The Banach Space-valued BMO, Carleson’s Condition, and Paraproducts

Hytönen, Tuomas; Weis, Lutz

doi:10.1007/s00041-009-9100-2

Cited by 16 publications

(33 citation statements)

References 19 publications

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“…Van Neerven and Portal [24], Hytönen and Weis [25], Kaiser [26], and Kaiser and Weis [27], amongst others. In this work we use γ-radonifying operators.…”

Section: Introductionmentioning

confidence: 99%

Square functions and spectral multipliers for Bessel operators in UMD spaces

Castro¹,

Rodríguez-Mesa²

2016

Banach J. Math. Anal.

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Abstract. In this paper we consider square functions (also called Littlewood-Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner Lebesgue spacewhere B is a UMD Banach space. As special cases we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operatorWe characterize the UMD property for a Banach space B by using L p ((0, ∞), B)-boundedness properties of g-functions defined by Bessel-Poisson semigroups. As a by product we prove that the fact that the imaginary power ∆ iω λ , ω ∈ R \ {0}, of the Bessel operator ∆ λ is bounded in L p ((0, ∞), B), 1 < p < ∞, characterizes the UMD property for the Banach space B. As applications of our results for square functions we establish the boundedness in L p ((0, ∞), B) of spectral multipliers m(∆ λ ) of Bessel operators defined by functions m which are holomorphic in sectors Σ ϑ .

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“…Van Neerven and Portal [24], Hytönen and Weis [25], Kaiser [26], and Kaiser and Weis [27], amongst others. In this work we use γ-radonifying operators.…”

Section: Introductionmentioning

confidence: 99%

Square functions and spectral multipliers for Bessel operators in UMD spaces

Castro¹,

Rodríguez-Mesa²

2016

Banach J. Math. Anal.

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“…This is precisely our notion of nondegenerateness, which fully answers the question we just raised above. We mention that the same class of test functions shows up in other contexts (e.g., [18]).…”

Section: Introductionmentioning

confidence: 62%

Multidimensional Tauberian theorems for vector-valued distributions

Pilipović

Vindas

2014

Publ Inst Math (Belgr)

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We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of $f$ is given by the integral transform $M^{f}_{\varphi}(x,y)=(f\ast\varphi_{y})(x),$ $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}_{+}$, with kernel $\varphi_{y}(t)=y^{-n}\varphi(t/y)$. We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on $\left\{x_0\right\}\times \mathbb R^m$. In addition, we present a new proof of Littlewood's Tauberian theorem.Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1012.509

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“…Assume now that

F \in L^{p} (R^{n}) \otimes B

. According to [, Lemma 2.3] we have

{∥ F ∥}_{L^{p} (R^{n}, boldB)} = \underset{\begin{matrix} g \in L^{p^{'}} ({boldR}^{n}) \otimes B^{*} \\ {∥ g ∥}_{L^{p^{'}} ({boldR}^{n}, {boldB}^{*})} \leq 1 \end{matrix}}{trueprefixsup} |\int_{{boldR}^{n}} {⟨ g (x), F (x) ⟩}_{B^{*}, boldB} d x| .

Then, since

{boldB}^{*}

is also a UMD Banach space, by using Proposition , [, Proposition 2.2] and we obtain

\begin{matrix} true \\ right {∥ F ∥}_{L^{p} (R^{n}, boldB)} & left = \frac{2^{2 r}}{Γ (2 r)} sup_{\begin{matrix} g \in L^{p^{'}} (R^{n}) \otimes {boldB}^{*} \\ {∥ g ∥}_{L^{p^{'}} (R^{n}, B^{*})} \leq 1 \end{matrix}} (||, \int_{0}^{\infty}, \int_{R^{n}}, ⟨ {scriptG}_{H, B^{*}}^{r} (g) (x, t), {scriptG}_{H, B}^{r} (F) (x, t) ⟩) \end{matrix}

…”

Section: Proof Of Theorem For the Hermite Operatormentioning

confidence: 99%

UMD Banach spaces and square functions associated with heat semigroups for Schrödinger, Hermite and Laguerre operators

Castro

Fariña

Rodríguez-Mesa

2015

Mathematische Nachrichten

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In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schrödinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L p -boundedness properties for the square functions to our Banach valued setting by using γ -radonifying operators. We also prove that these L p -boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.

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The Banach Space-valued BMO, Carleson’s Condition, and Paraproducts

Cited by 16 publications

References 19 publications

Square functions and spectral multipliers for Bessel operators in UMD spaces

Square functions and spectral multipliers for Bessel operators in UMD spaces

Multidimensional Tauberian theorems for vector-valued distributions

UMD Banach spaces and square functions associated with heat semigroups for Schrödinger, Hermite and Laguerre operators

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