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Abstract

Let $1 \lt p \lt \infty $ be a fixed exponent. Suppose that $(f_n)_{n\geq 0}$, $(g_n)_{n\geq 0}$ are real-valued martingales satisfying $f_0\equiv x$, $g_0\equiv y$, $\|(f_n)_{n\geq 0}\|_p=F$ and $g_n-g_{n-1}=v_n(f_n-f_{n-1})$, $n=1,2,\ldots ,$ for some predictable sequence $(v_n)_{n\geq 1}$ taking values in $[-1,1]$. The purpose of this paper is to determine the optimal (i.e. the smallest) constant $B=B_{1,p}(x,y,F)$ for which $ \|(g_n)_{n\geq 0}\|_1\leq B_{1,p}(x,y,F)$.