arXiv subjects
Publications and source records attributed to Emiel Lorist.
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.
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 $\mathcal{R}$- or $γ$-boundedness of sets of operators, but, for example, any operator ideal from the Euclidean space $\ell^2_n$ 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 classical factorization and extension theorems. Furthermore we use these Euclidean structures to build vector-valued function spaces and define an interpolation method based on these spaces, 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 define generalizations of the classical square function estimates in $L^p$-spaces and establish, via the $H^\infty$-calculus, a version of Littlewood-Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our results for sectorial operators lead to some sophisticated counterexamples.
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 weighted estimates are not sharp for the second and higher order commutators.
In this paper 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 $A_2$-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 $\mathbb{R}^d$ and smooth and angular domains.
For every cube $Q \subset \mathbb{R}^n$ we let $X_Q$ be a quasi-Banach function space over $Q$ such that $\|χ_Q\|_{X_Q} \simeq 1$, and for $X= \{X_Q\}$ define \begin{align*} \|f\|_{\mathrm{BMO}_X} &:=\sup_Q \,\|f-{\textstyle\frac{1}{|Q|}\int_Qf} \|_{X_Q},\\ \|f\|_{\mathrm{BMO}_X^*} &:=\sup_Q \,\inf_c\, \|f-c\|_{X_Q}. \end{align*} We study necessary and sufficient conditions on $X$ such that $$ \mathrm{BMO} = \mathrm{BMO}_X = \mathrm{BMO}_{X}^*. $$ In particular, we give a full characterization of the embedding $\mathrm{BMO} \hookrightarrow \mathrm{BMO}_X$ in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding $\mathrm{BMO}_X^* \hookrightarrow \mathrm{BMO}$. Our main theorems recover and improve all previously known results in this area.
We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$, where $x\in {\mathbb R}^n$ and $Q$ is a cube in ${\mathbb R}^n$. When applied to operators, this result recovers our recent works. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalized Poincaré-Sobolev inequalities, tent spaces, and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localizable in the sense of our previous works, as we will demonstrate in an application to vector-valued square functions.
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 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 \[T:L^{p_1}(w_1^{p_1})\times\cdots\times L^{p_m}(w_m^{p_m})\to L^p(w^p) \] 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 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 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 $\ell^1$-sum in the sparse operator is replaced by an $\ell^r$-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the $A_2$-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 prove the $\ell^s$-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 $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia.
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 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 new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.
In this paper we prove an $\ell^s$-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 $L^p$-regularity for parabolic problems with time-dependent generator is developed.