On Homogeneous Polynomials on a Complex Ball

Abstract

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