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 L2L^2 and L∞L^\infty Widom-factor problems on an analytic Jordan arc have equivalent extremal descriptions and related extremal functions. In particular, the H2H^2-extremizer is obtained from the H∞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 LqL^q spaces, leaving the relationship between these problems unresolved for other exponents.

References

It is not clear, however, how to extend this correspondence to Lq-minimizing polynomials for values of q other than 2 and \infty.

— Chebyshev polynomials on a Jordan arc  (2608.13445 - Buchecker et al., 13 Aug 2026) in Remark following Theorem 2.4, Section 2, Discussion

In , Nazarov and Shcheglova conjectured that

\lambda_3(n,1,p,1)

2\lambda_3(n,0,p,\infty), \qquad n\geq2,\quad 1\leq p\leq\infty.

Moreover, the corresponding extremal functions coincide and are symmetric with respect to $x=\frac12$.

— On the Nazarov--Shcheglova Conjecture for Sharp Sobolev Inequalities: The Case (n,p)=(3,2)  (2609.34516 - Jleli et al., 28 Sep 2026) in Section 1, Introduction