A discrete framework for the interpolation of Banach spaces

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...

Bloom weighted bounds for sparse forms associated to commutators

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...

Banach function spaces done right

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...

Stein interpolation for the real interpolation method

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...

Sparse domination implies vector-valued sparse domination

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...

Vector-valued harmonic analysis with applications to SPDE

In this dissertation we develop vector-valued harmonic analysis methods. Particular emphasis is put on the study of stochastic singular integral operators, which arise naturally in the study of SPDE.

Singular stochastic integral operators

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...

On Pointwise ℓ^r -Sparse Domination in a Space of Homogeneous Type

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<sup>r<...

Rescaled extrapolation for vector-valued functions

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.

Sparse domination for the lattice Hardy–Littlewood maximal operator

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.

Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

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...

The ℓ ^s-boundedness of a family of integral operators on UMD banach function spaces

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...

Fourier Multipliers in Banach Function Spaces with UMD Concavifications

We prove various extensions of the Coifman-Rubio de Francia-Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call $ {\ell ^{r}(\ell ^{s})}$-boundedness, which implies $ \mathcal {R}$-boundedness in many cases. The proofs are based on...

On the ℓs-boundedness of a family of integral operators

In this paper we prove an ℓs-boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal Lp-regularity for parabolic problems with time...