$HW^{2,2}_{\rm loc}$-regularity for $p$-harmonic functions in Heisenberg groups

Here HW 1,p loc (Ω) is the collection of functions u ∈ L p loc (Ω) with their distributional horizontal derivatives Xu = (X1u, · · · ,X2nu) ∈ Lploc (Ω,R2n). In the linear case p = 2, 2-harmonic functions are exactly harmonic functions in Ω ⊂ Hn, and their C∞-regularity follows from a result by Hörmander [6]. In the quasilinear case p 6= 2, the study of regularity of p-harmonic functions in Hn attracted a lot of attention in past decades. In particular, their Hölder regularity was established by Lu [8] and Capogna [1]. Recently, their Lipshictz regularity and also the Hölder regularity of their horizontal gradients were proved by Domokos and Manfredi [4], Manfredi and Mingione [9], Mingione, Zatorska-Goldstein and Zhong [11], Zhong [14], and Mukherjee and Zhong [12]. On the other hand, if p > 2, Capogna [1] proved the C∞-regularity of p-harmonic functions u under an additional assumption that |Xu| is strictly bounded from above and also away from 0. In general, |Xu| may vanish in some set and u is also not necessarily smooth. In 2005, Domokos and Manfredi [3] established an interesting second order differentiability: if