Second-Order Regularity for Degenerate Parabolic Quasi-Linear Equations in the Heisenberg Group

In the Heisenberg group H-n, we obtain the local second-order HWloc2,2 -regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations partial derivative(t)u = Sigma(2n)(i=1) X(i)A(i)(Xu) modeled on the parabolic p-Laplacian equation. Specifically, when 2 <= p <= 4, we demonstrate the integrability of (partial derivative(t)u)(2), namely, partial derivative(t)u is an element of L-loc(2); when 2 <= p < 3, we demonstrate the HWloc2,2 -regularity of u, namely, XXu is an element of L-loc(2). For the HWloc2,2-regularity, when p >= 2, the range of p is optimal compared to the Euclidean case.