arXiv subjects
Publications and source records attributed to Blanca Radillo-Murguia.
We provide a condition on a set of directions $Ω\subset \mathbb{S}^1$ ensuring that the associated directional maximal operator $M_Ω$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. The techniques of proof extend ideas of Bateman and Katz involving probabilistic construction of Kakeya-type sets involving sticky maps and Bernoulli percolation.
$K_σ$ sets involving sticky maps $σ$ have been used in the theory of differentiation of integrals to probabilistically construct Kakeya-type sets that imply certain types of directional maximal operators are unbounded on $L^p(\mathbb{R}^2)$ for all $1 \leq p < \infty$. We indicate limits to this approach by showing that, given $ε> 0$ and a natural number $N$, there exists a tree $\mathcal{T}_{N, ε}$ of finite height that is lacunary of order $N$ but such that, for \emph{every} sticky map $σ: \mathcal{B}^{h(\mathcal{T}_{N, ε})} \rightarrow \mathcal{T}_{N, ε}$, one has $|K_σ \cap ((1,2) \times \mathbb{R})| \geq 1 - ε$.
For a graph $G$ and an abelian group $A$, a labeling of the vertices of $G$ induces a labeling of the edges via the sum of adjacent vertex labels. Hovey introduced the notion of an $A$-cordial vertex labeling when both the vertex and edge labels are as evenly distributed as possible. Much work has since been done with trees, hypertrees, paths, cycles, ladders, prisms, hypercubes, and bipartite graphs. In this paper we show that all trees are $\mathbb{Z}_2^2$-cordial except for $P_4$ and $P_5$. In addition, we give numerous results relating to $\mathbb{Z}_m$-cordiality of the friendship graph $F_n$. The most general result shows that when $m$ is an odd multiple of $3$, then $F_n$ is $\mathbb{Z}_m$-cordial for all $n$. We also give a general conjecture to determine when $F_n$ is $\mathbb{Z}_m$-cordial.