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I am trying to touch the half Laplace operator from the opposite direction. In particular, we define the following singular integral operator:$$ Kf = C_n\mathop{\lim}_{\epsilon\to0}\int_{|y-x|>\epsilon}\frac{f(x)-f(y)}{|x-y|^{n+1}}\,dy $$ where $C_n$ is some constant dependent of $n$. My questions is the following: By the definition above, how can one show that this $K$ is exactly $\sqrt{-\Delta}$ for $f\in C^2\cap L^{\infty}$? For example if I want to show that $K\circ Kf=\Delta f$, how am I supposed to proceed to verify this in a straightforward way? I know that the fractional Laplacian is better treated under the Sobolev space setting, but what about the twice differentiable bounded functions?

Any comments on how to proceed, or suggestions of helpful references will be appreciated. Thanks in advance!

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    I am not sure if this is even well defined. The part of the integral $\int_{|y|>\epsilon}\frac{1}{y^2}dy$ clearly diverges in $\mathbb{R}^{1}$ case, assuming $x=0$ and $f(0)=1$. Maybe I am confused with something basic at here. – 2017-02-02

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