A counter example

Question

I have this two spaces $$C_{\theta}=\{u\in C(\overline{\Omega}), \sup (|x|^{\theta} |u(x)|)<\infty\}$$ with the norm $\displaystyle\|u \|_{\theta}=\sup_{\Omega}(|x|^{\theta} |u(x)|)$

and $$L_{1}^{p^*}=\{u ~~\text{measurable};~~\int_{\Omega} (|x|\cdot|u(x)|)^{p^*} dx<\infty\}$$ with the norm $\|u\|_{L^{p^*}_{1}}^{p^*}=\int_{\Omega} (|x|\cdot|u(x)|)^{p^*}dx$

Where $\Omega\subset\mathbb{R}^N$ is bounded with $0\in \Omega$ and $\theta>\frac{N}{p}>1,$ $p^*=\frac{pN}{N-p}$ and $N>p.$

I need a counter example to say that $C_{\theta}$ is not continuously embeded in $L^{p^*}_{1}$, so i'm searching a function which is in $C_{\theta}$ but not in $L^{p^*}_1$

Or a sequence $u_n$ which converge to $u$ un $C_{\theta}$ but not in $L^{p^*}_{1}$

Please help me thank you

Answer 1

Let's try a canonical test function, $u(x)=|x|^\alpha$ and see if some value of $\alpha$ works. Suppose $\Omega=B(0,1)$. Then $$ \sup_{x\in\Omega}|x|^\theta|u(x)|=1\tag{1} $$ as long as $\alpha+\theta\ge0$. $$ \int_\Omega|xu(x)|^{p^*}\,\mathrm{d}x =\omega_{N-1}\int_0^1r^{(1+\alpha)p^*}r^{N-1}\,\mathrm{d}r\tag{2} $$ which is finite when $0\lt(1+\alpha)p^*+N=N\left[\frac{(1+\alpha)p}{N-p}+1\right]$. This is equivalent to $$ \frac{(1+\alpha)p}{N-p}\gt-1\iff\alpha\gt-\frac Np\tag{3} $$ For a counterexample, we would need $$ -\theta\le\alpha\le-\frac Np\tag{4} $$ Since $\frac Np\lt\theta$ by hypothesis, we can find an $\alpha$ strictly between $-\theta$ and $-\frac Np$. Thus, for $\alpha$ satisfying $(4)$, $u=|x|^\alpha$ and $\Omega=B(0,1)$ satisfy $(1)$, yet fail to have $(2)$ converge.

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Since the question also wants $u\in C(\overline{\Omega})$, $C_\theta$ is not closed. We have to limit our functions to be in $C(\overline{\Omega})$. So, define $$ u_n(x)=\left\{\begin{array}{} |x|^\alpha&\text{if }|x|\ge\frac1n\\ \frac1{n^\alpha}&\text{if }|x|\lt\frac1n \end{array}\right. $$ So each $u_n$ is in $C(\overline{\Omega})$ and satisfies $(1)$. However, $u_n\to u$ monotonically as $n\to\infty$; thus, by Monotone Convergence, we have that $\|u_n\|_{\large L_{p^\ast}^1}\to\infty$.