Extension to intermediate Lq extremal problems

Determine whether the correspondence between the \(L^2\)- and \(L^\infty\)-extremal problems for polynomials on analytic Jordan arcs extends to \(L^q\)-minimizing polynomials for values of \(q\) other than 2 and \(\infty\).

Background

The paper explains that the limiting $L^2$ and $L^\infty$ Widom-factor problems on an analytic Jordan arc have equivalent extremal descriptions and related extremal functions. In particular, the $H^2$-extremizer is obtained from the $H^\infty$-extremizer by taking a square root.

The authors explicitly note that they do not know how to extend this correspondence to polynomial norm minimization in intermediate $L^q$ spaces, leaving the relationship between these problems unresolved for other exponents.