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

Abstract

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