Parabolic Lipschitz truncation and caloric approximation

We develop an improved version of the parabolic Lipschitz truncation, which allows qualitative control of the distributional time derivative and the preservation of zero boundary values. As a consequence, we establish a new caloric approximation lemma. We show that almost p-caloric functions are close to p-caloric functions. The distance is measured in terms of spatial gradients as well as almost uniformly in time. Both results are extended to the setting of Orlicz growth.