Establish an $L_\infty$ estimate for the off-diagonal good component

Establish an $L_\infty(\mathcal N)$ estimate for the off-diagonal good component $g_{\mathrm{off}}$ arising in the modified noncommutative Calderón–Zygmund decomposition for operator-valued functions on arbitrary, including unbounded, Vilenkin groups, complementing the available $L_2(\mathcal N)$ estimate.

Background

To prove the weak type $(1,1)$ estimate for Vilenkin partial-sum operators on arbitrary Vilenkin groups, the paper introduces a modified noncommutative Calderón–Zygmund decomposition adapted to a finer regular filtration. This decomposition writes the function as the sum of a diagonal good part $g_{\mathrm d}$, an off-diagonal good part $g_{\mathrm{off}}$, and diagonal and off-diagonal bad parts.

The diagonal good part satisfies both $L_1(\mathcal N)$ and $L_\infty(\mathcal N)$ bounds, while the off-diagonal good part is controlled only in $L_2(\mathcal N)$ by an estimate of the form $$\|g_{\mathrm{off}}\|_{L_2(\mathcal N)}^2\leq c\lambda\|f\|_{L_1(\mathcal N)}$$. The authors explicitly state that they are unable to establish the corresponding $L_\infty$ estimate; the unresolved task is therefore to obtain such a bound, although the existing $L_2$ estimate suffices for the results proved in the paper.