We extend recent results by Pisier on$K$-subcouples, i.e. subcouples of an interpolation couple that preserve the$K$-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinterpretation of the Adamyan-Arov-Krein theorem for Hankel operators.

Let σ be the geodesic inversion on a Heisenberg type group$N$with homogeneous dimension$Q$, and denote by$S$the jacobian of σ. We prove that, for $$ - \frac{1}{2}Q< \alpha< \frac{1}{2}Q$$ , the operators $$T_\alpha :f \mapsto S^{1/2 - \alpha /Q} (f \circ \sigma )$$ are bounded on certain homogeneous Sobolev spaces $$\mathcal{H}^\alpha (N)$$ if and only if$N$is an Iwasawa$N$-group.

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