We fix $p \in [1, \infty)$. We have for every $f \in L^p (\mathbb R^d)$ that $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$. I wonder if there is an estimate of above decay, i.e.,
Is there a measurable function $\varphi : \mathbb R_+ \to \mathbb R_+$ and constants $c, \delta \ge 0$ (possibly depending on $d, p$) such that $\lim_{r \to \infty} \varphi (r) = \infty$ and that $$ \|1_{B(z, 1)} f\|_{L^p} \le \frac{c(1+\|f\|_{L^{p+\delta}})}{\varphi (|z|)}, \quad \forall f \in L^{p+\delta} (\mathbb R^d), z \in \mathbb R^d? $$
Thank you so much for your elaboration!
