In previous work we have defined and studied holomorphic functions in tubeswhich generalize the HardyHρ(TB)spaces; the baseBof the tubeTBis a proper open subset ofIf B=C, an open convex cone inand 1 < ρ ⩽ 2, we have proved that our new functions have distributional boundary values in the strong topology ofpas. In this paper we continue to study these new functions and extend the existence of boundary values to the cases 2 < ρ < ∞ We prove that our functions obtain boundary values in the strong topology ofPas, whereCis a polygonal cone or a regular cone, for the cases 2 < ρ < ∞.