On Burkholder function for orthogonal martingales and zeros of Legendre polynomials
On Burkholder function for orthogonal martingales and zeros of Legendre polynomials
Burkholder obtained a sharp estimate of ${\bf E}|W|^p$ via ${\bf E}|Z|^p$, for martingales $W$ differentially subordinated to martingales $Z$. His result is that ${\bf E}|W|^p\le (p^*-1)^p{\bf E}|Z|^p$, where $p^* =\max \big(p, {p\over p-1}\big)$. What happens if the martingales have an extra property of being orthogonal martingales? This property is an …