For R  d   or T  d  , a strong converse inequality of type A (in the terminology of Ditzian and Ivanov (J. Anal. Math. 61 (1993) 61)) is obtained for the high order averages on balls and the  K -functionals generated by the high order Laplacian, which answers a problem raised by Ditzian and Runovskii (J. Approx. Theory 97 (1999) 113).
['Feng Dai', 'Kunyang Wang']
A note on the equivalences between the averages and the K-functionals related to the Laplacian