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We study a characterization of slice Carleson measures and of Carlesonmeasures for the both the Hardy spaces $H^p(\mathbb B)$ and the Bergman spaces$\mathcal A^p(\mathbb B)$ of the quaternionic unit ball $\mathbb B$. In thecase of Bergman spaces, the characterization is done in terms of the axiallysymmetric completion of a pseudohyperbolic disc in a complex plane. We alsoshow that a characterization in terms of pseudohyperbolic balls is notpossible.

Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball