Let $\phi\in\mathcal{S}(\mathbb{R}^{n+1})$, $\phi=\phi(x,y)$, $x\in\mathbb{R}^n$, $y\in\mathbb{R}$, let $\mathcal{F}$ the Fourier transform wrt $x\in\mathbb{R}^n$. Is true that: $$ \partial_y\mathcal{F}(\phi(\cdot,y))=\mathcal{F}(\partial_y\phi(\cdot,y))?$$ I have no idea. I don't know even to start. Any help is appreciated.
2026-04-24 16:02:42.1777046562
Fourier transform commute with partial derivative
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