Joint continuity of the local times of fractional Brownian sheets

Let $B^H=\{B^H(t),t\in {\mathbb{R}}_{+}^N\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,\mathrm{.}\mathrm{.}\mathrm{.},H_N)\in (0,1{)}^N$ defined by $B^H(t)=(B_1^H(t),\mathrm{.}\mathrm{.}\mathrm{.},B_d^H(t))(t\in {\mathbb{R}}_{+}^N),$ where $B_1^H,\mathrm{.}\mathrm{.}\mathrm{.},B_d^H$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<{\sum }_{\mathrm{\ell }=1}^N{H}_{\mathrm{\ell }}^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global H"{o}lder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.