The Mazur product map on Hardy-type sequence spaces

We study certain Hardy-type sequence spaces H p and H 0 p , 1 ⩽ p ⩽ ∞ , which are analogues of ℓ ∞ and c 0 , respectively. We show that the Mazur product ⊠ : H 0 p × H q → c 0 is not onto for every p ∈ ( 1 , ∞ ) with q = p ( p − 1 ) −1 . We present corollaries for spaces defined via weighted ℓ p seminorms and for c 0 . The latter corollary provides a new solution of Mazurʹs Problem 8 in the Scottish Book.