Martingale inequalities for spline sequences

Abstract

We show that D. L\'{e}pingle's $L_1(\ell_2)$-inequality \begin{equation*} \Big\| \big( \sum_n \mathbb E[f_n | \mathscr F_{n-1}]^2 \big)^{1/2}\Big\|_1 \leq 2\cdot \Big\| \big( \sum_n f_n^2 \big)^{1/2} \Big\|_1, \qquad f_n\in\mathscr F_n, \end{equation*} extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that $f_n$ is contained in a suitable spline space $\mathscr S(\mathscr F_n)$. This is done provided the filtration $(\mathscr F_n)$ satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in $\mathscr S(\mathscr F_n)$. As a by-product, we also obtain a spline version of $H_1$-$BMO$ duality under this assumption.