Self-improving properties of generalized poincaré type inequalities through rearrangements

We prove, within the context of spaces of homogeneous type, L^p and exponential type self-improving properties for measurable functions satisfying the following Poincaré type inequality: inf _α((f - α)_ΧB)*_μ (λμ(B)) ≤ c_λa(B). Here, f*_μ denotes the non-increasing rearrangement of f, and a is a functional acting on balls B, satisfying appropriate geometric conditions. Our main result improves the work in [11], [12] as well as [2], [3] and [14]. Our method avoids completely the "good-λ" inequality technique and any kind of representation formula.