Decay rate of a functional norm.

Question

I’ve been working on a project lately that has resulted in needing a following fact: if $\chi_{B_r(x)}$ denotes the characteristic function of the ball of radius $r$ about $x$ in $\mathbb{R}^2$, then for $f$ in the Lorentz space $L^{2,1}(\mathbb{R}^2)$ then one has $$\|f\chi_{B_r(x)}\|_{L^{2,1}}\leq \lambda(r)$$ where $\lambda(r)$ shrinks faster than $r$ independently of $x$. That is $$\lambda(r)/r\to0$$ as $r\to 0$. I’m curious to know if one can prove this without any additional assumptions on $f$. Alternatively one can use the embeddings $W^{1,1}\hookrightarrow BV\hookrightarrow L^{2,1}$ in dimension $2$ to maybe instead prove the sufficient condition that the $W^{1,1}$ or $BV$ norm decays like the above. If anyone happens to know anything like this I’d love to hear it.