Equivalence of measures of smoothness in $L_{p}(S^{d-1})$, 1 < p < ∞

F. Dai; Z. Ditzian; Hongwei Huang

Equivalence of measures of smoothness in

L_{p} (S^{d - 1})

, 1 < p < ∞

F. Dai ; Z. Ditzian ; Hongwei Huang

Studia Mathematica, Tome 196 (2010), p. 179-205 / Harvested from The Polish Digital Mathematics Library

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Résumé

Suppose Δ̃ is the Laplace-Beltrami operator on the sphere $S^{d - 1}, Δ_{ρ}^{k} f (x) = Δ_{ρ} Δ_{ρ}^{k - 1} f (x)$ and $Δ_{ρ} f (x) = f (ρ x) - f (x)$ where ρ ∈ SO(d). Then $ω^{m} {(f, t)}_{L_{p} (S^{d - 1})} \equiv s u p ∥ Δ_{ρ}^{m} f ∥_{L_{p} (S^{d - 1})} : ρ \in S O (d), m a x_{x \in S^{d - 1}} ρ x \cdot x \geq c o s t$ and $K ̃ ₘ {(f, t^{m})}_{p} \equiv i n f ∥ f - g ∥_{L_{p} (S^{d - 1})} + t^{m} ∥ {(- Δ ̃)}^{m / 2} g ∥_{L_{p} (S^{d - 1})} : g \in ({(- Δ ̃)}^{m / 2})$ are equivalent for 1 < p < ∞. 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.

Publié le : 2010-01-01

Zbl 1230.42014

EUDML-ID : urn:eudml:doc:285537

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F. Dai; Z. Ditzian; Hongwei Huang. Equivalence of measures of smoothness in $L_{p}(S^{d-1})$, 1 < p < ∞. Studia Mathematica, Tome 196 (2010) pp. 179-205. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-2-5/