MODULUS OF CONTINUITY OF THE DIRICHLET SOLUTIONS

Let D be a bounded domain in Rn with n >= 2. For a function f on ∂D we denote by HDf the Dirichlet solution of f over D. It is classical that if D is regular, then HD maps the family of continuous boundary functions to the family of harmonic functions in D continuous up to the boundary ∂D. We show that the better continuity of a boundary function f ensures the better continuity of HDf in the context of general modulus of continuity