All Questions

It is well-known that if $1\leq p\leq q\leq \infty $ then $$ L^p(X)\cap L^q(X)\subset L^r(X)\quad\quad \text{whenever $r\in [p,q]$}\tag{I}\label{Eq}.$$ Indeed let $u\in L^p(X)\cap L^q(X)$. For some $...

Fix a non-empty open domain $\Omega\subseteq \mathbb{R}^d$ with compact closure, and a finite Borel measure $\mu$ on its closure $\overline{\Omega}$. In Halmos' book it is shown that: Classical ...

I am bearing seeking for a sequential decomposition of the norm in Orlicz space. Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$. Given $u\in L^p(\Bbb R^d)$ let $$n\...