Lipschitz Properties of Harmonic and Holomorphic Functions

We prove two results in this dissertation, one concerning Lipschitz harmonic functions and the other concerning Lipschitz holomorphic functions.

Let B be a regular majorant. We show that a harmonic function, in a smoothly bounded domain Ω in ℝⁿ, that is Lipschitz-B along a family of curves transversal to bΩ is Lipschitz-B in Ω (i.e., Lipschitz-B in all directions in Ω).

Let Ω be a smoothly bounded domain in ℂⁿ (n > 1). Let P ∈ bΩ and let ν_P be the outward unit normal to bΩ at P. Fix P ∈ bΩ and a unit vector v⃗ ∈ ℂⁿ. For δ > 0, we define R(P_δ ; v⃗), where P_δ = P−δν_P, to be the radius of a complex disc centred at P_δ in the v⃗ direction that fits inside Ω̅ satisfying some additional properties. We show that a Lipschitz-B holomorphic function in Ω has a Lipschitz gain along complex discs centred at P_δ in the v⃗ direction. This gain is given by the inverse of R(P_t ; v⃗) as function of t. Some examples including an application to convex domains of finite type in ℂⁿ are discussed.