Interpolation of Paley–Wiener spaces for balls

Determine whether the Paley–Wiener spaces \(\PW^{p_1}(B)\) and \(\PW^{p_2}(B)\), associated with a ball \(B\subset\mathbb{R}^n\), form a complex interpolation scale for exponents \(1<p_1<p_2\).

Background

The paper establishes a complex interpolation identity for Paley–Wiener spaces associated with polyhedra, which is a key ingredient in the complete characterization of boundary-weighted Fourier inequalities for those domains. For balls, however, the authors require a different analytic-family argument to obtain the sharp range when the Fourier exponent satisfies $p\geq 2$.

The obstruction is that Fefferman’s theorem rules out a bounded projection from $L^p(\mathbb{R}^n)$ onto $\PW^p(B)$ unless $p=2$, so the standard retraction argument for interpolation cannot be used. The unresolved interpolation question would clarify whether analogous methods could extend the Fourier-inequality results to the range $p<2$.