Poincare-Sobolev and related inequalities for submanifolds of RN

Abstract

We prove Poincaré-Sobolev and related inequalities for rectifiable varifolds in RN. In particular, all our results apply to properly immersed submanifolds of RN. Suppose M ⊂ BR = BR(0) ⊂ RN = Rn+k for some R > 0, and V = υ(M, θ) is a countably n-rectifiable varifold in BR with generalised mean curvature vector H. µ is the weight measure defined by µ = θHn ⌊M. h: M → R is a Lipschitz function. In Theorem 1 we prove a Poincaré-Sobolev result for non-negative h in case µ{ξ: h(ξ) > 0} < ωnRn and h ∈ W1, p(µ) for some p < n. This generalises a Poincare result of Leon Simon; but in addition the relevant constant here does not depend on µ(BR). Theorem 2 is an Orlicz space result in case p = n. The proofs of Theorems 1 and 2 use a covering argument to obtain weak LP type estimates on µ{ξ: h(ξ) > s}. Theorems 3 and 4 are generalisations of Theorems 1 and 2 in case there is no restriction on µ{ξ: h(ξ) ≠ 0} (again the constants in the estimates do not depend on µ(BR)). The conclusion of Theorem 4 is analogous to the conclusion of the John-Nirenberg theorem for functions of bounded mean oscillation. Show more