Suppose that $\{f_n\}$ is a sequence of measurable real valued functions on $\Bbb{R^3}$ and satisfies
i) For any compact set $K \subset\Bbb{R^3},$ $$\lim_{n\rightarrow\infty}\int_{K}{|f_n(x)|dx}=0.$$ ii) For any ${3 \over 2}\le p\le3,$ $$\sup_{n\ge1}{||f_n||_p<\infty} .$$ Then, Show that $$ \lim_{n\rightarrow\infty}{\int_{\Bbb{R^3}}{|f_n(x)|\over|x|^{3/2}} dx }=0 .$$
my thoughts : I have my doubts about $3/2\le p\le3. $ I think that only $ p=3/2,3$.
proof. Let $ \epsilon >0.$ and let $ g_n(x)=|f_n(x)|/|x|^{3/2}. $
First, For $ \epsilon$ and any $ n$ $$\exists R>0 ; \int_{|x|>R} {g_n(x) dx<\epsilon.}$$ Indeed, $$ \int_{|x|>R} g_n(x) dx<\left[ \int_{|x|>R} |f_n(x)|^{3/2}\right]^{3/2} \left[ \int_{|x|>R} {1 \over |x|^{9/2}} dx\right]^{1/3} \;\;\; (by \;H\ddot o lder) $$ $$ <4M_{3/2}\pi \int_R^{\infty} r^{-5/2} dr \rightarrow 0\;\;(R\rightarrow 0). $$
Similarly ( change the role of two functions ) , we have $$ \exists \delta>0;\int_{|x|<\delta}g_n(x) dx<\epsilon.$$ therefore, (i) completes the solution of the problem.