Multiparameter extensions of the Christ-Kiselev maximal theorem: strong variational bounds

For a linear operator $T$ bounded from $L^p(Y)$ to $L^q(X)$, the Christ-Kiselev theorem gives $L^p \to L^q$ bounds for the maximal function $T^{*}$ associated to filtrations on $Y$. This result has been extended by establishing bounds for the maximal function associated to a product of filtrations, also known as the multiparameter extension of the Christ-Kiselev theorem. In this note, we strengthen the multiparameter theorem by proving the $r$-variational bounds for the multiparameter trunctations when $r>p$. Furthermore, we replace $T$ by a multilinear operator to obtain a strong variational, multilinear, multiparameter extension of the Christ-Kiselev theorem.