Approximating By Compactly Supported Functions in Weak Convergence Arguements

I'm working on the following question. I'm having trouble with the solution presented in the textbook - specifically weak convergence.

Let $$f \in L^p(\mathbb{R}^N), 1 with $$\alpha>0$$ real, define $$f_n = n^\alpha f(nx)$$ for $$n = 1,2,3,\dots$$. Does $$\{f_n\}$$ strongly converge? Does it weakly converge?

It can be shown that for $$\alpha < N/p$$ the sequence strongly converges, otherwise it does not strongly converge. Clearly, for $$\alpha < N/p$$ the sequence also weakly converges (I got this far). Now consider $$\alpha = N/p$$. Suppose $$f$$ vanishes for $$|x| >K$$...(some analysis to conclude weak convergence). For $$f$$ arbitrary, given $$\epsilon > 0$$ and $$\phi \in L^q(\mathbb{R}^n)$$, let $$g$$ be a function vanishing outside a compact set such that $$\|f-g\|_p \leq \epsilon/2\|\phi\|_q$$.

1. Why do they consider $$f$$ compact first?
2. How do we get that estimate between $$f$$ and the compactly supported $$g$$?

I think you want $$\mathbb{R}^N$$ rather than $$\mathbb{R}^d$$, so your later $$N/p$$ makes sense.
1. The analysis that you omitted should tell you why. Essentially when $$f$$ is compactly supported, eventually $$f_n$$ lives inside a very small ball centered 0 so you can estimate the integrals by turning the condition onto your $$\phi\in L^q$$, i.e., $$\lvert\int\phi f_n\rvert\leq \lVert f_n\rVert_p\cdot\lVert \phi 1_{\lvert x\rvert.
2. Consider $$g_R=f\cdot 1_{|x|. As $$R\to\infty$$ we have $$g_R\to f$$ in $$L^p$$ so there is an $$R$$ where $$\lVert f-g\rVert_p$$ satisfies this estimate.
• yes I did mean $N$ not $d$ -- thanks! – yoshi May 23 at 15:54