Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of fractional Laplacian, the first one, $\big(s\in (0,1)\big)$
$H_{\Omega}^s$ is defined by $$(H_{\Omega}^s(u), u)_{L^2}:= c\int_{\mathbf R^n} |\xi|^{2s}|\hat{u}|^2d\xi=c\int \int _{\mathbf R^n\times\mathbf R^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}dxdy,$$ with the domain $Dom=\{u\in L^2: \text{supp}~u \in \Omega, (H_{\Omega}^s(u), u)_{L^2}< \infty\}.$
The second one is just $(-\Delta_{\Omega})^s$, which is the $s$-power of functional $-\Delta_{\Omega}$ in the sense of the functional calculus. In particular, when $\Omega$ is a smooth bounded domain, $-\Delta_{\Omega}$ admits eigenvalues and eigenfunctions $\{\lambda_i, \phi_i\}$, and we have $$(-\Delta_{\Omega})^s\phi_i = \lambda_i^s\phi_i .$$
We call two operators $A, B$ which are bounded from below, we write $A\leq B$ if their quadratic forms $a,b$ with form domains $Dom(A)$ and $Dom(B)$ satisfy $Dom(B)\subset Dom(A)$ and $a(u) \leq b(u)$ for every $u\in Dom(B)$. Theorem 5.2.3 in Frank's paper gives that $$H_{\Omega}^s \leq (-\Delta_{\Omega})^s.$$ And he sketches the idea of the proof based on the following observation:
If $A$ is a non-negative operator in a Hilbert space with a trivial kernel, $P$ an orthogonal projection, and $\varphi$ an operator monotone function on $\mathbf R^+$ (for example, $\varphi(x)=x^s$), then $$ P\varphi(PAP)P \geq P\varphi(A)P.\label{1}\tag{1} $$ The operator inequality is closely related to the Sherman–Davis inequality, that is, when $A$ is a matrix in $\mathbf R^n$ and $\psi$ is a convex function, then $$ P\psi(PAP)P \leq P\psi(A)P.\label{2}\tag{2} $$ If we take $A=-\Delta_{R^n}$ and $P=\text{multiplication by the characteristic function of} ~\Omega$ and $\varphi(x)=x^s$, formally I think $$ P\varphi(PAP)P=(-\Delta_{\Omega})^s ~\text{and}~ P\varphi(A)P=H^s_{\Omega}.\label{3}\tag{3} $$ And in Frank's paper $A^s$ admits an integral representation $$ A^s= \frac{\sin(\pi s)}{\pi}\int^{\infty}_0 t^{s-1}\frac{A}{(A+t)}dx.\label{4}\tag{4} $$
I am seriously curious about operator inequalities as \eqref{1} and \eqref{2}, and integral representation of operators as \eqref{4}. But since I'm not majoring in these fields, I feel trouble reading the original papers. And I fail to check \eqref{4} and $H^s_{\mathbf R^n}=(-\Delta_{\mathbf R^n})^s.$
I'm also interested in the relation of $A$'s kernel and $\varphi(A)$'s kernel. Could anyone give me some help or apply some references which suit PDE researchers reading?
