On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients

We consider fractional operators of the form $${\mathcal{H}}^s=({\mathrm{\partial }}_t-{\mathrm{d}\mathrm{i}\mathrm{v}}_x(A(x,t){\mathrm{\nabla }}_x){)}^s,(x,t)\in {\mathbb{R}}^n\times \mathbb{R},$$ where $s\in (0,1)$ and $A=A(x,t)=\{A_{i,j}(x,t){\}}_{i,j=1}^n$ is an accretive, bounded, complex, measurable, $n\times n$-dimensional matrix valued function. We study the fractional operators ${\mathcal{H}}^s$ and their relation to the initial value problem $$({\lambda }^{1-2s}{\mathrm{u}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}(\lambda )={\lambda }^{1-2s}\mathcal{H}\mathrm{u}(\lambda ),\lambda \in (0,\mathrm{\infty }),$$ $$\mathrm{u}(0)=u,$$ in ${\mathbb{R}}_{+}\times {\mathbb{R}}^n\times \mathbb{R}$. Exploring this type of relation, and making the additional assumption that $A=A(x,t)=\{A_{i,j}(x,t){\}}_{i,j=1}^n$ is real, we derive some local properties of solutions to the non-local Dirichlet problem \mathcal{H}^su=(\partial_t -\mathrm{div}_{x} ( A(x,t)\nabla_{x}))^s u=0\ for $(x,t)\in \Omega \times J$, u=f\ for $(x,t)\in \mathbb R^{n+1\setminus (\Omega \times J)$}. Our contribution is that we allow for non-symmetric and time-dependent coefficients.