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We obtain harmonic extensions to the upper half-space of distributions in the weighted spaces w n+1 D L 1 , which according to [1] are the optimal spaces of tempered distributions S -convolvable with the classical euclidean version of the Poisson kernel . We also characterize the class of harmonic functions in the upper half-space with boundary * Partially supported by PAPIIT-IN105801. values in w n+1 D L 1 , generalizing in this way a classical result in the theory of Hardy spaces. Some facts concerning harmonic extensions of distributions in D L p , 1 < p ≤ ∞, are also approached in this paper, as well as natural relations among these spaces and the weighted spaces w n+1 D L 1 . We can also obtain n-harmonic extensions of appropriate weighted integrable distributions associated to a natural product domain version of the Poisson kernel.

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