Questions tagged [orlicz-spaces]

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

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3
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0answers
23 views

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 $k \in M_+(R^n)$ ...
2
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0answers
14 views

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-...
1
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1answer
25 views

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 \...
2
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0answers
22 views

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\...
1
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1answer
44 views

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 \...
2
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0answers
28 views

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 ...
1
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0answers
13 views

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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0answers
17 views

Upper-bound on $\inf_{(X,X')} P(\|X-X'\| > 2t)$ over all couplings $(X,X')$ of $P_1$ and $P_2$

Preamble: I've been struggling with the problem below (and similar problems https://mathoverflow.net/q/351317/78539) for a while now. Any kind of help would be very useful. Thanks in advance! So, let ...
3
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1answer
53 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
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0answers
100 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
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0answers
18 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
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1answer
48 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
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0answers
34 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
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0answers
35 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 ...
1
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1answer
308 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
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1answer
122 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
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0answers
32 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
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0answers
46 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
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1answer
88 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
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2answers
455 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....
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0answers
48 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
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1answer
204 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\...
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0answers
35 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
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2answers
82 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
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0answers
56 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
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0answers
62 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 $\partial\Omega$ ?
5
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0answers
498 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
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0answers
260 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
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0answers
220 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
1answer
58 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
0answers
129 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,\...
5
votes
1answer
266 views

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 $\...