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Let $B^H=\{B^H(t),t\in{{\mathbb{R}}_+^N}\}$ be an $(N,d)$-fractional Browniansheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by$B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where$B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Browniansheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then thelocal times of $B^H$ are jointly continuous. This verifies a conjecture of Xiaoand Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharplocal and global H\"{o}lder conditions for the local times of $B^H$. Theseresults are applied to study analytic and geometric properties of the samplepaths of $B^H$.

Joint continuity of the local times of fractional Brownian sheets