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The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, $T_b^m$, on Herz spaces $K{̇}_q^{\alpha ,p}\left(ℝⁿ\right)$ and $K_q^{\alpha ,p}\left(ℝⁿ\right)$, and on a new class of Herz-type Hardy spaces $HK{̇}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$ and $H{K}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$, where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

In this paper, we obtain some strong and weak type continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator on Hardy spaces, Hardy type spaces and weak Hardy type spaces.