Interpolation of Morrey Spaces on Metric Measure Spaces

Abstract

In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasimetric measure spaces, which generalizes some known results on ℝn.

Interpolation of Morrey-Campanato and related smoothness spaces

“…For proofs of (i) and (ii) of Theorem 2.5, we refer the reader to Lu et al [56]. Part (ii) of Theorem 2.5 is explicitly stated therein, and part (i) of Theorem 2.5 can be found in [56,Formula (2.3)]. Part (iii) of Theorem 2.5 follows from [46] (see Subsection 4.1 for more details).…”

“…where X 1−Θ 0 X Θ 1 denotes the Calderón product of X 0 and X 1 (see [56,111]). Concerning the real interpolation method with u and p as in (1.4), one knows that…”

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

“…The following interpolation results on classical Morrey spaces can be found in [24], Theorem 1.2 (see also [39]).…”

“…Lemarié-Rieusset in [23] further pointed out explicitly that Morrey spaces have no interpolation properties if the parameters λ of two Morrey spaces are different. Very recently, it was proved in [24] (see also [39]) that, via the ± interpolation method, the interpolation space of two Morrey spaces with the same λ on quasi-metric measure space is also a Morrey space. In view of this, it is natural to ask whether we can interpolate generalized grand Morrey spaces as well.…”

Homogeneous Campanato–Sobolev classes

“…[33,2,22,24] and the references therein for the relevant information) we have to perform a direct demonstration based on a fractional derivative analysis.…”