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We study the boundary regularity in the Dirichlet problem of the differential operators formula math. Our main result is: if γ > -1/2 is neither an integer nor a half-integer not less than n/2 - 1, one cannot expect global smooth solutions of Δ γ u = 0; if u ∈ C∞(B n ) satisfies Δ γ u = 0, then u must be either a polynomial of degree at most 2γ + 2 - n or a polyharmonic function of degree γ + 1.

Boundary regularity in the Dirichlet problem for the invariant Laplacians $\Delta _\gamma $ on the unit real ball