Remarks on the Semivariation of Vector Measures with Respect to Banach Spaces
نویسنده
چکیده
Let L(ν)⊗̂γqY = Lq(ν, Y ) and X⊗̂∆pL(μ) = Lp(μ,X). It is shown that any Lp(μ)-valued measure has finite L2(ν)-semivariation with respect to the tensor norm L(ν)⊗̂∆pL(μ) for 1 ≤ p < ∞ and finite Lq(ν)semivariation with respect to the tensor norm L(ν)⊗̂γqL(μ) 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 L(ν)⊗̂γqL(μ) 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 L(ν)⊗̂γqL(μ) if q < p.
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