Extrapolation of weights revisited: new proofs and sharp bounds (Q630791)

In this paper various extrapolation theorems are proven for weighted \(L^p\) spaces. A new proof is given of Rubio de Francia's extrapolation theorem, with sharp bounds that depend only on the \(A_p\) constant of the weight and the weighted \(L^p\)-norm of the Hardy-Littlewood maximal operator. The proof uses a different way of factorizing weights. One application is to show that if \(T\) is a Calderon-Zygmund operator, then \(\parallel T f \parallel_{L^p(w)} \leqslant C [w]_{A_q} \parallel f \parallel_{L^p(w)}\) for all \(w\in A_q\) and \(1 \leqslant q < p < \infty\). Extensions are given to off diagonal, partial range and multi-variable extrapolation. The introduction contains a good summary of the history of extrapolation theorems.