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Remarks on the semivariation of vector measures with respect to Banach spaces.

  • Oscar Blasco (a1)

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Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.

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References

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[1] 1Arregui, J.L. and Blasco, O., ‘(p, q)-summing sequences’, J. Math. Anal. Appl. 247 (2002), 812827.
[2]Bartle, R., ‘A general bilinear vector integral’, Studia Math. 15 (1956), 337351.
[3]Blasco, O. and Gregori, P., ‘Lorentz spaces of vector-valued measures’, J. London Math. Soc. 67 (2003), 739751.
[4]Dinculeanu, N., Vector measures, International Series of Monographs in Pure and Applied Mathematics 95 (Pergamon Press, Oxford, New York, Toronto, 1967).
[5]Diestel, J., Jarchow, H. and Tonge, A., Absolutely summing operators (Cambridge Univ. Press, Cambridge, 1995).
[6]Diestel, J. and Uhl, J.J., Vector measures, Math. Surveys 15 (Amer. Math. Soc., Providence, R.I., 1997).
[7]Jefferies, B. and Okada, S., ‘Bilinear integration in tensor products’, Rocky Mountain J. Math. 28 (1998), 517545.
[8]Jefferies, B. and Okada, S., ‘Semivariation in L p-spaces’, Comment. Math. Univ. Carolin. 44 (2005), 425436.
[9]Jefferies, B., Okada, S. and Rodrigues-Piazza, L., ‘L p-valued measures without finite X-semivariation for 2 < p < ∞ ’, (preprint).
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  • Oscar Blasco (a1)

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