Author:
MacDonald Sullivan F.,Rodney Scott
Abstract
Abstract
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.
Publisher
Canadian Mathematical Society
Reference7 articles.
1. [3] Mailhot, A. , On the limit of Orlicz norms. Undergraduate thesis, Cape Breton University, 2022. http://faculty.cbu.ca/srodney/pdf/theses/thesisam.pdf.
2. Orlicz–Sobolev spaces and imbedding theorems;Trudinger;J. Funct. Anal.,1971
3. Bounded weak solutions to elliptic PDE with data in Orlicz spaces