N. K. Agbeko | Searcharxiv

arXiv2006

Studies on concave Young-functions

We succeeded to isolate a special class of concave Young-functions enjoying the so-called \emph{density-level property}. In this class there is a proper subset whose members have each the so-called degree of contraction denoted by $c^{\ast}$, and map bijectively the interval $[ c^{\ast}, \infty) $ onto itself. We constructed the fixed point of each of these functions. Later we proved that every positive number $b$ is the fixed point of a concave Young-function having $b$ as degree of contraction. We showed that every concave Young-function is square integrable with respect to a specific Lebesgue measure. We also proved that the concave Young-functions possessing the density-level property constitute a dense set in the space of concave Young-functions with respect to the distance induced by the $L^{2}$-norm.

math.AP↗

arXiv2006

Bijections and metric spaces induced by some collective properties of concave Young-functions

For each ${\small b\in(0, \infty)}$ we intend to generate a decreasing sequence of subsets $(\mathcal{Y}_{b}^{(n)}) \subset Y_{\mathrm{conc}}$ depending on $b$ such that whenever $n\in\mathbb{N}$, then $\mathcal{A}\cap\mathcal{Y}_{b}^{(n)}% $ is dense in $\mathcal{Y}_{b}^{(n)}$ and the following four sets $\mathcal{Y}_{b}^{(n)}$, $\mathcal{Y}_{b}^{(n) }\backslash(\mathcal{A}\cap\mathcal{Y}_{b}^{(n)}) $, $\mathcal{A}\cap\mathcal{Y}_{b}^{(n)}$ and $\mathcal{Y}_{\mathrm{conc}}$ are pairwise equinumerous. Among others we also show that if $f$ is any measurable function on a measure space $(Ω,\mathcal{F},λ) $ and $p\in[ 1,\infty) $ is an arbitrary number then the quantities $\left\Vert f\right\Vert_{L^{p}}$ and $\sup_{Φ\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}}(Φ(1)) ^{-1}\left\Vert Φ\circ| f| \right\Vert_{L^{p}}$ are equivalent, in the sense that they are both either finite or infinite at the same time.

math.GM↗