Self-improving properties of generalized Poincaré type inequalities through rearrangements - Lerner, Andrei K.; Pérez, Carlos (MATHEMATICA SCANDINAVICA, 2005)
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: 26733 \inf_{α}\bigl((f-α)χ_{B}\bigr)_{μ}^*\bigl(λμ(B)\bigr) \le c_{λ}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.