In this survey, we discuss the definition of a (quasi-)Banach function space. We advertise the original definition by Zaanen and Luxemburg, which does not have various issues introduced by other, subsequent definitions. Moreover, we prove versions of well-known basic properties of Banach function spaces in the setting of quasi-Banach function spaces.

In this paper we give growth estimates for ‖Tⁿ‖ for n→∞ in the case T is a strongly Kreiss bounded operator on a UMD Banach space X. In several special cases we provide explicit growth rates. This includes known cases such as Hilbert and L^p-spaces, but also intermediate UMD spaces such as non-commutative L^p-spaces and variable Lebesgue spaces.

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 consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner–Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón–Zygmund operators, the previously known quantitative Bloom bounds are not sharp for the second and higher order commutators.

We develop a discrete framework for the interpolation of Banach spaces, which contains the well-known real and complex interpolation methods, but also more recent methods like the Rademacher, γ- and ℓ^q-interpolation methods. Our framework is based on a sequential structure imposed on a Banach space, which allows us to deduce properties of interpolation methods from properties of sequential structures. Our framework has a formulation modelled after both the real and the complex interpolation methods. This enables us to extend various results, previously known only for either the real or the complex interpolation method, to all interpolation methods that fit into our framework. As applications, we prove an interpolation result for analytic operator families and an interpolation result for intersections.

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ on a Hilbert space. Our assumption on Γ is expressed in terms of α-boundedness for a Euclidean structure α on the underlying Banach space X. This notion is originally motivated by R- or γ-boundedness of sets of operators, but for example any operator ideal from the Euclidean space l²_n to X defines such a structure. Therefore, our method is quite flexible. Conversely we show that Γ has to be α-bounded for some Euclidean structure α to be representable on a Hilbert space. By choosing the Euclidean structure α accordingly, we get a unified and more general approach to the Kwapień-Maurey factorization theorem and the factorization theory of Maurey, Nikišin and Rubio de Francia. This leads to an improved version of the Banach function space-valued extension theorem of Rubio de Francia and a quantitative proof of the boundedness of the lattice Hardy-Littlewood maximal operator. Furthermore, we use these Euclidean structures to build vectorvalued function spaces. These enjoy the nice property that any bounded operator on L² extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem, we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We for example extend and refine the known theory based on R-boundedness for the joint and operator-valued H^∞-calculus. Moreover, we extend the classical characterization of the boundedness of the H^∞- calculus on Hilbert spaces in terms of BIP, square functions and dilations to the Banach space setting. Furthermore we establish, via the H^∞-calculus, a version of Littlewood-Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the geometry of X, such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 on a closed subspace of L^p for 1 < p < ∞ with a bounded H^∞-calculus with optimal angle ω_H^∞(A) > 0.

We prove a complex formulation of the real interpolation method, showing that the real and complex interpolation methods are not inherently real or complex. Using this complex formulation, we prove Stein interpolation for the real interpolation method. We apply this theorem to interpolate weighted L^p-spaces and the sectoriality of closed operators with the real interpolation method.

We obtain a sparse domination principle for an arbitrary family of functions Formula Presented, where Formula Presented and Q is a cube in Formula Presented. When applied to operators, this result recovers our recent works [37, 39]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincaré-Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [39], as we will demonstrate in an application to vector-valued square functions.

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.

We introduce Calderón-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove L ^p-extrapolation results under a Hörmander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the A2-conjecture. The results are applied to obtain p-independence and weighted bounds for stochastic maximal L ^p-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on (Formula presented) and smooth and angular domains.

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual ℓ¹-sum in the sparse operator is replaced by an ℓ^r-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the A₂-theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

We study the domination of the lattice Hardy–Littlewood maximal operator by sparse operators in the setting of general Banach lattices. We prove that the admissible exponents of the dominating sparse operator are determined by the q-convexity of the Banach lattice.

We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood-Paley-Rubio de Francia-type estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.

We prove the ℓ^s-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the ℓ^s-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of ℓ^s-boundedness as weighted boundedness by Rubio de Francia.

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.