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In this paper, we obtain remainder term representation formulae for thehigher-order Steklov inequality for vector fields which imply short and directproofs of the sharp (classical) Steklov inequalities. The obtained resultsdirectly imply sharp Steklov type inequalities for some vector fieldssatisfying H\"ormander's condition, for example. We also give representationformulae for the $L^{2^{m}}$-Friedrichs inequalities for vector fields.

Representation formulae for the higher-order Steklov and $L^{2^{m}}$-Friedrichs inequalities