# Questions tagged [orlicz-spaces]

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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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### 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 ...
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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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### 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 ...
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) ...
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 $\widehat{A}$ ...