A Proof of the Marcinkiewicz-Zygmund Inequality on the Bi-Sphere

In this technical report we present a complete proof of the Marcinkiewicz-Zygmund inequality on the bi-Sphere from [1, Theorem 4.5]. We use the same notation as in [1]. The Theorem 4.5 of [1] reads as follows: Theorem 1. Let N ∈ N2 and R be a admissible decomposition of S2 × S2 according to a sampling set X such that ‖R‖ ≤ η 21 Ñ , (1) where η ∈ (0, 1) is arbitrarily fixed. Then for r = 1 or r =∞ and any p ∈ ΠN , we have (1− η) ‖p‖S2×S2,r ≤ ‖p‖X ,r ≤ (1 + η) ‖p‖S2×S2,r. (2) We apply subsequently the following inequalities: Lemma 2. The following estimates are valid, (i) max t∈[0,π] |vN (cos t)| ≤ 1 π N, (ii) ∫ ξ ξ− j ∣∣ d dt vN (cos t) ∣∣ dt ≤ 2 π N ‖R‖, (iii) ∫ π−‖R‖ ‖R‖ ∣∣ d dt vN (cos t) ∣∣ sin t dt ≤ 3 2π (5N + 1). Proof of Lemma 2: From [2, equations (3.13), (3.14)], we obtain immediately the inequality (i). Using the Bernstein inequality ‖T ‖∞ ≤ 2N‖T‖∞ for vN (cos t) and (i) we get (ii). The integral in (iii) can be estimated as in the proof of Theorem 4.2 in [2]. Proof of Theorem 1: We start with the case r =∞. For proving the left-hand side let (x,y) ∈ S2 × S2 an arbitrary fixed point and assume that (u,v) ∈ X is a point with d((x,y), (u,v)) = d((x,y),X ) = δX . We conclude by [1, Lemma 4.4] |p(x,y)| ≤ |p(x,y)− p(u,v)|+ |p(u,v)| ≤ 2Ñ d ((x,y), (u,v)) ‖p‖S2×S2,∞ + max j=1,...,M |p(xj ,yj)| .