# Riesz potential and homogeneous Sobolev spaces

Consider the Bessel potential and Riesz potential spaces $$L^{s,p}=\{f:f=(I-\Delta)^{-s/2}g,\, g\in L^p\},$$ $$\dot{L}^{\alpha,p}=\{f:f=(-\Delta)^{-s/2}g,\, g\in L^p\},$$ where $$(I-\Delta)^{s/2}$$ and $$(-\Delta)^{s/2}$$ are the Fourier multipliers with symbol $$(1+|\xi|^2)^{s/2}$$ and $$|\xi|^s$$, respectively.

There is a result by A.P. Calderón which states that $$L^{s,p}\cong W^{s,p}$$ when $$s\in\mathbb{N}$$ and $$1. My question is, does a similar result hold for $$\dot{L}^{s,p}$$ and $$\dot{W}^{s,p}$$?

Actually, the only thing I need to know right now is whether $$\dot{L}^{1,2}\cong \dot{H}^{1}$$. Formally, it makes all the sense in the world to me, considering Plancherel theorem and how the Fourier transform interacts with derivatives, but I often get surprised by how much the devil is in the details when studying functional analysis.

• You're right, thank you for pointing it out. Commented Jun 7, 2022 at 9:23
• Just edited it, thanks once again. Commented Jun 7, 2022 at 15:54

Sure. You can even deduce the result for the dotted norms by the non-dotted one, via scaling. Anyway, the Fourier transform argument is sufficient in this case. Namely, if $$u$$ is Schwartz class then of course you have $$\|u\|_{\dot H^1}^2\simeq \|\xi\widehat{u}\|_{L^2}\simeq \||\xi|\widehat{u}\|_{L^2}\simeq \|u\|_{\dot L^{1,2}}.$$ Then by density you obtain the equivalence of spaces. The only devilish detail I can think of is when you try to do the same for higher order Sobolev spaces; if you consider $$\dot H^s$$ with $$s>n/2$$ you run into the difficulty that the space $$\dot H^s$$ is not naturally embedded into temperate distributions and you must quotient away polynomials (but the previous equivalence of norms for good functions still holds).
• Just to add to Piero’s answer: The difficulty with “modulo polynomials” already holds for $s=n/2$, e.g., $\dot H^1(R^2)$. There, it is possible to approximate a constant function in $\| \cdot \|_{\dot H^1(R^2)}$ by compactly supported, smooth functions. Commented Jun 6, 2022 at 13:00