We study the optimal fractional Trudinger–Moser inequalities on \( \mathbb {R} \) when the integrands have the form \(\left( e^{\pi u^{2}}-1\right) \left| u\right| ^{2a}\) for some \(a\ge 0\). The equivalence of the subcritical and critical fractional Trudinger–Moser inequalities is set up in the spirit of Lam, Lu and Zhang. The existence of optimizers for the sharp subcritical fractional Trudinger–Moser inequalities is also investigated.

中文翻译：

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一维分数阶Trrudinger-Moser型不等式

当被积数的形式为\（\ left（e ^ {\ pi u ^ {2}}-1 \ right）\ left | u时，我们研究\（\ mathbb {R} \）上的最优分数Trudinger-Moser不等式\ right | ^ {2a} \）表示某些\（a \ ge 0 \）。亚临界和临界分数的Trudinger-Moser不等式的等价性是根据Lam，Lu和Zhang的精神建立的。还研究了尖锐的亚临界分数Trudinger-Moser不等式的优化器的存在。