We present an $L_q(L_p)$-theory for the equation${\partial }_t^{\alpha }u=\varphi (\mathrm{\Delta })u+f,t>0,x\in {\mathbb{R}}^d;u(0,\cdot )=u_0.$ Here $p,q>1$, $\alpha \in (0,1)$, ${\partial }_t^{\alpha }$ is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: $\exists {\delta }_0\in (0,1]$ and $c>0$ such that(0.1)$c{\left(\frac{R}{r}\right)}^{{\delta }_0}\le \frac{\varphi (R)}{\varphi (r)},0<r<R<\infty .$ We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove${\Vert |{\partial }_t^{\alpha }u|+|u|+|\varphi (\mathrm{\Delta })u|\Vert }_{L_q([0,T];L_p)}\le N({\Vert f\Vert }_{L_q([0,T];L_p)}+{\Vert u_0\Vert }_{B_{p,q}^{\varphi ,2-2/\alpha q}}),$ where $B_{p,q}^{\varphi ,2-2/\alpha q}$ is a modified Besov space on ${\mathbb{R}}^d$ related to ϕ.

Our approach is based on BMO estimate for $p=q$ and vector-valued Calderón-Zygmund theorem for $p\ne q$. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.

我们提出一个 ${\mathrm{大号}}_q（{\mathrm{大号}}_p）$方程的理论${\partial }_{Ť}^{\alpha }ü=\varphi （\mathrm{\Delta }）ü+F，Ť>0，X\in {\mathbb{[R}}^d;ü（0，\cdot ）={ü}_0。$ 这里 $p，q>\mathrm{1个}$， $\alpha \in （0，\mathrm{1个}）$， ${\partial }_{Ť}^{\alpha }$是α阶的Caputo分数阶导数，并且ϕ是满足以下条件的伯恩斯坦函数：$\exists {\delta }_0\in （0，\mathrm{1个}]$ 和 $C>0$这样（0.1）$C{（\frac{\mathrm{[R}}{\mathrm{[R}}）}^{{\delta }_0}\le \frac{\varphi （\mathrm{[R}）}{\varphi （\mathrm{[R}）}，0<\mathrm{[R}<\mathrm{[R}<\infty 。$我们证明了Sobolev空间中的唯一性和存在性结果，并获得了该解的最大规则性结果。特别是，我们证明${\Vert |{\partial }_{Ť}^{\alpha }ü|+|ü|+|\varphi （\mathrm{\Delta }）ü|\Vert }_{{\mathrm{大号}}_q（[0，Ť];{\mathrm{大号}}_p）}\le ñ（{\Vert F\Vert }_{{\mathrm{大号}}_q（[0，Ť];{\mathrm{大号}}_p）}+{\Vert {ü}_0\Vert }_{{乙}_{p，q}^{\varphi ，\mathrm{2个}-\mathrm{2个}/\alpha q}}），$ 在哪里 ${乙}_{p，q}^{\varphi ，\mathrm{2个}-\mathrm{2个}/\alpha q}$ 是上的修改后的Besov空间 ${\mathbb{[R}}^d$与ϕ有关。

我们的方法基于BMO的估算 $p=q$ 和向量值Calderón-Zygmund定理 $p\ne q$。Littlewood-Paley理论还用于处理非零初始数据问题。我们的证明依赖于基本解决方案的导数估计，该估计是根据概率理论在本文中获得的。