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## Equivalence of measures of smoothness in $L_p(S^{d-1})$, $1< p< \infty$

### Volume 196 / 2010

Studia Mathematica 196 (2010), 179-205 MSC: 42B15, 41A17, 41A63. DOI: 10.4064/sm196-2-5

#### 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.

#### Authors

• F. DaiDepartment of Mathematical
and Statistical Sciences
University of Alberta
Edmonton, Alberta
e-mail
• Z. DitzianDepartment of Mathematical
and Statistical Sciences
University of Alberta
Edmonton, Alberta
e-mail
• Hongwei HuangSchool of Mathematical Sciences
Xiamen University
361005, Xiamen, Fujian
China
e-mail

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