# Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space

It is known that for $\alpha\in(0,1)$ and $p>1$,

the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by $$W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^{n+\alpha p}}dx dy<\infty\},$$ and the Bessel space $H^{\alpha,p}(R^n)$ is defined by $$H^{\alpha,p}(R^n):=\{f\in L^p(R^n):\Delta^{\alpha/2}f\in L^p(R^n)\}.$$ There is also another kind of space known as Hajłasz–Sobolev space which is defined by $$M^{\alpha,p}(R^n):=\{f\in L^p(R^n):|f(x)-f(y)|\leq|x-y|^\alpha[g(x)+g(y)],a.e. x,y\in R^n, where\,\, g\in L^p(R^n)\,\, is\,\, nonnegative\}.$$

Is there any relationship between $M^{\alpha,p}$ with $H^{\alpha,p}$ or $W^{\alpha,p}$?