MODULUS OF CONTINUITY OF NORMAL DERIVATIVE OF A HARMONIC FUNCTIONS AT A BOUNDARY POINT

Abstract

We give sufficient conditions which ensure that harmonic
extension $u = P[f]$ to the upper half space $\{(x; y) | x \in R^n; y > 0\}$ of a function $f \in L^p(R^n)$ satisfies estimate $\partial u/\partial y (x, y)
\leq  C \omega(y)/y$ for every $x \in E \subset R^n$, where $\omega$ is a majorant. The conditions are expressed in
terms of behaviour of the Riesz transforms $R_j f$ of $f$ near points in $E$.
We brie y investigate related questions for the cases of harmonic and
hyperbolic harmonic functions in the unit ball.