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 Poincare type inequality: 26733 \inf_{\alpha}\bigl((f-\alpha)\chi_{B}\bigr)_{\mu}^*\bigl(\lambda\mu(B)\bigr) \le c_{\lambda}a(B). 26733 Here, $f_{\mu}^*$ 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 [4]. Our method avoids completely the "good-$\lambda$" inequality technique and any kind of representation formula.