# Questions tagged [orlicz-spaces]

For questions about Orlicz spaces, which are a generalization of classical Lebesgue spaces $\mathbb L^p$.

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### Definition and norm of an Orlicz space

Let me define the notions first, (1)$\textbf{Young Function:}$ A convex function $\Phi:\mathbb{R}\to \mathbb{R^+}$ is said to be a Young function if the following conditions satisfy : (a) $\Phi(0)=0$ (...
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### Example of a Young function which does not satisfy $\Delta_{2}$ condition

Let me define first (1) A convex function $\Phi \colon \mathbb{R}\to \mathbb{R}^+$ which satisfies the conditions, (a) $\Phi(0)=0$ (b)$\Phi(-x)=\Phi(x)$ (c) $\lim_{x \to \infty}\Phi(x)=+\infty$, is ...
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### Condition on the Young's function

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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### In question (ii) why we need the condition that $\Phi$ to be delta 2 function to prove that space is vector space.

Question Image Can we prove it without that condition where is the problem?
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### Example of a Young's function under Lorentz-Shimogaki condition

I am reading about convolution operators and Orlicz spaces. And I would like to contract some example and stuck at some point. Let $\Psi(t)=\int_0^t\psi(s)$ be a Young's function. Then, $\Psi$ ...
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### Orlicz space property for locally compact discrete group $G$

A function $\varphi:\mathbb{R}\to[0,\infty]$ is called a Young function if $\varphi$ is convex, even,and left continuous with $\varphi(0)=0$, also let $G$ denote a locally compact group with a left ...
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Let's say that random variable $X$ is $\sigma$-subGaussian about a point $c \in \mathbb R$ if $\mathbb E[\Psi_2(\sigma |X-c|)] \le 1$, where $\Psi_2(t):=e^{t^2}-1$. Now, suppose the random variable $... 3 votes 1 answer 105 views ### Vallée Poussin's Theorem on Uniform Integrablity I've started to read Rao's Theory of Orlicz Spaces book. There are two points I could not get well at proof of Vallée Poussin's uniform integrablity theorem. Please find the theorem above. The two ... 6 votes 0 answers 164 views ### Luxemburg norm as argument of Young's function:$\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 ... 2 votes 0 answers 32 views ### Sub-exponential and conditional expectation Let$X$be a real sub-exponential r.v. then we can easily show that, for any$t>0$, there exists$K>0$such that$E[X | X \geq t] \leq t + K $. My question: is there an equivalence between ... 1 vote 1 answer 92 views ### Boundedness of a sequence in Orlicz Space By the Definition of Orlicz function$M$, we know that$M(0)=0$,$M(x)>0$for$x>0$and$M(x)\rightarrow \infty$as$x\rightarrow \infty$. And the Orlicz-Luxemborg norm is given by $$\|x\|=\inf\... 1 vote 0 answers 55 views ### Uniform convexity of Orlicz spaces Let \rho\ge0 be a uniform convex function on R^n, that is, \rho(x)-\frac{c}2|x|^2 is convex function for some c>0. Further assume that \rho(0)=0 and \rho satisfies the \Delta_2 ... 1 vote 0 answers 52 views ### Maximal growth condition for embedding of Orlicz spaces Let \Phi:\mathbb{R}^+\to \mathbb{R}^+ be a Young's function, and let L^2_\Phi(0,1) denote the Orlicz space. Consider the continuous embedding H^2_0(0,1)\hookrightarrow L^2_\Phi(0,1). Should ... 2 votes 1 answer 567 views ### The diagonal extraction procedure? What do we mean by " The diagonal extraction procedure " in the extraction of sequence in the following proof taking from: http://leonard.perso.math.cnrs.fr/papers/Leonard-Orlicz%20spaces.... 4 votes 1 answer 270 views ### Generalization of Lp; L_{\phi}(\mu)=\{f: \exists M>0 \int \phi(\frac{f(x)}{M})d\mu < +\infty\} Let (\Omega, \Sigma, \mu) a measure space and \phi(t) defined on [0,+\infty] a continuous, increasing, and convex function with \phi(0)=0. We can define the space of the measurable functions ... 1 vote 0 answers 35 views ### An optimal embedding result concerning Orlicz spaces I am looking for the optimal embedding result H_0^2(0,1)\hookrightarrow L_{\Phi}(0,1). This includes finding the largest possible Yong function \Phi(x) (the smallest space L_\Phi(0,1)) for which ... 1 vote 0 answers 58 views ### Compactness for Orlicz spaces I would like to have a feedback and a different approach to my problem: I consider the embedding W_0^{1,N} into the Orlicz space defined by e^{\alpha |u|^{N/(N-1)}}. Here N is the dimension, we ... 2 votes 1 answer 143 views ### Sequences in L^p and Orlicz spaces Let (\Omega,\mathcal{F},P) be a probability space. Let 1\leq p,q<\infty be Hölder conjugates. Let L^p:=L^p(\Omega,\mathcal{F},P), L^q:=L^q(\Omega,\mathcal{F},P). Suppose that (x_n)\... 3 votes 2 answers 850 views ### How to prove that the centering inequality for the sub-gaussian norm does not hold Specifically, define the sub-gaussian norm for a r.v. X as$$ \|X\|_{\psi_2}=\inf\{t>0:e^{X^2/t^2}\leq 2\}. $$How do we prove that the centering inequality with C=1 does not hold in general? i.... 1 vote 0 answers 83 views ### A question on the Orlicz function whose complementary function need not satisfy \Delta_2 condition. Is there an Orlicz function which satisfies the \Delta_2 condition whereas its complementary function doesn't satisfy the \Delta_2 condition?. 4 votes 1 answer 272 views ### How do I prove that this space is a Banach space? We define the following space$$ L^{\Phi}(\Omega)=\left\{u:\Omega\rightarrow \mathbb{R}~\text{measurable};~\int_{\Omega}\Phi\left(\frac{u}{\lambda}\right) dx<+\infty, ~\text{for any}~\lambda>0\... 1 vote 0 answers 42 views ### is$C^\infty$_0 dense in the space of probability distributions with finite entropy For any$M>0$, let$\mathcal{H}_M(\mathbb{R}^d)$be the set of all probability density functions with differential entropy less than$M$in absolute value: $$\mathcal{H}_M(\mathbb{R}^d)=\left\{f\in ... -3 votes 2 answers 88 views ### How we use the convexity to prove that the limit is 0? (In Orlicz space) Hello please i have that w_{\rho}(x)=h_{\rho}(x)w(x) where$$w\in L^{\Phi}(\mathbb{R}^N)=\{u\in L^1(\mathbb{R}^N); \int_{\mathbb{R}^N}\Phi(\frac{|u|}{\lambda})dx<+\infty~\text{for some}~\lambda&... 1 vote 0 answers 76 views ### Limit behaviour of the norm in variable exponent Lebesgue spaces Just a necessary recap of the definitions: For a finite measure space$(\Omega,dx)$and a function$p \in L_\infty(\Omega),$we definite the modular $$\rho_{p(\cdot)}(f)= \int_\Omega |f(x)|^{p(x)}dx,... 0 votes 0 answers 66 views ### Definition of Orlicz-Sobolev space W^{1,A}_0(\Omega) i have this definition with this definition can i say that u\in W^{1,A}_0(\Omega) impies that u=0 on \partial\Omega ? 9 votes 1 answer 759 views ### Tighter tail bounds for subgaussian random variables Let X be a random variable on \mathbb{R} satisfying \mathbb{E}\left[e^{tX}\right] \leq e^{t^2/2} for all t \in \mathbb{R}. What is the best explicit upper bound we can give on \mathbb{P}[X \... 1 vote 0 answers 295 views ### Pre-compactness in L \log L As far as I know Zygmund class of Orlicz spaces or "L \log L" is defined as an Orlicz space with the Young function Q(t) = t \sqrt{\ln(1+t)} (or something similar to this in different references).... 5 votes 0 answers 242 views ### Lebesgue differentiation theorem for Orlicz spaces If f\in L_{p}^{\rm loc}(\mathbb{R}^{n}) and 1\leq p<\infty, then a stronger version of Lebesgue differentiation theorem holds:$$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\... 2 votes 1 answer 82 views ### Perturbation of a function in the Orlicz class by a constant ($\int_\Omega\Phi(|u|)<\infty$implies$\int_\Omega \Phi(|u|+\alpha)<\infty$?). Assume that$\Phi:[0,\infty)\to [0,\infty)$is an N-function, i.e.,$\Phi(0)=0$,$\Phi$is convex, strictly increasing,$\Phi(t)/t\to 0$if$t\to 0$and$\Phi(t)/t\to \infty$if$t\to \infty$, or ... 2 votes 0 answers 158 views ### Does the Orlicz Norm always make the corresponding integral 1? Let$\Psi: [0,\infty] \to [0,\infty]$so that$\Psi$is convex, and strictly increasing with$\Psi(0) = 0$and$\Psi(\infty) = \infty.$If$(X,A,\mu)$is a measure space, then we define$L^{\Psi}(X,A,\...
It is well known that $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}=|f(x)|$$ for almost all $x\in\mathbb{R}^{n}$. Here \$\...