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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Prove that Orlicz space is a Vector space

Let me first define the Orlicz space, Let $(\Omega,\Sigma,\mu)$ be a measure space,where $\Omega$ is set,$\Sigma$ is a sigma-algebra and $\mu$ is a measure.Then the space $L^\Phi(\mu)$={$f\colon \...
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Intuition behind defining the $\Delta_{2}$ condition for Young function

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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Please explain this theorem,I mean what this theorem is saying ,can you give any example to explain this

I am reading Orlicz space from M.Rao's book and this is the first theorem that says the equivalence between the statements. Let me define the Uniform Integrability Let $F$={$f_{\alpha}\colon \Omega \...
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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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Question about Lebesgue space for $0<p<1$

We have defined the $L^p$ space for $p\geq1$.One thing I know is that for $0<p<1$ the norm will not satisfy the triangle inequality. $L^p(A)$={$f \colon A\to R$ (measurable) such that $\int|f|^...
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What is the role of $x^p$ for defining the Lebesgue Integral?

We define the Lebesgue Space by $L^p(A)$={$f \colon A\to R$ such that $\int|f|^pd\nu<\infty$} I was reading one paper on orlicz spaces and there they have written that "For generalization of ...
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Why do we need convex functions to define the Orlicz space?

I am reading Theory of Orlicz spaces by M.Rao. My question is that : why do we need a convex function to define Orlicz spaces ? Can't we take any other type of function? Definition of Orlicz space: L$^...
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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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Luxemburg norm of p-th power of a function

Let $f>1$ on $\Omega\subset \mathbb{R}^n$. Does the inequality, $$ \|f^p\|_{L^{G}(\Omega)} \leq \|f\|_{L^{G}(\Omega)}^p $$ for any $p>0$, hold? Here $L^{G}(\Omega)$ denotes the Orlicz space on $\...
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Laplace and Orlicz characterizations of sub-Gaussianity

Many different characterizations of sub-Gaussianity for a centred r.v. X exist including the "Laplace formulation" where $\exists \sigma > 0$ such that $$ \mathbb{E} \exp(tX) \le \exp(t^2 ...
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Dual of a order continuous Banach Lattice $\subseteq L^1$

I have recently read a theorem in a paper (porbably) indicating that if we take a Banach Lattice $E \subseteq L^1$ with order continuous norm, then we can identify its dual $E'$ with another space $F \...
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How approach to demonstrate that the inequality $V(K)\le V(L)$ of compact set volumes of an Orlicz intersection body $K$ and a star body $L$ holds?

Even though a proof would be fantastic, please don't take this question as a request for a complete proof, but as a collector of ideas on how to deal with the stated inequality (visualizations, drafts ...
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Decomposing the Orlicz norm in sequential norm.

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\...
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Condition on kernel convolution operator

I am studying a about O'Neil's convolution inequality. It is stated that for $\Phi_1$ and $\Phi_2$ be $N$-functions, with $\Phi_i(2t)\approx \Phi_i(t), \quad i=1,2$ with $t\gg 1$ and $k \in M_+(R^n)$ ...
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Example when Kantorovich conditions would not hold

Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator $$ (T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+. $$ Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
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Concentration of the Norm for Sub-gaussians

I am reading Theorem 3.1.1 in HDP book by Vershynin. The theorem states that $ \text{Let } X=\left(X_1,\ldots,X_n \right) \text{be a random vector with independent, sub-gaussian coordinates } X_i \...
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calculation of an Orlicz norm

I would need, if possible, some help with the calculation of the Orlicz norm of a random variable $X$. I have this function: $\phi(x)=x\ln(1+x)$, and I need to compute $\inf c>0$ s.t. $E \left[\phi\...
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For what kind of random vectors do we have $\sup_{p \ge 1}\|X\|_p < \infty$?

Let $X$ be a random vector on $\mathbb R^m$ (assumed to have zero mean, for simplicity). For $p \in [1,\infty)$, define $e_p(X):=\mathbb E\sum_{j=1}^m|X_j|^p \in [0,\infty]$. Finally, define $\|X\|_p \...
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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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On non-centered subGaussian random variables

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 $...
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3 votes
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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 ...
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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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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 ...
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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\...
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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 $\Delta_2$ ...
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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 ...
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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....
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4 votes
1 answer
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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 ...
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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 ...
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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 ...
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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)\...
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3 votes
2 answers
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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....
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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?.
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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\...
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1 vote
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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 ...
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2 answers
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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&...
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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,...
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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 $\partial\Omega$ ?
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9 votes
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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 \...
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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)....
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5 votes
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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}(\...
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2 votes
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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 ...
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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,\...
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5 votes
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Analogue of Lebesgue differentiation theorem in Orlicz spaces

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 $\...
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