Does the weak $L^p$-norm of $f_n$ tending to $0$ imply $\int_Bf_n\phi→0$? Let $f_n \in L^p(B)$ be a sequence where $B$ is some ball in $\mathbb{R}^n$. Assume that  $\|f_n\|_{L^p(B)} \rightarrow 0$ when $n\rightarrow \infty$, then for any $\phi \in C^\infty_0(B)$, applying Hölder we get
$$
\int_B f_n \phi \le \|f_n\|_{L^p(B)}\|\phi\|_{L^{p/(p-1)}(B)}.
$$
Thus, $\int_B f_n \phi \rightarrow 0$ when $n\rightarrow \infty$.
Does the result remains true if we replace the condition $\|f_n\|_{L^p(B)} \rightarrow 0$ by $\|f_n\|_{L^p_w(B)} \rightarrow 0$? Here, $\|f\|_{L^p_w(B)}$ is the infimum of all constant $C$ such that
$$ \mathcal{L}^n \{ x \in B: |f(x) | > \alpha \} \le \dfrac{C^p}{\alpha^p} \quad (\forall \alpha >0)$$
is the weak $L^p$-norm of $f$ in $B$. This is. Do we have  $\int_{B} f_n \phi \rightarrow 0$ when $n\rightarrow \infty$?
 A: Assume that $p>1$. Let $1<q<p$ and $f\in L^{p,\infty}$. Then 
$$\int_B|f(x)|^q\mathrm dx=q\int_0^{+\infty}s^{q-1}\mathcal L\{t:|f(t)>s\}\mathrm dx\leqslant q\lVert f\rVert_{p,\infty}\left(\int_{1}^{\infty}s^{q-p-1}\mathrm ds+\mathcal L(B)\right).$$
In particular, the inclusion $L^{p,\infty}\to L^q$ is continuous. 
In this context, we have $\lVert f_n\rVert_{L^q}\to 0$, hence we can use the same argument as in the OP.
A: $\def\d{\mathrm{d}}\def\peq{\mathrel{\phantom{=}}{}}$Denote $C_n = \|f_n\|_{L_{\mathrm{w}}^p}\ (n \geqslant 1)$. For any bounded and measurable $φ$ on $B$, suppose $M = \|φ\|_{L^∞}$. For $n \geqslant 1$,\begin{align*}
&\peq \left| \int\limits_B f_nφ \,\d x \right| \leqslant M \int\limits_B |f_n| \,\d x = M \int_0^{+∞} m(\{x \in B \mid |f_n(x)| > t\}) \,\d t\\
&= M\int_0^{C_n} m(\{x \in B \mid |f_n(x)| > t\}) \,\d t + M \int_{C_n}^∞ m(\{x \in B \mid |f_n(x)| > t\}) \,\d t\\
&\leqslant M \int_0^{C_n} m(B) \,\d t + M \int_{C_n}^∞ \frac{C_n^p}{t^p} \,\d t = M\left( m(B) + \frac{1}{p - 1} \right) · C_n,
\end{align*}
thus$$
\lim_{n → ∞} C_n = 0 \Longrightarrow \lim_{n → ∞} \int\limits_B f_nφ \,\d x = 0.
$$
