# Questions tagged [orlicz-spaces]

The orlicz-spaces tag has no usage guidance.

29
questions

2
votes

0
answers

47
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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

0
answers

80
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 ...

2
votes

2
answers

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 ...

2
votes

0
answers

64
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 ...

3
votes

1
answer

48
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 $p \to \infty$. ...

2
votes

0
answers

34
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}=\...

3
votes

0
answers

70
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 ...

4
votes

0
answers

55
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 $M : L^1 \to L^{(1,\infty)}$ ...

3
votes

1
answer

154
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 \...

1
vote

1
answer

166
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-...

6
votes

1
answer

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 ...

3
votes

0
answers

44
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})} &...

4
votes

1
answer

173
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 $\lim_{t \to 0} \frac{\phi(t)}{t} = 0, \lim_{t\to \infty} \frac{\phi(t)}{t} = \infty$)....

4
votes

1
answer

152
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 $...

2
votes

0
answers

59
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
vote

0
answers

40
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
votes

0
answers

150
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 ...

0
votes

1
answer

376
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{...

1
vote

0
answers

117
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 ...

8
votes

1
answer

757
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 ...

3
votes

1
answer

244
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}...

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 \...

4
votes

1
answer

577
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 ...

3
votes

0
answers

92
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 ...