Mapping properties of a projection related to the Helmholtz equation (Q1879330)

Let \(H^2\) be the Hilbert space of functions \(u\) on \(\{x \in \mathbb R^2: | x| > 1 \}\) whose norm \[ \int_{| x| >1} (| u(x)| ^2 + | u_r(x)| ^2 + | u_\theta(x)| ^2) | x| ^{-3}\,dx \] is finite. It was shown by \textit{J. Alvarez, M. Folch-Gabayet} and \textit{S. Pérez-Esteva} [J. Fourier Anal. Appl. 7, No. 1, 49--62 (2001; Zbl 0989.46013)] that an entire solution to the Helmholtz equation \(\Delta u + u = 0\) is the Fourier transform of an \(L^2(S^1)\) measure if and only if it lies in this Hilbert space, and furthermore the Fourier extension map is an isometry. This yields an orthogonal projection \(\Pi\) from \(H^2\) to the solutions of the Helmholtz equation. The authors develop an analogous space \(H^p\), but show that the projection map \(\Pi\) cannot be extended to \(H^p\) for any \(p \neq 2\); this is because a certain Bessel function integral operator is similarly unounded on \(L^p\) for any \(p \neq 2\). However, boundedness is restored if one takes an \(L^2\) average in the angular variables. The proof uses the well-known Besicovitch construction, similar in spirit to \textit{C. Fefferman's} demonstration [Ann. Math. (2) 94, 330--336 (1971; Zbl 0234.42009)] that the disc multiplier is unbounded on \(L^p\) for \(p \neq 2\).