Spherical harmonics and spherical averages of Fourier transforms (Q1878654)

Let \(\theta\) denote the area measure on the unit sphere \(S^{n-1}\) in \({\mathbb{R}}^n\) for \(n\geq 2\) and set \[ \sigma(f)( R ):= \int_{S^{n-1}}| \hat{f}(R\xi)| ^2 \,d\theta(\xi) \] where \(f\in L^1({\mathbb{R}}^n)\). Let \[ \beta(\alpha):=\sup \left\{\beta: \exists C >0 \text{ such that } \sigma (f ) ( R ) \leq C R^{-\beta} \int _{{\mathbb{R}} ^n} | \hat f (\xi) | ^2 | \xi| ^{\alpha - n} d\xi\right\} \] where \(R >1\) and \(0 < \alpha \leq n\). When \(f\) is the finite sum of products of radial functions with spherical harmonics of fixed degree, and further, those radial functions are \(C^\infty\) and vanish both in a neighbourhood of the origin and outside the unit ball, then the author establishes that \(\beta(\alpha) = \alpha\) for \(0< \alpha \leq n-1\) and that \(\beta(\alpha) = n-1\) for \(n-1 < \alpha \leq n\). The result generalises that of the author in [Ann. Acad. Sci. Fenn., Math. 22, No. 1, 227--236 (1997; Zbl 0865.42007)].