Ultraproducts of real interpolation spaces between L p -spaces
Fecha
2006-06-01
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SPRINGER
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Let {(L p0 (Od, µd), L p1 (Od, µd)) , d e D}, 1 = p0 < p1 < 8, be a family of compatible couples of L p -spaces. We show that, given a countably incomplete ultrafilter U in D, the ultraproduct (L p0 (Od, µd), L p1 (Od, µd))? q)U , 0< ? < 1,1 = q < 8 of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type (L p0 (O1, ? 1 L p1 (O1, ? 1)) ?, q , an intermediate Köthe space between l p0 (O2, ? 2) and l p1 (O2, ? 2), (O2, ? 2)being a purely atomic measure space, and a Köthe function space K(O3) defined on some purely non atomic measure space (O3, ? 3) in such a way that O2 O3 ? ?. © Springer-Verlag Berlin Heidelberg 2006.
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Interpolation spaces, Ultraproducts