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Regularity of subelliptic eigenfunction on characteristic domain

Background: Consider the Hörmander vector fields $X=(X_1,\cdots,X_m)$ on $\mathbb{R}^n$, and the associated Dirichlet eigenvalue problem $$-\Delta u:=\sum_{i=1}^mX_i^*X_iu=\lambda u~~\text{on}~\Omega,~...
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