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.2010 Mathematics Subject Classification. 42B20, 42B25.

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 operatorfor 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.2010 Mathematics Subject Classification. Primary: 42B25; Secondary: 42B15, 46E30.Vector-valued extensions of the extrapolation theory have also been considered. Through an argument using Fubini's Theorem, the initial estimate (1.1) immediately implies not only the estimate (1.2) for all p ∈ (1, ∞), but also for extensions of the operator T to functions taking values in the sequence spaces ℓ s or more generally Lebesgue spaces L s for s ∈ (1, ∞). Moreover, Rubio de Francia showed in [60, Theorem 5] that one can take this even further. Indeed, this result states that assuming (1.1) holds for some p 0 ∈ (1, ∞) and for all weights w ∈ A p 0 , then for each Banach function space X with the UMD property, T extends to an operator T on the Bochner space L p (X) which satisfiesfor all p ∈ (1, ∞) and all f ∈ L p (X) . Recently, it was shown by Amenta, Veraar, and the first author in [2] that given p − ∈ (0, ∞), if (1.1) holds for p 0 ∈ (p − , ∞) and all weights w ∈ A p 0 /p − , then for each Banach function space X such that X p − has the UMD property, T extends to an operator T on the Bochner space L p (w; X) and satisfiesfor all p ∈ (p − , ∞), all weights w ∈ A p/p − and all f ∈ L p (w; X). Here X p − is the p − -concavification of X, see Section 2 for the definition.Vector-valued estimates in harmonic analysis have been actively developed in the past decades. Important for the mentioned vector-valued extrapolation are the equivalence of the boundedness of the vector-valued Hilbert transform on L p (X) and the UMD property of X for a Banach space X (see [9,13]) and the fact that for a Banach function space X the UMD property implies the boundedness of the lattice Hardy-Littlewood maximal operator on L p (X) (see [10,61]). For recent results in vector-valued harmonic analysis in UMD Banach function spaces, see for example [8,25,36,41,66].

We consider operators T satisfying a sparse domination property with averaging exponents . We prove weighted strong type boundedness for and use new techniques to prove weighted weak type boundedness with quantitative mixed – estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case 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 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.

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We prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination without the use of a Rubio de Francia type extrapolation result.We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.