Null Sets for Doubling and Dyadic Doubling Measures
Null Sets for Doubling and Dyadic Doubling Measures
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发表时间：
1993
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通讯作者：
A. I. MathematicaVolumen
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作者：
A. I. MathematicaVolumen
本文研究了实线上的集合对于R上的所有加倍测度为零，或者对于R上的所有二元加倍测度为零，给出了前者的一些充分条件，后者的一个检验，并给出了一些例子。我们的工作的动机是由feerman， Kenig和piphher[5]的二元加倍测度的表征，以及对某些非线性a算子的多孔集合和总a调和测度为零的集合的Martio 8]的结果。R上的一个测度具有常数倍性，当I和J是两个相邻的相同长度的区间时，则(I) (J)；用D（）表示所有倍增量的集合，D = 1 D（）。当I和J是两个长度相同的并进相邻区间且I J也是一个并进区间时，R上的测度具有恒定的并进倍增性；用D D（）和D D表示相应的双进倍增测度集合。
In this note, we study sets on the real line which are null with respect to all doubling measures on R, or with respect to all dyadic doubling measures on R. We give some suucient conditions for the former, a test for the latter, and some examples. Our work is motivated by a characterization of dyadic doubling measures by Feeerman, Kenig and Pipher 5], and by a result of Martio 8] on porous sets and sets of total A-harmonic measure zero for certain class of nonlinear A-operators. A measure on R is said to have the doubling property with constant if, whenever I and J are two neighboring intervals of same length then (I) (J); denote by D() the collection of all doubling measures with constant , and D = 1 D(). A measure on R has the dyadic doubling property with constant if (I) (J) whenever I and J are two dyadic neighboring intervals of same length and I J is also a dyadic interval; denote by D d () and D d the corresponding collections of dyadic doubling measures.