A weighted theory for multilinear fractional integral operators and maximal functions is presented. Sufficient conditions for the two weight inequalities of these operators are found, including "power and logarithmic bumps" and an A ∞ condition. For one weight inequalities a necessary and sufficient condition is then obtained as a consequence of the two weight inequalities. As an application, Poincaré and Sobolev inequalities adapted to the multilinear setting are presented.

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.

We establish two-weight norm inequalities for singular integral operators defined on spaces of homogeneous type. We do so first when the weights satisfy a double bump condition and then when the weights satisfy separated logarithmic bump conditions. Our results generalize recent work on the Euclidean case, but our proofs are simpler even in this setting. The other interesting feature of our approach is that we are able to prove the separated bump results (which always imply the corresponding double bump results) as a consequence of the double bump theorem.

Abstract. We investigate the weighted bounds for multilinear maximal functions and Calderón-Zygmund operators from, where 1 < p1, · · · , pm < ∞ with 1/p1+· · ·+1/pm = 1/p and w is a multiple A P weight. We prove the sharp bound for the multilinear maximal function for all such p1, . . . , pm and prove the sharp bound for m-linear Calderón-Zymund operators when p ≥ 1.

In the 1970s Muckenhoupt and Wheeden made several conjectures relating two weight norm inequalities for the Hardy-Littlewood maximal operator to such inequalities for singular integrals. Using techniques developed for the recent proof of the A 2 conjecture we prove a related pair of conjectures linking the Riesz potential and the fractional maximal operator. As a consequence we are able to prove a number of sharp one and two weight norm inequalities for the Riesz potential.1991 Mathematics Subject Classification. 42B25, 42B30, 42B35.