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One can define Fourier multipliers on a Banach function space by using thedirect and inverse Fourier transforms on $L^2(\mathbb{R}^n)$ or by using thedirect Fourier transform on $S(\mathbb{R}^n)$ and the inverse one on$S'(\mathbb{R}^n)$. In the former case, one assumes that the Fouriermultipliers belong to $L^\infty(\mathbb{R}^n)$, while in the latter one thisrequirement may or may not be included in the definition. We provide sufficientconditions for those definitions to coincide as well as examples when theydiffer. In particular, we prove that if a Banach function space$X(\mathbb{R}^n)$ satisfies a certain weak doubling property, then the space ofall Fourier multipliers $\mathcal{M}_{X(\mathbb{R}^n)}$ is continuouslyembedded into $L^\infty(\mathbb{R}^n)$ with the best possible embeddingconstant one. For weighted Lebesgue spaces $L^p(\mathbb{R}^n,w)$, the weakdoubling property is much weaker than the requirement that $w$ is a Muckenhouptweight, and our result implies that$\|a\|_{L^\infty(\mathbb{R}^n)}\le\|a\|_{\mathcal{M}_{L^p(\mathbb{R}^n,w)}}$for such weights. This inequality extends the inequality for $n=1$ from\cite[Theorem~2.3]{BG98}, where it is attributed to J.~Bourgain. We show thatalthough the weak doubling property is not necessary, it is quite sharp. Itallows the weight $w$ in $L^p(\mathbb{R}^n,w)$ to grow at any subexponentialrate. On the other hand, the space $L^p(\mathbb{R},e^x)$ has plenty ofunbounded Fourier multipliers.

When does the norm of a Fourier multiplier dominate its $L^\infty$ norm?