(untitled) (Analysis in Theory and Applications, 2021)
We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R^2_+=R×(0,∞), where (Df)(x)=f'(x)+(λ/x)[f(x)-f(-x)] for given λ0. A C^2 function u in R^2_+ is said to be λ-harmonic if (D_x^2+_y^2)u=0. For a λ-harmonic function u in R^2_+ and for a subset E of R^2_+=R symmetric about y-axis, we prove that the following three assertions are equivalent: (i) u has a finite non-tangential limit at (x,0) for a.e. xE; (ii) u is non-tangentially bounded for a.e. xE; (iii) (Su)(x)