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We study strongly harmonic functions in Carnot-Carath\'eodory groups definedvia the mean value property with respect to the Lebesgue measure. For suchfunctions we show their Sobolev regularity and smoothness. Moreover, we provethat strongly harmonic functions satisfy the sub-Laplace equation for theappropriate gauge norm and that the inclusion is sharp. We observe thatspherical harmonic polynomials in $\mathbb{H}_1$ are both strongly harmonic andsatisfy the sub-Laplace equation. Our presentation is illustrated by examples.

Mean value property and harmonicity on Carnot-Carath\'eodory groups