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Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case. (English) Zbl 1447.35125
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.
MSC:
 35H20 Subelliptic equations 35B65 Smoothness and regularity of solutions to PDEs 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
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