Some new estimates for the Helgason-Fourier transform on rank 1 symmetric spaces (Q723473)

In this paper, useful estimates are proved for the Fourier transform in the space of square integrable functions on certain classes of functions, characterized by the generalized continuity modulus, employing a translation operator. It is observed that the Helgason-Fourier transform on an arbitrary Riemannian symmetric space of noncompact type is defined by spherical functions, see [\textit{S. Helgason}, Differential geometry and symmetric spaces. Übersetzung aus dem Englischen von A. L. Oniscik. Moskau: Verlag `Mir'. 533 S. (1964; Zbl 0122.39901)]. Definitions and fundamental regarding the Helgason-Fourier transform are given in Section 2, whereas the main results, in the form of three theorems, proposed estimation for square integrable functions. Reviewer's remark: It may be noticed that reference by \textit{S. Helgason} [Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80. New York-San Francisco-London: Academic Press. XV, 628 p. (1978; Zbl 0451.53038)] is not A. V. Abilov et al. as typed by the authors in the first line of Section 1 (Introduction). There are two other references for V. A. Abilov et al. An errata may be useful for readers to avoid confusion.