On homogeneous polynomials on a complex ball

Abstract

We prove that there exist n n -homogeneous polynomials p n {p_n} on a complex d d -dimensional ball such that ‖ p n ‖ ∞ = 1 {\left \| {{p_n}} \right \|_\infty } = 1 and ‖ p n ‖ 2 ⩾ π 2 − d {\left \| {{p_n}} \right \|_2} \geqslant \sqrt \pi {2^{- d}} . This enables us to answer some questions about H p {H_p} and Bloch spaces on a complex ball. We also investigate interpolation by n n -homogeneous polynomials on a 2 2 -dimensional complex ball.