Summary: We study regularity results for solutions \(u\in H W^{1,p}(\Omega )\) to the obstacle problem \[ \int _{\Omega } \mathcal {A}(x, \nabla _{\mathbb H} u)\nabla _{\mathbb H}(v-u)\,dx \geq 0 \quad \forall v\in \mathcal {K}_{\psi ,u}(\Omega ) \] such that \(u\geq \psi \) a.e. in \(\Omega \), where \(\mathcal {K}_{\psi ,u}(\Omega )= \{v\in HW^{1,p}(\Omega )\: v-u\in HW_{0}^{1,p}(\Omega ) v\geq \psi \text{ a.e. in } \Omega \}\), in Heisenberg groups \(\mathbb {H}^n\). In particular, we obtain weak differentiability in the \(T\)-direction and horizontal estimates of Calderon-Zygmund type, i.e. \[ T\psi \in HW^{1,p}_{\text{loc}}(\Omega )\Rightarrow Tu\in L^p_{\text{loc}}(\Omega ), \]
\[ | \nabla _{\mathbb H}\psi | ^p\in L^{q}_{\text{loc}}(\Omega )\Rightarrow | \nabla _{\mathbb H} u| ^p \in L^q_{\text{loc}}(\Omega ), \] where \(2<p<4\), \(q>1\).