Let n = v + z be an H-type group, and let n + a be the harmonic semidirect product of n with a similar or equal to R. Let N A be the corresponding simply connected Lie group. If dim v = m and dim z = k, denote Q = m/2 + k. We prove that the spherical Fourier transform is a topological isomorphism between the p-Schwartz space L(p)(N A)(#), (0 < p less than or equal to 2), and the space of holomorphic rapidly decreasing functions on the strip {s is an element of C:\Re(s)\ < epsilon Q/2} with epsilon = 2/p - 1
DI BLASIO, B. (1997). Paley-Wiener type theorems on harmonic extensions of H-type groups. MONATSHEFTE FÜR MATHEMATIK, 123(1), 21-42 [10.1007/BF01316934].
Paley-Wiener type theorems on harmonic extensions of H-type groups
DI BLASIO, BIANCA
1997
Abstract
Let n = v + z be an H-type group, and let n + a be the harmonic semidirect product of n with a similar or equal to R. Let N A be the corresponding simply connected Lie group. If dim v = m and dim z = k, denote Q = m/2 + k. We prove that the spherical Fourier transform is a topological isomorphism between the p-Schwartz space L(p)(N A)(#), (0 < p less than or equal to 2), and the space of holomorphic rapidly decreasing functions on the strip {s is an element of C:\Re(s)\ < epsilon Q/2} with epsilon = 2/p - 1
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