Dunkl spectral multipliers with values in UMD lattices

Deleaval, Luc; Kriegler, Christoph

doi:10.1016/j.jfa.2016.12.013

Cited by 13 publications

(8 citation statements)

References 61 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For an overview of these topics we refer to [31], and for useful applications to parabolic PDEs see for example [13,36,43]. Recent work on vector-valued harmonic analysis in UMD Banach function spaces includes [4,12,16,17,27,30,34,42,51,54].…”

Section: Introductionmentioning

confidence: 99%

Rescaled extrapolation for vector-valued functions

Amenta¹,

Lorist²,

Veraar³

2019

Publicacions Matemàtiques

View full text Add to dashboard Cite

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 Franciatype estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.is well-defined and strongly measurable. If X is p 0 -convex and X p0 has the UMD property, then T extends to a bounded operator on L p (R d , w; X) for all p ∈ (p 0 , ∞) and w ∈ A p/p0 .

show abstract

Section: Introductionmentioning

confidence: 99%

Rescaled extrapolation for vector-valued functions

Amenta¹,

Lorist²,

Veraar³

2019

Publicacions Matemàtiques

View full text Add to dashboard Cite

show abstract

“…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].…”

Section: Introductionmentioning

confidence: 99%

Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

Lorist

Nieraeth

2019

J Fourier Anal Appl

View full text Add to dashboard Cite

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].

show abstract

“…Remark 8.7. In a forthcoming paper [17], we will also show a Hörmander theorem for A ⊗ id Y on X = L p (Ω; Y ) for many self-adjoint A such that exp(−tA) has Gaussian estimates, where Y is any UMD Banach lattice.…”

Section: Assumption 82mentioning

confidence: 95%

Spectral multiplier theorems via $$H^\infty $$ H ∞ calculus and R-bounds

Kriegler

Weis

2017

Math. Z.

Self Cite

View full text Add to dashboard Cite

We prove spectral multiplier theorems for Hörmander classes H α p for 0-sectorial operators A on Banach spaces assuming a bounded H ∞ (Σ σ ) calculus for some σ ∈ (0, π) and norm and certain R-bounds on one of the following families of operators: the semigroup e −zA on C + , the wave operators e isA for s ∈ R, the resolvent (λ − A) −1 on C\R, the imaginary powers A it for t ∈ R or the Bochner-Riesz means (1 − A/u) α + for u > 0. In contrast to the existing literature we neither assume that A operates on an L p scale nor that A is self-adjoint on a Hilbert space. Furthermore, we replace (generalized) Gaussian or Poisson bounds and maximal estimates by the weaker notion of R-bounds, which allow for a unified approach to spectral multiplier theorems in a more general setting. In this setting our results are close to being optimal. Moreover, we can give a characterization of the (R-bounded) H α 1 calculus in terms of R-boundedness of Bochner-Riesz means.

show abstract

Dunkl spectral multipliers with values in UMD lattices

Cited by 13 publications

References 61 publications

Rescaled extrapolation for vector-valued functions

Rescaled extrapolation for vector-valued functions

Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

Spectral multiplier theorems via $$H^\infty $$ H ∞ calculus and R-bounds

Contact Info

Product

Resources

About