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 $$(\lambda^{1-2s}\mathrm{u}')'(\lambda) =\lambda^{1-2s}\mathcal{H} \mathrm{u}(\lambda), \quad \lambda\in (0, \infty),$$ $$\mathrm{u}(0) = u,$$ 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 $$\mathcal{H}^su=(\partial_t -\mathrm{div}_{x} ( A(x,t)\nabla_{x}))^s u=0\ \mbox{ for $(x,t)\in \Omega \times J$},$$ $$ u=f\ \mbox{ for $(x,t)\in \mathbb R^{n+1}\setminus (\Omega \times J)$}. $$ Our contribution is that we allow for non-symmetric and time-dependent coefficients.