Complex interpolation of predual spaces of general local Morrey-type spaces. (English) Zbl 1412.46040

The author studies the interpolation properties of the predual spaces of generalized local Morrey spaces. The local or central Morrey space \(LM^\lambda_p({\mathbb R}^n)\) is the set of all functions \(f\in L_p^{loc}({\mathbb R}^n)\) such that \(\|f|LM^\lambda_p\| := \sup_{r>0} \|f\chi_{B(r)}\|_p < \infty\), where \(B(r)\) denotes the open ball centred at the origin of radius \(r > 0\). The condition that defines its generalized version \(LM_{p, q,w}({\mathbb R}^n)\) is given by \(\|f|LM_{p, q,w}\| := \|w(r) \|f\chi_{B(r)}\|\, \|_q < \infty\), where \(w\) is a non-negative measurable function on \(]0,1[\) and \(1< p, q\leq\infty\). The author describes the predual spaces to the generalized Morrey-type spaces \(LM_{p, q,w}({\mathbb R}^n)\). The spaces are the generalized version of so-called local block spaces. It is proved that the local block spaces behave well with respect to the complex Calderón interpolation method. Two interpolation formulas are proved with some technical assumption concerning the weight function \(w\) and the coefficients \(p\) and \(q\).


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B70 Interpolation between normed linear spaces
42B35 Function spaces arising in harmonic analysis
46M35 Abstract interpolation of topological vector spaces
Full Text: DOI Euclid


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