Growth of A-harmonic functions and Carnot groups

Craig A. Nolder

The order of growth of a harmonic function is determined by the growth of its gradient and conversely. We extend these results to solutions of certain subelliptic equations in John domains in Carnot groups. The modulus of the gradient is replaced by a local average of the horizontal gradient. In the harmonic case these quantities are equivalent. The proof uses recent integral inequalities associated with work on potential theory in Carnot groups. We also obtain results on the mutual growth of related A-harmonic functions which generalize corresponding results for conjugate harmonic functions.