Bessel potential space

Remarks

The operator $J^{-s} = (I - \Delta)^{-s/2}$ is the Bessel potential of order $s$. For $s > 0$ it is convolution with an integrable kernel $G_s$ whose Fourier transform is $(1 + \abs{\xi}^2)^{-s/2}$. So for $s > 0$, $H^{s,p}(\R^n)$ is the set of functions $G_s * g$ with $g \in L^p(\R^n)$, and $\norm{G_s * g}_{H^{s,p}} = \norm{g}_{L^p}$.

Example: for $s = 0$ the multiplier is $1$, so $J^0 f = f$ and $H^{0,p}(\R^n) = L^p(\R^n)$ with the same norm.

Non-example: the constant function $f = 1$ is a tempered distribution, but it is not in any $H^{s,p}(\R^n)$. Its Fourier transform is a multiple of the Dirac mass at $0$, and the multiplier equals $1$ at $\xi = 0$. So $J^s 1 = 1$, which is not in $L^p(\R^n)$ for $p < \infty$.

For an integer $k \ge 0$ and $1 < p < \infty$, $H^{k,p}(\R^n)$ equals the Sobolev space $W^{k,p}(\R^n)$ with an equivalent norm. The proof of this uses the boundedness of the Riesz transforms on $L^p$. For $p = 2$, Plancherel's theorem gives $H^{s,2}(\R^n) = H^s(\R^n)$, the fractional Sobolev space. Other notations are $L^p_s$, $H^s_p$ and $W^{s,p}$.