Ultraproducts of real interpolation spaces between L^p-spaces

Abstract.

Let ${\left\{ {{\left( {L^{{p_{0} }} {\left( {\Omega _{d} ,\mu _{d} } \right)},L^{{p_{1} }} {\left( {\Omega _{d} ,\mu _{d} } \right)}} \right)},d \in \mathfrak{D}} \right\}},1 \leqslant p_{0} < p_{1} < \infty$, be a family of compatible couples of L^p-spaces. We show that, given a countably incomplete ultrafilter ${\user1{\mathcal{U}}}$ in $\mathfrak{D}$, the ultraproduct ${\left( {{\left( {L^{{p_{0} }} {\left( {\Omega _{d} ,\mu _{d} } \right)},L^{{p_{1} }} {\left( {\Omega _{d} ,\mu _{d} } \right)}} \right)}_{{\theta ,q}} } \right)}_{{\user1{\mathcal{U}}}} ,0 < \theta < 1,{\kern 1pt} {\kern 1pt} 1 \leqslant q < \infty$of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type ${\left( {L^{{p_{0} }} {\left( {\Omega _{1} ,\upsilon _{1} } \right)},L^{{p_{1} }} {\left( {\Omega _{1} ,\upsilon _{1} } \right)}} \right)}_{{\theta ,q}}$, an intermediate Köthe space between ${\ell }^{{p_{0} }} {\left( {\Omega _{2} ,\upsilon _{2} } \right)}$ and ${\ell }^{{p_{1} }} {\left( {\Omega _{2} ,\upsilon _{2} } \right)},{\left( {\Omega _{2} \upsilon _{2} } \right)}$ being a purely atomic measure space, and a Köthe function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅.