All Questions
Tagged with orlicz-spaces fa.functional-analysis
18 questions
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 ...
- 193
5
votes
1
answer
764
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
4
votes
1
answer
196
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 $...
- 1,101
4
votes
0
answers
179
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
3
votes
1
answer
162
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 ...
- 5,405
3
votes
1
answer
355
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 \...
- 363
3
votes
0
answers
81
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
3
votes
0
answers
68
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
$$\|f\|_{H_p} = \sup_{0 < r < 1} \|f(r\cdot)\|_{L_p(\mathbb{T})} &...
- 565
3
votes
0
answers
97
views
Boudedness of linear operator between generalized Orlicz spaces
I am using the notations, definitions, and results of the Section X of [1] on generalized Orlicz spaces.
We say that $\varphi : \mathbb{R} \rightarrow \mathbb{R}^+$ is a $\varphi$-function if it is ...
- 2,306
2
votes
1
answer
352
views
Young’s complement of $ x \mapsto x \, {\log^{+}}(x) $, $ N $-functions and Orlicz spaces
The function $ \Phi: \mathbb{R} \to \mathbb{R} $ is an $ N $-function if and only if it is continuous, even and convex with:
$ \displaystyle \lim_{x \to 0} \frac{\Phi(x)}{x} = 0 $.
$ \displaystyle \...
- 259
2
votes
1
answer
109
views
Dual space and conditions for weak convergence in Orlicz Space not having $\Delta_{2}$ property
I am interested in conditions for weak convergence on Orlicz spaces where the corresponding Young function, $\Phi:[0,\infty) \rightarrow [0,\infty)$, does not have the $\Delta_{2}$ condition, i.e. ...
- 21
2
votes
0
answers
76
views
Fractional integration in Orlicz spaces
I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
- 343
2
votes
0
answers
39
views
Example when Lorentz-Shimogaki condition satisfied with a specific Young's function
Let $\Psi(t)=\int_0^t\psi(s)$ be a Young's function. Then, $\Psi$ satisfies the Lorentz-Shimogaki condition if
$$
\int_0^{\infty}\frac{\Psi(st)}{v(t)^2}\psi(t)dt< \infty.
$$
Denote $\rho_{\Psi}=\...
- 343
2
votes
0
answers
62
views
Decomposition of the Orlicz norm into sequential norm
I am bearing seeking for a sequential decomposition of the norm in Orlicz space.
Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$.
Given $u\in L^p(\Bbb R^d)$ let
$$n\...
- 1,101
2
votes
0
answers
182
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 ...
- 4,432
1
vote
0
answers
45
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
$$
\|f\|_M:=\inf\left\{\lambda &...
- 5,405
0
votes
1
answer
546
views
Orlicz–Sobolev spaces
Let $A$ be an N-function and suppose that
$$\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{...
0
votes
1
answer
128
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) ...
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