On the Optimality of Compactness Results for Interpolation Methods Associated to Polygons

Abstract

Compactness results of Cobos and Peetre [3] guarantee that the interpolated operator is compact assuming that all but two restrictions of the operator (located in adjacent vertices) are compact. Comparing these results with others in the literature, one might expect that the same conclusion holds under weaker compactness conditions. However, the Cobos-Peetre results are best possible as we show by means of counterexamples.