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Let $f\in \dot{H}^s$ with $s\in (0,1)$ and $g\in \dot{H}^{-s}$, where $\dot{H}^s$ denote the fractional homogeneous Sobolev space $\dot{W}^{p,s}$with $p=2$. Does one have the following Cauchy-Schwarz type inequality? $$ (f,g)_{L^2}\leq \| f \|_{\dot{H}^s} \| g \|_{\dot{H}^{-s}}, $$ where $(.,.)_{L^2}$ is the $L^2$-inner product and on the right are the seminorms corresponding to the homogeneous spaces (given by the Gagliardo seminorms).

I have seen an inequality of this type for fractional Sobolev spaces but without any proof. Does one have a reference (also for the case above)?

Thanks for your help!

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1 Answer 1

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Let $\hat f, \hat g$ be the Fourier transforms of $f,g$. By Plancherel and Cauchy-Schwarz,

$$(f,g)_{L^2} = (\hat f, \hat g)_{L^2} = \int (\hat f(x) |x|^s)( |x|^{-s} \hat g(x)) dx \leq \||\cdot|^s\hat f \|_{L^2} \||\cdot|^{-s}\hat g\|_{L^2} = \|f\|_{\dot H^s}\|g\|_{\dot H^{-s}}.$$

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