= Homogeneous Sobolev space
{title2=$\dot H^s$}
A homogeneous Sobolev space controls derivatives without an independent low-frequency <L2 norm>: formally $\|f\|_{\dot H^s}^2=\int |\xi|^{2s}|\widehat f(\xi)|^2\,d\xi$, with the normalization chosen according to the <Fourier transform>. Completions or quotient conventions must be specified when polynomial ambiguities occur. In three dimensions $\dot H^1$ is represented by $L^6$ functions with <gradient> in $L^2$, and $\|f\|_6\leq C\|\nabla f\|_2$.
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