On Weighted Estimates of Non-increasing Rearrangements

Let ω be a weight satisfing Muckenhoupt’s condition A∞. In present paper the estimate of rearrangement f∗ ω(t) was obtained f∗ ω(t) ≤ 2(M λ f)ω(2t) + f∗ ω(2t) (0 < t < ∞), where f is any measurable function, M λ f is local sharp maximal function due to John [12] and Strömberg [18]. Before (Bennett, DeVore and Sharpley [3], Bagby and Kurtz [1]) the similar estimates were expressed in terms of Fefferman-Stein sharp-function f which is sufficiently larger then M λ f . In paper several applications of this estimate were pointed out.