Some new results related to Lorentz \(G\Gamma\)-spaces and interpolation. (English) Zbl 1442.46014
Interpolation questions are studied for the Lebesgue spaces,
grand and small Lebesgue spaces and their generalizations as
well as the Lorentz \(G\Gamma\)-spaces. The real interpolation
method is involved and the corresponding \(K\)-functional is
constructed. In many cases the question of interpolation is
completely solved and the corresponding interpolation spaces
are described. In some cases interpolation inequalities are
derived and some properties of interpolation spaces are
exposed. The result of interpolation is always a
\(G\Gamma\)-space. The duality theorem for the interpolation
spaces is derived as well. The results are applicable to the
study of regularity of the so-called very weak solutions of a linear differential equation in a domain. Examples of
such applications for solutions to the Poisson equation are
displayed.
Reviewer: Sergey G. Pyatkov (Khanty-Mansiysk)
MSC:
| 46B70 | Interpolation between normed linear spaces |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46N20 | Applications of functional analysis to differential and integral equations |
Keywords:
grand and small Lebesgue spaces; classical Lorentz spaces; interpolation; very weak solutionReferences:
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