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Recently it was proved for 1 < p < ∞ that ω(f, t)p, a modulus of smoothness on the unit sphere, and e Km(f, t)p, a K-functional involving the LaplaceBeltrami operator, are equivalent. It will be shown that the range 1 < p <∞ is optimal; that is, the equivalence ω(f, t)p ≈ e Km(f, t)p does not hold either for p = ∞ or for p = 1.

Optimality of the range for which equivalence between certain measures of smoothness holds