How we use the convexity to prove that the limit is $0$? (In Orlicz space)

Question

Hello please i have that $w_{\rho}(x)=h_{\rho}(x)w(x)$ where

$$w\in L^{\Phi}(\mathbb{R}^N)=\{u\in L^1(\mathbb{R}^N); \int_{\mathbb{R}^N}\Phi(\frac{|u|}{\lambda})dx<+\infty~\text{for some}~\lambda>0\}$$

and $h_{\rho}\in C^{\infty}(\mathbb{R}^N,[0,1])$ such that $h_{\rho}(x)\equiv 1~\text{on}~B_{\rho}(0),~\text{and}~ supp(h_{\rho})\subset B_{2\rho}(0)$

How to prove that $$ \int_{\mathbb{R}^N}\Phi(|w_{\rho_n}(x)-w(x)|)dx \overset{n\rightarrow+\infty}{\longrightarrow}0 $$

$(\rho_n)$ is a real sequence where $\rho_n\rightarrow +\infty,~\text{whene}~ n\rightarrow+\infty$

where $\Phi$ is a real positive convex function

Answer 1

the statement is wrong.

Consider the constant function $\Phi=1$ which is convex. Then the integral $$ \int_{\mathbb R^n} 1 \,\mathrm dx = \infty $$ and thus does not converge to $0$.

Edit: OP made same changes to the question and added non-obvious assumptions, but i am leaving my original answer up here.

Answer 2

The convex image of an integrable function need not be integrable. As an example of this, let $\sqrt 2< a< 2$ and consider $w:\mathbb R\to \mathbb R$ given by \begin{equation*} w(x) = \begin{cases} a^j & \text{ if } 2j - 1< x< 2j-1+2^{1- j}\\ 0 & \text{ otherwise}, \end{cases} \end{equation*} where $j= 1, 2, \ldots$. Let $\Phi(r) = r^2$. On one hand, since $a< 2$ we have \begin{equation*} \int_{\mathbb R} w(x)\; dx = \sum_{j = 1}^\infty a^j 2^{1-j}< \infty. \end{equation*} On the other hand, since $a>\sqrt 2$ we have \begin{equation*} \int_{\mathbb R} \Phi(w(x))\; dx = \int_{\mathbb R} w^2(x)\; dx = \sum_{j = 1}^\infty a^{2j}2^{1 - j} = +\infty. \end{equation*}