Are Sobolev spaces $W^{k,1}(\mathbb R^d)$ and $H^{k,1}(\mathbb R^d)$ the same?
Solution 1:
The spaces $W^{k,1}(\mathbb{R}^{d})$ and $H^{k,1}(\mathbb{R}^{d})$ coincide for $k$ even and $d=1$, while $W^{k,1}(\mathbb{R}^{d})\subset H^{k,1} (\mathbb{R}^{d})$ for $k$ even and $d>1$. For $k$ odd there is no relation between $W^{k,1}(\mathbb{R}^{d})$ and $H^{k,1}(\mathbb{R}^{d})$. It's a guided exercise in Stein's book on singular integrals (see Exercise 6.6 on page 160). The notation is different (he uses $L^{k,p}(\mathbb{R}^{d})=W^{k,p} (\mathbb{R}^{d})$ and $\mathcal{L}^{k,p}(\mathbb{R}^{d})=H^{k,p} (\mathbb{R}^{d})$).