A note on weighted bounds for rough singular integrals

We show that the $L^2(w)$ operator norm of the composition $M\!\circ T_{\Omega}$, where $M$ is the maximal operator and $T_{\Omega}$ is a rough homogeneous singular integral with angular part $\Omega\in L^{\infty}(S^{n-1})$, depends quadratically on $[w]_{A_2}$, and this dependence is sharp

Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy

We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. It is written down discrete equations which naturally generalize the first discrete Painlev\'e equation $\mathrm{dP}_{\rm I}$ in a sense that autonomous version of these equations admit the limit to the first Painlev\'e equation. It is shown that each of these equations describes B\"acklund transformations of Veselov-Shabat periodic dressing lattices with odd period known also as Noumi-Yamada systems of type $A_{2(n-1)}^{(1)}$.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA