The packing indices for some Lévy processes

Let X={Xt,t≥0} be a Lévy process, and γ0=sup{α≥0:lim infa→0a−αET(a,1)<∞}, where T(a,1)=∫01I{|Xt|≤a}dt. Taylor (1986) showed that the packing dimension of the trajectory of XX is γ0γ0.An open question from Pruitt and Taylor (1996) is: Is it true that γ0=inf{α≥0:a−αT(a,1)→∞ a.s.  as  a→0}?γ0=inf{α≥0:a−αT(a,1)→∞ a.s.  as  a→0}? In this paper, we give some answers to this question.