and then apply these inequalities to nonlin-ear parabolic problems (see, for instance, [47, 46, 84, 19, 20, 21, 69, 94]). ... 11, 92, 52, 47, 46, 84, 19, 20, 21, 69, 94] andmany more), but also allows us to establish Lq-Lr-regularity estimates, 1 q,r ,