Operator-valued Fourier Haar multipliers

Girardi, Maria

doi:10.1016/j.jmaa.2006.01.007

Cited by 8 publications

(2 citation statements)

References 14 publications

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“…We verify equation (17), exploiting again the compactness properties of 𝑇 red diag = (𝑅 𝐿 ) in tandem with the probabilistic estimates of Lemma 5.3 and Lemma 5.4.…”

Section: The Present Papermentioning

confidence: 99%

“…This recent formula of Semenov and Uksusov is a very elegant characterisation of boundedness on Haar multipliers on 𝐿 1 . In that spirit, Girardi studied related operators on 𝐿 𝑝 and 𝐿 𝑝 (𝑋) of a multiplier type [17]. Wark has since simplified the proof of Semenov-Uksusov [38] as well as extended the formula to the vector-valued case [39,40].…”

Section: Haar Multipliers On 𝐿mentioning

confidence: 99%

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The space is primary for 1 < p < ∞

Lechner,

Motakis,

Müller

et al. 2022

Forum of Mathematics, Sigma

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The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which $$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$ We show that $L_1(L_p) (1 < p < \infty )$ is primary, meaning that whenever $L_1(L_p) = E\oplus F$ , where E and F are closed subspaces of $L_1(L_p)$ , then either E or F is isomorphic to $L_1(L_p)$ . More generally, we show that $L_1(X)$ is primary for a large class of rearrangement-invariant Banach function spaces.

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“…We verify equation (17), exploiting again the compactness properties of 𝑇 red diag = (𝑅 𝐿 ) in tandem with the probabilistic estimates of Lemma 5.3 and Lemma 5.4.…”

Section: The Present Papermentioning

confidence: 99%

Section: Haar Multipliers On 𝐿mentioning

confidence: 99%

The space is primary for 1 < p < ∞

Lechner,

Motakis,

Müller

et al. 2022

Forum of Mathematics, Sigma

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On the Multipliers of Fourier Series in the Generalized Haar System

Tleukhanova,

Sarybekova,

Orynbek

2024

Trends in Mathematics

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Multipliers on bi-parameter Haar system Hardy spaces

Lechner,

Motakis,

Müller

et al. 2024

Math. Ann.

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Let $$(h_I)$$ ( h I ) denote the standard Haar system on [0, 1], indexed by $$I\in \mathcal {D}$$ I ∈ D , the set of dyadic intervals and $$h_I\otimes h_J$$ h I ⊗ h J denote the tensor product $$(s,t)\mapsto h_I(s) h_J(t)$$ ( s , t ) ↦ h I ( s ) h J ( t ) , $$I,J\in \mathcal {D}$$ I , J ∈ D . We consider a class of two-parameter function spaces which are completions of the linear span $$\mathcal {V}(\delta ^2)$$ V ( δ 2 ) of $$h_I\otimes h_J$$ h I ⊗ h J , $$I,J\in \mathcal {D}$$ I , J ∈ D . This class contains all the spaces of the form X(Y), where X and Y are either the Lebesgue spaces $$L^p[0,1]$$ L p [ 0 , 1 ] or the Hardy spaces $$H^p[0,1]$$ H p [ 0 , 1 ] , $$1\le p < \infty $$ 1 ≤ p < ∞ . We say that $$D:X(Y)\rightarrow X(Y)$$ D : X ( Y ) → X ( Y ) is a Haar multiplier if $$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ D ( h I ⊗ h J ) = d I , J h I ⊗ h J , where $$d_{I,J}\in \mathbb {R}$$ d I , J ∈ R , and ask which more elementary operators factor through D. A decisive role is played by the Capon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ C : V ( δ 2 ) → V ( δ 2 ) given by $$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ C h I ⊗ h J = h I ⊗ h J if $$|I|\le |J|$$ | I | ≤ | J | , and $$\mathcal {C} h_I\otimes h_J = 0$$ C h I ⊗ h J = 0 if $$|I| > |J|$$ | I | > | J | , as our main result highlights: Given any bounded Haar multiplier $$D:X(Y)\rightarrow X(Y)$$ D : X ( Y ) → X ( Y ) , there exist $$\lambda ,\mu \in \mathbb {R}$$ λ , μ ∈ R such that $$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$ λ C + μ ( Id - C ) approximately 1-projectionally factors through D , i.e., for all $$\eta > 0$$ η > 0 , there exist bounded operators A, B so that AB is the identity operator $${{\,\textrm{Id}\,}}$$ Id , $$\Vert A\Vert \cdot \Vert B\Vert = 1$$ ‖ A ‖ · ‖ B ‖ = 1 and $$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$ ‖ λ C + μ ( Id - C ) - A D B ‖ < η . Additionally, if $$\mathcal {C}$$ C is unbounded on X(Y), then $$\lambda = \mu $$ λ = μ and then $${{\,\textrm{Id}\,}}$$ Id either factors through D or $${{\,\textrm{Id}\,}}-D$$ Id - D .

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Operator-valued Fourier Haar multipliers

Abstract: Criteria are given to ensure the boundedness of Fourier Haar multiplier operators from L p ([0, 1]where the Fourier Haar multiplier sequences come not from R, as in the classical setting, but rather from the space of bounded linear operators from a Banach space X into a Banach space Y .

Cited by 8 publications

References 14 publications

The space is primary for 1 < p < ∞

The space is primary for 1 < p < ∞

On the Multipliers of Fourier Series in the Generalized Haar System

Multipliers on bi-parameter Haar system Hardy spaces

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