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 $W^{\alpha,p}$?

Many thanks for the answer!


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