$\int_{\mathbb R} \frac{1}{(1+ |t-y|)^r} \|f(y)\|_{L^1} dy \leq \frac{C}{r-1} \|f\|_{C(\mathbb R, L^1(\mathbb R))}$ for some $r>0$?

Question

Let $f:\mathbb R \to L^1(\mathbb R):t\mapsto f(t)\in L^1(\mathbb R).$ [So $f(t):\mathbb R \to \mathbb C: x\mapsto f(t)(x)$] Assume that $f$ is continuous with time variable $t$. Assume that $f$ is nice

For $t>0,$ Can we say $\int_{\mathbb R} \frac{1}{(1+ |t-y|)^r} \|f(y)\|_{L^1} dy \leq \frac{C}{r-1} \|f\|_{C(\mathbb R, L^1(\mathbb R))}$ for some $r>0$?

Where $C(\mathbb R, L^1(\mathbb R))$ denotes class of continuous functions $f:\mathbb R \to L^1(\mathbb R).$

Answer 1

By the norm on the continuous function mapping $\mathbb{R} \to L^1(\mathbb{R})$ I assume you mean the maximum-norm, i.e. $$ \|f\|_{C(\mathbb{R},L^1(\mathbb{R}))} = \sup_{t\in \mathbb{R}} \|f(t)\|_{L^1(\mathbb{R})} $$

If that is the case, then the answer to your question is yes. Indeed, for any $t\in \mathbb{R}$ and $r > 1$ one has the estimate:

$$\int_{\mathbb R} \frac{1}{(1+ |t-y|)^r} \|f(y)\|_{L^1} dy \leq \frac{2}{r-1} \|f\|_{C(\mathbb R, L^1(\mathbb R))}.$$