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Recall the Stein Derivative, $$\mathcal{D}^{s} f(x)=\left(\int \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}} d y\right)^{1 / 2}.$$ I want to show that, $$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \mathcal{D}^{s} g\right\|_{2}+\left\|g \mathcal{D}^{s} f\right\|_{2}$$ holds for $s\in(0,1)$ and $f,g\in \mathcal{S}(\mathbb{R}^n).$

I tried the look at the quantity $|f(x)g(x)-f(y)g(y)|^2$ and estimate it but there are extra terms that I am not sure how to control. Any suggestions/remarks will be much appreciated.

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    $\begingroup$ $|f(x)g(x)-f(y)g(y)|\leqslant A(x,y)+B(x,y)$, where $A=|f(x)|\cdot |g(x)-g(y)|$, $B=|g(y)|\cdot |f(x)-f(y)|$. Now use the triange inequality bound $\|A\|+\|B\|\geqslant \|A+B\|$, where the norm is taken in the weighted $L^2(\mathbb{R}^n\times \mathbb{R}^n,1/|x-y|^{n+2s})$. $\endgroup$
    – Fedor Petrov
    Commented Jan 19, 2022 at 21:09
  • $\begingroup$ Aah, of course. Thanks! $\endgroup$
    – Student
    Commented Jan 19, 2022 at 21:50

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