The ℓ s-Boundedness of a Family of Integral Operators on UMD Banach Function Spaces

We prove the mixed-norm L p L q -boundedness of a general class of singular integral operators having a multi-parameter singularity and acting on vector-valued (UMD Banach lattice-valued) functions. Moreover, families of such operators with uniform assumptions are shown to be not only uniformly bounded but R-bounded, a genuinely stronger property that is often needed in applications. Previous results of this nature only dealt with convolution-type or slightly more general paraproduct-free singular integrals. In contrast, our analysis specifically targets the array of different partial paraproducts that arise in the multi-parameter setting by interpreting them as paraproduct-valued one-parameter operators. This new point-of-view provides a conceptual simplification over the existing representation results for multi-parameter operators, which is a key to the proof of the boundedness of these operators.

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“…A weighted version of this in

R^{n}

is proved in . See also and . Again, the point made in Remark applies here.…”

Section: Definitions and Preliminariesmentioning

confidence: 86%

“…An easy to read reference for this section is (see also and ). A normed space

E

is a Banach function space (or a function lattice) if the following four conditions hold.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Multi‐parameter estimates via operator‐valued shifts

Hytönen

Martikainen

Vuorinen

2019

Proceedings of London Math Soc

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“…The following is an analogue of [HvNVW18, Proposition 2.3.9 p. 104] and [Gra14, p. 82 and p. 84]. See also [Lor19] for related results. Recall that the radially decreasing majorant R ϕ is defined in (3.4).…”

Section: R-boundedness Of Some Family Of Convolution Operatorsmentioning

confidence: 90%

Boundedness of H ∞ Functional Calculus of Hodge-Dirac Operators

Arhancet¹,

Kriegler

2022

Lecture Notes in Mathematics

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For any non-archimedean local field K and any integer n ⩾ 1, we show that the Taibleson operator admits a bounded H ∞ (Σ θ ) functional calculus for any angle θ > 0 on the Banach space L p (K n ), where Σ θ = {z ∈ C * : | arg z| < θ} and 1 < p < ∞, and even a bounded Hörmander functional calculus of order 3 2 (with striking contrast to the Euclidean Laplacian on R n ). In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups establishing the R-boundedness of a family of convolution operators. Our results enhance the understanding of functional calculi of operators acting on L p -spaces associated to totally disconnected spaces and have implications for the maximal regularity of the fundamental evolution equations associated to the Taibleson operator, relevant in various physical models. Contents 1 Introduction 1 2 Preliminaries 5 3 Locally compact Spector-Vilenkin groups and radial functions 10 4 Maximal inequalities 15 5 R-boundedness of some family of convolution operators 17 6 Application to functional calculus 20 Bibliography 27 2020 Mathematics subject classification: 46S10, 47S10, 11S80.

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“…• In [Lor19] it was shown that the UMD property is sufficient for the ℓ 2 -boundedness of a quite broad class of convolution operators on L p (R d ; X). Using a similar proof as the one presented in Theorem 8.7 one can show that also for the ℓ 2 -boundedness of these operators the UMD property of the Banach function space X is necessary.…”

Section: Define For Every Interval I ∈ D the Haar Function H I Bymentioning

confidence: 99%

Euclidean structures and operator theory in Banach spaces

Kalton,

Lorist,

Weis

2019

Preprint

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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 ℓ 2 n to X (like the γradonifying or the 2-summing operator ideal) defines such a structure. Therefore our method is quite general and flexible and allows to unify the approach to seemingly unrelated theorems. Conversely we show that Γ has to be α-bounded for some Euclidean structure α for it 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 theorems like the Kwapień-Maurey factorization theorem, an improved version of the Banach function space-valued extension theorem of Rubio de Francia, a quantitative proof of the boundedness of the lattice Hardy-Littlewood maximal operator and the equivalence of the UMD and the dyadic UMD + property on Banach function spaces. Furthermore we use these Euclidean structures to build vector-valued function spaces, which enjoy the nice property that any bounded operator on L 2 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 quite general 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 extend and refine the known theory based on R-boundedness for the joint and operator-valued H ∞ -calculus and the "sum of operators" theorem for commuting sectorial operators. 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 Banach space 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 ωH∞ (A) > 0.

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The ℓ s-Boundedness of a Family of Integral Operators on UMD Banach Function Spaces

Cited by 4 publications

References 36 publications

Multi‐parameter estimates via operator‐valued shifts

Multi‐parameter estimates via operator‐valued shifts

Boundedness of H ∞ Functional Calculus of Hodge-Dirac Operators

Euclidean structures and operator theory in Banach spaces

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