Nikol'skij-Besov differential functional classes on a compact symmetric space of rank one. (Q1432135)

This article contains the following main result: Suppose that \(r>0\) and the functional \({\mathcal H}\) preserves bounds and is almost semi-additive. Then, for any nonnegative integer \(\rho<r\), positive integer \(k>\alpha -\rho\), and nonnegative integer \(\nu<r\), the following assertions are valid. Let \({\mathcal H} \in{\mathcal F}_{0+} \cap{\mathcal J}_{0+}\). (i) If the functional \({\mathcal H}\) is almost homogeneous and finite on the characteristic functions of half-intervals \((t_0, \pi]\) for any \(t_0>0\), then \[ \| f\|+ {\mathcal H}^\omega\bigl[t^\rho \omega_k (t;f_*^{(\rho)})\bigr] \sim\| f\|+{\mathcal H}^\omega \bigl[t^\nu E_{\pi/t} (f^{(\nu)}_*)\bigr] \] for (a) both moduli of smoothness instead of \(\omega_k(t;f^{(\rho)}_*)\) and the best approximations in the \(M\)-norm; (b) all (four) moduli of smoothness and the best approximations in the TM-norm; and (c) all (four) moduli of smoothness and the best approximations in both norms if they are equivalent on \(M\). (ii) For the maximal moduli of smoothness, we have \[ {\mathcal H}^\omega \bigl[t^\rho \omega_k(t;f_*^{(\rho)}) \bigr]\sim {\mathcal H}^\omega \bigl[ t^\nu E_{\pi/t} (f_*^{(\nu)})\bigr] \] (a) in the \(M\)-norm, (b) in the \(TM\)-norm, and (c) in both norms if they are equivalent on \(M\). If \({\mathcal H}\in {\mathcal F}_{0+}\), then the pre-norms \({\mathcal H}^\omega [t^\rho \omega_k (t;f_*^{(\rho)})]\) are equivalent to each other with maximal moduli of smoothness and the pre-norms \({\mathcal H}^\omega[t^\nu E_{\pi/t} (f_*^{(\nu)})]\) are equivalent to each other in both cases where the former equivalence holds. In addition, \[ t^\rho\omega_k (t;f_*^{(\rho)}) \asymp\omega(t) \Leftrightarrow n^-E_n(f_*^{(\nu)}) \asymp \omega \left(\frac{\pi} {n}\right), \] the symbol \(\asymp\) denotes the inequalities \(\ll\) in both directions.