Let u* be the Fefferman-Stein sharp function of u, and for 1 < r < oo, let Mru be an appropriate version of the Hardy-Littlewood maximal function of u. If A is a (not necessarily homogeneous) pseudodifferential operator of order 0, then there is a constant c > 0 such that the pointwise estimate (Au)*(x) < cMru(x) holds for all x G R" and all Schwartz functions u. This estimate implies the boundedness of 0-order pseudodifferential operators on weighted Lp spaces whenever the weight function

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... gs to Muckenhoupt's class Ap. Having established this, we construct weighted Sobolev spaces of fractional order in R" and on a compact manifold, prove a version of Sobolev's theorem, and exhibit coercive weighted estimates for elliptic pseudodifferential operators.