Estimates of p-harmonic functions in planar sectors

Suppose that p is an element of(1, infinity], nu is an element of[1/2, infinity), S-nu = {(x(1), x(2))is an element of R-2\{(0, 0)}:|phi|< pi/2 nu}, where phi is the polar angle of (x(1), x(2)). Let R>0 and w(p)(x) be the p-harmonic measure of partial derivative B(0, R)boolean AND nS(nu) at x with respect to B(0, R)boolean AND S-nu. We prove that there exists a constant C such C-1 (vertical bar x vertical bar/R)(k(nu,p)) <= w(p)(x) <= C (vertical bar x vertical bar/R)(k(nu,p)) whenever x is an element of B(0, R)boolean AND S-2 nu and where the exponent k(nu, p) is given explicitly as a function of nu 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 Phragmen-Lindelof theorem for p-subharmonic functions in the unbounded sector S-nu. Moreover, if p=infinity then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in R-n. Finally, when nu is an element of(1/2, infinity) and p is an element of(1, infinity) we prove uniqueness (modulo normalization) of positive p-harmonic functions in S nu vanishing on partial derivative S-nu.