9 problems

Let $d\geq2$, and let $\mathcal{E}$ be a $\frac12$-sparse antichain of $d$-dimensional dyadic rectangles. Let $\operatorname{sh}(\mathcal{E})$ be the shadow of $\mathcal{E}$. Antic…

Let $d\geq2$, and let $\mathcal{E}$ be a $\frac12$-sparse antichain of dyadic $d$-dimensional rectangles. Let $h_{\mathcal{E}}$ denote the associated multiplicity function and…

Let $\Phi:\mathbb{Z}^{d-1}\to\mathbb{Z}$ be monotone increasing in each argument, and let $\mathcal{Z}_d$ be the collection of dyadic rectangles in $\mathbb{R}^d$ whose sidelengths…

Let $T^2$ be a finite bi-tree with boundary $\partial T^2$, let $f$ and $g$ be non-negative functions on $T^2$, and let $\mathbb{I}$ denote summation over all ancestors. Assume … a…

Let $Q\subset\mathbb{R}^d$ be a cube, let $\mathrm{BMO}_{\varepsilon}^{\mathrm{dyad}}(Q)$ denote the dyadic BMO class with parameter $\varepsilon$, and define … An extension…

Let $N\in\mathbb{N}$ with $N\geq 2$ and let $0\leq \ell\leq N-2$. For the dyadic collection ${\mathcal E}^N_{\ell,0}$ and the postorder rearrangement operator $\tau_N$, write…

Let $\mathcal{D}$ be the dyadic intervals on $\mathbb{R}$, let $\mathsf{P}_{b}^{(0,1)}$, $\mathsf{P}_{d}^{(0,1)}$, $\mathsf{P}_{d}^{(1,0)}$, and $\mathsf{P}_{b}^{(1,0)}$ denote the…

Let $M=\{m_j:j\geq1\}$ be a shift, and set $N_j(M)$ for the associated Semenov counting quantity. A sequence is called decomposable when it satisfies the hypothesis of Definition 1…

Let $\t$ be a rearrangement of the dyadic intervals, and write $\cF_1,\dots,\cF_N$ for subcollections of the dyadic intervals. Suppose there is a constant $C$ such that, for every…