Fubini theorems for Orlicz spaces of Lebesgue-Bochner measurable functions

Abstract:

Let $(X,V,v)$ be the volume space formed as the product of the volume spaces $({X_i},{V_i},{v_i})\;(i = 1,2)$. Let $(p,q)$ be a pair of complementary (continuous) Young’s functions, let $Y,Z,{Z_1},{Z_2}$, be Banach spaces, let w be a multilinear continuous operator on $Y \times {Z_1} \times {Z_2} \to W$. Let ${L_p}(v,Y)$ be the Orlicz space of Lebesgue-Bochner measurable functions generated by p, and let ${K_q}(v,Z)$ be the associated space of finitely additive Z-valued set functions. The principal result of this paper is as follows: Let $f \in {L_p}(v,Y), {\mu _2} \in {K_q}({v_2},{Z_2})$. Then (a) the function $f({x_1}, \cdot )$ is ${v_2}$-Orlicz summable ${v_1}$-a.e.; (b) the operator $r(f,{\mu _2})$ defined by the expression $r(f,{\mu _2})({x_1}) = \smallint {w_1}(f({x_1},{x_2}),{\mu _2}(d{x_2}))\;{v_1}$-a.e. is bilinear and continuous from ${L_p}(v,Y) \times {K_q}({v_2},{Z_2})$ into ${L_p}({v_1},{Y_1})/N$, where ${w_1}(y,{z_2}) = w(y,{z_2})$, where ${Y_1}$ is the Banach space of bounded linear operators from ${Z_1}$ into W, and where N is the set of ${Y_1}$-valued ${v_1}$-measurable functions of zero seminorm; (c) the equality $\smallint w(f,d{\mu _1},d{\mu _2}) = \smallint {w_0}(r(f,{\mu _2}),d{\mu _1})$ holds for all $f \in {L_p}(v,Y),{u_i} \in {K_q}({v_i},{Z_i})\;(i = 1,2)$, where ${w_0}({y_1},{z_1}) = {y_1}({z_1})$ for all ${y_1} \in {Y_1},{z_1} \in {Z_1}$.