Equivalence of measures of smoothness in $L_p(S^{d-1})$, $1< p< \infty $
Volume 196 / 2010
Abstract
Suppose $\widetilde{\mit\Delta} $ is the Laplace–Beltrami operator on the sphere $S^{d-1},$ $\Delta ^k_\rho f(x) = \Delta _\rho \Delta ^{k-1}_\rho f(x)$ and $ \Delta _\rho f(x) = f(\rho x) - f(x)$ where $\rho \in SO(d) .$ Then $$ \omega ^m (f,t)_{L_p(S^{d-1})} \equiv \sup\{\Vert \Delta ^m_\rho f\Vert _{L_p(S^{d-1})}: \rho \in SO(d), \, \max_{x\in S^{d-1}} \rho x\cdot x \ge \cos t\} $$ and $$ \widetilde K_m(f,t^m)_p\equiv \inf \{\Vert f-g\Vert _{L_p(S^{d-1})} + t^m\Vert (-\widetilde{\mit\Delta} )^{m/2}g\Vert _{L_p(S^{d-1})} :g\in {\cal D}((-\widetilde{\mit\Delta} )^{m/2})\} $$ are equivalent for $1< p< \infty .$ We note that for even $m$ the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for $L_p(S^{d-1})$ given in this paper plays a significant role in the proof.
