Weak convergence, $f\in L^p(\Bbb R^d) $ and $g\in L^q(\Bbb R^d) $ then, $\lim_{j\to\infty} \int_{\Bbb R^d}f_j(x)g(x)dx = 0$ Assume $f\in L^p(\Bbb R^d) $ and  $g\in L^q(\Bbb R^d) $ 
Where, $1&lt;p&lt;\infty$ and $1&lt;q&lt;\infty$ are dual ecxponents namely, $$\frac1p+\frac1q =1$$
Then for every $s\in\Bbb R$ such that, $sp\le d$ show that, 
$$\lim_{j\to\infty} \int_{\Bbb R^d}f_j(x)g(x)dx = 0$$

Where, $$f_j(x) = j^sf(jx)~~~$$

My attempt,

I Applied Holder inequality, and use the change of variables $u =jx$ to get the following,

\begin{align}\left|\int_{\Bbb R^d}f_j(x)g(x)dx\right| &amp;\le \left(\int_{\Bbb R^d}|f_j(x)|^pdx\right)^{1/p} \left(\int_{\Bbb R^d}|g(x)|^qdx\right)^{1/q}\\&amp;=\frac{1}{j^{\frac dp-s}}\left(\int_{\Bbb R^d}|f(x)|^pdx\right)^{1/p} \left(\int_{\Bbb R^d}|g(x)|^qdx\right)^{1/q} \\&amp;=\frac{1}{j^{\frac dp-s}}\|f\|_{p}\|f\|_{q}\end{align}

If I suppose that, $\color{blue}{d&gt;sp}$ then $\frac{1}{j^{\frac dp-s}}\to 0$
and the result follows. 


  Question: How do I prove the case where $sp= d$?
Note it's not true for $p=1$. Assume $p&gt;1$. Come to think of it, it's also false for $p=\infty$. Assume $p&lt;\infty$.

Let $\epsilon&gt;0$. Choose $A&gt;0$ so that $$\left(\int_{|x|\le A}|g(x)|^q\right)^{1/q}&lt;\epsilon.$$(This is where $q&lt;\infty$ is needed.)
An explicit calculation shows that $$\left(\int_{|x|&gt;A}|f_j(x)|^p\right)^{1/p}\to0.$$(This uses $p&lt;\infty$.)

Edit: What's above will look like a solution to many readers. Others may be wondering how those two lines prove what's asserted. The (standard) argument goes like so: Assume $\epsilon$ and $A$ are as above. Choose $N$ so that $$\left(\int_{|x|&gt;A}|f_j(x)|^p\right)^{1/p}&lt;\epsilon\quad(j&gt;N).$$
Noting that $||f_j||_p=||f||_p$ (since $sp=d$), for $j&gt;N$ we have $$\left|\int f_jg\right|\le\left|\int_{|x\le A}f_jg\right|+\left|\int_{|x|&gt;A}f_jg\right|\le||f||_p\epsilon+\epsilon||g||_q.$$
This is just a reminiscent of the this:
How to prove that, $\lim\limits_{k \rightarrow \infty} \int_{\Bbb{R}} f_k(x) g(x) dm(x) = 0$.

If $sp =d$ then, Proceed as follows:


  Lemma: Continuous function of compact support are dense in $L^r(\Bbb R^d)$ for, $1&lt;r&lt;\infty$.


For,  $\varepsilon&gt;0$ we know that there exist a $f_\varepsilon,g_\varepsilon\in C_c(\Bbb R)$ a continuous function of compact support such that, 
$$ \|g-g_\varepsilon\|_q\le \varepsilon \|f\|_p^{-1} $$
and 
$$ \|f-f_\varepsilon\|_p\le \varepsilon \|g_\varepsilon\|_q^{-1} $$

Proceeding like here with   Cauchy Schwartz inequality replaced by Holder inequality, one easily arrives at

$$\left|\int_\mathbb R f_j(x) g(x) dx\right| \le\left[\varepsilon+\varepsilon j^{s-d}+ j^{s-d}\left|\int_{\text{supp }f_\varepsilon}f_\varepsilon(x) g_\varepsilon(\frac{x}{j}) dm(x)\right| \right]\\= \left[\varepsilon+\varepsilon j^{s(1-p)}+ j^{s(1-p)}\left|\int_{\text{supp }f_\varepsilon}f_\varepsilon(x) g_\varepsilon(\frac{x}{j}) dm(x)\right| \right]$$

Moreover, $f_\varepsilon$ $h_\varepsilon$ are continuous functions  with compact support, we get
 $$\color{blue}{\lim_{j\to \infty}\int_{\text{supp }f_\varepsilon}f_\varepsilon(x) g_\varepsilon(\frac{x}{j}) dx=\int_{\text{supp }f_\varepsilon}f_\varepsilon(x) g_\varepsilon(0) dx}$$

Whence, 
$$ \lim_{j\to \infty} \left|\int_\mathbb R f_j(x) g(x) dx\right| \le\lim_{j\to \infty} \left[\varepsilon+\varepsilon j^{s-d}+ j^{s-d}\left|\int_{\text{supp }f_\varepsilon}f_\varepsilon(x) g_\varepsilon(\frac{x}{j}) dx\right| \right]=\varepsilon $$

Thus, for all $\varepsilon&gt;0,$ we have, $$ \limsup_{j\to \infty} \left|\int_\mathbb R f_j(x) g(x) dx\right| \le \varepsilon $$
letting  $\varepsilon\to 0$  yields the require result.

$$ \color{red}{\lim_{j\to \infty}  \int_\mathbb R f_j(x) g(x) dx =0.} $$