Remarks on Generalized Hardy Algebras
Year: 2018
Type: preprint
Abstract: For a measure space $(Ω, Σ, μ)$ with a positive finite measure $μ$, and a positive real number $p$, we define the space $L_p^{+}(μ)=L_p^{+}$ of all (equivalence classes of) $Σ$-measurable complex functions $f$ defined on $Ω$ such that the function $\left(\log^+|f|\right)^p$ is integrable with respect to $μ$.We define the metric $d_p$ on $L^{+}_p$ which generalizes the metric introduced by Gamelin and Lumer in [G] for the case $p=1$. It is shown that the space $L^{+}_p$ is a topological algebra. On the other hand, one can define on the space $L_p^{+}$ an equivalent $F$-norm $| \cdot|_p$ that makes $L_p^{+}$ into an Orlicz space. For the case of the normalized Lebesgue's measure $dt/2π$ on $[0,2π)$, it follows that the class $N^p(1
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2796398402