Let $\alpha \in [0, 1)$ and $\Omega$ be an open connected subset of $\mathbb{R}^{n \geq 2}$. This paper shows that the $Q_{\alpha}$-restriction problem $Q_{\alpha} \vert {}_{\Omega} = \mathscr{Q}_{\alpha} (\Omega)$ is solvable if and only if $\Omega$ is an Ahlfors $n$-regular domain; i.e., $\operatorname{vol} \bigl ( B(x, r) \cap \Omega \bigr ) \gtrsim r^n$ for any Euclidean ball $B(x, r)$ with center $x \in \Omega$ and radius $r \in \bigl ( 0, \operatorname{diam} (\Omega) \bigr ) $ , thereby not only yielding an exponential $Q_{\alpha}$-integrability as a proper adjustment of the John–Nirenberg type inequality for $Q_{\alpha}$ conjectured in [3, Problem 8.1, (8.2)] but also resolving the quasiconformal extension problem for $Q_{\alpha}$ posed in [3, Problem 8.5].

中文翻译：

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$ Q_ {\ alpha} $限制问题

令[\，0]中的\\ alpha \和$ \ Omega $是$ \ mathbb {R} ^ {n \ geq 2} $的开放连接子集。本文显示$ Q _ {\ alpha} $约束问题$ Q _ {\ alpha} \ vert {} _ {\ Omega} = \ mathscr {Q} _ {\ alpha}（\ Omega）$可解决仅在$ \ Omega $是Ahlfors $ n $常规域时；即$ \ operatorname {vol} \ bigl（B（x，r）\ cap \ Omega \ bigr）\ gtrsim r ^ n $对于任何欧氏球$ B（x，r）$以$ x \ in \ Omega为中心$和半径$ r \ in \ bigl（0，\ operatorname {diam}（\ Omega）\ bigr）$，从而不仅产生了指数性的$ Q _ {\ alpha} $可积性，而且是对John–Nirenberg的适当调整在[3，问题8.1，（8.2）]中推测的$ Q _ {\ alpha} $的不等式类型，但也解决了在[3，问题8.5]中提出的$ Q _ {\ alpha} $的拟保形扩展问题。