# Questions tagged [orlicz-spaces]

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### Orlicz-Sobolev Spaces

let $A$ an N-function and suppose that $$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty$$ we denote by $\widehat{A}$ an N-function equal to A near infinity and $\widehat{A}$ ...
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### Lyapounov's inequality for Orlicz norms

When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$. This inequality is a special (or rather limit) case of ...
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### Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$

Let $\Phi$ be a Youngs's function, i.e. $$\Phi(t) = \int_0^t \varphi(s) \,\mathrm d s$$ for some $\varphi$ satifying $\varphi:[0,\infty)\to[0,\infty]$ is increasing $\varphi$ is lower semi ...
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I have been reading from Weak Convergence and Empirical Processes, and came across the following: Let $a_1,\ldots,a_n$ be constants and $\epsilon_1,\ldots,\epsilon_n\sim$Rademacher. Then $\mathbb{P}... 1answer 75 views ### Reference Request:$L^p(x)$/(Musielak–Orlicz space) analogue of classical$L^p$result Fix a non-empty open domain$\Omega\subseteq \mathbb{R}^d$with compact closure, and a finite Borel measure$\mu$on its closure$\overline{\Omega}$. In Halmos' book it is shown that: Classical ... 1answer 114 views ### Which Orlicz functions$f$make the function$f^{-1}\left(\frac{\sum_{j=1}^s f(x_j)}{s}\right)$convex? Let$f:\mathbb{R}_+\to\mathbb{R}_+$be an Orlicz function, or sometimes referred to as an Young function, i.e. it is a convex, non-decreasing function such that$f(0)=0$. I am trying to study the ... 0answers 55 views ### Concavification of Orlicz function In the Handbook of the Geometry of Banach Spaces, vol 1, page 855, Johnson and Schechtman say that if$M$is an Orlicz function, the following are equivalent: The unit vector basis of the Orlicz ... 1answer 96 views ### Definition of an Orlicz modular space In Nowak (1989), a modular$\rho$on a vector lattice is defined by the following properties (N1)$\rho(x)=0\implies x=0$; (N2)$\lvert x\rvert \le \lvert y\rvert\implies \rho(x) \le \rho(y)$; (N3) ... 0answers 90 views ### Modular which is metrizing but does not satisfy the$\Delta_2$condition Let$\Phi$be a nice Young function (N-function) and$(\Omega,\mathcal{F},P)$a probability space such that either$P$is diffuse on a set of non-zero probability or$P$is purely atomic and there are ... 1answer 220 views ### Young’s complement of$ x \mapsto x \, {\log^{+}}(x) $,$ N $-functions and Orlicz spaces The function$ \Phi: \mathbb{R} \to \mathbb{R} $is an$ N $-function if and only if it is continuous, even and convex with:$ \displaystyle \lim_{x \to 0} \frac{\Phi(x)}{x} = 0 $.$ \displaystyle \...
I am using the notations, definitions, and results of the Section X of [1] on generalized Orlicz spaces. We say that $\varphi : \mathbb{R} \rightarrow \mathbb{R}^+$ is a $\varphi$-function if it is ...