Rough hypoellipticity for the heat equation in Dirichlet spaces

Abstract This paper aims at proving the local boundedness and continuity of solutions heat equation in context Dirichlet spaces under some rather weak additional assumptions. We consider symmetric regular forms, which satisfy mild assumptions concerning (1) existence cut‐off functions, (2) a ultracontractivity hypothesis, (3) off‐diagonal upper bound. In this setting, equation, their time derivatives, are shown to be locally bounded; they further continuous, if semigroup admits continuous density function. Applications results provided including discussions on bounded kernel; structure for ancient (local weak) equation.