Questions tagged [orlicz-spaces]
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37 questions
0 votes
1 answer
117 views
Reflexivity of the Orlicz space associated with the Huber function
We consider the Huber function given by $$H_\delta(x) = \begin{cases} \frac{x^2}{2} & \text{if } x \leq \delta\,, \\ \delta \cdot ( |x| - \frac{1}{2}\delta) & \text{otherwise}\,. \end{cases}$...
- 2,602
0 votes
1 answer
173 views
On the extreme points of the unit ball associated to the Huber function
The Huber function is defined as follows $$H(x) = \begin{cases} \frac{x^2}{2} & \text{if } x \leq \delta \\ \delta \cdot ( |x| - \frac{1}{2}\delta) & \text{otherwise} \end{cases}$$ The ...
- 2,602
2 votes
0 answers
72 views
Dual operators for non-$\Delta_{2}$ Orlicz spaces
I am studying adjoint operators on non-$\Delta_{2}$ Orlicz spaces. Let $\Phi(t) = e^{|t|}-1$ be a Young function, noting it does not meet the $\Delta_{2}$ condition: $$ \sup_{t \in [0,\infty)} \dfrac{...
- 81
2 votes
1 answer
145 views
Functions not approximable by simple functions in non $\Delta_{2}$ Orlicz space
I am attempting to understand elements of non-$\Delta_{2}$ Orlicz spaces that are not approximable by simple functions. Suppose $L^{\Phi}(dx)$ is an Orlicz space with on the unit interval $[0,1]$ with ...
- 81
2 votes
1 answer
256 views
Dual space and conditions for weak convergence in Orlicz Space not having $\Delta_{2}$ property
I am interested in conditions for weak convergence on Orlicz spaces where the corresponding Young function, $\Phi:[0,\infty) \rightarrow [0,\infty)$, does not have the $\Delta_{2}$ condition, i.e. ...
- 81
0 votes
0 answers
391 views
Lower bounds for sub-Gaussians?
For a random variable $X$, define $$\lVert X\rVert_{\psi_2} =\inf \{k>0\mid \mathbb{E}[\exp((X/k)^2)]\leq 2\}$$ and for a random vector $\vec X$, define $$\lVert \vec X\rVert_{\psi_2} = \sup_{\...
- 2,456
1 vote
1 answer
125 views
Improved bounds on $\lVert XY\rVert_{\psi_2}$ via concentration data of the (bounded) random variable $Y$?
Throughout I will use the language of Orlicz norms associated with the family of functions $\psi_a(x) = \exp(x^a)-1$ for $a\in[1,\infty)$, and $$\psi_\infty(x) = \begin{cases}\infty & x>1\\1 &...
- 2,456
1 vote
1 answer
185 views
Is the space of bounded $\psi_\infty$ Orlicz norm random variables equal to $L^\infty$?
Let $\psi_\alpha(x) = \exp(x^\alpha)-1$ for $\alpha\geq 1$. Define $$ \psi_\infty(x) = \begin{cases}\infty & x>1\\1& x = 1\\ 0 & x <1 \end{cases} $$ to be such that for any $x>0$ $...
- 2,456
0 votes
1 answer
179 views
Is the product of sub-Gaussian polynomials in $\mathbb{R}[x]/(x^n-1)$ sub-Gaussian?
Let $\psi_\alpha(x) := \exp(x^\alpha)-1$. It is well-known that for $\alpha\geq 1$ that $$\lVert X\rVert_{\psi_\alpha} = \inf\{k>0\mid \mathbb{E}[\psi_\alpha(|X|/k)] \leq 1\}$$ defines an Orlicz ...
- 2,456
2 votes
0 answers
83 views
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 ...
- 161
1 vote
0 answers
151 views
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 ...
- 161
2 votes
2 answers
250 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 ...
2 votes
0 answers
92 views
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 ...
- 343
3 votes
1 answer
107 views
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 $p \to \infty$. ...
- 203
2 votes
0 answers
55 views
Example when Lorentz-Shimogaki condition satisfied with a specific Young's function
Let $\Psi(t)=\int_0^t\psi(s)$ be a Young's function. Then, $\Psi$ satisfies the Lorentz-Shimogaki condition if $$ \int_0^{\infty}\frac{\Psi(st)}{v(t)^2}\psi(t)dt< \infty. $$ Denote $\rho_{\Psi}=\...
- 343
3 votes
0 answers
107 views
Example of the bounded convolution operator when Sharpley's conditions does not hold
I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference ...
- 97
4 votes
0 answers
117 views
Maximal function in Orlicz space
Consider the maximal operator defined for a function $f\in L^1_{loc}$: $$ Mf : x\mapsto \sup_{r>0} \frac{1}{|B(x,r)|} \int\limits_{B(x,r)} f. $$ It is well know that $M : L^1 \to L^{(1,\infty)}$ ...
- 625
3 votes
1 answer
491 views
Hölder inequality between different Orlicz spaces
If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satisfying $1/r+1/s=1$, then we know that $fg\in L^1$. But if $g$ is a little bit more than $L^s$, say $L^s \...
- 625
1 vote
1 answer
408 views
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-...
- 143
6 votes
1 answer
954 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 ...
- 5,465
3 votes
0 answers
91 views
Relationship between Hardy-Orlicz space and the corresponding Orlicz space
For $p \in [1, \infty]$ the Hardy space $H_p$ is defined as the space of all analytic functions $f$ on the open disk satisfying $$\|f\|_{H_p} = \sup_{0 < r < 1} \|f(r\cdot)\|_{L_p(\mathbb{T})} &...
- 565
6 votes
1 answer
349 views
Weak concentration bounds for averages of independent random variables in Orlicz spaces
Let $\phi$ be an $N$-function, (i.e. $\phi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ is convex and satisfies $\lim_{t \to 0} \frac{\phi(t)}{t} = 0, \lim_{t\to \infty} \frac{\phi(t)}{t} = \infty$)....
- 1,450
4 votes
1 answer
245 views
On the intersection of two Orlicz spaces
It is well-known that if $1\leq p\leq q\leq \infty $ then $$ L^p(X)\cap L^q(X)\subset L^r(X)\quad\quad \text{whenever $r\in [p,q]$}\tag{I}\label{Eq}.$$ Indeed let $u\in L^p(X)\cap L^q(X)$. For some $...
- 1,155
2 votes
0 answers
82 views
Decomposition of the Orlicz norm into sequential norm
I am bearing seeking for a sequential decomposition of the norm in Orlicz space. Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$. Given $u\in L^p(\Bbb R^d)$ let $$n\...
- 1,155
1 vote
0 answers
59 views
Generalizations of the Wiener Tauberian Theorem to Musielak-Orlicz spaces
Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm $$ \|f\|_M:=\inf\left\{\lambda &...
- 4,942
4 votes
0 answers
198 views
Condition on kernel convolution operator
I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$ \Phi_i(2t)\approx \Phi_i(t), \quad i=1,2 $$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
- 101
0 votes
1 answer
644 views
Orlicz–Sobolev spaces
Let $A$ be an N-function and suppose that $$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty. $$ We denote by $\widehat{A}$ an N-function equal to $A$ near infinity and $\widehat{...
2 votes
0 answers
228 views
Lyapounov's inequality for Orlicz norms
When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$. This inequality is a special (or rather limit) case of ...
- 4,706
9 votes
1 answer
1k views
Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$
Let $\Phi$ be a Youngs's function, i.e. $$ \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 continuous ...
- 193
3 votes
1 answer
481 views
Hoeffding to bound Orlicz norm
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}...
- 33
3 votes
1 answer
194 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 ...
- 4,942
3 votes
1 answer
203 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
102 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 space ...
- 565
0 votes
1 answer
166 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) ...
user113782
2 votes
1 answer
403 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 \...
- 259
5 votes
1 answer
828 views
Looking for a reference for a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces
Please I need a reference where I can find a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces analogous to Theorem 1.2.7 in Semilinear Elliptic Equations for ...
- 277
3 votes
0 answers
118 views
Boudedness of linear operator between generalized Orlicz spaces
I am using the notations, definitions, and results of the Section X of [1] on generalized Orlicz spaces. We say that $\varphi : \mathbb{R} \rightarrow \mathbb{R}^+$ is a $\varphi$-function if it is ...
- 2,602