The following boundary value
result is obtained: If ϕ is a quasiregular function on a plane domain U with non-polar
complement and ϕ satisfies a growth condition analogue to the classical Hp-condition
for analytic functions, then there exists a uniformly elliptic diffusion Xt such that for
a.a. η ∈ ∂U with respect to its elliptic-harmonic measure the limit of ϕ along the
η-conditional Xt-paths exists a.s.
It is proved that if U is the unit disc then convergence along the η-conditional
Xt-paths implies the classical non-tangential convergence. Therefore the result above
is a generalization of the classical Fatou theorem. As an application, using known
properties of elliptic-harmonic measure we obtain that there exists α > 0 (depending
on ϕ) such that for every interval J ⊂ ∂D there is a subset F ⊂ J of positive
α-dimensional Hausdorff measure such that the non-tangential limit of ϕ exists at
every point of F.
