We give sufficient conditions which ensure that harmonic extension u = P[ f ] to the upper half space {(x, y) | x ∈ Rn, y > 0} of a function f ∈ Lp(Rn) satisfies estimate ∂u ∂y (x, y) ≤ Cω(y)/y for every x in E ⊂ Rn, where ω is a majorant. The conditions are expressed in terms of behaviour of the Riesz transforms Rj f of f near points in E. We briefly investigate related questions for the cases of harmonic and hyperbolic harmonic functions in the unit ball.