Resumen de Self-improving properties of generalized Poincare type inequalities through rearrangements

A. K. Lerner

• We prove, within the context of spaces of homogeneous type, Lp and exponential type self-improving properties for measurable functions satisfying the following Poincaré 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∗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 [4]. Our method avoids completely the "good-λ" inequality technique and any kind of representation formula.