Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces

We will prove that for 1 < p < ∞ and 0 < λ < n, the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mγc equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. When p = 1 and 0 < λ < n, it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mγc equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. Moreover, the same results are true for the truncated uncentered Hardy-Littlewood maximal operator. Our work extends the previous results of Lebesgue spaces to Morrey spaces.