Determine whether a function $f$ satisfying $\int_{B_{2r}\setminus B_r}|f(x)|^3dx<Cr$, for every $r$ is $L^1(B_1)$

Solution 1:

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.

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