We consider a non-local operator $L_{{ \alpha}}$ which is the sum of a fractional Laplacian $\triangle^{\alpha/2} $, $\alpha \in (0,1)$, plus a first order term which is measurable in the time variable and locally $\beta$-H\"older continuous in the space variables. Importantly, the fractional Laplacian $\Delta^{ \alpha/2} $ does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition $\alpha + \beta >1$. Thus, the constant appearing in the Schauder estimates is in fact independent of the $L^{\infty}$-norm of the first order term. In our approach we do not use the so-called extension property and we can replace $\triangle^{\alpha/2} $ with other operators of $\alpha$-stable type which are somehow close, including the relativistic $\alpha$-stable operator. Moreover, when $\alpha \in (1/2,1)$, we can prove Schauder estimates for more general $\alpha$-stable type operators like the singular cylindrical one, i.e., when $\triangle^{\alpha/2} $ is replaced by a sum of one dimensional fractional Laplacians $\sum_{k=1}^d (\partial_{x_k x_k}^2 )^{\alpha/2}$.

中文翻译：

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超临界情况下漂移分数算子的 Schauder 估计

我们考虑一个非局部运算符 $L_{{ \alpha}}$，它是分数拉普拉斯算子 $\triangle^{\alpha/2} $、$\alpha \in (0,1)$ 和一个在时间变量中可测量的一阶项和局部 $\beta$-H\"在空间变量中更老连续。重要的是，分数拉普拉斯算子 $\Delta^{\alpha/2} $ 并不支配一阶项. 我们表明，即使在这种情况下，全局抛物线 Schauder 估计在自然条件 $\alpha + \beta >1$ 下也成立。因此，出现在 Schauder 估计中的常数实际上与 $L^{\infty}$ 无关- 一阶项的范数。在我们的方法中，我们不使用所谓的扩展属性，我们可以用其他 $\alpha$-stable 类型的运算符替换 $\triangle^{\alpha/2} $关闭，包括相对论 $\alpha$ 稳定运算符。此外，当 $\alpha \in (1/2,1)$ 时，我们可以证明 Schauder 估计对于更一般的 $\alpha$-stable 类型算子，如奇异圆柱算子，即当 $\triangle^{\alpha/ 2} $ 由一维分数拉普拉斯算子 $\sum_{k=1}^d (\partial_{x_k x_k}^2 )^{\alpha/2}$ 的和代替。