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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 $$ \int_1^\infty\frac{dr}{r^2 L(r)}<+\infty. $$