

 Let $\nu$ be an
arbitrary finite measure on $\mathbf{R}^n$ of finite 1-energy. We
choose a constant $\varsigma$ and a sequence $\{x_k\}$ described
as in Condition ${\rm (C^{\log\log})}$. We assume without loss of
generality that $x_1>2$. We will show below that $
\lim_{\lambda\rightarrow\infty}\int_{\{B(z)\ge
N_{x_1}\}}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^2dz=0$.


For $\lambda>1$, we define
$$
g(\lambda):=\sum_{k=1}^{\infty}\frac{1_{\{N_{x_k+\frac{1}{2}}\le\lambda<
N_{x_k+\frac{3}{4}}\}}}{x_k\log\lambda}.
$$
Then, we have
\begin{eqnarray}\label{wei51}
& &\int_{N_{x_1}}^{\infty}\frac{g(\lambda)d\lambda}{\lambda
}\int_{\{B(z)\ge
N_{x_1}\}}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^2dz\nonumber\\
&=&\int_{\{B(z)\ge
N_{x_1}\}}|\hat\nu(z)|^2dz\int_{N_{x_1}}^{\infty}\frac{g(\lambda)d\lambda}{
\lambda^2+B^2(z)}\nonumber\\
&=&\int_{\{B(z)\ge N_{x_1}\}}|\hat\nu(z)|^2dz\int_{
\frac{N_{x_1}}{A(z)}}^{\infty}\frac{g(A(z)\eta)A(z)d\eta}{
A^2(z)\eta^2+B^2(z)}\nonumber\\
&=&\int_{\{B(z)\ge
N_{x_1}\}}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\int_{
\frac{N_{x_1}}{A(z)}}^{\infty}\frac{g(A(z)\eta)(\frac{B(z)}{A(z)})^2d\eta}{
\eta^2+(\frac{B(z)}{A(z)})^2}\nonumber\\
&\le&\int_{\{B(z)\ge
N_{x_1}\}}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\int_{\{\frac{N_{x_k+\frac{1}{2}}}{A(z)}\le
\eta<
\frac{N_{x_k+\frac{3}{4}}}{A(z)}\}}\frac{(\frac{B(z)}{A(z)})^2d\eta}{x_k\log(N_{x_k+\frac{1}{2}})
(\eta^2+(\frac{B(z)}{A(z)})^2)}\nonumber\\
&=&\int_{\{B(z)\ge
N_{x_1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{x_k\log(N_{x_k+\frac{1}{2}})}\nonumber\\
&=&\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{x_k\log(N_{x_k+\frac{1}{2}})}\nonumber\\
& &+\sum_{l=1}^{\infty}\int_{\{N_{(x_l)+1}\le B(z)<
N_{x_{(l+1)}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{k=1}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{x_k\log(N_{x_k+\frac{1}{2}})}\nonumber\\
  &=&\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\left\{\sum_{k=1}^{l-1}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{x_k\log(N_{x_k+\frac{1}{2}})}\right.\nonumber\\
& &\ \ \ \
\left.+\sum_{k=l}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{x_k\log(N_{x_k+\frac{1}{2}})}\right\}\nonumber\\
&&+\sum_{l=1}^{\infty}\int_{\{N_{(x_l)+1}\le B(z)<
N_{x_{(l+1)}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\left\{\sum_{k=1}^{l}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{x_k\log(N_{x_k+\frac{1}{2}})}\right.\nonumber\\
& &\ \ \ \
\left.+\sum_{k=l+1}^{\infty}\frac{\arctan\frac{N_{x_k+\frac{3}{4}}}{B(z)}-
\arctan\frac{N_{x_k+\frac{1}{2}}}{B(z)}}{x_k\log(N_{x_k+\frac{1}{2}})}\right\}\nonumber\\
&\le&\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\left\{\frac{N_{x_{(l-1)}+\frac{3}{4}}}{B(z)}\sum_{k=1}^{l-1}\frac{1}{x_k\log(N_{x_k+\frac{1}{2}})}\right.\nonumber\\
& &\ \ \ \
\left.+\frac{\pi}{2x_l\log(N_{x_l+\frac{1}{2}})}+\sum_{k=l+1}^{\infty}
\frac{\frac{B(z)}{N_{x_k+\frac{1}{2}}}}{x_k\log(N_{x_k+\frac{1}{2}})}\right\}\nonumber\\
&&+\sum_{l=1}^{\infty}\int_{\{N_{(x_l)+1}\le B(z)<
N_{x_{(l+1)}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\left\{\frac{N_{(x_{l})+\frac{3}{4}}}{B(z)}
\sum_{k=1}^{l}\frac{1}{x_k\log(N_{x_k+\frac{1}{2}})}\right.\nonumber\\
& &\ \ \ \
\left.+\sum_{k=l+1}^{\infty}\frac{\frac{B(z)}{N_{x_k+\frac{1}{2}}}}{x_k\log(N_{x_k+\frac{1}{2}})}\right\}\nonumber\\
&\le&\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz
\left(\frac{1}{\varsigma^{x_1+\frac{1}{2}}(\log\varsigma)(1-\frac{1}{\varsigma})(B(z))^{1-\varsigma^{-\frac{1}{4}}}}\right.\nonumber\\
& &\ \ \ \ \left.+\frac{\pi(\log \varsigma)
\varsigma^{\frac{1}{2}}}{\log
(B(z))[\log\log(B(z))]}+\frac{1}{\varsigma^{x_1+\frac{1}{2}}(\log\varsigma)(1-\frac{1}{\varsigma})(B(z))^{\varsigma^{\frac{1}{2}}-1}}\right)\nonumber\\
&&+\sum_{l=1}^{\infty}\int_{\{N_{(x_l)+1}\le B(z)<
N_{x_{(l+1)}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz
\left(\frac{1}{\varsigma^{x_1+\frac{1}{2}}(\log\varsigma)(1-\frac{1}{\varsigma})(B(z))^{1-\varsigma^{-\frac{1}{4}}}}\right.\nonumber\\
& &\ \ \ \ \left.+\frac{1}{\varsigma^{x_1+\frac{1}{2}}(\log\varsigma)(1-\frac{1}{\varsigma})(B(z))^{\varsigma^{\frac{1}{2}}-1}}\right)\nonumber\\
&\le&D_1\int_{\mathbf{R}^n}
\frac{1}{B(z)\log(2+B(z))[\log\log(2+B(z))]^{1+\delta}}|\hat
\nu(z)|^2dz\nonumber\\
&&+ {\pi(\log \varsigma)
\varsigma^{\frac{1}{2}}}\sum_{l=1}^{\infty}\int_{\{N_{x_{l}}\le
B(z)<
N_{(x_{l})+1}\}}\frac{1}{B(z)\log(B(z))[\log\log(B(z))]}|\hat\nu(z)|^2dz,\
\ \ \ \ \ \ \
\end{eqnarray}
where $D_1>0$ and $\delta>0$ are constants depending only on
$\varsigma$. The fact that $\frac{\pi}{2}-\arctan x\le\frac{1}{x}$
for $x>0$ has been used above to get the second inequality.

Since $\int_{N_{x_1}}^{\infty}\frac{g(\lambda)d\lambda}{\lambda
}=\infty$ , we obtain by  (\ref{op1}), (\ref{new345}),
(\ref{wei51}) and Theorem \ref{slight2} that
$\lim_{\lambda\rightarrow\infty}\int_{\{B(z)\ge
N_{x_1}\}}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^2dz=0$. The
proof of Assertion (i) is complete by Proposition \ref{thmnb}.

\noindent \textbf{Assertion (ii):}

Suppose $X$ satisfies (H). Let $\nu$ be an arbitrary finite
measure on $\mathbf{R}^n$ of finite 1-energy and $\varsigma>1$ be
a constant.  We choose a sequence of increasing natural numbers
$\{p_k\}$ satisfying
\begin{equation}\label{dfgo}
c(\lambda)\le\frac{1}{2^k},\ \ {\rm if}\ \lambda\ge p_k.
\end{equation}
For $m=1,2\dots$, we set $x_{m,1}=p_m+2$ and choose $k_m$ such
that
\begin{equation}\label{dec}
1\le\frac{1}{p_m+2}+\frac{1}{p_m+4}+\cdots+\frac{1}{p_m+2k_m}\le
2.
\end{equation}
Define $x_{m,l_m}=p_m+2l_m$ for $1\le l_m\le k_m$. We can require
without loss of generality that
\begin{equation}\label{qguai}
x_{1,1}>1-\frac{2\log\log\varsigma}{\log\varsigma},
\end{equation}
and
$$
N^{\varsigma}_{x_{1,1}}>e\ \ {\rm and}\ \ x_{m,k_m}<p_{m+1},\ \
m=1,2,\dots
$$
Denote
$$ F:=\{z\in\mathbf{R}^n:B(z)\ge 2A(z)\}.
$$
We will show below that
$$ \sum_{m=1}^{\infty}\sum_{l_m=1}^{k_m}\int_{\{z\in F:N_{x_{m,l_m}}\le B(z)< N_{x_{m,l_m}+1}\}}
\frac{1}{B(z)\log(B(z))[\log\log(B(z))]}|\hat \nu(z)|^2dz<\infty.
$$

We define
$$
f(\lambda)=k(\log(N_k)-\log(N_{k-1})),\ \ {\rm when}\ N_{k-1}\le
\lambda<N_k,\ k\ge 2.
$$
Set $$
\Lambda:=\bigcup_{m=1}^{\infty}\bigcup_{l_m=1}^{k_m}\{\lambda:N_{x_{m,l_m}}\le
\lambda<N_{x_{m,l_m}+1}\}.
$$
By (\ref{dfgo}) and (\ref{dec}), we get
\begin{eqnarray}\label{wsd}
\infty&>&\int_{\Lambda}\frac{d\lambda}{\lambda
f(\lambda)}\int_{\mathbf{R}^n}\frac{\lambda}{\lambda^2+B^2(z)}|\hat\nu(z)|^21_{\{A(z)\le\lambda\}}dz\nonumber\\
&=&\int_{\mathbf{R}^n}|\hat\nu(z)|^2dz\int_{\Lambda\cap\{A(z)\le\lambda\}}\frac{d\lambda}{f(\lambda)
(\lambda^2+B^2(z))}\nonumber\\
&=&\int_{\mathbf{R}^n}|\hat\nu(z)|^2dz\int_{\frac{\Lambda}{A(z)}\cap\{1\le\eta\}}\frac{A(z)d\eta}{f(A(z)\eta)
(A^2(z)\eta^2+B^2(z))}\nonumber\\
&=&\int_{\mathbf{R}^n}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\int_{\frac{\Lambda}{A(z)}\cap\{1\le\eta\}}\frac{(\frac{B(z)}{A(z)})^2d\eta}{f(A(z)\eta)
(\eta^2+(\frac{B(z)}{A(z)})^2)}\nonumber\\
&=&\int_{\mathbf{R}^n}\frac{A(z)}{B^{2}(z)}|\hat\nu(z)|^2dz\sum_{m=1}^{\infty}\sum_{l_m=1}^{k_m}
\int_{\{1\vee\frac{N_{x_{m,l_m}}}{A(z)}\le \eta<
\frac{N_{x_{m,l_m}+1}}{A(z)}\}}\nonumber\\
& &\ \ \ \ \
\cdot\frac{(\frac{B(z)}{A(z)})^2d\eta}{({x_{m,l_m}+1})(\log(N_{x_{m,l_m}+1})-\log(N_{{x_{m,l_m}}}))
(\eta^2+(\frac{B(z)}{A(z)})^2)}\nonumber\\
&=&\int_{\mathbf{R}^n}\frac{1}{B(z)}|\hat\nu(z)|^2dz\sum_{m=1}^{\infty}\sum_{l_m=1}^{k_m}\frac{\arctan\frac{N_{x_{m,l_m}+1}}{B(z)}-
\arctan\frac{N_{{x_{m,l_m}}}\vee A(z)}{B(z)}}{({x_{m,l_m}+1})(\log(N_{x_{m,l_m}+1})-\log(N_{{x_{m,l_m}}}))}\nonumber\\
&\ge&\sum_{m=1}^{\infty}\sum_{l_m=1}^{k_m}\int_{\{z\in
F:\,N_{x_{m,l_m}}\le B(z)<
N_{x_{m,l_m}+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\nonumber\\
& &\ \ \ \ \ \cdot\frac{\arctan\frac{N_{x_{m,l_m}+1}}{B(z)}-
\arctan\frac{N_{x_{m,l_m}}}{B(z)}\vee\frac{1}{2})}{({x_{m,l_m}+1})(\log(N_{x_{m,l_m}+1})-\log(N_{{x_{m,l_m}}}))}\nonumber\\
&=&\sum_{m=1}^{\infty}\sum_{l_m=1}^{k_m}\left\{\int_{\{z\in
F:\,N_{x_{m,l_m}}\le B(z)<
\frac{N_{x_{m,l_m}+1}}{2}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\right.\nonumber\\
& &\ \ \ \ \ \cdot\left.\frac{\arctan\frac{N_{x_{m,l_m}+1}}{B(z)}-
\arctan(\frac{N_{x_{m,l_m}}}{B(z)}\vee\frac{1}{2})}{({x_{m,l_m}+1})(\log(N_{x_{m,l_m}+1})-\log(N_{{x_{m,l_m}}}))}\right.\nonumber\\
&&+\left.\int_{\{z\in F:\,\frac{N_{x_{m,l_m}+1}}{2}\le B(z)<
N_{x_{m,l_m}+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\right.\nonumber\\
& &\ \ \ \ \ \cdot\left.\frac{\arctan\frac{N_{x_{m,l_m}+1}}{B(z)}-
\arctan(\frac{N_{x_{m,l_m}}}{B(z)}\vee\frac{1}{2})}{({x_{m,l_m}+1})(\log(N_{x_{m,l_m}+1})-\log(N_{{x_{m,l_m}}}))}\right.\nonumber\\
&\ge&\sum_{m=1}^{\infty}\sum_{l_m=1}^{k_m}\left\{\int_{\{z\in
F:\,N_{x_{m,l_m}}\le B(z)<
\frac{N_{x_{m,l_m}+1}}{2}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\right.\nonumber\\
& &\ \ \ \ \ \cdot\left.\frac{\arctan2-
\arctan1}{({x_{m,l_m}+1})(\log(N_{x_{m,l_m}+1})-\log(N_{{x_{m,l_m}}}))}\right.\nonumber\\
&&+\left.\int_{\{z\in F:\,\frac{N_{x_{m,l_m}+1}}{2}\le B(z)<
N_{x_{m,l_m}+1}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\right.\nonumber\\
& &\ \ \ \ \ \cdot\left.\frac{\arctan1-
\arctan\frac{1}{2}}{({x_{m,l_m}+1})(\log(N_{x_{m,l_m}+1})-\log(N_{{x_{m,l_m}}}))}\right\}\nonumber\\
&\ge&\sum_{m=1}^{\infty}\sum_{l_m=1}^{k_m}\int_{\{z\in
F:\,N_{x_{m,l_m}}\le B(z)<
{N_{x_{m,l_m}+1}}\}}\frac{1}{B(z)}|\hat\nu(z)|^2dz\nonumber\\
& &\ \ \ \ \ \cdot\frac{\arctan1-
\arctan\frac{1}{2}}{({x_{m,l_m}+1})(\log(N_{x_{m,l_m}+1})-\log(N_{{x_{m,l_m}}}))}\nonumber\\
&\ge&\frac{(\log \varsigma)(\arctan1-
\arctan\frac{1}{2})}{2\varsigma}
\sum_{m=1}^{\infty}\sum_{l_m=1}^{k_m}\int_{\{z\in
F:\,N_{x_{m,l_m}}\le B(z)< N_{x_{m,l_m}+1}\}}\nonumber\\
& &\ \ \ \ \ \cdot \frac{1}{B(z)\log(B(z))[\log\log(B(z))]}|\hat
\nu(z)|^2dz,
\end{eqnarray}
where (\ref{qguai}) has been used to obtain the last inequality.

Set $\{x_k\}=\{x_{m,l_m}:1\le l_m\le k_m\}$. Then, we obtain by
(\ref{wsd}) that
$$
\sum_{k=1}^{\infty}\int_{\{z\in F:\,N_{x_k}\le B(z)< N_{x_k+1}\}}
\frac{1}{B(z)\log(B(z))[\log\log(B(z))]}|\hat \nu(z)|^2dz<\infty.
$$
Therefore the proof of Assertion (ii) is complete by noting
(\ref{energy}). \hfill\fbox


\bigskip

{ \noindent {\bf\large Acknowledgments} \vskip 0.1cm  \noindent We
acknowledge the support of NSERC and thank Prof. Z.C. Hu for
comment on Proposition \ref{prop11}.}


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\bibitem{FK} Fitzsimmons P.J. and Kanda M. (1992)  On Choquet's
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\bibitem{G83} Glover J. (1983)  Topics in energy and potential
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\bibitem{GR86} Glover J. and Rao M. (1986)  Hunt's hypothesis (H) and
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\bibitem{Hart} Hartman P. and Wintner A. (1942) On the infinitesimal generators of
integral convolutions. Amer. J. Math. \textbf{64}, 273-298

\bibitem{Haw} Hawkes J. (1979)  Potential theory of L\'evy processes. Proc. London Math. Soc. {\bf 38}, 335-352

\bibitem{HS11} Hu Z.C. and Sun W. (2012) Hunt's hypothesis (H) and
Getoor's conjecture for L\'{e}vy processes. Stoch. Proc. Appl.
\textbf{122}, 2319-2328

\bibitem{HSZ} Hu Z.C., Sun W. and
Zhang J. (2014)  New results on Hunt's hypothesis (H) for L\'evy
processes. Potential Anal., to appear, DOI:
10.1007/s11118-014-9446-1

\bibitem{Ka76} Kanda M. (1976)  Two theorems on capacity for Markov processes
with stationary independent increments. Z. Wahrsch. verw. Gebiete
{\bf 35}, 159-165

\bibitem{Ka78} Kanda M. (1978)  Characterisation of semipolar sets
for processes with stationary independent increments. Z. Wahrsch.
verw. Gebiete {\bf 42}, 141-154

%\bibitem{Ke69} Kesten H.:
%Hitting probabilities of single points for processes with
%stationary independent increments. Memoirs of the American
%Mathematical Society, No. 93, American Mathematical Society,
%Providence, R.I. (1969).

%\bibitem{PS69} Port S.C. and Stone C.J.:  The asymmetric Cauchy
%process on the line. Ann. Math. Statist. {\bf 40}, 137-143 (1969).


\bibitem{R77}Rao M. (1977)  On a result of M. Kanda. Z. Wahrsch. verw.
Gebiete {\bf 41}, 35-37

%\bibitem{R87} Rao M.:  On polar sets for L\'evy processes. J. London Math. Soc. \textbf{35},
%569-576 (1987).

\bibitem{R88} Rao M. (1988)  Hunt's hypothesis for L\'{e}vy
processes. Proc. Amer. Math. Soc. {\bf 104},  621-624

\end{thebibliography}


\end{document}

By virtue of the results obtained in \cite{HS11}, when we consider
Getoor's conjecture, we may restrict to considering pure jump
L\'evy process.

1406.2013.extracted_bib
{B Bertoin J. (1996) L\'evy Processes. Cambridge Univ. Press.
1406.2013.extracted_bib
{BG61 Blumenthal R.M. and Getoor R.K. (1961) Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. {10, 493-516
1406.2013.extracted_bib
{BG68 Blumenthal R.M. and Getoor R.K. (1968) Markov Processes and Potential Theory. Academic Press, New York and London
1406.2013.extracted_bib
{BG70 Blumenthal R.M. and Getoor R.K. (1970) Dual processes and potential theory. Proc. 12th Biennial Seminar of the Canadian Math. Congress, 137-156 %
1406.2013.extracted_bib
{Br71 Bretagnolle J.: R\'{esults de Kesten sur les processus \`{a %accroissements ind\'{ependants. S\'{eminare de Probabilit\'{es %V, Lect. Notes in Math., Vol. 191, Springer-Verlag, Berlin, 21-36 %(1971).
1406.2013.extracted_bib
{FK Fitzsimmons P.J. and Kanda M. (1992) On Choquet's dichotomy of capacity for Markov processes. Ann. Probab. {20, 342-349
1406.2013.extracted_bib
{F75 Forst G. (1975) The definition of energy in non-symmetric translation invariant Dirichlet spaces. Math. Ann. {216, 165-172
1406.2013.extracted_bib
{G83 Glover J. (1983) Topics in energy and potential theory. Seminar on Stochastic Processes, 1982, Birkh\"{auser, 195-202
1406.2013.extracted_bib
{GR86 Glover J. and Rao M. (1986) Hunt's hypothesis (H) and Getoor's conjecture. Ann. Probab. {14, 1085-1087
1406.2013.extracted_bib
{Hart Hartman P. and Wintner A. (1942) On the infinitesimal generators of integral convolutions. Amer. J. Math. 64, 273-298
1406.2013.extracted_bib
{Haw Hawkes J. (1979) Potential theory of L\'evy processes. Proc. London Math. Soc. {38, 335-352
1406.2013.extracted_bib
{HS11 Hu Z.C. and Sun W. (2012) Hunt's hypothesis (H) and Getoor's conjecture for L\'{evy processes. Stoch. Proc. Appl. 122, 2319-2328
1406.2013.extracted_bib
{HSZ Hu Z.C., Sun W. and Zhang J. (2014) New results on Hunt's hypothesis (H) for L\'evy processes. Potential Anal., to appear, DOI: 10.1007/s11118-014-9446-1
1406.2013.extracted_bib
{Ka76 Kanda M. (1976) Two theorems on capacity for Markov processes with stationary independent increments. Z. Wahrsch. verw. Gebiete {35, 159-165
1406.2013.extracted_bib
{Ka78 Kanda M. (1978) Characterisation of semipolar sets for processes with stationary independent increments. Z. Wahrsch. verw. Gebiete {42, 141-154 %
1406.2013.extracted_bib
{Ke69 Kesten H.: %Hitting probabilities of single points for processes with %stationary independent increments. Memoirs of the American %Mathematical Society, No. 93, American Mathematical Society, %Providence, R.I. (1969). %
1406.2013.extracted_bib
{PS69 Port S.C. and Stone C.J.: The asymmetric Cauchy %process on the line. Ann. Math. Statist. {40, 137-143 (1969).
1406.2013.extracted_bib
{R77Rao M. (1977) On a result of M. Kanda. Z. Wahrsch. verw. Gebiete {41, 35-37 %
1406.2013.extracted_bib
{R87 Rao M.: On polar sets for L\'evy processes. J. London Math. Soc. 35, %569-576 (1987).
1406.2013.extracted_bib
{R88 Rao M. (1988) Hunt's hypothesis for L\'{evy processes. Proc. Amer. Math. Soc. {104, 621-624