We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.

In this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite. This extends the recent extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued estimates including ℓ ^∞ spaces and, in particular, we are able to reprove all the vector-valued bounds for the bilinear Hilbert transform obtained through the helicoidal method of Benea and Muscalu. Moreover, our result is quantitative and, in particular, allows us to extend quantitative estimates obtained from sparse domination in the Banach space setting to the quasi-Banach space setting. Our proof does not rely on any off-diagonal extrapolation results and we develop a multilinear version of the Rubio de Francia algorithm adapted to the multisublinear Hardy–Littlewood maximal operator. As a corollary, we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak-type and BMO spaces.

We give an extension of Rubio de Francia’s extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an m-(sub)linear operator T:Lp1(w1p1)×⋯×Lpm(wmpm)→Lp(wp) for a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces L ^p (w ^p ; X) for a wide class of Banach function spaces X, which includes certain Lebesgue, Lorentz and Orlicz spaces. We apply the extrapolation result to various operators, which yields new vector-valued bounds. Our examples include the bilinear Hilbert transform, certain Fourier multipliers and various operators satisfying sparse domination results.

We consider operators T satisfying a sparse domination property (Formula presented.)with averaging exponents (Formula presented.). We prove weighted strong type boundedness for (Formula presented.) and use new techniques to prove weighted weak type (Formula presented.) boundedness with quantitative mixed (Formula presented.)–(Formula presented.) estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case (Formula presented.) we improve upon their results as we do not make use of a Hörmander condition of the operator T. Moreover, we also establish a dual weak type (Formula presented.) estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.