Asymptotic estimates for double-coverings

A collection of finite sets $\{A_1, A_2,\ldots, A_{p}\}$ is said to be a double-covering if each $a\in \cup_{k=1}^{p}A_k$ is included in exactly two sets of the collection. For fixed integers $l$ and $p$, let $\mu_{l,p}$ be the number of equivalency classes of double-coverings with $\#(A_k)=l$, $k=1,2,\ldots,p$. We characterize the asymptotic behavior of the quantity $\mu_{l,p}$ as $p\to \infty$. The results are applied to give an alternative approach to the Bonami-Kiener hypercontraction inequality.Comment: 19 page

On generalizations of Fatou's theorem for the integrals with general kernels

We define $\lambda(r)$-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson-Stiltjes integrals for general kernels $\{\varphi_r\}$, forming an approximation of identity. We prove that the bound \md0 \limsup_{r\to 1}\lambda(r) \|\varphi_r\|_\infty<\infty \emd is necessary and sufficient for almost everywhere $\lambda(r)$-convergence of the integrals \md0 \int_\ZT \varphi_r(t-x)d\mu(t). \emdComment: 14 page