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# Questions tagged [orlicz-spaces]

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### Well-posedness or existence for a Poisson problem in Orlicz spaces

I know that the problem \begin{equation} \Delta_p u = f \end{equation} make sense if $f \in L^q$ with $n/p<q<n$ and that is there a existence theory for $$u_t -\Delta_p u = f$$ For a given ...
1 vote
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### When is there an inclusion between regular Orlicz Spaces?

It is a classical result that $L^p(\Omega) \subset L^q(\Omega)$ when $q<p$ and $|\Omega| < \infty$. I'd like to know if there is an Orlicz version of this fact. In other words, let $L^{G_1}$ and ...
166 views

### Elementary convexity example

I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...
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### Fractional integration in Orlicz spaces

I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley. And I would like to understand one question: Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
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### When do Orlicz norms tend to the uniform norm?

It is well known that the $p$-norms tend to the $\infty$-norm, in that if $\lVert f \rVert_q < \infty$ for some $q \ge 1$ then $\lVert f \rVert_p \to \lVert f \rVert_\infty$ as $p \to \infty$. ...
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1 vote
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### Independent Sums and Orlicz Norms

Let $X_{i}$ be a collection of iid random variables of cardinality $n$, and let $S_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i}.$ Let $|| X||:=\inf_{B}\{E[\exp(X/B)-1]\leq 1\}$. This is the so-called sub-...
349 views

### An $L^1$ function but (really) no better?

Question: For a smooth, bounded domain $\Omega\subset \mathbb R^d$, does there exist a function $u\in L^1(\Omega)$ such that $u\not\in L^\Phi(\Omega)$ for any Orlicz space $\Phi$? For the definition ...
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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 $\mathbb{P}... 3 votes 1 answer 130 views ### Reference Request:$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}$. In Halmos' book it is shown that: Classical ... 3 votes 1 answer 147 views ### Which Orlicz functions$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$. I am trying to study the ... 2 votes 0 answers 63 views ### Concavification of Orlicz function In the Handbook of the Geometry of Banach Spaces, vol 1, page 855, Johnson and Schechtman say that if$M$is an Orlicz function, the following are equivalent: The unit vector basis of the Orlicz ... 0 votes 1 answer 112 views ### Definition of an Orlicz modular space In Nowak (1989), a modular$\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)$; (N3) ... 1 vote 0 answers 101 views ### Modular which is metrizing but does not satisfy the$\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$is purely atomic and there are ... 2 votes 1 answer 300 views ### Young’s complement of$ 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  on generalized Orlicz spaces. We say that $\varphi : \mathbb{R} \rightarrow \mathbb{R}^+$ is a $\varphi$-function if it is ...