Given three Sobolev-Slobodeckii spaces with interpolating
parameters, we study the embedding of one of them in the sum of the
other two. We focus on the pathological case when the embedding does not
hold. In this case, we indicate the construction of counterexamples. We
also show the connection of our results with the Gagliardo-Nirenberg
non-inequalities and the so called "sum-intersection property".

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