Let $1<p<\infty$. Suppose $a_i\geq 0$, and $\{B_{r_i}(x_i)\}$ is a sequence of open balls in $\mathbb{R}^n$ centered at $x_i$ with radius $r_i$.
Let $g\in L^q(\mathbb{R}^n)$ where $\frac 1p+\frac 1q=1$ and define the maximal function $$g^*(y)=\sup\{\frac{1}{|B|}\int_B|g|\,dx:B\ \text{is any open ball containing}\ y\}.$$
For convenience, denote $B_i=B_{r_i}(x_i)$ and $3B_i=B_{3r_i}(x_i)$.
Question 1) Prove that there exists $C_0>0$ such that $$\int_{\mathbb{R}^n}\sum_{i}a_i\chi_{3B_i}(x)|g(x)|\,dx\leq\int_{\mathbb{R}^n}C_0\sum_ia_i\chi_{B_i}(x)g^*(x)\,dx.$$
Question 2) Hence prove that there exists $C>0$ (independent of $a_i$) such that $$\|\sum_{i}a_i\chi_{3B_i}\|_{L^p}\leq C\|\sum_ia_i\chi_{B_i}\|L^p$$
Thanks for any help.
My attempt:
Upon seeing the "$3B_i$", I would think that Vitali Covering Lemma would be useful, since that is the only other place I have seen "$3B_i"$.
I tried proving a simplified version, where the index set only has one element, i.e. $\{a_i\}=\{a_1\}$ and $\{B_{r_i}(x_i)\}=\{B_{r_1}(x_1)\}$. However, it didn't quite work out:
\begin{align*} LHS&=\int_{\mathbb{R}^n}a_1\chi_{3B_1}(x)|g(x)|\,dx\\ &=\int_{3B_i}a_1|g(x)|\,dx\\ &=3^n|B_1|\frac{1}{|3B_1|}\int_{3B_1}a_1|g(x)|\,dx\\ &\leq 3^n|B_1|a_1 g^*(x) \end{align*}
Something is not quite right at the end...
Some other miscellaneous thoughts: Holder's inequality should be useful, since there is $\frac 1p+\frac 1q=1$.