Locally solvable vector fields and Hardy spaces

Abstract

We characterize the boundary value of homegeneous solutions of planar one-sided locally solvable vector fields with analytic coefficients with the property that the L p norm of their traces is locally uniformly bounded, 0 < p ⩽ 1 . For p ≠ 1 / n , n = 1 , 2 , … , the boundary value must locally belong to the local Hardy space h p ( R ) of Goldberg while for p = 1 / n , n = 1 , 2 , … , the answer calls for a new class of atomic Hardy spaces if the vector field is of infinite type at some boundary point.