The matrix-weighted dyadic convex body maximal operator is not bounded

Abstract

The convex body maximal operator is a natural generalization of the Hardy–Littlewood maximal operator. In this paper we are considering its dyadic version in the presence of a matrix weight. To our surprise it turns out that this operator is not bounded. This is in a sharp contrast to a Doob's inequality in this context. At first, we show that the convex body Carleson Embedding Theorem with matrix weight fails. We then deduce the unboundedness of the matrix-weighted convex body maximal operator.