Let $B_r=\{x\in\mathbb{R}^2:|x|<r\}$ denote the ball in $\mathbb{R}^2$ centered at the origin or radius $r$. Let $f$ and $g$ be measurable functions on $\mathbb{R}^2$ satisfying:
There exists a constant $C>0$ such that for every $r>0$: $$\int_{B_{2r}\setminus B_r}|f(x)|^3dx<Cr\;\;\;\; \text{ and }\;\;\;\;\int_{B_{2r}\setminus B_r}|g(x)|^4dx<Cr^{-7}$$
Determine whether $f\in L^1(B_1)$ and whether $g\in L^1(B_1)$. Further, in addition, if $f$ and $g$ are assumed to be continuous, then show that $fg\in L^1(\mathbb{R}^2)$.
I'm unable to see the relationship between $\int_{B_1}|f|$ and the given property that $f$ satisfies. Likewise for $g$. I'm looking for hints only. Thanks for your time.
Hölder's inequality with the exponents $p=3$ and $q=\frac32$ gives $$ \int_{B_{2r}\setminus B_r}|f|\cdot1 \leq \big(\int_{B_{2r}\setminus B_r}|f|^3\big)^{\frac13} \big(\int_{B_{2r}\setminus B_r}1\big)^{\frac23} \leq cr^{1/3}\cdot r^{4/3}. $$ Then the question is about summability of these.