If $f \in L^{q}$ for all $1 \leq q \leq \infty$ and $g \in L^{p}$ for some $1<p \leq 2$, how is it possible to get that \begin{equation} \mathcal{F}^{-1}(f \mathcal{F}(g))(x)=\mathcal{F}^{-1}f*g(x)? \end{equation} I know the result for $g \in \mathscr{S}$ using tempered distributions but I can't figure it out in this case. I don't even know if this accounts make sense in this spaces.
2026-04-01 10:44:51.1775040291
Inverse Fourier Transform of product of two functions in $L^p$
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