# Questions tagged [orlicz-spaces]

For questions about Orlicz spaces, which are a generalization of classical Lebesgue spaces $\mathbb L^p$.

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### Condition on kernel convolution operator

I am studying a about O'Neil's convolution inequality. It is stated that for $\Phi_1$ and $\Phi_2$ be $N$-functions, with $\Phi_i(2t)\approx \Phi_i(t), \quad i=1,2$ with $t\gg 1$ and $k \in M_+(R^n)$ ...
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### Example when Kantorovich conditions would not hold

Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator $$(T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+.$$ Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
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### Upper-bound on $\inf_{(X,X')} P(\|X-X'\| > 2t)$ over all couplings $(X,X')$ of $P_1$ and $P_2$

Preamble: I've been struggling with the problem below (and similar problems https://mathoverflow.net/q/351317/78539) for a while now. Any kind of help would be very useful. Thanks in advance! So, let ...
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### Vallée Poussin's Theorem on Uniform Integrablity

I've started to read Rao's Theory of Orlicz Spaces book. There are two points I could not get well at proof of Vallée Poussin's uniform integrablity theorem. Please find the theorem above. The two ...
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### 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 ...
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### Sub-exponential and conditional expectation

Let $X$ be a real sub-exponential r.v. then we can easily show that, for any $t>0$, there exists $K>0$ such that $E[X | X \geq t] \leq t + K$. My question: is there an equivalence between ...
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### Perturbation of a function in the Orlicz class by a constant ($\int_\Omega\Phi(|u|)<\infty$ implies $\int_\Omega \Phi(|u|+\alpha)<\infty$?).
Assume that $\Phi:[0,\infty)\to [0,\infty)$ is an N-function, i.e., $\Phi(0)=0$, $\Phi$ is convex, strictly increasing, $\Phi(t)/t\to 0$ if $t\to 0$ and $\Phi(t)/t\to \infty$ if $t\to \infty$, or ...