Asymptotics of Lʳ extremal polynomials for 0<r≤∞ on C¹⁺ Jordan regions

We study strong asymptotics of Lʳ-extremal polynomials for measures supported on Jordan regions with C¹⁺ boundary for 0<r<∞. Using the results for $r=2$, we derive asymptotics of weighted Chebyshev and residual polynomials for upper-semicontinuous weights supported on a C¹⁺ Jordan region corresponding to r=∞. As an application, we show how strong asymptotics for extremal polynomials in the Ahlfors problem on a C¹⁺ Jordan region can be obtained from that for the weighted residual polynomials. Based on the results we pose a conjecture for asymptotics of weighted Chebyshev and residual polynomials for a C¹⁺ arc.