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Ultraproducts of real interpolation spaces between L p -spaces

Authors:	J. A. López Molina M. E. Puerta M. J. Rivera

Affiliation:	(1) E.T.S. Ingenieros Agrónomos, Universidad Politécnica de Valencia, Camino de Vera, 46072 Valencia, SPAIN;(2) Universidad Eafit, Carrera 49 7Sur-50, 3300 Medellín, COLOMBIA

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 in , 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 K?the 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 K?the function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.

Keywords:	ultraproducts interpolation spaces
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