Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space ${\mathsf{h}}_p$ for all $0<p<2$ by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of ${\mathsf{h}}_p$ for all $0<p<1$, and provide a constructive proof of the atomic decomposition for $p=1$ which resolves a main problem on the subject left open for the last twelve years. We also study ${(p,\infty )}_c$-atoms, and show that every ${(p,2)}_c$-atom can be decomposed into a sum of ${(p,\infty )}_c$-atoms; consequently, for every $0<p\le 1$, the ${(p,q)}_c$-atoms lead to the same atomic space for all $2\le q\le \infty$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space ${\mathsf{h}}_p$ ($0<p<1$) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.     

Note on rainbow cycles in edge-colored graphs

Let G be a graph of order n with an edge-coloring c, and let ${\delta }^c(G)$ denote the minimum color-degree of G. A subgraph F of G is called rainbow if all edges of F have pairwise distinct colors. There have been a lot of results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if ${\delta }^c(G)>\frac{2n?1}{3}$, then every vertex of G is contained in a rainbow triangle; (ii) if ${\delta }^c(G)>\frac{2n?1}{3}$ and $n\ge 13$, then every vertex of G is contained in a rainbow $C_4$; (iii) if G is complete, $n\ge 7k?17$ and ${\delta }^c(G)>\frac{n?1}{2}+k$, then G contains a rainbow cycle of length at least k, where $k\ge 5$.