Estimates of p-harmonic functions in planar sectors.

Suppose that p∈(1,∞], ν∈[1/2,∞), S_ν = (x₁, x₂)∈R² \{(0, 0)}: |φ|< π/2ν, where φ is the polar angle of (x₁, x₂). Let R>0 and ω_p(x) be the p-harmonic measure of ∂B(0, R)∩S_ν at x with respect to B(0, R)∩S_ν. We prove that there exists a constant C such that C⁻¹ (|x|/R)^k(ν,p) ≤ ω_p(x) ≤ C (|x|/R)^k(ν,p) whenever x∈B(0, R)∩S_2ν and where the exponent k(ν, p) is given explicitly as a function of ν and p. Using this estimate we derive local growth estimates for p-sub- and p-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of p-harmonic measure we also derive a sharp Phragmén-Lindelöf theorem for p-subharmonic functions in the unbounded sector S_ν. Moreover, if p=∞ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in Rn. Finally, when ν∈(1/2, ∞) and p∈(1,∞) we prove uniqueness (modulo normalization) of positive p-harmonic functions in Sν vanishing on ∂S_ν. [ABSTRACT FROM AUTHOR]