Following Christian Remling suggestion, it seems that
$$
\Vert u\Vert_{L^\infty(\mathbb R^d)}\le \gamma(d)\bigl\{
\Vert u\Vert_{L^2(\mathbb R^d)}+
\Vert \vert D\vert^{d/2} L(\vert D\vert) u\Vert_{L^2(\mathbb R^d)}
\bigr\}
$$
with
$$
\int_1^\infty\frac{dr}{r^2 L(r)}<+\infty.
$$