Hunt's hypothesis (H) and Getoor's conjecture for L\'{e}vy Processes

In this paper, Hunt’s hypothesis (H) and Getoor’s conjecture for L'{e}vy processes are revisited. Let $X$ be a L'{e}vy process on ${\mathbf{R}}^n$ with L'{e}vy-Khintchine exponent $(a,A,\mu )$. {First, we show that if $A$ is non-degenerate then $X$ satisfies (H). Second, under the assumption that $\mu ({\mathbf{R}}^n\mathrm{\backslash }\sqrt{A}{\mathbf{R}}^n)<\mathrm{\infty }$, we show that $X$ satisfies (H) if and only if the equation $$\sqrt{A}y=-a-{\int }_{\{x\in {\mathbf{R}}^n\mathrm{\backslash }\sqrt{A}{\mathbf{R}}^n:\mathrm{\mid }x\mathrm{\mid }<1\}}x\mu (dx),y\in {\mathbf{R}}^n,$$ has at least one solution. Finally, we show that if $X$ is a subordinator and satisfies (H) then its drift coefficient must be 0.}