Local times and sample path properties of the Rosenblatt process

Let $Z=(Z_t{)}_{t\ge 0}$ be the Rosenblatt process with Hurst index $H \in (1/2, 1)$\mathrm{.}Weprovejointcontinuityforthelocaltimeof$Z$, and establish H"older conditions for the local time. These results are then used to study the irregularity of the sample paths of $Z$. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in the second chaos.