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

Abstract

We consider fractional operators of the form $${\mathcal{H}}^s=({\partial }_t-{\mathrm{div}}_x(A(x,t){\nabla }_x){)}^s,(x,t)\in {\mathbb{R}}^n\times \mathbb{R},$$ 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 $$({\lambda }^{1-2s}{\mathrm{u}}^{\prime }{)}^{\prime }(\lambda )={\lambda }^{1-2s}\mathcal{H}\mathrm{u}(\lambda ),\lambda \in (0,\infty ),$$ $$\mathrm{u}(0)=u,$$ 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 $$\mathcal{H}^su=(\partial_t -\mathrm{div}_{x} ( A(x,t)\nabla_{x}))^s u=0\ \mbox{ for $(x,t)\in Ω\times J$},$$ $$ u=f\ \mbox{ for $(x,t)\in \mathbb R^{n+1}\setminus (Ω\times J)$}. $$ Our contribution is that we allow for non-symmetric and time-dependent coefficients.