All Questions
Tagged with orlicz-spaces banach-spaces
9 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
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
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
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
$$n\...
- 1,101
2
votes
0
answers
70
views
Concavification of Orlicz function
In the Handbook of the Geometry of Banach Spaces, vol 1, page 855, Johnson and Schechtman say that if $M$ is an Orlicz function, the following are equivalent:
The unit vector basis of the Orlicz ...
- 565
1
vote
0
answers
45
views
Generalizations of the Wiener Tauberian Theorem to Musielak-Orlicz spaces
Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm
$$
\|f\|_M:=\inf\left\{\lambda &...
- 5,405