= Fractional Dirichlet domain scale
{title2=$H_\alpha=D(A^\alpha)$}
For the positive <Dirichlet Laplacian> $A$ on a bounded smooth domain, with normalized <eigenfunctions> $e_j$ and positive <eigenvalues> $\mu_j$, define $A^\alpha u=\sum_j\mu_j^\alpha(u,e_j)e_j$ on the vectors for which $\sum_j\mu_j^{2\alpha}|(u,e_j)|^2<\infty$. The resulting <Hilbert space> embeds in the <Sobolev space> $H^{2\alpha}$, with the boundary conditions appropriate to the exponent. In particular $H_\alpha$ embeds in $C^{0,\beta}$ whenever $0<\beta<\min(1,2\alpha-n/2)$. High powers impose further boundary compatibility; ordinary smoothness alone does not imply membership in every fractional domain.
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