二重权的并矢逆Hölder常数和Muckenhoupt常数的相互估计
Mutual estimates for the dyadic Reverse Hölder and Muckenhoupt constants for the dyadically doubling weights
摘要：
Muckenhoupt和Reverse Hölder权类在调和分析、偏微分方程和拟共形映射中起着重要的作用。1974年，Coifman和Festerman证明了一个权属于Muckenhoupt类$A_p$，当且仅当它属于一个反向Hölder类$RH_q$，对于某些$1<p<\infty$，当且仅当它属于一个反向Hölder类$RH_q$，对于某些$1<q<\infty$。2009年Vasyunin使用Bellman函数方法发现了$p$，$q$和相应的权重特征之间的精确依赖关系。Coifman和Frederick的结果适用于二元类的权重，但需要额外假设权重是二元加倍的。我们将Vasyunin的结果推广到并元Reverse Hölder类和Muckenhoupt类，得到了$p$，$q$，倍常数与相应的权特征之间的依赖关系.更准确地说，给定一个二元加倍的重量在$RH_p^d$在一个给定的二元区间$I$，我们发现一个上限估计的平均值的函数$w^{q}$在区间$I$。从这个平均值的界限，我们可以得出结论，例如，$w$属于相应的$A_{s_1}^d$类，或者对于$s_i$的某些值，$w^p$属于$A_{s_2}^d$类。我们使用Bellman函数的方法得到我们的结果。
英文摘要：
Muckenhoupt and Reverse Hölder classes of weights play an important role in harmonic analysis, PDE's and quasiconformal mappings. In 1974 Coifman and Fefferman showed that a weight belongs to a Muckenhoupt class $A_p$ for some $1<p<\infty$ if and only if it belongs to a Reverse Hölder class $RH_q$ for some $1<q<\infty$. In 2009 Vasyunin found the exact dependence between $p$, $q$ and the corresponding characteristic of the weight using the Bellman function method. The result of Coifman and Fefferman works for the dyadic classes of weights under an additional assumption that the weights are dyadically doubling. We extend the Vasyunin's result to the dyadic Reverse Hölder and Muckenhoupt classes and obtain the dependence between $p$, $q$, the doubling constant and the corresponding characteristic of the weight. More precisely, given a dyadically doubling weight in $RH_p^d$ on a given dyadic interval $I$, we find an upper estimate on the average of the function $w^{q}$ over the interval $I$. From the bound on this average we can conclude, for example, that $w$ belongs to the corresponding $A_{s_1}^d$ class or that $w^p$ is in $A_{s_2}^d$ for some values of $s_i$. We obtain our results using the method of Bellman functions.
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