Sobolev domains of powers of an elliptic Dirichlet operator ($X_k=D(A_D^{k/2})$)

For a strictly positive Dirichlet realization of an elliptic operator with smooth coefficients and smooth boundary, put $X_0=L^2(U)$. For each integer $k\ge 1$,These are the domains $D(A_D^{k/2})$, with norms equivalent to the indicated Sobolev space norms. The bilinear forms $((u,v){)}_{2l}=(L^{l}u,L^{l}v{)}_{L^2}$ and $((u,v){)}_{2l+1}=B[L^{l}u,L^{l}v]$ are inner products on $X_{2l}$ and $X_{2l+1}$. The higher boundary conditions are essential: for $L=-d^2/d{x}^2$ and $u=x(\pi -x)$, $u$ is smooth and zero at the boundary but $L^{2}u=0$, so $\Vert L^{2}u{\Vert }_2$ is not a norm on all of $H^4\cap H_0^1$. Repeated elliptic regularity proves $A_D:X_{k+2}\to X_k$ is an isomorphism, which gives the norm equivalence inductively from $X_0$ and $X_1=H_0^1$.