超临界情形下漂移分数次算子的Schauder估计
Schauder estimates for drifted fractional operators in the supercritical case
摘要：
我们考虑一个非局部算子$L_{ α}$，它是一个分数拉普拉斯算子$\triangle^{α/2} $，$α\in (0,1)$，加上一个在时间变量上可测且在空间变量上局部$β$ -Hölder连续的一阶项的和.重要的是，分数拉普拉斯$Δ^{ α/2} $不支配一阶项。我们表明，全球抛物线Schauder估计举行，即使在这种情况下，在自然条件$α+ β>1$。因此，在Schauder估计中出现的常数实际上与一阶项的$L^{\infty}$ -范数无关。在我们的方法中，我们不使用所谓的扩展属性，我们可以取代$\triangle^{α/2} $与其他运营商的$α$ -稳定的类型是某种程度上接近，包括相对论性$α$ -稳定的运营商。此外，当$α\in (1/2,1)$，我们可以证明Schauder估计更一般的$α$ -稳定型算子，如奇异圆柱，即，当$\triangle^{α/2} $被替换为一维分数拉普拉斯算子之和$\sum_{k=1}^d (\partial_{x_k x_k}^2 )^{α/2}$时。
英文摘要：
We consider a non-local operator $L_{ α}$ which is the sum of a fractional Laplacian $\triangle^{α/2} $, $α\in (0,1)$, plus a first order term which is measurable in the time variable and locally $β$-Hölder continuous in the space variables. Importantly, the fractional Laplacian $Δ^{ α/2} $ does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition $α+ β>1$. 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^{α/2} $ with other operators of $α$-stable type which are somehow close, including the relativistic $α$-stable operator. Moreover, when $α\in (1/2,1)$, we can prove Schauder estimates for more general $α$-stable type operators like the singular cylindrical one, i.e., when $\triangle^{α/2} $ is replaced by a sum of one dimensional fractional Laplacians $\sum_{k=1}^d (\partial_{x_k x_k}^2 )^{α/2}$.
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