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 Str¨omberg [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.