Spaces of integrable functions with respect to a vector measure and factorizations through <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:math> and Hilbert spaces

Fernández, A.; Mayoral, Fernando; Naranjo, Francisco; Sáez, C.; Pérez, Enrique Alfonso Sánchez

doi:10.1016/j.jmaa.2006.07.107

Cited by 9 publications

(2 citation statements)

References 8 publications

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“…. , n}, h ∈ L 0 ( µ j ), 1 ≤ r ≤ ∞ and [8] define the measure η := Kd µ. We define the Lebesgue space related to K, which we denote L r K ( µ j ), as the space of µ j -measurable functions such that the norm…”

Section: Factorization For Multilinear Kernel Operatorsmentioning

confidence: 99%

“…The L p -spaces of vector measures have two main properties. First, they represent (order isometrically) any p-convex order continuous Banach function space defined over a finite measure (see [8,Proposition 2.4]). Second, they represent the maximal space of extension of a µ-determined operator between a Banach function space over µ and a Banach space, among all p-convex order continuous Banach function spaces of finite measures (see, for instance, [19,Theorem 5.7]).…”

Section: Introductionmentioning

confidence: 99%

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Factorizing Multilinear Kernel Operators Through Spaces of Vector Measures

Galdames-Bravo¹

2020

J. Aust. Math. Soc.

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We consider a multilinear kernel operator between Banach function spaces over finite measures and suitable order continuity properties, namely $T:X_{1}(\,\unicode[STIX]{x1D707}_{1})\times \cdots \times X_{n}(\,\unicode[STIX]{x1D707}_{n})\rightarrow Y(\,\unicode[STIX]{x1D707}_{0})$ . Then we define, via duality, a class of linear operators associated to the $j$ -transpose operators. We show that, under certain conditions of $p$ th power factorability of such operators, there exist vector measures $m_{j}$ for $j=0,1,\ldots ,n$ so that $T$ factors through a multilinear operator $\widetilde{T}:L^{p_{1}}(m_{1})\times \cdots \times L^{p_{n}}(m_{n})\rightarrow L^{p_{0}^{\prime }}(m_{0})^{\ast }$ , provided that $1/p_{0}=1/p_{1}+\cdots +1/p_{n}$ . We apply this scheme to the study of the class of multilinear Calderón–Zygmund operators and provide some concrete examples for the homogeneous polynomial and multilinear Volterra and Laplace operators.

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Section: Factorization For Multilinear Kernel Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Factorizing Multilinear Kernel Operators Through Spaces of Vector Measures

Galdames-Bravo¹

2020

J. Aust. Math. Soc.

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On the Structure of L1 of a Vector Measure via its Integration Operator

Calabuig

Rodríguez

Pérez

2009

Integr. equ. oper. theory

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Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L 1 (m). In this paper we study the cases of the integration operator being: (i) p-concave on L p (m), or (ii) positive p-summing on L 1 (m) (where 1 ≤ p < ∞). We prove that (i) is equivalent to saying that L 1 (m) contains continuously the L p space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L 1 (m) is order isomorphic to the L 1 space of a non-negative scalar measure. (2000). 46E30, 46G10. Mathematics Subject Classification

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Factorizing Kernel Operators

Galdames-Bravo

Pérez

2012

Integr. Equ. Oper. Theory

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Abstract. Consider an operator T : X(µ) → Y (µ) between Banach function spaces having adequate order continuity and Fatou properties. Assume that T can be factorized through a Banach space as T = R • S, where R and the adjoint of S are p-th power and q-th power factorable, respectively. Then a canonical factorization scheme can be given for T . We show that it provides a tool for analyzing T that becomes specially useful for the case of kernel operators. In particular, we show that this square factorization scheme for T is equivalent to some inequalities for the bilinear form defined by T . Kernel operators are studied from this point of view.Primary 46E30, Secondary 47B38, 46B42, 46B28 Banach function spaces, Köthe duality, p-th power factorable operators, factorization, kernel operators.

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Spaces of integrable functions with respect to a vector measure and factorizations through Lp and Hilbert spaces

Cited by 9 publications

References 8 publications

Factorizing Multilinear Kernel Operators Through Spaces of Vector Measures

Factorizing Multilinear Kernel Operators Through Spaces of Vector Measures

On the Structure of L1 of a Vector Measure via its Integration Operator

Factorizing Kernel Operators

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