Regularity results for a class of obstacle problems in Heisenberg groups

We study regularity results for solutions u ∈ HW ^1,p (Ω) to the obstacle problem $$int_Omega mathcal{A} left( {x,nabla _{mathbb{H}^u } } right)nabla _mathbb{H} left( {v - u} right)dx geqslant 0 forall v in mathcal{K}_{psi ,u} left( Omega right)$$ such that u ? ψ a.e. in Ω, where $xxx$ , in Heisenberg groups ? ⁿ . In particular, we obtain weak differentiability in the T-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$begin{gathered} Tpsi in HW_{loc}^{1,p} left( Omega right) Rightarrow Tu in L_{loc}^p left( Omega right), hfill left| {nabla _{mathbb{H}psi } } right|^p in L_{loc}^q left( Omega right) Rightarrow left| {nabla _{mathbb{H}^u } } right|^p in L_{loc}^q left( Omega right), hfill end{gathered}$$ where 2 < p < 4, q > 1.