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

Abstract We consider fractional operators of the form $$\begin{aligned} {\mathcal {H}}^s=(\partial _t -\text {div}_{x} ( A(x,t)\nabla _{x}))^s,\ (x,t)\in {\mathbb {R}}^n\times {R}}, \end{aligned}$$ H s = ( ? t - div x A , ) ? ? R n × where $$s\in (0,1)$$ 0 1 and $$A=A(x,t)=\{A_{i,j}(x,t)\}_{i,j=1}^{n}$$ { i j } is an accretive, bounded, complex, measurable, $$n\times n$$ -dimensional matrix valued function. study $${{\mathcal {H}}}^s$$ their relation to initial value problem \begin{aligned} (\lambda ^{1-2s}\textrm{u}')'(\lambda )&=\lambda ^{1-2s}{\mathcal {H}}\textrm{u}(\lambda ), \quad \lambda \in (0, \infty \\ \textrm{u}(0)&= u, \end{aligned} ? 2 u ? ? u in $${\mathbb {R}}_+\times {R}}$$ + . Exploring relation, making additional assumption that real, we derive some local properties solutions non-local Dirichlet {H}}^su=(\partial _{x}))^su&=0\hbox { for}\ \Omega \times J,\nonumber u&=f \text{ for } {R}}^{n+1}\setminus (\Omega J). for ? J f \ . Our contribution allow non-symmetric time-dependent coefficients.