# Functions $f_n:\mathbb{R}\to\mathbb{R}$ with $f_n \to 0 \text{ in } L^3, f_n\rightharpoonup 0 \text{ in } L^2, f_n \nrightarrow 0 \text{ in } L^2$

Give a sequence of functions $$f_n:\mathbb{R}\to\mathbb{R}$$ such that $$f_n \to 0 \text{ in } L^3(\mathbb{R}), f_n\rightharpoonup 0 \text{ in } L^2(\mathbb{R}), f_n \nrightarrow 0 \text{ in } L^2(\mathbb{R}).$$

I took $$f_n(x)=n^{1/2}I_{[0,1/n]}$$, then $$||f_n||_{L^2}=1$$ and the third condition holds. For $$\phi \in C_c^{\infty}(\mathbb{R})$$ dense in $$L^2(\mathbb{R})$$ we have$$\int_0^{1/n} f_n\phi=\int_0^{1/n} n^{1/2}\phi =n^{-1/2}\phi(1/n)-\int_0^{1/n} n^{1/2}x\phi'\leq n^{-1/2}\phi(1/n)-\frac12n^{-3/2}\sup |\phi'|\to 0$$ which shows the second condition. But I I cannot the satisfy first one as well. Tweaking the exponents makes one of the other conditions fail…

Since $$L^3 \hookrightarrow L^2$$ on bounded domains, the support of the functions $$f_n$$ cannot be uniformly bounded. Otherwise (1) and (3) contradict.
Define $$f_n = \chi_{[0,n]} n^{-1/2}.$$ Then $$\|f_n\|_{L^3}^3=n^{-1/2}$$, $$\|f_n\|_{L^2}^2=1$$. In addition, $$\int_{\mathbb R} f_n \phi \to0$$ for all $$\phi \in C_c(\mathbb R)$$. Such functions are dense in $$L^2(\mathbb R)$$, hence $$f_n \rightharpoonup 0$$ in $$L^2(\mathbb R)$$.
• Do we have that $\phi(n)=0$? We get a boundary term $[n^{-1/2}x\phi(x)]_0^n$ from integrating by parts? – user30523 Feb 14 '20 at 10:41
• ?? there is no integration by parts: $\int |f_n\phi| \le n^{-1/2} \|\phi\|_{L^\infty}\to0$. – daw Feb 14 '20 at 12:08