12
votes
Accepted
An $L^1$ function but (really) no better?
There is a much more general result of Vallée-Poussin from which a negative answer to your question follows.
Let $(X,\mu)$ be a measure space. We say that a family of function $\mathcal{F}\subset L^1(...
- 28k
7
votes
Accepted
Weak concentration bounds for averages of independent random variables in Orlicz spaces
In general, the answer is no. Moreover, the answer is no even if
\begin{equation}
\phi(t)=t\ln(1+t). \tag{1}
\end{equation}
Indeed, suppose that $P(Z_i=0)=1-2p$ and $P(Z_i=b)=p=P(Z_i=-b)$ for all $...
- 128k
6
votes
Accepted
Looking for a reference for a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces
This is Theorem 1.4 in Bennet and Sharpley's "Interpolation of Operators" (page 3). It actually holds for Banach spaces of functions equipped with what they call "function norms" and not only Orlicz ...
- 12.7k
5
votes
Accepted
Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$
The conjectured inequality does not hold.
For a counterexample, consider $\Phi(t)=\max(t^2,t^3)$ and $\Omega=(0,1)$.
Let $f=a\chi_{(0,b)}$ for $a,b\in (0,1)$.
It can be calculated that $\|f\|_{L^\Phi}...
- 215
4
votes
Accepted
When do Orlicz norms tend to the uniform norm?
$\newcommand{\ep}{\varepsilon}\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}$A natural generalization of the fact that the $p$-norm converges to the $\infty$-norm as $p\to\infty$ is as follows.
...
- 128k
3
votes
On the intersection of two Orlicz spaces
It's many years ago that I read it, but I think that some of the most general interpolation type results for Orlicz spaces were contained in O’Neil, Richard, Integral transforms and tensor products on ...
- 1,869
3
votes
Accepted
Hoeffding to bound Orlicz norm
As I understand these notations, $\Psi_2(x)=\exp(x^2)-1$ and
$\|Z\|_{\Psi}=\inf \{c>0 : \mathbb{E} (\Psi(|Z| / c)) \leqslant 1\}$.
So the inequality $\|Z\|_{\Psi}\leqslant c$ means that $\mathbb{...
- 109k
3
votes
Accepted
Reference Request: $L^p(x)$/(Musielak–Orlicz space) analogue of classical $L^p$ result
Yes this is true (in the variable exponent's case which was the objective of the question).
See Theorem 3.4.12 of this book: "Lebesgue and Sobolev spaces with variable exponents" by "Lars Diening, ...
- 5,405
2
votes
Accepted
Which Orlicz functions $f$ make the function $f^{-1}\left(\frac{\sum_{j=1}^s f(x_j)}{s}\right)$ convex?
It follows from Hardy--Littlewood--Polya (see section 3.16, page 86), that if for $x>0$ we have $f, f', f''>0$, and $f'/f''$ is concave, and $d\mu$ is a probability measure on the probability ...
- 3,946
2
votes
Young’s complement of $ x \mapsto x \, {\log^{+}}(x) $, $ N $-functions and Orlicz spaces
That space should be $L_{\exp}$. Check Bennett and Sharpley's "Interpolation of Operators" Chapter 4.6 or look in Rao and Ren's "Theory of Orlicz Spaces".
- 12.7k
2
votes
Accepted
Hölder inequality between different Orlicz spaces
Yes, we can say so. Indeed, let us show that the conditions $f\in L^r$ and $g\in L^s\ln L$ imply $fg\in L\ln^t L$ for $t:=1/s$. Moreover, we shall show that the value $t=1/s$ here is optimal, as it ...
- 128k
2
votes
Accepted
Elementary convexity example
Here is how to remove the assumption that $p-2+\delta\ge0$.
Let
\begin{equation*}
s:=p-1+\delta.
\end{equation*}
The conditions $p>1$ and $\delta>0$ imply $s\ge0$. No other conditions on $...
- 128k
2
votes
Accepted
Is the space of bounded $\psi_\infty$ Orlicz norm random variables equal to $L^\infty$?
For each real $k>0$,
\begin{equation}E\psi_\infty(|X|/k)=\infty\,P(|X|>k)+P(|X|=k) \\
=\left\{\begin{aligned}\infty\text{ if } P(|X|>k)>0,\\
P(|X|=k)\le1\text{ if } P(|X|>k)=0.
\end{...
- 128k
1
vote
Accepted
Improved bounds on $\lVert XY\rVert_{\psi_2}$ via concentration data of the (bounded) random variable $Y$?
This is not possible.
Take $X \sim \mathcal{N}(0, 1)$, and $Z \sim \mathcal{N}(0,1)$, with $Y = \sqrt{B} Z$ when $|Z| < \sqrt{B}$ and $Y = B \mathop{sgn}(Z)$ otherwise. Clearly $\|X\|_{\psi_2} \leq ...
- 1,190
1
vote
Is the product of sub-Gaussian polynomials in $\mathbb{R}[x]/(x^n-1)$ sub-Gaussian?
I had known that
Or (in general) via some stronger Orlicz-norm bound on $f(x)g(x)$
than the above $\psi_1$ bound?
was plausibly related to the Bennett-Orlicz norm (or perhaps Bernstein-Orlicz norm), ...
- 2,183
1
vote
Elementary convexity example
Here is a much simpler proof, actually of the more general fact that
$$f(x):=x^p(1+\ln^+ x)^s$$
is convex in $x\ge0$ for any real $p\ge1$ and $s\ge0$, where $\ln^+ x:=\ln\max(1,x)$.
For the left and ...
- 128k
1
vote
Accepted
Independent Sums and Orlicz Norms
For the equality $\|S_n\|_{L^2}=\|X_i\|_{L^2}$ you need the zero-mean condition -- that $EX_i=0$.
Let $X,X_1,\dots,X_n$ be any random variables with the same norm: $\|X\|=\|X_1\|=\cdots=\|X_n\|$.
...
- 128k
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