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    Blasco, O. and Calabuig, J. M. 2009. Fourier analysis with respect to bilinear maps. Acta Mathematica Sinica, English Series, Vol. 25, Issue. 4, p. 519.


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  • Bulletin of the Australian Mathematical Society, Volume 78, Issue 3
  • December 2008, pp. 411-430

p-VARIATION OF VECTOR MEASURES WITH RESPECT TO BILINEAR MAPS

  • O. BLASCO (a1) and J. M. CALABUIG (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972708000798
  • Published online: 01 December 2008
Abstract
Abstract

We introduce the spaces Vp(X) (respectively 𝒱p(X)) of the vector measures ℱ:Σ→X of bounded (p,ℬ)-variation (respectively of bounded (p,ℬ)-semivariation) with respect to a bounded bilinear map ℬ:X×YZ and show that the spaces Lp(X) consisting of functions which are p-integrable with respect to ℬ, defined in by Blasco and Calabuig [‘Vector-valued functions integrable with respect to bilinear maps’, Taiwanese Math. J. to appear], are isometrically embedded in Vp(X). We characterize 𝒱p(X) in terms of bilinear maps from Lp′×Y into Z and Vp(X) as a subspace of operators from Lp′(Z*) into Y*. Also we define the notion of cone absolutely summing bilinear maps in order to describe the (p,ℬ)-variation of a measure in terms of the cone-absolutely summing norm of the corresponding bilinear map from Lp′×Y into Z.

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[8]J. Diestel and J. J. Uhl Jr , Vector Measures, Mathematical Surveys, 15 (American Mathematical Society, Providence, RI, 1977), xiii+322 pp.

[9]N. Dinculeanu , Vector Measures, International Series of Monographs in Pure and Applied Mathematics, 95 (Pergamon Press, Oxford/VEB Deutscher Verlag der Wissenschaften, Berlin, 1967).

[10]M. Girardi and L. Weis , ‘Integral operators with operator-valued kernels.’, J. Math. Anal. Appl. 290(1) (2004), 190212.

[11]B. Jefferies and S. Okada , ‘Bilinear integration in tensor products’, Rocky Mountain J. Math. 28(2) (1998), 517545.

[12]R. S. Phillips , ‘On linear transformations’, Trans. Amer. Math. Soc. 48(290) (1940), 516541.

[13]R. A. Ryan , Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics (Springer, London, 2002).

[14]H. H. Schaefer , Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, 215 (Springer, Heidelberg, 1974).

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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