On a function space related to the Hardy-Littlewood inequality for Riemannian symmetric spaces (Q1917220)

\(L_p\)-Schwartz spaces \({\mathcal I}_p (G)\) of bi-\(K\)-invariant functions on a semisimple Lie group \(G\) with finite center were defined by Harish-Chandra for \(0 < p \leq 2\), \(K\) being the maximal compact subgroup of \(G\). The space \({\mathcal I}_p (G)\) is a subspace of \(I_p (G) = L_p (K \backslash G/K)\) and the spherical transform of a function belonging to \({\mathcal I}_p (G)\) can be extended to a holomorphic function on a tube domain \(T_p\). The author defines an \(L_q\)-Schwartz space \({\mathcal J}_q (G)\), \(2 \leq q < \infty\), of bi-\(K\)-invariant functions that correspond to the Schwartz space with weight \(|x |^{n (q - 2)}\) on \(\mathbb{R}^n\). The weights were used in the Hardy-Littlewood theorem, which was proved for symmetric spaces by Eguchi and Kumahara. The last theorem says that if a function \(f\) belongs to a weighted function space \(J_q (X)\) then its spherical transform belongs to \(L_q\). Let \(p\) and \(q\) be conjugated, \(1/p + 1/q = 1\). Some inclusion properties of the introduced spaces are proved, e.g. \({\mathcal I}_p (G) \subset {\mathcal J}_q (G) \subset I(G)\) if \(2 > q\) and \(1/p - 1/q < 1/r < 1/p\). The spherical transform of an \(L_q\)-Schwartz function can be extended to a holomorphic function on \(T_p\). If \(2 \leq q < 4\), then the coincidence theorem \({\mathcal J}_q (G) = {\mathcal I}_p (G)\) is proved. If \(\text{rank} (G,K) = 1\), then some of the above results are improved.