Sjölin, Per Spherical harmonics and spherical averages of Fourier transforms. (English) Zbl 1048.42013 Rend. Semin. Mat. Univ. Padova 108, 41-51 (2002). 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)]. Reviewer: D. L. Salinger (Leeds) MSC: 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type Keywords:Fourier transforms; spherical averages Citations:Zbl 0865.42007 PDFBibTeX XMLCite \textit{P. Sjölin}, Rend. Semin. Mat. Univ. Padova 108, 41--51 (2002; Zbl 1048.42013) Full Text: EuDML References: [1] J. BOURGAIN, Hausdorff dimension and distance sets, Israel J. Math., 87 (1994), pp. 193-201. Zbl0807.28004 MR1286826 · Zbl 0807.28004 [2] P. MATTILA, Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika, 34 (1987), pp. 207-228. Zbl0645.28004 MR933500 · Zbl 0645.28004 [3] P. MATTILA, Geometry of sets and measures in Euclidean spaces, Cambridge studies in advanced mathematics, 44, Cambridge University Press, 1995. Zbl0819.28004 MR1333890 · Zbl 0819.28004 [4] P. SJÖLIN, Estimates of spherical averages of Fourier transforms and dimensions of sets, Mathematika, 40 (1993), pp. 322-330. Zbl0789.28006 MR1260895 · Zbl 0789.28006 [5] P. SJÖLIN, Estimates of averages of Fourier transforms of measures with finite energy, Ann. Acad. Sci. Fenn. Math., 22 (1997), pp. 227-236. Zbl0865.42007 MR1430401 · Zbl 0865.42007 [6] P. SJÖLIN - F. SORIA, Some remarks on restriction of the Fourier transform for general measures, Publicacions Matemàtiques, 43 (1999), pp. 655-664. Zbl0958.42011 MR1744623 · Zbl 0958.42011 [7] P. SJÖLIN - F. SORIA, Estimates of averages of Fourier transforms with respect to general measures, Manuscript 2000. Zbl1051.42011 · Zbl 1051.42011 [8] E.M. STEIN - G. WEISS, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, 1971. Zbl0232.42007 MR304972 · Zbl 0232.42007 [9] T. WOLFF, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices (1999), pp. 547-567. Zbl0930.42006 MR1692851 · Zbl 0930.42006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.