We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $alphain(0,2]$, to include both linear multifractional Brownian sheets ($alpha=2$) and linear multifractional stable sheets ($alpha<2$). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS. The main results are Theorems 2.9 and 2.11, which provide a sufficient and necessary condition for the existence of local times and a weaker sufficient condition for the joint continuity of local times of LMSS, respectively. We also prove a local Holder condition for the local time in the set variable in Theorem 3.1. All these theorems improve significantly the existing results for the local times of multifractional Brownian sheets and linear fractional stable sheets in the literature.
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