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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 = v(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
Poincaré 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.
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