= Spectral characterization of elliptic Dirichlet domains
{title2=$u\in X_k\iff\sum_m\lambda_m^k|(u,w_m)|^2<\infty$}
Let $(w_m)$ be the <orthonormal basis> of Dirichlet <eigenfunctions> for a strictly positive <Dirichlet realization of an elliptic operator>, with <eigenvalues> $\lambda_m$. Then
$$
u\in D(A_D^{k/2})\iff\sum_m\lambda_m^k|(u,w_m)_{L^2}|^2<\infty.
$$
For integers $k$, these domains are the <Sobolev domains of powers of an elliptic Dirichlet operator>. At $k=0$ the condition is just <Parseval identity> and imposes no boundary condition. At $k=1,2$ it describes $H_0^1$ and $H^2\cap H_0^1$, respectively. Higher $k$ impose traces of powers of $L$, not only the trace of $u$. For $L=-d^2/dx^2$ on $(0,\pi)$, the smooth Dirichlet function $u=x(\pi-x)$ has sine coefficients proportional to $m^{-3}$ for odd $m$, so the weighted sum diverges at $k=3$ despite its ordinary $H^3$ regularity.
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