A multiparameter Garsia–Rodemich–Rumsey inequality and some applications

We extend the classical Garsia–Rodemich–Rumsey inequality to the multiparameter situation. The new inequality is applied to obtain some joint Hölder continuity along the rectangles for fractional Brownian fields W(t,x)W(t,x) and for the solution u(t,y)u(t,y) of the stochastic heat equation with additive white noise.