Nonlocal Quasilinear Parabolic Equations in Heisenberg Group: Local Boundedness with an Optimal Tail

We prove local boundedness for a quasilinear parabolic equation on the Heisenberg group $${\mathrm{\partial }}_{t}u(\xi ,t)+\text{p.v.}{\int }_{{\mathbb{H}}^N}\frac{\mathrm{\mid }u(\xi ,t)-u(\eta ,t){\mathrm{\mid }}^{p-2}(u(\xi ,t)-u(\eta ,t))}{\mathrm{\mid }{\eta }^{-1}\circ \xi {\mathrm{\mid }}^{Q+sp}}d\eta =0,$$ with optimal regularity assumption on the tail term. We also prove interpolation inequalities and an extension theorem for fractional Sobolev spaces on the Heisenberg group.