Suppose $f\in \mathscr S(\mathbb R^d)$ is in Schwarz space, where functions and their all (partial) derivatives could be controlled by polynomials. E is a finite Borel-measurable set, $|E|$ is its Lebesgue measure. If $\text{supp}(\hat{f})\subseteq E\subseteq \mathbb R^d$, then $$\|f\|_q\leqslant |E|^{\frac 1p-\frac 1q}\|f\|_p$$ for arbitrary $1\leqslant p\leqslant q$. I've seen the power of $|E|$ by scaling, but I have no clues on the construction of this inequality.
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Your question is similar to https://math.stackexchange.com/questions/66029/lp-and-lq-space-inclusion and https://math.stackexchange.com/questions/390227/proving-that-lp-subset-lq-when-1-le-q-le-p – rafilou2003 Dec 04 '24 at 14:37
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@rafilou2003 Sorry, but I don't think they're the same because in general these two spaces can't contain each other when considering the $\mathbb R^d$ space. Here $E$ contains the support of Fourier transform of $f$, not $f$ itself. – Robert Dec 04 '24 at 14:46
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You're correct, didn't notice the "hat" on top of $f$ – rafilou2003 Dec 04 '24 at 14:48
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Are you familiar with this: https://en.wikipedia.org/wiki/Plancherel_theorem ? – Sine of the Time Dec 04 '24 at 21:51
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@SineoftheTime I think I know this theorem but how to connect $|f|_2$ to $\f|_p$ for general $p\geqslant 1$. – Robert Dec 05 '24 at 01:10