We consider weighted Besicovitch sets which are defined in terms of the weighted frequency of digit 1 in the dyadic expansion of real numbers. Explicit formulas for their Hausdorff dimensions are given.
Let X be an Ahlfors d-regular space and rn a d-regular measure on X. We prove that a measure μ on X is d-homogeneous if and only if μ is mutually absolutely continuous with respect to m and the derivative Dmμ（x） ...Let X be an Ahlfors d-regular space and rn a d-regular measure on X. We prove that a measure μ on X is d-homogeneous if and only if μ is mutually absolutely continuous with respect to m and the derivative Dmμ（x） is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.展开更多
基金Supported by the Special Fund for Major State Basic Research Projects (Grant No. 60472041)
文摘We consider weighted Besicovitch sets which are defined in terms of the weighted frequency of digit 1 in the dyadic expansion of real numbers. Explicit formulas for their Hausdorff dimensions are given.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10971056 and 10771164)
文摘Let X be an Ahlfors d-regular space and rn a d-regular measure on X. We prove that a measure μ on X is d-homogeneous if and only if μ is mutually absolutely continuous with respect to m and the derivative Dmμ（x） is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.
