In this paper we consider properties of medians as they pertain to the continuity and vanishing oscillation of a function. Our approach is based on the observation that medians are related to local sharp maximal functions restricted to a cube of R n .

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A local median decomposition is used to prove that a weighted mean of a function is controlled locally by the weighted mean of its local sharp maximal function. Together with the estimate M 0 , s ♯ ( T f ) ( x ) ≤ c M f ( x ) M^{\sharp }_{0,s}(Tf)(x) \le c\,Mf(x) for Calderón-Zygmund singular integral operators, this allows us to express the local weighted control of T f Tf by M f Mf . Similar estimates hold for T T replaced by singular integrals with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and M M replaced by an appropriate maximal function M T M_T . Using sharper bounds in the local median decomposition we prove two-weight, L v p − L w q L^p_v-L^q_w estimates for the singular integral operators described above for 1 > p ≤ q > ∞ 1>p\le q>\infty and a range of q q . The local nature of the estimates leads to results involving weighted generalized Orlicz-Campanato and Orlicz-Morrey spaces.

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