"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits

We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in ${\displaystyle {ℝ}^N}$ which is dense in the space of all harmonic functions in ${\displaystyle {ℝ}^N}$ and lim_{{‖x‖→∞} {x ∈ S}} ‖x‖^{μ}D^{α}v(x) = 0 for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if μ ≥ N+1, then every function v ∈ M satisfies ∫_H vdλ =0 for every (N-1)-dimensional hyperplane H, where λ denotes the (N-1)-dimensional Lebesgue measure. On the other hand, we prove that there exists a linear manifold M of harmonic functions in the unit ball of ${\displaystyle {ℝ}^N}$, which is dense in the space of all harmonic functions and each function in M has zero nontangential limit at every point of the boundary of .