Families of Young functions and limits of Orlicz norms
Canadian mathematical bulletin, Tome 67 (2024) no. 1, pp. 26-39
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Given a $\sigma $-finite measure space $(X,\mu )$, a Young function $\Phi $, and a one-parameter family of Young functions $\{\Psi _q\}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L^\Phi (X,\mu )$ to satisfy $$\begin{align*}\lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}. \end{align*}$$The constant C is independent of f and depends only on the family $\{\Psi _q\}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.
MacDonald, Sullivan F.; Rodney, Scott. Families of Young functions and limits of Orlicz norms. Canadian mathematical bulletin, Tome 67 (2024) no. 1, pp. 26-39. doi: 10.4153/S0008439523000449
@article{10_4153_S0008439523000449,
author = {MacDonald, Sullivan F. and Rodney, Scott},
title = {Families of {Young} functions and limits of {Orlicz} norms},
journal = {Canadian mathematical bulletin},
pages = {26--39},
year = {2024},
volume = {67},
number = {1},
doi = {10.4153/S0008439523000449},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439523000449/}
}
TY - JOUR AU - MacDonald, Sullivan F. AU - Rodney, Scott TI - Families of Young functions and limits of Orlicz norms JO - Canadian mathematical bulletin PY - 2024 SP - 26 EP - 39 VL - 67 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/S0008439523000449/ DO - 10.4153/S0008439523000449 ID - 10_4153_S0008439523000449 ER -
%0 Journal Article %A MacDonald, Sullivan F. %A Rodney, Scott %T Families of Young functions and limits of Orlicz norms %J Canadian mathematical bulletin %D 2024 %P 26-39 %V 67 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4153/S0008439523000449/ %R 10.4153/S0008439523000449 %F 10_4153_S0008439523000449
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