We define $(α_n)$ -regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and in particular give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold in the real-line.
In this paper, we are concerned with the relationship among the lower Assouad type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as $θ$ tends to 1 equals to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as $θ$ tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to…
In this article we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$, the sets of all $p \in [1, \infty]$ for which the weak and strong type $(p,p)$ inequalities hold for the centered and non-centered modified Hardy--Littlewood maximal operators, $M^{\rm c}_k$ and $M_k$, $k \geq 1$. For any fixed $k$ we describe the necessary conditions that $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$ must satisfy in general and illu…
In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize rectifiable pointwise doubling measures in Hilbert space. Given a measure $μ$, we construct a multiresolution family $\mathscr{C}^μ$ of windows, and then we use a weighted Jones' function $\hat{J}_2(μ, x)$ to record how well lines approximate the distribution of mass in each window. We show that when $μ$ is rectifiable, the mass is sufficiently concentrated around a lines at each scale and that the convers…
Characterization of the atomic space $H^1$ for non doubling measures in terms of a grand maximal operator
设$μ$是$R^d$上的Radon测度，它可以是非加倍的。对于所有$x,r$和某些固定的$0<n\leq d$，$μ$必须满足的唯一条件是$μ(B(x,r))\leq C r^n$。最近，我们引入了$BMO(μ)$和$H^1(μ)$型空间，它们被证明在不假设加倍条件的情况下对研究Calderón-Zygmund算子的$L^p(μ)$有界性是有用的.本文用极大算子$M_Φ$刻画了新的原子空间$H^1(μ)$。结果表明，$f$属于$H^1(μ)$，当且仅当$f\in L^1(μ)$、$\int f dμ=0$和$M_Φ(f)\in L^1(μ)$与通常的加倍情况相同。$μ$缺乏任何正则性条件，除了上述的大小条件，是一个主要的困难，出现时，试图将经典的论点，目前的情况。
The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and only if it is doubling. Motivated by recent progress on both the Assouad dimension of measures that satisfy a strong separation condition and the quasi-Assouad dimension of metric spaces, we introduce the notion of the quasi-Assouad dimension of a mea…
We consider several (related) notions of geometric regularity in the context of limit sets of geometrically finite Kleinian groups and associated Patterson-Sullivan measures. We begin by computing the upper and lower regularity dimensions of the Patterson-Sullivan measure, which involves controlling the relative measure of concentric balls. We then compute the Assouad and lower dimensions of the limit set, which involves controlling local doubling properties. Unlike the Hausdorff, packing, and box-counting dimensions, we show that the Assouad and lower dimensions are not necessarily given by t…
