Aim of the authors is to establish two partial Hölder continuity results, Theorem 1.1 and Theorem 1.2. Firstly an appropriate Sobolev-Poincaré inequality, which plays an important part on proving Hölder regularity, is established. Then, an A-harmonic approximation lemma, and a prior estimate for weak solution \(h \in HW^{1,1}\) to the constant coefficient homogeneous sub-elliptic systems are given.
In the sequel the authors prove the first partial regularity result (Theorem 1.1) under sub-quadratic controllable structure assumptions by several steps. Step 1 is to gain a suitable Caccioppoli-type inequality which is an essential tool to get partial regularity. An appropriate linearization strategy is given in the second step. Then, one can achieve that solutions are approximately A-harmonic by the linearization procedure, and an excess improvement estimate for a functional, called \(\psi\), is obtained under two smallness condition assumptions, by combining with A-harmonic approximation lemma in the third steps. Once the excess improvement is established, the iteration for the \(\psi\)-excess and the \(C_y\)-excess can be acquired in Step 4. Finally, the authors show boundedness of the Campanato-type excess which leads immediately to desired Hölder continuity and Morrey regularity of Theorem 1.1. The last section shows the results of Theorem 1.2 under sub-quadratic natural structure assumptions. In such a case, the authors establish appropriate estimates just for the natural growth term, and the rest procedure is similar to the proof of Theorem 1.1.