We study the fractional Fokker-Planck-Kolmogorov equation with the fractional index \(\alpha \in [1,2)\) and use a vector-valued Calderón-Zygmund theorem to obtain the existence and uniqueness of \(L^p([0,T];{{\mathcal {C}}}_b^{\alpha +\beta }({{\mathbb {R}}}^d))\cap W^{1,p}([0,T];{{\mathcal {C}}}_b^\beta ({{\mathbb {R}}}^d))\) solution under the assumptions that the drift coefficient and nonhomogeneous term are in \(L^p([0,T];{{\mathcal {C}}}_b^{\beta }({{\mathbb {R}}}^d))\) with \(p\in [\alpha /(\alpha -1),+\infty ]\) and \(\beta \in (0,1)\). As applications, we prove the unique strong solvability as well as Davie’s type uniqueness of time inhomogeneous stochastic differential equation with the drift in \(L^p([0,T];{{\mathcal {C}}}_b^{\beta }({\mathbb R}^d;{{\mathbb {R}}}^d))\) and driven by the \(\alpha \)-stable process for \(\beta > 1-\alpha /2\) and \(p>2\alpha /(\alpha +2\beta -2)\).

我们研究分数阶指数为 \(\alpha \in [1,2)\) 的分数 Fokker-Planck-Kolmogorov 方程，并使用向量值 Calderón-Zygmund 定理来获得 \(L^p( [0,T];{{\mathcal {C}}}_b^{\alpha +\beta }({{\mathbb {R}}}^d))\cap W^{1,p}([0 ,T];{{\mathcal {C}}}_b^\beta ({{\mathbb {R}}}^d))\) 假设漂移系数和非齐次项在 \(L^ 中) 的解p([0,T];{{\mathcal {C}}}_b^{\beta }({{\mathbb {R}}}^d))\) 与 \(p\in [\alpha /( \alpha -1),+\infty ]\) 和 \(\beta \in (0,1)\)。作为应用，我们证明了漂移为 \(L^p([0,T];{{\mathcal {C}}}_b^{\) 的时间非齐次随机微分方程的唯一性强可解性以及 Davie 型唯一性beta }({\mathbb R}^d;{{\mathbb {R}}}^d))\) 并由 \(\alpha \) 稳定过程驱动 \(\beta > 1-\alpha / 2\) 和 \(p>2\alpha /(\alpha +2\beta -2)\)。