Time-smoothing for parabolic variational problems in metric measure spaces

In 2013, Masson and Siljander determined a method to prove that the minimal p-weak upper gradient $$g_{f_\varepsilon }$$ for time mollification $$f_\varepsilon $$ , $$\varepsilon >0$$ of parabolic Newton–Sobolev function $$f\in L^p_\mathrm {loc}(0,\tau ;N^{1,p}_\mathrm {loc}(\Omega ))$$ with $$\tau $$\Omega open domain in doubling metric measure space $$(\mathbb {X},d,\mu )$$ supporting weak (1, p)-Poincaré inequality, $$p\in (1,\infty is such $$g_{f-f_\varepsilon }\rightarrow 0$$ as \rightarrow $$L^p_\mathrm _\tau being cylinder :=\Omega \times (0,\tau . Their approach involved use Cheeger’s differential structure, therefore exhibited some limitations; here, we shall see definition formal properties Sobolev spaces themselves allow find more direct show convergence, which relies on gradients only valid regardless structural assumptions ambient space, also limiting case when $$p=1$$