WEIGHTED LOCAL ESTIMATES FOR SINGULAR INTEGRAL OPERATORS

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(not equal) (Tf)(x) = <= c Mf(x) for Caldereon-Zygmund singular integral operators, this allows us to express the local weighted control of Tf by Mf. Similar estimates hold for T replaced by singular integrals with kernels satisfying Hormander-type conditions or integral operators with homogeneous kernels, and M replaced by an appropriate maximal function M-T. Using sharper bounds in the local median decomposition we prove two-weight, L-v(p) - L-w(q) estimates for the singular integral operators described above for 1 < p <= q < infinity and a range of q. The local nature of the estimates leads to results involving weighted generalized Orlicz-Campanato and Orlicz-Morrey spaces.