Marcinkiewicz–Zygmund measures on manifolds

Let XX be a compact, connected, Riemannian manifold (without boundary), ρρ be the geodesic distance on XX, μμ be a probability measure on XX, and {ϕk}{ϕk} be an orthonormal (with respect to μμ) system of continuous functions, ϕ0(x)=1ϕ0(x)=1 for all x∈Xx∈X, {ℓk}k=0∞ be a nondecreasing sequence of real numbers with ℓ0=1ℓ0=1, ℓk↑∞ℓk↑∞ as k→∞k→∞, ΠL:=span{ϕj:ℓj≤L}, L≥0L≥0. We describe conditions to ensure an equivalence between the LpLp norms of elements of ΠLΠL with their suitably discretized versions. We also give intrinsic criteria to determine if any system of weights and nodes allows such inequalities. The results are stated in a very general form, applicable for example, when the discretization of the integrals is based on weighted averages of the elements of ΠLΠL on geodesic balls rather than point evaluations.