LORENTZ SPACES OF VECTOR-VALUED MEASURES

Abstract

Given a non-atomic, finite and complete measure space $(\Omega,\Sigma,\mu)$ and a Banach space $X$, the modulus of continuity for a vector measure $F$ is defined as the function $\omega_F(t) = \sup_{\mu(E)\le t} |F|(E)$ and the space ${V}^{p,q}(X)$ of vector measures such that $t^{-1/p^\prime}\wf (t)\in L^q((0,\mu(\Omega)],dt/t)$ is introduced. It is shown that $V^{p,q}(X)$ contains isometrically $\lpq(X)$ and that $L^{p,q}(X)=V^{p,q}(X)$ if and only if $X$ has the Radon–Nikodym property. It is also proved that ${V}^{p,q}(X)$ coincides with the space of cone absolutely summing operators from $L^{p^\prime,q^\prime}$ into $X$ and the duality ${V}^{p,q}(X^*)= (L^{p^\prime,q^\prime}(X))^*$ where $1/p+1/p^\prime= 1/q+1/q^\prime=1$. Finally, $V^{p,q}(X)$ is identified with the interpolation space obtained by the real method $(V^{1}(X)$, $V^{\infty}(X))_{1/p',q}$. Spaces where the variation of $F$ is replaced by the semivariation are also considered.