Variants of the Maurey–Rosenthal Theorem for Quasi Köthe Function Spaces

Variants of the Maurey–Rosenthal Theorem for Quasi Köthe Function Spaces The Maurey–Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some Lp(ν)(0<p<r<∞) which satisfies a vector-valued norm inequality $$\left\| {\left( {\sum {\left| {Tx_k } \right|} ^r } \right)^{1/r} } \right\|_{L_p } \leqslant \left( {\sum {\left\| {x_k } \right\|_E^r } } \right)^{1/r} {\text{ for all }}x_1 , \ldots ,x_n \in E,$$ even allows a weighted norm inequality: there is a function 0≤w∈L 0(ν) such that $$\left( {\int {\frac{{\left| {Tx} \right|^r }}{w}dv} } \right)^{1/r} \leqslant \left\| x \right\|_E {\text{ for all }}x \in E.$$ Continuing the work of Garcia-Cuerva and Rubio de Francia we give several scalar and vector-valued variants of this fundamental result within the framework of quasi Köthe function spaces X(ν) over measure spaces. They are all special cases of our main result (Theorem 2) which extends the Maurey–Rosenthal cycle of ideas to the case of homogeneous operators between vector spaces being homogeneously representable in quasi Köthe function spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Variants of the Maurey–Rosenthal Theorem for Quasi Köthe Function Spaces

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2001 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1011466509838
Publisher site
See Article on Publisher Site

Abstract

The Maurey–Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some Lp(ν)(0<p<r<∞) which satisfies a vector-valued norm inequality $$\left\| {\left( {\sum {\left| {Tx_k } \right|} ^r } \right)^{1/r} } \right\|_{L_p } \leqslant \left( {\sum {\left\| {x_k } \right\|_E^r } } \right)^{1/r} {\text{ for all }}x_1 , \ldots ,x_n \in E,$$ even allows a weighted norm inequality: there is a function 0≤w∈L 0(ν) such that $$\left( {\int {\frac{{\left| {Tx} \right|^r }}{w}dv} } \right)^{1/r} \leqslant \left\| x \right\|_E {\text{ for all }}x \in E.$$ Continuing the work of Garcia-Cuerva and Rubio de Francia we give several scalar and vector-valued variants of this fundamental result within the framework of quasi Köthe function spaces X(ν) over measure spaces. They are all special cases of our main result (Theorem 2) which extends the Maurey–Rosenthal cycle of ideas to the case of homogeneous operators between vector spaces being homogeneously representable in quasi Köthe function spaces.

Journal

PositivitySpringer Journals

Published: Oct 3, 2004

References

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