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VECTOR-VALUED FUNCTIONS INTEGRABLE WITH RESPECT TO BILINEAR MAPS

2008
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Taiwanese journal of mathematics
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Let (Ω, Σ, µ) be a σ−finite measure space, 1 ≤ p < ∞, X be a Banach space X and B : X × Y → Z be a bounded bilinear map. We say that an X-valued function f is p−integrable with respect to B whenever sup{ R Ω B(f (w), y) p dµ : y = 1} is finite. We identify the spaces of functions integrable with respect to the bilinear maps arising from Hölder's and Young's inequalities. We apply the theory to give conditions on X-valued kernels for the boundedness of integral operators T B (f )(w) = R Ω B(k(w,

doi:10.11650/twjm/1500405186
fatcat:wg44cqihnfcunplz35qgz6x2ue