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: 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.

Parking Functions and Noncrossing Partitions

A parking function is a sequence $(a_1,\dots,a_n)$ of positive integers such that, if $b_1\leq b_2\leq \cdots\leq b_n$ is the increasing rearrangement of the sequence $(a_1,\dots, a_n),$ then $b_i\leq i$. A noncrossing partition of the set $[n]=\{1,2,\dots,n\}$ is a partition $\pi$ of the set $[n]$ with the property that if $a < b < c < d$ and some block $B$ of $\pi$ contains both $a$ and $c$, while some block $B'$ of $\pi$ contains both $b$ and $d$, then $B=B'$. We establish some connections between parking functions and noncrossing partitions. A generating function for the flag $f$-vector of the lattice NC$_{n+1}$ of noncrossing partitions of $[{\scriptstyle n+1}]$ is shown to coincide (up to the involution $\omega$ on symmetric function) with Haiman's parking function symmetric function. We construct an edge labeling of NC$_{n+1}$ whose chain labels are the set of all parking functions of length $n$. This leads to a local action of the symmetric group ${S}_n$ on NC$_{n+1}$.