Poincare Inequalities

Poincare inequalities are a simple way to obtain lower bounds on the distortion of mappings X into Y. These are shown below to be sharp when we consider the Lp spaces. A Poincare inequality is one of the following type: suppose Ψ : [0, ∞) → [0, ∞) is a nondecreasing function and that au,v, bu,v are finite arrays of real numbers (for u, v ∈ X, and not all of the numbers 0). We say that functions from X to Y obey a Poincare inequality if for all f : X → Y we have that ∑u,v∈X au,vΨ(d( f (u), f (v))) ≥ ∑u,v∈X bu,vΨ(d( f (u), f (v))). Note that the space X here only comes into play when we look at the indices of these sums, all that X does is label.