VECTOR-VALUED FUNCTIONS INTEGRABLE WITH RESPECT TO BILINEAR MAPS

O. Blasco, J. M. Calabuig
2008 Taiwanese journal of mathematics  
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,
more » ... f )(w) = R Ω B(k(w, w ), f (w ))dµ (w ) from L p (Y ) into L p (Z), extending the results known in the operatorvalued case, corresponding to B : L(X, Y ) × X → Y given by B(T, x) = T x.
doi:10.11650/twjm/1500405186 fatcat:wg44cqihnfcunplz35qgz6x2ue