Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz space) and define $f_k(x) =f(x-k) \ (k\in \mathbb Z^d)$ (translation of $f$).
Question: For which kernel $h:\mathbb R^d \to \mathbb C,$ one can expect that $$\|f_k\ast h \|_{\ell ^1} = \sum_{k\in Z^d} |f_k \ast h(x)|\leq C$$ ( $C$ is some constant independent of $k$ but can be dependent on $f$ and $h$). (where * is the convolution in $\mathbb R^d$.)
Side notes: I'm interested in the singular kernel $h(x)= \frac{1}{|x|^{\gamma}} (\gamma >0)$